101098rspa20000582

Water vapour sonoluminescence andsonochemistry

By Brian D Storey an d Andr ew J Szeri

Department of Mechanical Engineering University of CaliforniaBerkeley CA 94720-1740 USA

Received 4 August 1999 revised 4 November 1999 accepted 20 January 2000

Sonoluminescence is the production of light from acoustically forced bubbles sono-chemistry is a related chemical processing technique The two phenomena share asensitive dependence on the liquid phase The present work is an investigation ofthe fate and consequences of water vapour in the interior of strongly forced argonmicro-bubbles Due to the extreme nonlinearity of the volume oscillations excesswater vapour is trapped in the bubble during a rapid inertial collapse Water vapouris prevented from exiting by relatively slow dinotusion and non-equilibrium conden-sation at the bubble wall By reducing the compression heating of the mixture andthrough primarily endothermic chemical reactions the water vapour reduces the tem-peratures within the bubble signishy cantly The quantity and disposition of hydroxylradicals produced within the bubble are studied in some detail as this is of keeninterest in sonochemistry It was recently shown by Moss and co-workers that lightemission from a sonoluminescence bubble depends sensitively on the water-vapourcontent The quantity of trapped water vapour determined in the present analysisis in excellent agreement with the amount found by Moss and co-workers to matchphoton yields and pulse widths of recent experiments

Keywords sonoluminescence sonochemistry bubble dynamics

1 Introduction

Single-bubble sonoluminescence (SBSL) results from the intense focusing of acousticenergy by a single cavitation bubble during the violent bubble collapse the gas insidethe bubble becomes hot enough to emit a brief regash of light (Barber et al 1997)Sonochemistry is based on similar principles where the collapses of many acousti-cally forced bubbles create high temperatures and pressures that promote chemicalactivity within or near the bubbles A number of chemical processing applicationswere recently reviewed (Mason 1999 Suslick et al 1999 Von Sonntag et al 1999)

In the present work we study the physical and chemical consequences of watervapour on these phenomena Water vapour has a profound enotect on the peak tem-peratures that can be achieved during violent bubble collapses and leads to theproduction of hydroxyl radicals

During intense volume oscillations of a bubble water is constantly evaporating andcondensing driven by dis-equilibrium between the partial pressure of water vapourinside the bubble and saturation pressure at the interface Hence water vapour shy lls

Proc R Soc Lond A (2000) 456 16851709

1685

creg 2000 The Royal Society

1686 B D Storey and A J Szeri

a bubble as it reaches a maximum radius During collapse water vapour tends tocondense at the wall

In an earlier work (Storey amp Szeri (1999) henceforth referred to as SampS) westudied the dinotusive segregation of noble-gas mixtures in collapsing bubbles It wasshown that the time-scale of mixture segregation due to thermal and pressure dif-fusion was much longer than the time-scale of the temperature peak caused by theinertial collapse and rebound As a result the inhomogeneous bubble compositionwas essentially frozen on the time-scale of the temperature peak In the presentpaper we reason along the same lines and shy nd that the bubble motions becomeso rapid on collapse that the water vapour at the centre has insumacr cient time toescape Therefore excess water vapour is left behind in the bubble interior throughthe violent collapse

The trapped water vapour subsequently undergoes rapid chemical reactions Theseprimarily endothermic reactions take up a signishy cant amount of the focused energyand are the basis for production of the OH radical that is of keen interest in sono-chemistry

Previous authors have considered these matters Work by Kamath et al (1993)separated the chemical kinetics from a RayleighPlesset model to estimate the pro-duction of OH radicals Gong amp Hart (1998) coupled chemical reactions with aRayleighPlesset equation and captured some trends observed in sonochemistryexperiments Yasui (1997b) accounted for chemical reactions and non-equilibriumphase change in a collapsing air bubble under SBSL conditions Colussi et al (1998)also coupled bubble motions non-equilibrium phase change and chemical reactionsin a sonochemistry model Sochard et al (1997) modelled free-radical productionin mildly forced bubbles by accounting for non-equilibrium phase change and gasvapour inter-dinotusion with a RayleighPlesset model assuming chemical equilibriumat all times

Each of these approaches makes dinoterent simplifying assumptions which may bevalid at mild forcings In this paper we will relax many of the usual assumptionsand investigate the physics and chemistry of more strongly forced bubbles in detailIn particular we will show that the inter-dinotusion of gas and vapour is crucial topredicting the amount of OH produced in the bubble and the peak temperatureachieved Recently Moss et al (1999) found that inclusion of water vapour is crucialto a correct determination of photon yields and light pulse widths in SBSL we revisitthis point at the end of the paper

In what follows we examine the transport of water vapour into and out of thebubble The NavierStokes equations of a gas mixture are coupled to a reactionmechanism to determine the fate and consequences of trapped water vapour

2 Formulation

We shall assume that the single bubble is spherical and is composed of argon watervapour and reaction products (H H2 O O2 OH HO2 H2O2) For conveniencethe argon is regarded as insoluble in the liquid water on the time-scale of the acousticforcing In SampS we showed that the regux of argon into the liquid has a negligible enotecton the gas dynamics as the total mass of argon contained in the bubble reguctuatesby only ca 1 over one cycle of forcing Radiation heat transfer and light emissionhave negligible enotects on the gas dynamics (Moss et al 1999)

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Water vapour sonoluminescence and sonochemistry 1687

(a) Gas dynamics

The starting point for the study is the balance of mass energy and momentum inspherical coordinates for an n-component gas mixture as given in Bird et al (1960)The formulation that follows dinoters subtly from SampS hence we state the governingequations of the current work

The problem is formulated in a Lagrangian way where r(a t) is the radial coor-dinate at time t of a gas `particlersquo that was at radial position a at t = 0 In thepresent work the Lagrangian marker particles move with the velocity of the argonie r=t = vA This reference velocity is chosen for convenience because the velocityof the bubble wall moves with the velocity of the (insoluble) argon We note thatthis formulation could be modishy ed to allow the argon to cross the moving interface

In the present Lagrangian formulation dinotusion should be thought of relativeto the background argon velocity which may be large on collapse Mathematicallythe transform from Eulerian to Lagrangian coordinates changes the usual Eulerianmaterial derivative as follows

t+ v

r=)

t+ (v iexcl vA)

a

r

a

where v is the mass average velocityWe make the equations dimensionless using the initial values of radius R0 density

raquo 0 pressure P0 and temperature T0 The velocity scale v0 is (P0=raquo 0)1=2 the massregux scale is raquo 0v0 and the internal energy and enthalpy scale is v2

0 We make theviscosity and thermal conductivity ( middot and k) dimensionless with the values at theinitial state The dinotusion coemacr cients (Dij) are made dimensionless with the aver-age of the binary coe cient matrix Davg Under these transformations the massconservation equations in dimensionless form become

raquo AJ = wA0 (21)

wi

t=

jA

wA0

r2

a2

wi

aiexcl 1

a2

wA

wA0

r2ji

a+

curren i

raquo (22)

where raquo is the density wi is the mass fraction curren i is the net rate of chemical produc-tion or destruction the subscript A denotes argon the subscript i denotes species iand the subscript 0 denotes the initial value The Jacobian of the map between thereference and current conshy gurations J is given by

J =r2

a2

r

a

The radial dinotusive mass regux of species i with respect to the mass average velocityji is

ji = iexcl DMi raquo

Mi

M

1

ReSc

a

r

microxi

a+

Z0M

raquo T M0(xi iexcl wi)

P

a+ KT

i

ln(T )

a

para

where Mi is the molecular weight of species i M is the mean molecular weight atthe local conditions xi is the molar fraction Z0 is the compressibility in the initialstate KT

i is the thermal-dinotusion ratio T is the temperature and P is the pressureThe Schmidt number Sc sup2 middot 0=raquo 0Davg is the ratio of the dinotusivity of momentum to

Proc R Soc Lond A (2000)

1688 B D Storey and A J Szeri

the dinotusivity of mass the Reynolds number is Re sup2 raquo 0R0v0=middot 0 The mean dinotusioncoemacr cient DM

i is a function of temperature pressure and composition It is deshy nedas

DMi =

1 iexcl xiPnj 6= i xj=Dij

where Dij is the binary dinotusion coemacr cient between species i and j The thermal-dinotusion factor is given as

KTi = xi

X

j 6= i

xj not ij

where not ij is the thermal-dinotusion factor between species i and jThe dinotusive mass regux written with the mean dinotusion coemacr cients is an approx-

imation in the interest of computational emacr ciency (Gardiner 1984 Hirschfelder etal 1954) The approximation is valid due to the fact that the other species (besidesargon and water) are present in trace amounts To ensure that the mass reguxes sumto zero as is required for mass conservation the regux of water vapour is computedas minus the sum of the reguxes of all other species rather than in primitive form

The balance of linear momentum written in dimensionless Lagrangian form is

v

t=

jA

wA0

r2

a2

v

aiexcl

wA

wA0

r2

a2

microP

a+ 3

frac12

r

r

a+

frac12

a

para (23)

where v is the mass average velocity and frac12 is the deviatoric stress given as

frac12 =iexcl 4 middot (T P xi)

3Re

sup3v

a

a

riexcl

v

r

acute

The energy equation in these variables is

E

t=

jA

wA0

r2

a2

E

aiexcl

wA

wA0

1

a2

a[r2(q + frac12 v + P v)] (24)

where the radial heat regux is

q = iexcl Cp0T0 raquo 0

P0

k(T P xi)

RePr

T

a

a

r+

nX

i= 1

jiHi

Here Hi is the partial molal enthalpy and Pr sup2 Cp0 middot 0=k0 is the Prandtl numberWe note that

Cp0T0 raquo 0

P0=

reg

reg iexcl 1

for a perfect gas where reg is the ratio of specishy c heats

(b) Transport properties and equation of state

The transport properties ( middot k not ij and Dij) are calculated from equations basedon the ChapmanEnskog theory using a LennardJones 126 potential with correc-tions for high temperatures and densities The individual molecular parameters arecombined in an empirical way to develop enotective mixture parameters for use in thesingle-reguid equations These equations are given in detail in the appendix of SampSbut the essential equations are found in Hirschfelder et al (1954)

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1689

There are a few minor modishy cations to the equations used in SampS In the presentpaper a correction is added in the combination rules for the interaction between polarand non-polar molecules The Eucken correction is used for the thermal conductivityto account for internal degrees of freedom (Hirschfelder et al 1954) Finally thethermal-dinotusion factors are computed using the approximation given by Fristromamp Monchik (1988) for computational emacr ciency We stress that our previous workshows that the results are relatively insensitive to the transport properties as longas the general trends with temperature and pressure are captured

The SoaveRedlichKwong equation of state (EOS) is used with combination rulesfor the empirical constants of the mixture (Reid et al 1987) We use the polynomialdata shy ts for ideal gas specishy c heats at 1 atm from Gardiner (1984) and computedepartures with the EOS While this approach may seem overly simplishy ed comparedwith the EOS used in SampS a similar EOS for water and its reaction products thatallows us to account for shy nite-rate chemistry is not available It was found in SampSand again in the present work that the EOS has no impact on the basic physics ofthe problem and only modishy es the shy nal quantitative answers

(c) Motion and energy transport in the liquid

Rather than solve the NavierStokes equations governing the motion of the liquidwe assume that a RayleighPlesset-type equation for the bubble radius is valid TheRayleighPlesset equation is an ordinary dinoterential equation for the bubble radiuswhich is coupled with the gas dynamics of the interior through the pressure and stressat the bubble wall The form of the equation we use is similar to the one derivedby Prosperetti amp Lezzi (1986) dinotering only in that the present work accounts forthe extra velocity at the interface due to evaporation or condensation The enotect ofevaporationcondensation on the RayleighPlesset equation is very small but theenotect is included for consistency The resulting equation is similar to ones derivedby Yasui (1997a) and by Fujikawa amp Teruaki (1980)

Rather than solve the full energy equation in the liquid we solve the boundary-layer form as described in detail by Fyrillas amp Szeri (1994) The boundary-layerassumption reduces the energy equation to the linear dinotusion equation in Lagrangianboundary-layer coordinates This approximation for the energy equation was used inVuong amp Szeri (1996) and in SampS

(d) Evaporation and condensation

The equation derived from kinetic theory given in Carey (1992) is used to computethe rate of evaporation and condensation at the bubble surface This is written indimensionless variables as

_m00kin =

rMH2 O Z0

2 ordm M0

sup3iexcl PH2 Op

Tin tiexcl P s atp

Tin t

acute

_m00act = frac14 _m00

kin

9gt=

gt(25)

Here _m00kin is the rate at which water molecules hit the interface (from kinetic theory)

frac14 is the accommodation coe cient (sticking probability) _m00act is the rate at which

water molecules hit the interface and undergo phase change PH2 O is the partial

Proc R Soc Lond A (2000)

1690 B D Storey and A J Szeri

pressure of water P s at is the saturation pressure at the temperature of the interfaceTin t M0 is the initial molecular mass of the bubble contents iexcl is a correction forbulk motion to the interface and remains close to one for low mass transfer Weemphasize that this equation is only valid for reasonable temperatures and pressuresand not at the extreme conditions found during a bubble collapse As we argue belowthe evaporation condition is of vanishing consequence during the terminal stages ofbubble collapse when (25) loses validity

There is some discord in measurements of the accommodation coe cient Thevalue we use frac14 = 04 is consistent with that used by Yasui (1997a) (developed frommolecular dynamics simulations) and those recommended by Eames et al (1997)The sensitivity of the results to the accommodation coemacr cient will be consideredlater

(e) Chemical kinetics

Chemical kinetics described by the reaction mechanism determines the produc-tion and destruction of the dinoterent species The reaction mechanism that we use isdescribed in Maas amp Warnatz (1988) The mechanism includes 19 forward and reverseelementary reactions and nine species (Ar H H2 O O2 OH HO2 H2O2) Argononly enters the mechanism as a third body

The net rate of progress of each elementary reaction is the dinoterence in the forwardand reverse rate of progress The forward (reverse) rate of progress is given as theproduct of the reaction rate and the concentrations of all species on the reactant(product) side of the equation The change in concentration of species i with timeis the sum of the net rate of progress of each elementary reaction that containsspecies i The details can be found in several texts (eg Gardiner 1984) In thepresent formulation the chemical kinetics lead to the net chemical production ordestruction term curren i in (22)

The reaction rates are given empirically with the reaction mechanism The Linde-mann form is used to take into account the pressure fall-onot in some of the three-bodyreactions (Gardiner 1984) The high-pressure limit of the reaction rates for availablereactions were taken from Bowman et al (1999)

The elevated pressure in the bubble upon collapse accelerates the rate of progressof the reactions and alters the equilibrium state to which the reactions tend so thatfor a given temperature there is less dissociation at higher pressures This change inequilibrium state with pressure is accounted for in the reaction mechanism becauseforward and reverse reaction rates are related through the equilibrium constantThere is considerable uncertainty in the reaction rates at the high pressures weshall encounter The mechanism and reaction rates we use seem to be the best onespresently available

The reaction mechanism does not include ionization reactions The onset of ion-ization is delayed due to the high pressure The degree of ionization at the conditionswe report is quite low as estimated by the Saha equation Moss et al (1997) putforth the criterion that the onset of ionization in a dense gas is at one-quarter ofthe ionization potential in the present case this temperature is ca 30 000 K Theresults we present are consistent with neglecting ionization based on either of thesecriteria

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1691

(f ) Mass transfer to the liquid

One of the most di cult quantities to determine accurately is the rate of transportof radical species into the liquid While general convectiondinotusionreaction equa-tions can be written for each species dissolved in the liquid this approach involvesmany unknowns For example data on the solubility of OH radicals in liquid watervary by two orders of magnitude (Sander 1998) We know of no solubility data forH or O Moreover there is the possibility of heterogeneous surface reactions (VonSonntag et al 1999) which are poorly understood especially at the time of collapsewhen the liquid very near the bubble interface is supercritical

To avoid these uncertainties we take a very simple approach To determine the rateof radical invasion into the liquid approximately we introduce the uptake coemacr cientpound

pound =Nin iexcl Nou t

Ncoll

(26)

where Nin is the number of molecules invading the liquid Nou t is the number ofmolecules ejected from the liquid and Ncoll is the number of molecules collidingwith the surface (from kinetic theory) This view of mass transfer is oversimplishy edone can easily see that pound is not an intrinsic quantity (Takami et al 1998 Chameides1984)

This approach has nevertheless been used to determine the absorption of radicalspecies onto liquid and solid surfaces The uptake coe cients have been measured forsome of the species in our system over aqueous surfaces (Takami et al 1998 Hansonet al 1992) We shall study the inreguence of uptake coe cient on the results of ourcalculations below

(g) Numerical method

The same Chebyshev collocation method is used that was described in detail inSampS The variables in the gas are approximated by expansions in Chebyshev poly-nomials in the argon Lagrangian coordinate a The temperature in the liquid isapproximated with an expansion in rational Chebyshev polynomials The equationsin the gas evolve with a straightforward explicit predictorcorrector scheme We usestandard operator splitting (Press et al 1988) to solve the chemical kinetics equa-tions separately with a semi-implicit method to avoid time-step restrictions causedby the stinotness of these equations The energy equation in the liquid is integrated intime with a fully implicit method to avoid time-step restrictions associated with theclose spacing of grid points near the bubble wall As an error check global balancesof mass and energy are monitored imbalances always remain below a fraction of aper cent

3 Results and discussion

We begin by presenting detailed results of one particular conshy guration Due to thecomplexity of the problem we shall attempt to present the results in a systematicway in order to demonstrate the importance of certain phenomena The cases to beinvestigated are as follows

Proc R Soc Lond A (2000)

1692 B D Storey and A J Szeri

Table 1 Summary of the direg erent cases investigated in detail

(In all cases the bubble is a 45 m radius argon bubble in water at 298 K acoustically forcedwith a 12 bar pressure amplitude at 265 kHz)

case phase change chemistry OH uptake Rm ax =Rm in ( m) Tm ax (K)

I no no 00 280080 20 900

II yes no 00 313070 9 700

III yes yes 00 317065 7 000

IV(a) yes yes 0001 317065 7 000

IV(b) yes yes 001 317065 7 000

Case I no water vapour allowed in the bubble

Case II water vapour allowed to evaporate and condense but no chemical reactions

Case III water vapour and chemical reactions but no mass transfer across thebubble interface

Cases IV(a) and IV(b) mass transfer of OH across the bubble interface at dif-ferent uptake coe cients

In all cases the bubble is a 45 m-radius argon bubble in 298 K water acousticallyforced with a 12 bar pressure amplitude at 265 kHz These cases are summarized intable 1 With respect to SBSL this bubble was found to be stable to mass exchangeand shape changes though would have minimal light emission (Hilgenfeldt et al 1996 Wu amp Roberts 1998a b) We describe the results for more strongly forcedbubbles below The data we obtain are for steady-state oscillations ie after thedecay of initial transients After presenting the details of these cases we considervariations in the parameters to illustrate interesting trends

(a) Excess water is trapped by the rapid collapse

We shall now argue that signishy cant amounts of water vapour become trapped inthe interior of the bubble when it collapses violently Water is trapped in the interiorbecause the bubble motions become so rapid that the vapour near the centre hasinsu cient time to dinotuse to the bubble wall Non-equilibrium phase change alsoplays a signishy cant role in the amount of water that is trapped All results in thissubsection apply to case II this case allows one to focus on the dynamics of evapora-tion condensation and dinotusion of water inside the bubble without the complicationof chemical reactions

In shy gure 1 we show the radial history of the bubble over one cycle of the acousticforcing in case II The essential features of the bubble dynamics for cases I and II arethe same but the inclusion of water vapour increases the maximum radius and delaysthe collapse The dinoterence is to be expected because case I has no vapour pressureThe compression ratio (Rm ax=Rm in ) for cases I and II is 35 and 44 respectively Thecompression ratio is an approximate way of separating out dinoterences in the resultsdue to dinoterences in the bubble dynamics Typically a higher compression ratio leadsto more compression heating and higher temperatures if all else is equal We returnto this point later

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1693

0 10 20 30

10

20

30

time (micros)

radi

us (

microm

)

Figure 1 Radius of the bubble versus time over one cycle of acoustic forcing in case IIThe essential features for cases I II III and IV are the same

0 10 20 30

10 8

10 9

1010

1011

num

ber

of m

olec

ules

H2O

Ar

time (micros)

Figure 2 Total number of water-vapour molecules contained in the bubble versus time over onecycle of acoustic forcing in case II Note that water comprises ca 14 of the bubble contents atthe end of the main collapse

Proc R Soc Lond A (2000)

1694 B D Storey and A J Szeri

0

4000

8000te

mpe

ratu

re (

K)

1

2

3

time relative to minimum radius (ns)

num

ber

of H

2O m

olec

ules

109

acute-

- 4 - 2 0 2 4

(b)

(a)

Figure 3 Temperature (a) and total number of water molecules contained in the bubble (b) versustime around the point of minimum radius (time = 0) in case II The temperature undergoesorder-of-magnitude changes on this time-scale while the total number of water molecules changesvery little

In shy gure 2 we show the total number of H2O molecules contained within thebubble over the acoustic cycle (case II) A large amount of water evaporates intothe bubble during the main expansion when the pressure is low Water condenses onthe bubble wall during the collapse as the gas pressure increases There is signishy cantwater vapour in the bubble at the main collapse ca 14 (molar basis) of the totalbubble contents is water vapour If the interface were at equilibrium and the speedof mass dinotusion were inshy nite there would be a negligible amount of water in thebubble on collapse

In shy gure 3 we show the temperature history of the centre and the total numberof water molecules in the bubble around the time of minimum radius (time = 0) incase II Over the 10 ns time-scale the total amount of water vapour contained in thebubble is nearly constant whereas the temperature changes by an order of magni-tude This recalls SampS in which it was shown that evolving mixture compositionswere essentially frozen on the time-scale of the temperature peak

The temperature peak may be related to the light emission The equation foroptical power given by Moss et al (1999) with a constant opacity provides an easyestimate for the optical power output as a function of time For case II this estimategives a light emission pulse width (FWHM) of 350 ps which is much shorter than theca 2000 ps temperature peak This optical-pulse-width estimate would be somewhatshortened if the temperature dependence of the opacity were taken into accountThe peak optical power is ca 01 mW although this number is very sensitive to theopacity Typical stable SBSL experiments (at higher forcings than the present case)yield light emission of the order of milliwatts per regash (Gaitan et al 1996 Hiller etal 1992)

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1695

0 5 10 15 20 25

02

04

06

08

10

- 843 ns

- 455 ns

- 299 ns

- 153 ns

- 41 ns

- 15 ns

0 ns - 5 ns

radial position (microm)

mol

ar f

ract

ion

Figure 4 Molar fraction of water vapour versus radius for selected times prior to the moment ofminimum radius (time = 0) in case II Note the development of the non-uniform composition asthe collapse accelerates toward minimum radius In the last 15 ns before minimum radius thecomposition changes by only a few per cent when viewed in the Lagrangian frame instead

There are two mechanisms that determine the excess vapour left behind at thecollapse shy nite-rate mass dinotusion and non-equilibrium phase change We examineeach mechanism and their interaction in turn

(i) Role of direg usion

The shy rst reason for excess water vapour remaining during the collapse is the rapidbubble collapse overwhelming slow mass dinotusion this was extensively discussedin SampS In SampS it was shown that there is a competition between two importanttime-scales the dynamic time-scale of the bubble motions and the mass dinotusiontime-scale These are given explicitly in dimensionless terms as

frac12 d yn = R= _R

frac12 d if = ReScR2

D MH2 O

ordm ReSc1

Rp

T

In the latter we made use of the fact that the dinotusion coe cient of water vapourin the gas mixture D M

H2 O scales as R3pT

The time-scales argument is clarishy ed upon examination of shy gure 4 which showsthe evolving bubble composition as a function of radius at selected times before thetime of minimum radius (case II) In the early part of the collapse the dynamic time-scale is longer than the dinotusion time-scale frac12 d yn frac34 frac12 d if this results in a uniformcomposition of the bubble (upper curves in shy gure 4) As the bubble accelerates into

Proc R Soc Lond A (2000)

1696 B D Storey and A J Szeri

the collapse frac12 d yn sup1 frac12 d if and the decrease in water fraction at the centre begins tolag the decrease in water fraction at the wall (middle curves of shy gure 4) Furtherinto the collapse frac12 d yn frac12 frac12 d if and the water vapour cannot dinotuse to the bubble wallon the time-scale of the rapid collapse This slow dinotusion results in a nearly shy xeddistribution of water vapour (three lowest curves of shy gure 4)

While the three lowest curves in shy gure 4 look quite dinoterent in this Eulerianrepresentation they dinoter by only a few per cent when viewed in the Lagrangianrepresentation (not shown) The curve at iexcl 15 ns is from a time when R = R0 andthe temperature at the centre is ca 750 K The curve at 0 ns is from the time ofminimum radius (015R0) when the temperature at the centre is 9700 K Consistentwith shy gure 3 over times when there are drastic changes in the bubble radius andtemperature the composition of the bubble is almost a material shy eld

(ii) Role of non-equilibrium phase change

Non-equilibrium phase change is the second mechanism by which vapour is trappedin the bubble interior during the collapse By adjusting the accommodation coe -cient one has control over the time-scale for the partial pressure of the water vapourto come to equilibrium with the saturation pressure at the interface The dimension-less condensation time-scale is

frac12 con d =R

frac14

r2 ordm MH2 O

9M0Tin t

When frac12 d yn frac34 frac12 con d the condensation is in quasi-equilibrium with respect to thebubble motions If frac12 d yn frac12 frac12 con d then the bubble collapse is so rapid that no masscan escape the bubble on the time-scale of the collapse

(iii) Interaction of the two mechanisms

It is the competition between the time-scales for mass dinotusion condensation andbubble dynamics that determines which mechanism traps water vapour in the bubbleOnce frac12 d yn frac12 frac12 d if or frac12 d yn frac12 frac12 con d the total amount of water vapour in the bubbleis shy xed Only one of these conditions needs to be met for vapour to be trapped inthe bubble Both mechanisms can however participate to determine the quantity oftrapped vapour

During the bubble collapse the condition frac12 d yn frac12 frac12 d if is reached well before frac12 d yn frac12frac12 con d in case II Therefore in this case (as well as the others presented herein) it isthe slow mass dinotusion that is responsible for trapping vapour in the bubble

For cases where slow mass dinotusion traps the vapour non-equilibrium phasechange helps to determine the amount trapped The accommodation coe cient rep-resents a resistance to condensation at the interface during collapse of the bubble Alower value of accommodation coe cient provides more resistance to phase changeclearly increasing the mass of water retained in the bubble at any time during thecollapse The importance of the non-equilibrium phase change can be estimated bycomparing frac12 d yn with frac12 con d at the time that frac12 d yn = frac12 d if The closer frac12 d yn is to frac12 con d the more important non-equilibrium phase change is in determining the amount ofwater trapped

Colussi et al (1998) developed a model including chemical reactions in a collapsingbubble using an accommodation coe cient of frac14 = 0001 but did not take into

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1697

account the shy nite rate of mass dinotusion The low value of frac14 which works to trapsignishy cant water vapour at collapse was necessary for their model to be consistentwith experimental sonochemistry data When such a low value for frac14 is used thecondition frac12 d yn frac12 frac12 con d is reached well before frac12 d yn frac12 frac12 d if (the opposite of our case)Such a low accommodation coe cient works to trap signishy cant amounts of watervapour in the bubble but in this case the hindered regux at the interface eliminatesmass transfer out of the bubble while the distribution of water vapour within isessentially uniform In the present work we argue that the signishy cant amounts ofwater vapour trapped in the bubble are a consequence of slow dinotusion as well asnon-equilibrium phase change

The temperature of the interface exceeds the critical point for a very brief time(from iexcl 340 ps to 1800 ps relative to the time of minimum radius) For the conditionswe explored the maximum thickness of the supercritical region is quite thin less than1 of the bubble radius ie ca 10 nm We neglect any changes in the liquid equationof state and transport properties when the interface exceeds the critical temperatureThis assumption is justishy ed based on the thinness of the region and the fact that thecritical temperature is reached very late in the collapse Any disturbances caused bythe supercritical interface have insumacr cient time to propagate into the bubble interioron the main collapse Although the evaporation equation is invalid for these shorttimes the total mass added to the bubble is negligible regardless of the magnitudeof evaporative mass regux The issue of the super-critical interface may require futureimprovements but is beyond the scope of the present work

As a shy nal note both thermal and pressure dinotusion also play a role in determiningthe distribution of water vapour during the collapse In the argonwater system thethermal-dinotusion coemacr cient is small and the molecules are of similar size hence pres-sure dinotusion is quite weak Little dinoterence is seen in calculations with and withoutthese other forms of mass dinotusion In a helium or xenon bubble in water thermaland pressure dinotusion are more pronounced and can cause a 1015 dinoterence inthe amount of trapped water vapour In the interest of brevity we will not discussthese results further here

(b) Gas-phase chemical reactions break apart H2O and reduce the temperature

Now we turn to an investigation of what transpires when chemical reactions aretaken into account (case III) We shall show that there is signishy cant time for reactionsto occur ie the time-scale for chemical reaction is similar to that of the bubblecollapse The primarily endothermic reactions absorb a signishy cant amount of energyand further reduce the peak temperatures from cases I and II The distribution ofspecies produced in the bubble is determined by dinotusive transport of water vapourprior to the collapse and heat transfer out of the bubble during the collapse Thesteep temperature gradient at the wall conshy nes the bulk of the chemical reactions tothe bubble interior

(i) Consequences of chemistry

The overall bubble dynamics of cases II and III are essentially the same with acompression ratio of 49 in case III (slightly higher than case II) A convenient pointof departure is to compare the temperature histories of the bubble centre aroundthe main collapse for the three cases we have considered (see shy gure 5) The inclusion

Proc R Soc Lond A (2000)

1698 B D Storey and A J Szeri

- 4 - 2 0 2 410

3

104

tem

pera

ture

(K

)

time relative to minimum radius (ns)

Figure 5 Temperature history of the centre of the bubble around the time of minimum radius(t = 0) in cases I II and III Note the reduction in temperature from case I ( ) to case II( ) as water is allowed to enter the bubble The chemical reactions added in case III (||)further reduce the temperatures

of water vapour (case II) signishy cantly reduces the peak temperature from case IThe decrease in temperature is due mainly to the decrease in the ratio of specishy cheats of the mixture which reduces the compression heating Upon comparison ofcases II and III one observes that the chemical reactions further reduce the peaktemperature due to the endothermic decomposition of the water vapour The centretemperature histories of cases II and III are nearly identical below 4000 K

It is important to understand that the temperature dinoterences in these threecases are not due to subtle dinoterences in bubble dynamics The compression ratio(Rm ax=Rm in ) of the three cases increases from case I to case II to case III whereasthe temperature decreases in this order This decrease in temperature occurs despitethe fact that a higher compression ratio will lead to more compression heating in thebubble collapse all other things being equal In all these cases there are no shocksinside the bubble Shocks will be considered in x 3 e

In shy gure 6 we show the history of the total number of molecules of the dinoterentspecies as a function of time on the same time-scale as in shy gure 5 The lowest curve isthe total amount of water vapour only The next curve is the total amount of watervapour plus the total amount of H2 Each curve represents the addition of one speciesas labelled until the uppermost curve denotes everything but argon We do not labelHO2 and H2O2 as they appear only in trace amounts One observes which speciesare most abundant during the time of the main collapse Note there is a relativelylarge amount of undissociated water considering the elevated temperatures This isprimarily a consequence of the high pressure suppressing the dissociation of H2O

From shy gures 5 and 6 one observes that the onset of signishy cant chemical activity in

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1699

- 4 - 2 0 2 4

109

1010

time relative to minimum radius (ns)

num

ber

of m

olec

ules

H2O

H2

O2

OH

O

H

Figure 6 Cumulative number of molecules of each species in case III The lowest curve is thetotal number of water molecules the next curve is total water + H2 and so on H2 O2 and HO2

are neglected in this plot as they appear in trace amounts

the bubble corresponds to a centre temperature of ca 4000 K Therefore the reducedtemperature of case III shown in shy gure 5 can be attributed to the endothermicchemical reactions Although the bubble dynamics are quite rapid the chemicalreactions keep pace as is shown by the similar time-scales of the changing quantitiesin shy gures 5 and 6 The rapid rate at which the reactions progress is due to the veryhigh density Recall that the rate of progress of each reaction is directly related tothe species concentration (mol cmiexcl3)

(ii) Dynamics and distribution of OH production

In sonochemistry applications the invasion of OH radicals into the liquid isthought to be responsible for much of the complex and useful chemistry that occursTherefore we shall focus on the behaviour of this species in particular In shy gure 7 weplot the total number of OH molecules contained in the bubble on two time-scalesover one acoustic cycle in shy gure 7a and around the time of collapse in shy gure 7b Theinitial concentrations of all species were selected so that the chemical production anddestruction balanced in one acoustic cycle The main collapse is associated with arapid creation of OH once the temperature exceeds 4000 K

Upon rebound the bubble rapidly cools and the OH and other products tend torecombine back into H2O These recombination reactions proceed more slowly thanthe dissociation reactions upon collapse The result is that the bubble is left with asuper-equilibrium amount of OH after it has cooled to ambient temperature Thissuper-equilibrium OH then slowly decays through the rest of the cycle with briefspurts of production and destruction at each of the after-bounces

Proc R Soc Lond A (2000)

1700 B D Storey and A J Szeri

0 10 20 30time (micros)

- 4 - 2 0 2 4

time relative to minimum radius (ns)

106

1010

104

106

108

108

num

ber

of O

H

radi

cals

nu

mbe

r of

OH

ra

dica

ls

Figure 7 Total number of OH molecules contained in the bubble versus time over one acousticcycle (a) and around the time of minimum radius (b) in case III The macrrst sharp peak in (a)is expanded in (b) Subsequent peaks in (a) correspond to local radius minima in the bubbledynamics The initial amounts of OH and all the other species were chosen for no net speciesproduction or destruction over an acoustic cycle

The distribution of OH in the bubble is characterized by a fairly uniform con-centration through much of the bubble interior and a very sharp gradient near thewall This sharp gradient is a result of heat transfer during the collapse Duringcompression conduction between the hot interior and the cool wall sets up a thintemperature boundary layer This boundary layer conshy nes the chemical reactions tothe interior away from the interface The cooler wall leads to a much lower concen-tration of OH at the interface than in the centre at the time of the main collapseIn shy gure 8 we show the molar concentration history of OH at the centre and at theinterface When the bubble is at minimum radius and the OH concentration at thecentre is at its maximum the concentration at the wall is three orders of magnitudelower

Because the chemistry time-scale is much faster than the dinotusion time-scale OHis produced where water vapour was trapped at high enough temperature The liquidinterface is insulated from the high concentration of OH generated deep inside in thebubble The only OH produced at the centre that reaches the bubble interface is theslowly decaying super-equilibrium OH that is left inside the bubble on re-expansion

(c) OH uptake by the liquid does not indeg uence the bubble interior

The inclusion of liquid uptake of OH (case IV) adds further complications wenow try to illuminate some of the main qualitative features We shall argue that OHuptake has little enotect on the gas dynamics and chemical production

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1701

- 4 - 2 0 2 410 - 10

10 - 8

10 - 6

10 - 4

10 - 2

time relative to minimum radius (ns)

conc

entr

atio

n of

OH

(m

ol c

m-3

)

Figure 8 Molar concentration of OH at the centre (||) and the interface ( ) versus timearound the point of minimum radius Note that the concentration at the centre is about threeorders of magnitude greater than at the interface at the point of minimum radius

Due to dinotering ideas on the proper value of the uptake coemacr cient the values areset at pound = 0001 and pound = 001 (cases IV(a) and IV(b)) consistent with the work ofothers to which we made reference earlier The bubble dynamics and temperatureproshy les for cases IV(a) and IV(b) are indistinguishable from those for case III Inshy gure 9 we show the total number of OH molecules in the bubble versus time ontwo dinoterent scales for cases III IV(a) and IV(b) The history for one acoustic cycleis plotted in shy gure 9a and the history concentrated at the main collapse is plottedin shy gure 9b

In the main expansion when the bubble is cool the OH resident in the bubblewill invade the liquid as can be seen by the decrease in the amount of OH at theend of the long expansion as the uptake increases When the bubble collapses thechemical production of OH from recently evaporated water rapidly overwhelms theOH uptake by the liquid As the bubble proceeds into the collapse the curves oftotal OH content for dinoterent uptakes become indistinguishable The total amountof OH in the bubble is dominated by the amount created in the interior away fromthe wall The total amount of OH lost to the liquid during a collapse is only afraction of the amount contained in the bubble Hence the peak amount of OH inthe bubble during the collapse is insensitive to the uptake coe cient of OH Similarstatements apply to uptake of other species

In shy gure 10 we show for the purposes of qualitative discussion the total massof OH exiting the bubble in cases IV(a) and IV(b) While the OH uptake coef-shy cient does not inreguence the total amount of OH contained in the bubble at themain collapse there are large dinoterences in the amount invading the liquid over

Proc R Soc Lond A (2000)

1702 B D Storey and A J Szeri

0 10 20 30

106

1010

102

time (micros)

- 4 - 2 0 2 4104

106

108

time relative to minimum radius (ns)

1010

(b)

(a)nu

mbe

r of

OH

ra

dica

ls

num

ber

of O

H

radi

cals

Figure 9 Comparison of total amount number of OH molecules in the bubble versus time overone acoustic cycle (a) and focused around the time of minimum radius (b) in cases III (||)IV(a) ( ) and IV(b) ( ) The uptake of OH plays a role during the slow expansionswhen the resident amount of OH left in the bubble is well above chemical equilibrium At thecollapse OH uptake does not indeg uence the total amount of OH production

a cycle Much of the OH uptake occurs during the main collapse and the subse-quent re-expansion when the concentration of OH at the wall is at its maximum(see shy gure 8) Further uptake occurs throughout the cycle at each of the after-bounces

(d ) Inclusion of water vapour alters the ereg ect of increased forcing

Next we shall compare the results of cases I II and III with all parameters shy xedexcept for the acoustic pressure amplitude which varies between 10 and 13 barWe that note this is experimentally awkward for SBSL because pressure amplitudeand ambient radius are dependent quantities through the mass exchange equilibriumof rectishy ed dinotusion (Hilgenfeldt et al 1996) The way we present these results isnonetheless useful as it demonstrates the importance of water vapour in the under-standing of sonoluminescence and sonochemistry

Because the bubble dynamics are altered by the inclusion of water vapour weuse the compression ratio as the distinguishing parameter In shy gure 11 we plot themaximum centre temperature versus the compression ratio for cases I II and III Thecalculations in shy gure 11 are devoid of strong shocks a matter to which we devotemore attention below If there are strong shocks then shy gure 11 will look the samein terms of a suitably averaged temperature rather than the centre temperature

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1703

0 10 20 30102

103

104

105

106

num

ber

of O

H

radi

cals

inva

ding

liqu

id

time (micros)

Figure 10 Comparison of the amount of OH absorbed by the liquid versus time for cases IV(a)(|| uptake coeplusmn cient pound = 0001) and IV(b) ( pound = 001) Most of the OH uptakeoccurs at the main collapse and the subsequent after-bounce

In case I the plot shows linear behaviour (on the loglog scale) One observes thatwhen water vapour is not taken into account stronger forcing yields more extremetemperatures

In case II the relationship between compression ratio and maximum temperature isvery dinoterent First we note that the amount of water trapped in the bubble increaseswith compression ratio If we revisit the water-trapping mechanism (x 3 a (iii)) thistrend is readily explained The increase in water-vapour content reduces the ratioof specishy c heats of the gasvapour mixture and hence reduces the compressionheating

At low compression ratios in case II the temperature increases with compressionratio in a manner very similar to case I At low compression ratios the amountof water at collapse is very small the bubble is nearly pure argon As the forcingis increased more water is trapped and the maximum temperature of the bubblebecomes nearly constant In this range of bubble dynamics there is an almost per-fectly compensating enotect (with respect to peak temperature) between the decreasein specishy c-heat ratio and the increase in compression Case III is very similar tocase II except that the plateau in temperature occurs at a lower temperature aconsequence of the endothermic chemical reactions

(e) Inclusion of water vapour promotes the formation of shocks

Water vapour is of great importance in the formation of shocks in very stronglyforced bubbles We shall brieregy consider a 60 m bubble forced with a pressureamplitude of 14 bar at a frequency of 206 kHz (Case A1 in Moss et al (1999))

Proc R Soc Lond A (2000)

1704 B D Storey and A J Szeri

10 1 10 2

10 3

10 4

10 5

Rmax R min

max

imum

tem

pera

ture

(K

)

Figure 11 Change in centre temperature versus compression ratio in cases I ( ) II ( curren )and III (+) In this plot all parameters were held constant except the pressure amplitudewhich was varied between 10 and 13 bar Case I shows a steady increase in temperature withcompression ratio Cases II and III reach a maximum temperature as the forcing is increased

When water vapour is neglected a compression wave is launched into the bubble oncollapse but the wave does not steepen into a shock When water vapour is takeninto account (with or without reactions) the compression wave steepens into a strongshock very near the centre of the bubble Vuong et al (1999) showed that the rateof steepening depends on the amount of compression heating before the compressionwave is launched

In very recent work Lin amp Szeri (2000) used the wavefront expansion technique todetermine the evolution of a compression or rarefaction wavefront that is propagatingspherically into a collapsing or expanding gas of non-uniform entropy This analyticalwork supports the idea that when the speed of sound is higher at the centre (dueto temperature and composition inhomogeneities) shock formation is delayed orprevented Furthermore this work demonstrated that the inward radial velocity alsohinders the steepening of compression waves into shocks

Consistent with the arguments made in Lin amp Szeri (2000) Vuong et al (1999)and Moss et al (1999) our results show that the decrease in the ratio of specishy cheats due to an increase in the water-vapour content can promote shock formationThe point we want to emphasize is that the question of shock formation cannot beundertaken without consideration of water vapour

With respect to total chemical production shocks are of negligible consequenceThe shock inreguences only about the inner 10 of the radius which is only 01 ofthe volume Therefore the increased production of OH at the centre does not anotectthe total amount produced in the bubble

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

1686 B D Storey and A J Szeri

a bubble as it reaches a maximum radius During collapse water vapour tends tocondense at the wall

In an earlier work (Storey amp Szeri (1999) henceforth referred to as SampS) westudied the dinotusive segregation of noble-gas mixtures in collapsing bubbles It wasshown that the time-scale of mixture segregation due to thermal and pressure dif-fusion was much longer than the time-scale of the temperature peak caused by theinertial collapse and rebound As a result the inhomogeneous bubble compositionwas essentially frozen on the time-scale of the temperature peak In the presentpaper we reason along the same lines and shy nd that the bubble motions becomeso rapid on collapse that the water vapour at the centre has insumacr cient time toescape Therefore excess water vapour is left behind in the bubble interior throughthe violent collapse

The trapped water vapour subsequently undergoes rapid chemical reactions Theseprimarily endothermic reactions take up a signishy cant amount of the focused energyand are the basis for production of the OH radical that is of keen interest in sono-chemistry

Previous authors have considered these matters Work by Kamath et al (1993)separated the chemical kinetics from a RayleighPlesset model to estimate the pro-duction of OH radicals Gong amp Hart (1998) coupled chemical reactions with aRayleighPlesset equation and captured some trends observed in sonochemistryexperiments Yasui (1997b) accounted for chemical reactions and non-equilibriumphase change in a collapsing air bubble under SBSL conditions Colussi et al (1998)also coupled bubble motions non-equilibrium phase change and chemical reactionsin a sonochemistry model Sochard et al (1997) modelled free-radical productionin mildly forced bubbles by accounting for non-equilibrium phase change and gasvapour inter-dinotusion with a RayleighPlesset model assuming chemical equilibriumat all times

Each of these approaches makes dinoterent simplifying assumptions which may bevalid at mild forcings In this paper we will relax many of the usual assumptionsand investigate the physics and chemistry of more strongly forced bubbles in detailIn particular we will show that the inter-dinotusion of gas and vapour is crucial topredicting the amount of OH produced in the bubble and the peak temperatureachieved Recently Moss et al (1999) found that inclusion of water vapour is crucialto a correct determination of photon yields and light pulse widths in SBSL we revisitthis point at the end of the paper

In what follows we examine the transport of water vapour into and out of thebubble The NavierStokes equations of a gas mixture are coupled to a reactionmechanism to determine the fate and consequences of trapped water vapour

2 Formulation

We shall assume that the single bubble is spherical and is composed of argon watervapour and reaction products (H H2 O O2 OH HO2 H2O2) For conveniencethe argon is regarded as insoluble in the liquid water on the time-scale of the acousticforcing In SampS we showed that the regux of argon into the liquid has a negligible enotecton the gas dynamics as the total mass of argon contained in the bubble reguctuatesby only ca 1 over one cycle of forcing Radiation heat transfer and light emissionhave negligible enotects on the gas dynamics (Moss et al 1999)

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1687

(a) Gas dynamics

The starting point for the study is the balance of mass energy and momentum inspherical coordinates for an n-component gas mixture as given in Bird et al (1960)The formulation that follows dinoters subtly from SampS hence we state the governingequations of the current work

The problem is formulated in a Lagrangian way where r(a t) is the radial coor-dinate at time t of a gas `particlersquo that was at radial position a at t = 0 In thepresent work the Lagrangian marker particles move with the velocity of the argonie r=t = vA This reference velocity is chosen for convenience because the velocityof the bubble wall moves with the velocity of the (insoluble) argon We note thatthis formulation could be modishy ed to allow the argon to cross the moving interface

In the present Lagrangian formulation dinotusion should be thought of relativeto the background argon velocity which may be large on collapse Mathematicallythe transform from Eulerian to Lagrangian coordinates changes the usual Eulerianmaterial derivative as follows

t+ v

r=)

t+ (v iexcl vA)

a

r

a

where v is the mass average velocityWe make the equations dimensionless using the initial values of radius R0 density

raquo 0 pressure P0 and temperature T0 The velocity scale v0 is (P0=raquo 0)1=2 the massregux scale is raquo 0v0 and the internal energy and enthalpy scale is v2

0 We make theviscosity and thermal conductivity ( middot and k) dimensionless with the values at theinitial state The dinotusion coemacr cients (Dij) are made dimensionless with the aver-age of the binary coe cient matrix Davg Under these transformations the massconservation equations in dimensionless form become

raquo AJ = wA0 (21)

wi

t=

jA

wA0

r2

a2

wi

aiexcl 1

a2

wA

wA0

r2ji

a+

curren i

raquo (22)

where raquo is the density wi is the mass fraction curren i is the net rate of chemical produc-tion or destruction the subscript A denotes argon the subscript i denotes species iand the subscript 0 denotes the initial value The Jacobian of the map between thereference and current conshy gurations J is given by

J =r2

a2

r

a

The radial dinotusive mass regux of species i with respect to the mass average velocityji is

ji = iexcl DMi raquo

Mi

M

1

ReSc

a

r

microxi

a+

Z0M

raquo T M0(xi iexcl wi)

P

a+ KT

i

ln(T )

a

para

where Mi is the molecular weight of species i M is the mean molecular weight atthe local conditions xi is the molar fraction Z0 is the compressibility in the initialstate KT

i is the thermal-dinotusion ratio T is the temperature and P is the pressureThe Schmidt number Sc sup2 middot 0=raquo 0Davg is the ratio of the dinotusivity of momentum to

Proc R Soc Lond A (2000)

1688 B D Storey and A J Szeri

the dinotusivity of mass the Reynolds number is Re sup2 raquo 0R0v0=middot 0 The mean dinotusioncoemacr cient DM

i is a function of temperature pressure and composition It is deshy nedas

DMi =

1 iexcl xiPnj 6= i xj=Dij

where Dij is the binary dinotusion coemacr cient between species i and j The thermal-dinotusion factor is given as

KTi = xi

X

j 6= i

xj not ij

where not ij is the thermal-dinotusion factor between species i and jThe dinotusive mass regux written with the mean dinotusion coemacr cients is an approx-

imation in the interest of computational emacr ciency (Gardiner 1984 Hirschfelder etal 1954) The approximation is valid due to the fact that the other species (besidesargon and water) are present in trace amounts To ensure that the mass reguxes sumto zero as is required for mass conservation the regux of water vapour is computedas minus the sum of the reguxes of all other species rather than in primitive form

The balance of linear momentum written in dimensionless Lagrangian form is

v

t=

jA

wA0

r2

a2

v

aiexcl

wA

wA0

r2

a2

microP

a+ 3

frac12

r

r

a+

frac12

a

para (23)

where v is the mass average velocity and frac12 is the deviatoric stress given as

frac12 =iexcl 4 middot (T P xi)

3Re

sup3v

a

a

riexcl

v

r

acute

The energy equation in these variables is

E

t=

jA

wA0

r2

a2

E

aiexcl

wA

wA0

1

a2

a[r2(q + frac12 v + P v)] (24)

where the radial heat regux is

q = iexcl Cp0T0 raquo 0

P0

k(T P xi)

RePr

T

a

a

r+

nX

i= 1

jiHi

Here Hi is the partial molal enthalpy and Pr sup2 Cp0 middot 0=k0 is the Prandtl numberWe note that

Cp0T0 raquo 0

P0=

reg

reg iexcl 1

for a perfect gas where reg is the ratio of specishy c heats

(b) Transport properties and equation of state

The transport properties ( middot k not ij and Dij) are calculated from equations basedon the ChapmanEnskog theory using a LennardJones 126 potential with correc-tions for high temperatures and densities The individual molecular parameters arecombined in an empirical way to develop enotective mixture parameters for use in thesingle-reguid equations These equations are given in detail in the appendix of SampSbut the essential equations are found in Hirschfelder et al (1954)

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1689

There are a few minor modishy cations to the equations used in SampS In the presentpaper a correction is added in the combination rules for the interaction between polarand non-polar molecules The Eucken correction is used for the thermal conductivityto account for internal degrees of freedom (Hirschfelder et al 1954) Finally thethermal-dinotusion factors are computed using the approximation given by Fristromamp Monchik (1988) for computational emacr ciency We stress that our previous workshows that the results are relatively insensitive to the transport properties as longas the general trends with temperature and pressure are captured

The SoaveRedlichKwong equation of state (EOS) is used with combination rulesfor the empirical constants of the mixture (Reid et al 1987) We use the polynomialdata shy ts for ideal gas specishy c heats at 1 atm from Gardiner (1984) and computedepartures with the EOS While this approach may seem overly simplishy ed comparedwith the EOS used in SampS a similar EOS for water and its reaction products thatallows us to account for shy nite-rate chemistry is not available It was found in SampSand again in the present work that the EOS has no impact on the basic physics ofthe problem and only modishy es the shy nal quantitative answers

(c) Motion and energy transport in the liquid

Rather than solve the NavierStokes equations governing the motion of the liquidwe assume that a RayleighPlesset-type equation for the bubble radius is valid TheRayleighPlesset equation is an ordinary dinoterential equation for the bubble radiuswhich is coupled with the gas dynamics of the interior through the pressure and stressat the bubble wall The form of the equation we use is similar to the one derivedby Prosperetti amp Lezzi (1986) dinotering only in that the present work accounts forthe extra velocity at the interface due to evaporation or condensation The enotect ofevaporationcondensation on the RayleighPlesset equation is very small but theenotect is included for consistency The resulting equation is similar to ones derivedby Yasui (1997a) and by Fujikawa amp Teruaki (1980)

Rather than solve the full energy equation in the liquid we solve the boundary-layer form as described in detail by Fyrillas amp Szeri (1994) The boundary-layerassumption reduces the energy equation to the linear dinotusion equation in Lagrangianboundary-layer coordinates This approximation for the energy equation was used inVuong amp Szeri (1996) and in SampS

(d) Evaporation and condensation

The equation derived from kinetic theory given in Carey (1992) is used to computethe rate of evaporation and condensation at the bubble surface This is written indimensionless variables as

_m00kin =

rMH2 O Z0

2 ordm M0

sup3iexcl PH2 Op

Tin tiexcl P s atp

Tin t

acute

_m00act = frac14 _m00

kin

9gt=

gt(25)

Here _m00kin is the rate at which water molecules hit the interface (from kinetic theory)

frac14 is the accommodation coe cient (sticking probability) _m00act is the rate at which

water molecules hit the interface and undergo phase change PH2 O is the partial

Proc R Soc Lond A (2000)

1690 B D Storey and A J Szeri

pressure of water P s at is the saturation pressure at the temperature of the interfaceTin t M0 is the initial molecular mass of the bubble contents iexcl is a correction forbulk motion to the interface and remains close to one for low mass transfer Weemphasize that this equation is only valid for reasonable temperatures and pressuresand not at the extreme conditions found during a bubble collapse As we argue belowthe evaporation condition is of vanishing consequence during the terminal stages ofbubble collapse when (25) loses validity

There is some discord in measurements of the accommodation coe cient Thevalue we use frac14 = 04 is consistent with that used by Yasui (1997a) (developed frommolecular dynamics simulations) and those recommended by Eames et al (1997)The sensitivity of the results to the accommodation coemacr cient will be consideredlater

(e) Chemical kinetics

Chemical kinetics described by the reaction mechanism determines the produc-tion and destruction of the dinoterent species The reaction mechanism that we use isdescribed in Maas amp Warnatz (1988) The mechanism includes 19 forward and reverseelementary reactions and nine species (Ar H H2 O O2 OH HO2 H2O2) Argononly enters the mechanism as a third body

The net rate of progress of each elementary reaction is the dinoterence in the forwardand reverse rate of progress The forward (reverse) rate of progress is given as theproduct of the reaction rate and the concentrations of all species on the reactant(product) side of the equation The change in concentration of species i with timeis the sum of the net rate of progress of each elementary reaction that containsspecies i The details can be found in several texts (eg Gardiner 1984) In thepresent formulation the chemical kinetics lead to the net chemical production ordestruction term curren i in (22)

The reaction rates are given empirically with the reaction mechanism The Linde-mann form is used to take into account the pressure fall-onot in some of the three-bodyreactions (Gardiner 1984) The high-pressure limit of the reaction rates for availablereactions were taken from Bowman et al (1999)

The elevated pressure in the bubble upon collapse accelerates the rate of progressof the reactions and alters the equilibrium state to which the reactions tend so thatfor a given temperature there is less dissociation at higher pressures This change inequilibrium state with pressure is accounted for in the reaction mechanism becauseforward and reverse reaction rates are related through the equilibrium constantThere is considerable uncertainty in the reaction rates at the high pressures weshall encounter The mechanism and reaction rates we use seem to be the best onespresently available

The reaction mechanism does not include ionization reactions The onset of ion-ization is delayed due to the high pressure The degree of ionization at the conditionswe report is quite low as estimated by the Saha equation Moss et al (1997) putforth the criterion that the onset of ionization in a dense gas is at one-quarter ofthe ionization potential in the present case this temperature is ca 30 000 K Theresults we present are consistent with neglecting ionization based on either of thesecriteria

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1691

(f ) Mass transfer to the liquid

One of the most di cult quantities to determine accurately is the rate of transportof radical species into the liquid While general convectiondinotusionreaction equa-tions can be written for each species dissolved in the liquid this approach involvesmany unknowns For example data on the solubility of OH radicals in liquid watervary by two orders of magnitude (Sander 1998) We know of no solubility data forH or O Moreover there is the possibility of heterogeneous surface reactions (VonSonntag et al 1999) which are poorly understood especially at the time of collapsewhen the liquid very near the bubble interface is supercritical

To avoid these uncertainties we take a very simple approach To determine the rateof radical invasion into the liquid approximately we introduce the uptake coemacr cientpound

pound =Nin iexcl Nou t

Ncoll

(26)

where Nin is the number of molecules invading the liquid Nou t is the number ofmolecules ejected from the liquid and Ncoll is the number of molecules collidingwith the surface (from kinetic theory) This view of mass transfer is oversimplishy edone can easily see that pound is not an intrinsic quantity (Takami et al 1998 Chameides1984)

This approach has nevertheless been used to determine the absorption of radicalspecies onto liquid and solid surfaces The uptake coe cients have been measured forsome of the species in our system over aqueous surfaces (Takami et al 1998 Hansonet al 1992) We shall study the inreguence of uptake coe cient on the results of ourcalculations below

(g) Numerical method

The same Chebyshev collocation method is used that was described in detail inSampS The variables in the gas are approximated by expansions in Chebyshev poly-nomials in the argon Lagrangian coordinate a The temperature in the liquid isapproximated with an expansion in rational Chebyshev polynomials The equationsin the gas evolve with a straightforward explicit predictorcorrector scheme We usestandard operator splitting (Press et al 1988) to solve the chemical kinetics equa-tions separately with a semi-implicit method to avoid time-step restrictions causedby the stinotness of these equations The energy equation in the liquid is integrated intime with a fully implicit method to avoid time-step restrictions associated with theclose spacing of grid points near the bubble wall As an error check global balancesof mass and energy are monitored imbalances always remain below a fraction of aper cent

3 Results and discussion

We begin by presenting detailed results of one particular conshy guration Due to thecomplexity of the problem we shall attempt to present the results in a systematicway in order to demonstrate the importance of certain phenomena The cases to beinvestigated are as follows

Proc R Soc Lond A (2000)

1692 B D Storey and A J Szeri

Table 1 Summary of the direg erent cases investigated in detail

(In all cases the bubble is a 45 m radius argon bubble in water at 298 K acoustically forcedwith a 12 bar pressure amplitude at 265 kHz)

case phase change chemistry OH uptake Rm ax =Rm in ( m) Tm ax (K)

I no no 00 280080 20 900

II yes no 00 313070 9 700

III yes yes 00 317065 7 000

IV(a) yes yes 0001 317065 7 000

IV(b) yes yes 001 317065 7 000

Case I no water vapour allowed in the bubble

Case II water vapour allowed to evaporate and condense but no chemical reactions

Case III water vapour and chemical reactions but no mass transfer across thebubble interface

Cases IV(a) and IV(b) mass transfer of OH across the bubble interface at dif-ferent uptake coe cients

In all cases the bubble is a 45 m-radius argon bubble in 298 K water acousticallyforced with a 12 bar pressure amplitude at 265 kHz These cases are summarized intable 1 With respect to SBSL this bubble was found to be stable to mass exchangeand shape changes though would have minimal light emission (Hilgenfeldt et al 1996 Wu amp Roberts 1998a b) We describe the results for more strongly forcedbubbles below The data we obtain are for steady-state oscillations ie after thedecay of initial transients After presenting the details of these cases we considervariations in the parameters to illustrate interesting trends

(a) Excess water is trapped by the rapid collapse

We shall now argue that signishy cant amounts of water vapour become trapped inthe interior of the bubble when it collapses violently Water is trapped in the interiorbecause the bubble motions become so rapid that the vapour near the centre hasinsu cient time to dinotuse to the bubble wall Non-equilibrium phase change alsoplays a signishy cant role in the amount of water that is trapped All results in thissubsection apply to case II this case allows one to focus on the dynamics of evapora-tion condensation and dinotusion of water inside the bubble without the complicationof chemical reactions

In shy gure 1 we show the radial history of the bubble over one cycle of the acousticforcing in case II The essential features of the bubble dynamics for cases I and II arethe same but the inclusion of water vapour increases the maximum radius and delaysthe collapse The dinoterence is to be expected because case I has no vapour pressureThe compression ratio (Rm ax=Rm in ) for cases I and II is 35 and 44 respectively Thecompression ratio is an approximate way of separating out dinoterences in the resultsdue to dinoterences in the bubble dynamics Typically a higher compression ratio leadsto more compression heating and higher temperatures if all else is equal We returnto this point later

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1693

0 10 20 30

10

20

30

time (micros)

radi

us (

microm

)

Figure 1 Radius of the bubble versus time over one cycle of acoustic forcing in case IIThe essential features for cases I II III and IV are the same

0 10 20 30

10 8

10 9

1010

1011

num

ber

of m

olec

ules

H2O

Ar

time (micros)

Figure 2 Total number of water-vapour molecules contained in the bubble versus time over onecycle of acoustic forcing in case II Note that water comprises ca 14 of the bubble contents atthe end of the main collapse

Proc R Soc Lond A (2000)

1694 B D Storey and A J Szeri

0

4000

8000te

mpe

ratu

re (

K)

1

2

3

time relative to minimum radius (ns)

num

ber

of H

2O m

olec

ules

109

acute-

- 4 - 2 0 2 4

(b)

(a)

Figure 3 Temperature (a) and total number of water molecules contained in the bubble (b) versustime around the point of minimum radius (time = 0) in case II The temperature undergoesorder-of-magnitude changes on this time-scale while the total number of water molecules changesvery little

In shy gure 2 we show the total number of H2O molecules contained within thebubble over the acoustic cycle (case II) A large amount of water evaporates intothe bubble during the main expansion when the pressure is low Water condenses onthe bubble wall during the collapse as the gas pressure increases There is signishy cantwater vapour in the bubble at the main collapse ca 14 (molar basis) of the totalbubble contents is water vapour If the interface were at equilibrium and the speedof mass dinotusion were inshy nite there would be a negligible amount of water in thebubble on collapse

In shy gure 3 we show the temperature history of the centre and the total numberof water molecules in the bubble around the time of minimum radius (time = 0) incase II Over the 10 ns time-scale the total amount of water vapour contained in thebubble is nearly constant whereas the temperature changes by an order of magni-tude This recalls SampS in which it was shown that evolving mixture compositionswere essentially frozen on the time-scale of the temperature peak

The temperature peak may be related to the light emission The equation foroptical power given by Moss et al (1999) with a constant opacity provides an easyestimate for the optical power output as a function of time For case II this estimategives a light emission pulse width (FWHM) of 350 ps which is much shorter than theca 2000 ps temperature peak This optical-pulse-width estimate would be somewhatshortened if the temperature dependence of the opacity were taken into accountThe peak optical power is ca 01 mW although this number is very sensitive to theopacity Typical stable SBSL experiments (at higher forcings than the present case)yield light emission of the order of milliwatts per regash (Gaitan et al 1996 Hiller etal 1992)

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1695

0 5 10 15 20 25

02

04

06

08

10

- 843 ns

- 455 ns

- 299 ns

- 153 ns

- 41 ns

- 15 ns

0 ns - 5 ns

radial position (microm)

mol

ar f

ract

ion

Figure 4 Molar fraction of water vapour versus radius for selected times prior to the moment ofminimum radius (time = 0) in case II Note the development of the non-uniform composition asthe collapse accelerates toward minimum radius In the last 15 ns before minimum radius thecomposition changes by only a few per cent when viewed in the Lagrangian frame instead

There are two mechanisms that determine the excess vapour left behind at thecollapse shy nite-rate mass dinotusion and non-equilibrium phase change We examineeach mechanism and their interaction in turn

(i) Role of direg usion

The shy rst reason for excess water vapour remaining during the collapse is the rapidbubble collapse overwhelming slow mass dinotusion this was extensively discussedin SampS In SampS it was shown that there is a competition between two importanttime-scales the dynamic time-scale of the bubble motions and the mass dinotusiontime-scale These are given explicitly in dimensionless terms as

frac12 d yn = R= _R

frac12 d if = ReScR2

D MH2 O

ordm ReSc1

Rp

T

In the latter we made use of the fact that the dinotusion coe cient of water vapourin the gas mixture D M

H2 O scales as R3pT

The time-scales argument is clarishy ed upon examination of shy gure 4 which showsthe evolving bubble composition as a function of radius at selected times before thetime of minimum radius (case II) In the early part of the collapse the dynamic time-scale is longer than the dinotusion time-scale frac12 d yn frac34 frac12 d if this results in a uniformcomposition of the bubble (upper curves in shy gure 4) As the bubble accelerates into

Proc R Soc Lond A (2000)

1696 B D Storey and A J Szeri

the collapse frac12 d yn sup1 frac12 d if and the decrease in water fraction at the centre begins tolag the decrease in water fraction at the wall (middle curves of shy gure 4) Furtherinto the collapse frac12 d yn frac12 frac12 d if and the water vapour cannot dinotuse to the bubble wallon the time-scale of the rapid collapse This slow dinotusion results in a nearly shy xeddistribution of water vapour (three lowest curves of shy gure 4)

While the three lowest curves in shy gure 4 look quite dinoterent in this Eulerianrepresentation they dinoter by only a few per cent when viewed in the Lagrangianrepresentation (not shown) The curve at iexcl 15 ns is from a time when R = R0 andthe temperature at the centre is ca 750 K The curve at 0 ns is from the time ofminimum radius (015R0) when the temperature at the centre is 9700 K Consistentwith shy gure 3 over times when there are drastic changes in the bubble radius andtemperature the composition of the bubble is almost a material shy eld

(ii) Role of non-equilibrium phase change

Non-equilibrium phase change is the second mechanism by which vapour is trappedin the bubble interior during the collapse By adjusting the accommodation coe -cient one has control over the time-scale for the partial pressure of the water vapourto come to equilibrium with the saturation pressure at the interface The dimension-less condensation time-scale is

frac12 con d =R

frac14

r2 ordm MH2 O

9M0Tin t

When frac12 d yn frac34 frac12 con d the condensation is in quasi-equilibrium with respect to thebubble motions If frac12 d yn frac12 frac12 con d then the bubble collapse is so rapid that no masscan escape the bubble on the time-scale of the collapse

(iii) Interaction of the two mechanisms

It is the competition between the time-scales for mass dinotusion condensation andbubble dynamics that determines which mechanism traps water vapour in the bubbleOnce frac12 d yn frac12 frac12 d if or frac12 d yn frac12 frac12 con d the total amount of water vapour in the bubbleis shy xed Only one of these conditions needs to be met for vapour to be trapped inthe bubble Both mechanisms can however participate to determine the quantity oftrapped vapour

During the bubble collapse the condition frac12 d yn frac12 frac12 d if is reached well before frac12 d yn frac12frac12 con d in case II Therefore in this case (as well as the others presented herein) it isthe slow mass dinotusion that is responsible for trapping vapour in the bubble

For cases where slow mass dinotusion traps the vapour non-equilibrium phasechange helps to determine the amount trapped The accommodation coe cient rep-resents a resistance to condensation at the interface during collapse of the bubble Alower value of accommodation coe cient provides more resistance to phase changeclearly increasing the mass of water retained in the bubble at any time during thecollapse The importance of the non-equilibrium phase change can be estimated bycomparing frac12 d yn with frac12 con d at the time that frac12 d yn = frac12 d if The closer frac12 d yn is to frac12 con d the more important non-equilibrium phase change is in determining the amount ofwater trapped

Colussi et al (1998) developed a model including chemical reactions in a collapsingbubble using an accommodation coe cient of frac14 = 0001 but did not take into

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1697

account the shy nite rate of mass dinotusion The low value of frac14 which works to trapsignishy cant water vapour at collapse was necessary for their model to be consistentwith experimental sonochemistry data When such a low value for frac14 is used thecondition frac12 d yn frac12 frac12 con d is reached well before frac12 d yn frac12 frac12 d if (the opposite of our case)Such a low accommodation coe cient works to trap signishy cant amounts of watervapour in the bubble but in this case the hindered regux at the interface eliminatesmass transfer out of the bubble while the distribution of water vapour within isessentially uniform In the present work we argue that the signishy cant amounts ofwater vapour trapped in the bubble are a consequence of slow dinotusion as well asnon-equilibrium phase change

The temperature of the interface exceeds the critical point for a very brief time(from iexcl 340 ps to 1800 ps relative to the time of minimum radius) For the conditionswe explored the maximum thickness of the supercritical region is quite thin less than1 of the bubble radius ie ca 10 nm We neglect any changes in the liquid equationof state and transport properties when the interface exceeds the critical temperatureThis assumption is justishy ed based on the thinness of the region and the fact that thecritical temperature is reached very late in the collapse Any disturbances caused bythe supercritical interface have insumacr cient time to propagate into the bubble interioron the main collapse Although the evaporation equation is invalid for these shorttimes the total mass added to the bubble is negligible regardless of the magnitudeof evaporative mass regux The issue of the super-critical interface may require futureimprovements but is beyond the scope of the present work

As a shy nal note both thermal and pressure dinotusion also play a role in determiningthe distribution of water vapour during the collapse In the argonwater system thethermal-dinotusion coemacr cient is small and the molecules are of similar size hence pres-sure dinotusion is quite weak Little dinoterence is seen in calculations with and withoutthese other forms of mass dinotusion In a helium or xenon bubble in water thermaland pressure dinotusion are more pronounced and can cause a 1015 dinoterence inthe amount of trapped water vapour In the interest of brevity we will not discussthese results further here

(b) Gas-phase chemical reactions break apart H2O and reduce the temperature

Now we turn to an investigation of what transpires when chemical reactions aretaken into account (case III) We shall show that there is signishy cant time for reactionsto occur ie the time-scale for chemical reaction is similar to that of the bubblecollapse The primarily endothermic reactions absorb a signishy cant amount of energyand further reduce the peak temperatures from cases I and II The distribution ofspecies produced in the bubble is determined by dinotusive transport of water vapourprior to the collapse and heat transfer out of the bubble during the collapse Thesteep temperature gradient at the wall conshy nes the bulk of the chemical reactions tothe bubble interior

(i) Consequences of chemistry

The overall bubble dynamics of cases II and III are essentially the same with acompression ratio of 49 in case III (slightly higher than case II) A convenient pointof departure is to compare the temperature histories of the bubble centre aroundthe main collapse for the three cases we have considered (see shy gure 5) The inclusion

Proc R Soc Lond A (2000)

1698 B D Storey and A J Szeri

- 4 - 2 0 2 410

3

104

tem

pera

ture

(K

)

time relative to minimum radius (ns)

Figure 5 Temperature history of the centre of the bubble around the time of minimum radius(t = 0) in cases I II and III Note the reduction in temperature from case I ( ) to case II( ) as water is allowed to enter the bubble The chemical reactions added in case III (||)further reduce the temperatures

of water vapour (case II) signishy cantly reduces the peak temperature from case IThe decrease in temperature is due mainly to the decrease in the ratio of specishy cheats of the mixture which reduces the compression heating Upon comparison ofcases II and III one observes that the chemical reactions further reduce the peaktemperature due to the endothermic decomposition of the water vapour The centretemperature histories of cases II and III are nearly identical below 4000 K

It is important to understand that the temperature dinoterences in these threecases are not due to subtle dinoterences in bubble dynamics The compression ratio(Rm ax=Rm in ) of the three cases increases from case I to case II to case III whereasthe temperature decreases in this order This decrease in temperature occurs despitethe fact that a higher compression ratio will lead to more compression heating in thebubble collapse all other things being equal In all these cases there are no shocksinside the bubble Shocks will be considered in x 3 e

In shy gure 6 we show the history of the total number of molecules of the dinoterentspecies as a function of time on the same time-scale as in shy gure 5 The lowest curve isthe total amount of water vapour only The next curve is the total amount of watervapour plus the total amount of H2 Each curve represents the addition of one speciesas labelled until the uppermost curve denotes everything but argon We do not labelHO2 and H2O2 as they appear only in trace amounts One observes which speciesare most abundant during the time of the main collapse Note there is a relativelylarge amount of undissociated water considering the elevated temperatures This isprimarily a consequence of the high pressure suppressing the dissociation of H2O

From shy gures 5 and 6 one observes that the onset of signishy cant chemical activity in

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1699

- 4 - 2 0 2 4

109

1010

time relative to minimum radius (ns)

num

ber

of m

olec

ules

H2O

H2

O2

OH

O

H

Figure 6 Cumulative number of molecules of each species in case III The lowest curve is thetotal number of water molecules the next curve is total water + H2 and so on H2 O2 and HO2

are neglected in this plot as they appear in trace amounts

the bubble corresponds to a centre temperature of ca 4000 K Therefore the reducedtemperature of case III shown in shy gure 5 can be attributed to the endothermicchemical reactions Although the bubble dynamics are quite rapid the chemicalreactions keep pace as is shown by the similar time-scales of the changing quantitiesin shy gures 5 and 6 The rapid rate at which the reactions progress is due to the veryhigh density Recall that the rate of progress of each reaction is directly related tothe species concentration (mol cmiexcl3)

(ii) Dynamics and distribution of OH production

In sonochemistry applications the invasion of OH radicals into the liquid isthought to be responsible for much of the complex and useful chemistry that occursTherefore we shall focus on the behaviour of this species in particular In shy gure 7 weplot the total number of OH molecules contained in the bubble on two time-scalesover one acoustic cycle in shy gure 7a and around the time of collapse in shy gure 7b Theinitial concentrations of all species were selected so that the chemical production anddestruction balanced in one acoustic cycle The main collapse is associated with arapid creation of OH once the temperature exceeds 4000 K

Upon rebound the bubble rapidly cools and the OH and other products tend torecombine back into H2O These recombination reactions proceed more slowly thanthe dissociation reactions upon collapse The result is that the bubble is left with asuper-equilibrium amount of OH after it has cooled to ambient temperature Thissuper-equilibrium OH then slowly decays through the rest of the cycle with briefspurts of production and destruction at each of the after-bounces

Proc R Soc Lond A (2000)

1700 B D Storey and A J Szeri

0 10 20 30time (micros)

- 4 - 2 0 2 4

time relative to minimum radius (ns)

106

1010

104

106

108

108

num

ber

of O

H

radi

cals

nu

mbe

r of

OH

ra

dica

ls

Figure 7 Total number of OH molecules contained in the bubble versus time over one acousticcycle (a) and around the time of minimum radius (b) in case III The macrrst sharp peak in (a)is expanded in (b) Subsequent peaks in (a) correspond to local radius minima in the bubbledynamics The initial amounts of OH and all the other species were chosen for no net speciesproduction or destruction over an acoustic cycle

The distribution of OH in the bubble is characterized by a fairly uniform con-centration through much of the bubble interior and a very sharp gradient near thewall This sharp gradient is a result of heat transfer during the collapse Duringcompression conduction between the hot interior and the cool wall sets up a thintemperature boundary layer This boundary layer conshy nes the chemical reactions tothe interior away from the interface The cooler wall leads to a much lower concen-tration of OH at the interface than in the centre at the time of the main collapseIn shy gure 8 we show the molar concentration history of OH at the centre and at theinterface When the bubble is at minimum radius and the OH concentration at thecentre is at its maximum the concentration at the wall is three orders of magnitudelower

Because the chemistry time-scale is much faster than the dinotusion time-scale OHis produced where water vapour was trapped at high enough temperature The liquidinterface is insulated from the high concentration of OH generated deep inside in thebubble The only OH produced at the centre that reaches the bubble interface is theslowly decaying super-equilibrium OH that is left inside the bubble on re-expansion

(c) OH uptake by the liquid does not indeg uence the bubble interior

The inclusion of liquid uptake of OH (case IV) adds further complications wenow try to illuminate some of the main qualitative features We shall argue that OHuptake has little enotect on the gas dynamics and chemical production

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1701

- 4 - 2 0 2 410 - 10

10 - 8

10 - 6

10 - 4

10 - 2

time relative to minimum radius (ns)

conc

entr

atio

n of

OH

(m

ol c

m-3

)

Figure 8 Molar concentration of OH at the centre (||) and the interface ( ) versus timearound the point of minimum radius Note that the concentration at the centre is about threeorders of magnitude greater than at the interface at the point of minimum radius

Due to dinotering ideas on the proper value of the uptake coemacr cient the values areset at pound = 0001 and pound = 001 (cases IV(a) and IV(b)) consistent with the work ofothers to which we made reference earlier The bubble dynamics and temperatureproshy les for cases IV(a) and IV(b) are indistinguishable from those for case III Inshy gure 9 we show the total number of OH molecules in the bubble versus time ontwo dinoterent scales for cases III IV(a) and IV(b) The history for one acoustic cycleis plotted in shy gure 9a and the history concentrated at the main collapse is plottedin shy gure 9b

In the main expansion when the bubble is cool the OH resident in the bubblewill invade the liquid as can be seen by the decrease in the amount of OH at theend of the long expansion as the uptake increases When the bubble collapses thechemical production of OH from recently evaporated water rapidly overwhelms theOH uptake by the liquid As the bubble proceeds into the collapse the curves oftotal OH content for dinoterent uptakes become indistinguishable The total amountof OH in the bubble is dominated by the amount created in the interior away fromthe wall The total amount of OH lost to the liquid during a collapse is only afraction of the amount contained in the bubble Hence the peak amount of OH inthe bubble during the collapse is insensitive to the uptake coe cient of OH Similarstatements apply to uptake of other species

In shy gure 10 we show for the purposes of qualitative discussion the total massof OH exiting the bubble in cases IV(a) and IV(b) While the OH uptake coef-shy cient does not inreguence the total amount of OH contained in the bubble at themain collapse there are large dinoterences in the amount invading the liquid over

Proc R Soc Lond A (2000)

1702 B D Storey and A J Szeri

0 10 20 30

106

1010

102

time (micros)

- 4 - 2 0 2 4104

106

108

time relative to minimum radius (ns)

1010

(b)

(a)nu

mbe

r of

OH

ra

dica

ls

num

ber

of O

H

radi

cals

Figure 9 Comparison of total amount number of OH molecules in the bubble versus time overone acoustic cycle (a) and focused around the time of minimum radius (b) in cases III (||)IV(a) ( ) and IV(b) ( ) The uptake of OH plays a role during the slow expansionswhen the resident amount of OH left in the bubble is well above chemical equilibrium At thecollapse OH uptake does not indeg uence the total amount of OH production

a cycle Much of the OH uptake occurs during the main collapse and the subse-quent re-expansion when the concentration of OH at the wall is at its maximum(see shy gure 8) Further uptake occurs throughout the cycle at each of the after-bounces

(d ) Inclusion of water vapour alters the ereg ect of increased forcing

Next we shall compare the results of cases I II and III with all parameters shy xedexcept for the acoustic pressure amplitude which varies between 10 and 13 barWe that note this is experimentally awkward for SBSL because pressure amplitudeand ambient radius are dependent quantities through the mass exchange equilibriumof rectishy ed dinotusion (Hilgenfeldt et al 1996) The way we present these results isnonetheless useful as it demonstrates the importance of water vapour in the under-standing of sonoluminescence and sonochemistry

Because the bubble dynamics are altered by the inclusion of water vapour weuse the compression ratio as the distinguishing parameter In shy gure 11 we plot themaximum centre temperature versus the compression ratio for cases I II and III Thecalculations in shy gure 11 are devoid of strong shocks a matter to which we devotemore attention below If there are strong shocks then shy gure 11 will look the samein terms of a suitably averaged temperature rather than the centre temperature

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1703

0 10 20 30102

103

104

105

106

num

ber

of O

H

radi

cals

inva

ding

liqu

id

time (micros)

Figure 10 Comparison of the amount of OH absorbed by the liquid versus time for cases IV(a)(|| uptake coeplusmn cient pound = 0001) and IV(b) ( pound = 001) Most of the OH uptakeoccurs at the main collapse and the subsequent after-bounce

In case I the plot shows linear behaviour (on the loglog scale) One observes thatwhen water vapour is not taken into account stronger forcing yields more extremetemperatures

In case II the relationship between compression ratio and maximum temperature isvery dinoterent First we note that the amount of water trapped in the bubble increaseswith compression ratio If we revisit the water-trapping mechanism (x 3 a (iii)) thistrend is readily explained The increase in water-vapour content reduces the ratioof specishy c heats of the gasvapour mixture and hence reduces the compressionheating

At low compression ratios in case II the temperature increases with compressionratio in a manner very similar to case I At low compression ratios the amountof water at collapse is very small the bubble is nearly pure argon As the forcingis increased more water is trapped and the maximum temperature of the bubblebecomes nearly constant In this range of bubble dynamics there is an almost per-fectly compensating enotect (with respect to peak temperature) between the decreasein specishy c-heat ratio and the increase in compression Case III is very similar tocase II except that the plateau in temperature occurs at a lower temperature aconsequence of the endothermic chemical reactions

(e) Inclusion of water vapour promotes the formation of shocks

Water vapour is of great importance in the formation of shocks in very stronglyforced bubbles We shall brieregy consider a 60 m bubble forced with a pressureamplitude of 14 bar at a frequency of 206 kHz (Case A1 in Moss et al (1999))

Proc R Soc Lond A (2000)

1704 B D Storey and A J Szeri

10 1 10 2

10 3

10 4

10 5

Rmax R min

max

imum

tem

pera

ture

(K

)

Figure 11 Change in centre temperature versus compression ratio in cases I ( ) II ( curren )and III (+) In this plot all parameters were held constant except the pressure amplitudewhich was varied between 10 and 13 bar Case I shows a steady increase in temperature withcompression ratio Cases II and III reach a maximum temperature as the forcing is increased

When water vapour is neglected a compression wave is launched into the bubble oncollapse but the wave does not steepen into a shock When water vapour is takeninto account (with or without reactions) the compression wave steepens into a strongshock very near the centre of the bubble Vuong et al (1999) showed that the rateof steepening depends on the amount of compression heating before the compressionwave is launched

In very recent work Lin amp Szeri (2000) used the wavefront expansion technique todetermine the evolution of a compression or rarefaction wavefront that is propagatingspherically into a collapsing or expanding gas of non-uniform entropy This analyticalwork supports the idea that when the speed of sound is higher at the centre (dueto temperature and composition inhomogeneities) shock formation is delayed orprevented Furthermore this work demonstrated that the inward radial velocity alsohinders the steepening of compression waves into shocks

Consistent with the arguments made in Lin amp Szeri (2000) Vuong et al (1999)and Moss et al (1999) our results show that the decrease in the ratio of specishy cheats due to an increase in the water-vapour content can promote shock formationThe point we want to emphasize is that the question of shock formation cannot beundertaken without consideration of water vapour

With respect to total chemical production shocks are of negligible consequenceThe shock inreguences only about the inner 10 of the radius which is only 01 ofthe volume Therefore the increased production of OH at the centre does not anotectthe total amount produced in the bubble

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1687

(a) Gas dynamics

The starting point for the study is the balance of mass energy and momentum inspherical coordinates for an n-component gas mixture as given in Bird et al (1960)The formulation that follows dinoters subtly from SampS hence we state the governingequations of the current work

The problem is formulated in a Lagrangian way where r(a t) is the radial coor-dinate at time t of a gas `particlersquo that was at radial position a at t = 0 In thepresent work the Lagrangian marker particles move with the velocity of the argonie r=t = vA This reference velocity is chosen for convenience because the velocityof the bubble wall moves with the velocity of the (insoluble) argon We note thatthis formulation could be modishy ed to allow the argon to cross the moving interface

In the present Lagrangian formulation dinotusion should be thought of relativeto the background argon velocity which may be large on collapse Mathematicallythe transform from Eulerian to Lagrangian coordinates changes the usual Eulerianmaterial derivative as follows

t+ v

r=)

t+ (v iexcl vA)

a

r

a

where v is the mass average velocityWe make the equations dimensionless using the initial values of radius R0 density

raquo 0 pressure P0 and temperature T0 The velocity scale v0 is (P0=raquo 0)1=2 the massregux scale is raquo 0v0 and the internal energy and enthalpy scale is v2

0 We make theviscosity and thermal conductivity ( middot and k) dimensionless with the values at theinitial state The dinotusion coemacr cients (Dij) are made dimensionless with the aver-age of the binary coe cient matrix Davg Under these transformations the massconservation equations in dimensionless form become

raquo AJ = wA0 (21)

wi

t=

jA

wA0

r2

a2

wi

aiexcl 1

a2

wA

wA0

r2ji

a+

curren i

raquo (22)

where raquo is the density wi is the mass fraction curren i is the net rate of chemical produc-tion or destruction the subscript A denotes argon the subscript i denotes species iand the subscript 0 denotes the initial value The Jacobian of the map between thereference and current conshy gurations J is given by

J =r2

a2

r

a

The radial dinotusive mass regux of species i with respect to the mass average velocityji is

ji = iexcl DMi raquo

Mi

M

1

ReSc

a

r

microxi

a+

Z0M

raquo T M0(xi iexcl wi)

P

a+ KT

i

ln(T )

a

para

where Mi is the molecular weight of species i M is the mean molecular weight atthe local conditions xi is the molar fraction Z0 is the compressibility in the initialstate KT

i is the thermal-dinotusion ratio T is the temperature and P is the pressureThe Schmidt number Sc sup2 middot 0=raquo 0Davg is the ratio of the dinotusivity of momentum to

Proc R Soc Lond A (2000)

1688 B D Storey and A J Szeri

the dinotusivity of mass the Reynolds number is Re sup2 raquo 0R0v0=middot 0 The mean dinotusioncoemacr cient DM

i is a function of temperature pressure and composition It is deshy nedas

DMi =

1 iexcl xiPnj 6= i xj=Dij

where Dij is the binary dinotusion coemacr cient between species i and j The thermal-dinotusion factor is given as

KTi = xi

X

j 6= i

xj not ij

where not ij is the thermal-dinotusion factor between species i and jThe dinotusive mass regux written with the mean dinotusion coemacr cients is an approx-

imation in the interest of computational emacr ciency (Gardiner 1984 Hirschfelder etal 1954) The approximation is valid due to the fact that the other species (besidesargon and water) are present in trace amounts To ensure that the mass reguxes sumto zero as is required for mass conservation the regux of water vapour is computedas minus the sum of the reguxes of all other species rather than in primitive form

The balance of linear momentum written in dimensionless Lagrangian form is

v

t=

jA

wA0

r2

a2

v

aiexcl

wA

wA0

r2

a2

microP

a+ 3

frac12

r

r

a+

frac12

a

para (23)

where v is the mass average velocity and frac12 is the deviatoric stress given as

frac12 =iexcl 4 middot (T P xi)

3Re

sup3v

a

a

riexcl

v

r

acute

The energy equation in these variables is

E

t=

jA

wA0

r2

a2

E

aiexcl

wA

wA0

1

a2

a[r2(q + frac12 v + P v)] (24)

where the radial heat regux is

q = iexcl Cp0T0 raquo 0

P0

k(T P xi)

RePr

T

a

a

r+

nX

i= 1

jiHi

Here Hi is the partial molal enthalpy and Pr sup2 Cp0 middot 0=k0 is the Prandtl numberWe note that

Cp0T0 raquo 0

P0=

reg

reg iexcl 1

for a perfect gas where reg is the ratio of specishy c heats

(b) Transport properties and equation of state

The transport properties ( middot k not ij and Dij) are calculated from equations basedon the ChapmanEnskog theory using a LennardJones 126 potential with correc-tions for high temperatures and densities The individual molecular parameters arecombined in an empirical way to develop enotective mixture parameters for use in thesingle-reguid equations These equations are given in detail in the appendix of SampSbut the essential equations are found in Hirschfelder et al (1954)

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1689

There are a few minor modishy cations to the equations used in SampS In the presentpaper a correction is added in the combination rules for the interaction between polarand non-polar molecules The Eucken correction is used for the thermal conductivityto account for internal degrees of freedom (Hirschfelder et al 1954) Finally thethermal-dinotusion factors are computed using the approximation given by Fristromamp Monchik (1988) for computational emacr ciency We stress that our previous workshows that the results are relatively insensitive to the transport properties as longas the general trends with temperature and pressure are captured

The SoaveRedlichKwong equation of state (EOS) is used with combination rulesfor the empirical constants of the mixture (Reid et al 1987) We use the polynomialdata shy ts for ideal gas specishy c heats at 1 atm from Gardiner (1984) and computedepartures with the EOS While this approach may seem overly simplishy ed comparedwith the EOS used in SampS a similar EOS for water and its reaction products thatallows us to account for shy nite-rate chemistry is not available It was found in SampSand again in the present work that the EOS has no impact on the basic physics ofthe problem and only modishy es the shy nal quantitative answers

(c) Motion and energy transport in the liquid

Rather than solve the NavierStokes equations governing the motion of the liquidwe assume that a RayleighPlesset-type equation for the bubble radius is valid TheRayleighPlesset equation is an ordinary dinoterential equation for the bubble radiuswhich is coupled with the gas dynamics of the interior through the pressure and stressat the bubble wall The form of the equation we use is similar to the one derivedby Prosperetti amp Lezzi (1986) dinotering only in that the present work accounts forthe extra velocity at the interface due to evaporation or condensation The enotect ofevaporationcondensation on the RayleighPlesset equation is very small but theenotect is included for consistency The resulting equation is similar to ones derivedby Yasui (1997a) and by Fujikawa amp Teruaki (1980)

Rather than solve the full energy equation in the liquid we solve the boundary-layer form as described in detail by Fyrillas amp Szeri (1994) The boundary-layerassumption reduces the energy equation to the linear dinotusion equation in Lagrangianboundary-layer coordinates This approximation for the energy equation was used inVuong amp Szeri (1996) and in SampS

(d) Evaporation and condensation

The equation derived from kinetic theory given in Carey (1992) is used to computethe rate of evaporation and condensation at the bubble surface This is written indimensionless variables as

_m00kin =

rMH2 O Z0

2 ordm M0

sup3iexcl PH2 Op

Tin tiexcl P s atp

Tin t

acute

_m00act = frac14 _m00

kin

9gt=

gt(25)

Here _m00kin is the rate at which water molecules hit the interface (from kinetic theory)

frac14 is the accommodation coe cient (sticking probability) _m00act is the rate at which

water molecules hit the interface and undergo phase change PH2 O is the partial

Proc R Soc Lond A (2000)

1690 B D Storey and A J Szeri

pressure of water P s at is the saturation pressure at the temperature of the interfaceTin t M0 is the initial molecular mass of the bubble contents iexcl is a correction forbulk motion to the interface and remains close to one for low mass transfer Weemphasize that this equation is only valid for reasonable temperatures and pressuresand not at the extreme conditions found during a bubble collapse As we argue belowthe evaporation condition is of vanishing consequence during the terminal stages ofbubble collapse when (25) loses validity

There is some discord in measurements of the accommodation coe cient Thevalue we use frac14 = 04 is consistent with that used by Yasui (1997a) (developed frommolecular dynamics simulations) and those recommended by Eames et al (1997)The sensitivity of the results to the accommodation coemacr cient will be consideredlater

(e) Chemical kinetics

Chemical kinetics described by the reaction mechanism determines the produc-tion and destruction of the dinoterent species The reaction mechanism that we use isdescribed in Maas amp Warnatz (1988) The mechanism includes 19 forward and reverseelementary reactions and nine species (Ar H H2 O O2 OH HO2 H2O2) Argononly enters the mechanism as a third body

The net rate of progress of each elementary reaction is the dinoterence in the forwardand reverse rate of progress The forward (reverse) rate of progress is given as theproduct of the reaction rate and the concentrations of all species on the reactant(product) side of the equation The change in concentration of species i with timeis the sum of the net rate of progress of each elementary reaction that containsspecies i The details can be found in several texts (eg Gardiner 1984) In thepresent formulation the chemical kinetics lead to the net chemical production ordestruction term curren i in (22)

The reaction rates are given empirically with the reaction mechanism The Linde-mann form is used to take into account the pressure fall-onot in some of the three-bodyreactions (Gardiner 1984) The high-pressure limit of the reaction rates for availablereactions were taken from Bowman et al (1999)

The elevated pressure in the bubble upon collapse accelerates the rate of progressof the reactions and alters the equilibrium state to which the reactions tend so thatfor a given temperature there is less dissociation at higher pressures This change inequilibrium state with pressure is accounted for in the reaction mechanism becauseforward and reverse reaction rates are related through the equilibrium constantThere is considerable uncertainty in the reaction rates at the high pressures weshall encounter The mechanism and reaction rates we use seem to be the best onespresently available

The reaction mechanism does not include ionization reactions The onset of ion-ization is delayed due to the high pressure The degree of ionization at the conditionswe report is quite low as estimated by the Saha equation Moss et al (1997) putforth the criterion that the onset of ionization in a dense gas is at one-quarter ofthe ionization potential in the present case this temperature is ca 30 000 K Theresults we present are consistent with neglecting ionization based on either of thesecriteria

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1691

(f ) Mass transfer to the liquid

One of the most di cult quantities to determine accurately is the rate of transportof radical species into the liquid While general convectiondinotusionreaction equa-tions can be written for each species dissolved in the liquid this approach involvesmany unknowns For example data on the solubility of OH radicals in liquid watervary by two orders of magnitude (Sander 1998) We know of no solubility data forH or O Moreover there is the possibility of heterogeneous surface reactions (VonSonntag et al 1999) which are poorly understood especially at the time of collapsewhen the liquid very near the bubble interface is supercritical

To avoid these uncertainties we take a very simple approach To determine the rateof radical invasion into the liquid approximately we introduce the uptake coemacr cientpound

pound =Nin iexcl Nou t

Ncoll

(26)

where Nin is the number of molecules invading the liquid Nou t is the number ofmolecules ejected from the liquid and Ncoll is the number of molecules collidingwith the surface (from kinetic theory) This view of mass transfer is oversimplishy edone can easily see that pound is not an intrinsic quantity (Takami et al 1998 Chameides1984)

This approach has nevertheless been used to determine the absorption of radicalspecies onto liquid and solid surfaces The uptake coe cients have been measured forsome of the species in our system over aqueous surfaces (Takami et al 1998 Hansonet al 1992) We shall study the inreguence of uptake coe cient on the results of ourcalculations below

(g) Numerical method

The same Chebyshev collocation method is used that was described in detail inSampS The variables in the gas are approximated by expansions in Chebyshev poly-nomials in the argon Lagrangian coordinate a The temperature in the liquid isapproximated with an expansion in rational Chebyshev polynomials The equationsin the gas evolve with a straightforward explicit predictorcorrector scheme We usestandard operator splitting (Press et al 1988) to solve the chemical kinetics equa-tions separately with a semi-implicit method to avoid time-step restrictions causedby the stinotness of these equations The energy equation in the liquid is integrated intime with a fully implicit method to avoid time-step restrictions associated with theclose spacing of grid points near the bubble wall As an error check global balancesof mass and energy are monitored imbalances always remain below a fraction of aper cent

3 Results and discussion

We begin by presenting detailed results of one particular conshy guration Due to thecomplexity of the problem we shall attempt to present the results in a systematicway in order to demonstrate the importance of certain phenomena The cases to beinvestigated are as follows

Proc R Soc Lond A (2000)

1692 B D Storey and A J Szeri

Table 1 Summary of the direg erent cases investigated in detail

(In all cases the bubble is a 45 m radius argon bubble in water at 298 K acoustically forcedwith a 12 bar pressure amplitude at 265 kHz)

case phase change chemistry OH uptake Rm ax =Rm in ( m) Tm ax (K)

I no no 00 280080 20 900

II yes no 00 313070 9 700

III yes yes 00 317065 7 000

IV(a) yes yes 0001 317065 7 000

IV(b) yes yes 001 317065 7 000

Case I no water vapour allowed in the bubble

Case II water vapour allowed to evaporate and condense but no chemical reactions

Case III water vapour and chemical reactions but no mass transfer across thebubble interface

Cases IV(a) and IV(b) mass transfer of OH across the bubble interface at dif-ferent uptake coe cients

In all cases the bubble is a 45 m-radius argon bubble in 298 K water acousticallyforced with a 12 bar pressure amplitude at 265 kHz These cases are summarized intable 1 With respect to SBSL this bubble was found to be stable to mass exchangeand shape changes though would have minimal light emission (Hilgenfeldt et al 1996 Wu amp Roberts 1998a b) We describe the results for more strongly forcedbubbles below The data we obtain are for steady-state oscillations ie after thedecay of initial transients After presenting the details of these cases we considervariations in the parameters to illustrate interesting trends

(a) Excess water is trapped by the rapid collapse

We shall now argue that signishy cant amounts of water vapour become trapped inthe interior of the bubble when it collapses violently Water is trapped in the interiorbecause the bubble motions become so rapid that the vapour near the centre hasinsu cient time to dinotuse to the bubble wall Non-equilibrium phase change alsoplays a signishy cant role in the amount of water that is trapped All results in thissubsection apply to case II this case allows one to focus on the dynamics of evapora-tion condensation and dinotusion of water inside the bubble without the complicationof chemical reactions

In shy gure 1 we show the radial history of the bubble over one cycle of the acousticforcing in case II The essential features of the bubble dynamics for cases I and II arethe same but the inclusion of water vapour increases the maximum radius and delaysthe collapse The dinoterence is to be expected because case I has no vapour pressureThe compression ratio (Rm ax=Rm in ) for cases I and II is 35 and 44 respectively Thecompression ratio is an approximate way of separating out dinoterences in the resultsdue to dinoterences in the bubble dynamics Typically a higher compression ratio leadsto more compression heating and higher temperatures if all else is equal We returnto this point later

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1693

0 10 20 30

10

20

30

time (micros)

radi

us (

microm

)

Figure 1 Radius of the bubble versus time over one cycle of acoustic forcing in case IIThe essential features for cases I II III and IV are the same

0 10 20 30

10 8

10 9

1010

1011

num

ber

of m

olec

ules

H2O

Ar

time (micros)

Figure 2 Total number of water-vapour molecules contained in the bubble versus time over onecycle of acoustic forcing in case II Note that water comprises ca 14 of the bubble contents atthe end of the main collapse

Proc R Soc Lond A (2000)

1694 B D Storey and A J Szeri

0

4000

8000te

mpe

ratu

re (

K)

1

2

3

time relative to minimum radius (ns)

num

ber

of H

2O m

olec

ules

109

acute-

- 4 - 2 0 2 4

(b)

(a)

Figure 3 Temperature (a) and total number of water molecules contained in the bubble (b) versustime around the point of minimum radius (time = 0) in case II The temperature undergoesorder-of-magnitude changes on this time-scale while the total number of water molecules changesvery little

In shy gure 2 we show the total number of H2O molecules contained within thebubble over the acoustic cycle (case II) A large amount of water evaporates intothe bubble during the main expansion when the pressure is low Water condenses onthe bubble wall during the collapse as the gas pressure increases There is signishy cantwater vapour in the bubble at the main collapse ca 14 (molar basis) of the totalbubble contents is water vapour If the interface were at equilibrium and the speedof mass dinotusion were inshy nite there would be a negligible amount of water in thebubble on collapse

In shy gure 3 we show the temperature history of the centre and the total numberof water molecules in the bubble around the time of minimum radius (time = 0) incase II Over the 10 ns time-scale the total amount of water vapour contained in thebubble is nearly constant whereas the temperature changes by an order of magni-tude This recalls SampS in which it was shown that evolving mixture compositionswere essentially frozen on the time-scale of the temperature peak

The temperature peak may be related to the light emission The equation foroptical power given by Moss et al (1999) with a constant opacity provides an easyestimate for the optical power output as a function of time For case II this estimategives a light emission pulse width (FWHM) of 350 ps which is much shorter than theca 2000 ps temperature peak This optical-pulse-width estimate would be somewhatshortened if the temperature dependence of the opacity were taken into accountThe peak optical power is ca 01 mW although this number is very sensitive to theopacity Typical stable SBSL experiments (at higher forcings than the present case)yield light emission of the order of milliwatts per regash (Gaitan et al 1996 Hiller etal 1992)

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1695

0 5 10 15 20 25

02

04

06

08

10

- 843 ns

- 455 ns

- 299 ns

- 153 ns

- 41 ns

- 15 ns

0 ns - 5 ns

radial position (microm)

mol

ar f

ract

ion

Figure 4 Molar fraction of water vapour versus radius for selected times prior to the moment ofminimum radius (time = 0) in case II Note the development of the non-uniform composition asthe collapse accelerates toward minimum radius In the last 15 ns before minimum radius thecomposition changes by only a few per cent when viewed in the Lagrangian frame instead

There are two mechanisms that determine the excess vapour left behind at thecollapse shy nite-rate mass dinotusion and non-equilibrium phase change We examineeach mechanism and their interaction in turn

(i) Role of direg usion

The shy rst reason for excess water vapour remaining during the collapse is the rapidbubble collapse overwhelming slow mass dinotusion this was extensively discussedin SampS In SampS it was shown that there is a competition between two importanttime-scales the dynamic time-scale of the bubble motions and the mass dinotusiontime-scale These are given explicitly in dimensionless terms as

frac12 d yn = R= _R

frac12 d if = ReScR2

D MH2 O

ordm ReSc1

Rp

T

In the latter we made use of the fact that the dinotusion coe cient of water vapourin the gas mixture D M

H2 O scales as R3pT

The time-scales argument is clarishy ed upon examination of shy gure 4 which showsthe evolving bubble composition as a function of radius at selected times before thetime of minimum radius (case II) In the early part of the collapse the dynamic time-scale is longer than the dinotusion time-scale frac12 d yn frac34 frac12 d if this results in a uniformcomposition of the bubble (upper curves in shy gure 4) As the bubble accelerates into

Proc R Soc Lond A (2000)

1696 B D Storey and A J Szeri

the collapse frac12 d yn sup1 frac12 d if and the decrease in water fraction at the centre begins tolag the decrease in water fraction at the wall (middle curves of shy gure 4) Furtherinto the collapse frac12 d yn frac12 frac12 d if and the water vapour cannot dinotuse to the bubble wallon the time-scale of the rapid collapse This slow dinotusion results in a nearly shy xeddistribution of water vapour (three lowest curves of shy gure 4)

While the three lowest curves in shy gure 4 look quite dinoterent in this Eulerianrepresentation they dinoter by only a few per cent when viewed in the Lagrangianrepresentation (not shown) The curve at iexcl 15 ns is from a time when R = R0 andthe temperature at the centre is ca 750 K The curve at 0 ns is from the time ofminimum radius (015R0) when the temperature at the centre is 9700 K Consistentwith shy gure 3 over times when there are drastic changes in the bubble radius andtemperature the composition of the bubble is almost a material shy eld

(ii) Role of non-equilibrium phase change

Non-equilibrium phase change is the second mechanism by which vapour is trappedin the bubble interior during the collapse By adjusting the accommodation coe -cient one has control over the time-scale for the partial pressure of the water vapourto come to equilibrium with the saturation pressure at the interface The dimension-less condensation time-scale is

frac12 con d =R

frac14

r2 ordm MH2 O

9M0Tin t

When frac12 d yn frac34 frac12 con d the condensation is in quasi-equilibrium with respect to thebubble motions If frac12 d yn frac12 frac12 con d then the bubble collapse is so rapid that no masscan escape the bubble on the time-scale of the collapse

(iii) Interaction of the two mechanisms

It is the competition between the time-scales for mass dinotusion condensation andbubble dynamics that determines which mechanism traps water vapour in the bubbleOnce frac12 d yn frac12 frac12 d if or frac12 d yn frac12 frac12 con d the total amount of water vapour in the bubbleis shy xed Only one of these conditions needs to be met for vapour to be trapped inthe bubble Both mechanisms can however participate to determine the quantity oftrapped vapour

During the bubble collapse the condition frac12 d yn frac12 frac12 d if is reached well before frac12 d yn frac12frac12 con d in case II Therefore in this case (as well as the others presented herein) it isthe slow mass dinotusion that is responsible for trapping vapour in the bubble

For cases where slow mass dinotusion traps the vapour non-equilibrium phasechange helps to determine the amount trapped The accommodation coe cient rep-resents a resistance to condensation at the interface during collapse of the bubble Alower value of accommodation coe cient provides more resistance to phase changeclearly increasing the mass of water retained in the bubble at any time during thecollapse The importance of the non-equilibrium phase change can be estimated bycomparing frac12 d yn with frac12 con d at the time that frac12 d yn = frac12 d if The closer frac12 d yn is to frac12 con d the more important non-equilibrium phase change is in determining the amount ofwater trapped

Colussi et al (1998) developed a model including chemical reactions in a collapsingbubble using an accommodation coe cient of frac14 = 0001 but did not take into

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1697

account the shy nite rate of mass dinotusion The low value of frac14 which works to trapsignishy cant water vapour at collapse was necessary for their model to be consistentwith experimental sonochemistry data When such a low value for frac14 is used thecondition frac12 d yn frac12 frac12 con d is reached well before frac12 d yn frac12 frac12 d if (the opposite of our case)Such a low accommodation coe cient works to trap signishy cant amounts of watervapour in the bubble but in this case the hindered regux at the interface eliminatesmass transfer out of the bubble while the distribution of water vapour within isessentially uniform In the present work we argue that the signishy cant amounts ofwater vapour trapped in the bubble are a consequence of slow dinotusion as well asnon-equilibrium phase change

The temperature of the interface exceeds the critical point for a very brief time(from iexcl 340 ps to 1800 ps relative to the time of minimum radius) For the conditionswe explored the maximum thickness of the supercritical region is quite thin less than1 of the bubble radius ie ca 10 nm We neglect any changes in the liquid equationof state and transport properties when the interface exceeds the critical temperatureThis assumption is justishy ed based on the thinness of the region and the fact that thecritical temperature is reached very late in the collapse Any disturbances caused bythe supercritical interface have insumacr cient time to propagate into the bubble interioron the main collapse Although the evaporation equation is invalid for these shorttimes the total mass added to the bubble is negligible regardless of the magnitudeof evaporative mass regux The issue of the super-critical interface may require futureimprovements but is beyond the scope of the present work

As a shy nal note both thermal and pressure dinotusion also play a role in determiningthe distribution of water vapour during the collapse In the argonwater system thethermal-dinotusion coemacr cient is small and the molecules are of similar size hence pres-sure dinotusion is quite weak Little dinoterence is seen in calculations with and withoutthese other forms of mass dinotusion In a helium or xenon bubble in water thermaland pressure dinotusion are more pronounced and can cause a 1015 dinoterence inthe amount of trapped water vapour In the interest of brevity we will not discussthese results further here

(b) Gas-phase chemical reactions break apart H2O and reduce the temperature

Now we turn to an investigation of what transpires when chemical reactions aretaken into account (case III) We shall show that there is signishy cant time for reactionsto occur ie the time-scale for chemical reaction is similar to that of the bubblecollapse The primarily endothermic reactions absorb a signishy cant amount of energyand further reduce the peak temperatures from cases I and II The distribution ofspecies produced in the bubble is determined by dinotusive transport of water vapourprior to the collapse and heat transfer out of the bubble during the collapse Thesteep temperature gradient at the wall conshy nes the bulk of the chemical reactions tothe bubble interior

(i) Consequences of chemistry

The overall bubble dynamics of cases II and III are essentially the same with acompression ratio of 49 in case III (slightly higher than case II) A convenient pointof departure is to compare the temperature histories of the bubble centre aroundthe main collapse for the three cases we have considered (see shy gure 5) The inclusion

Proc R Soc Lond A (2000)

1698 B D Storey and A J Szeri

- 4 - 2 0 2 410

3

104

tem

pera

ture

(K

)

time relative to minimum radius (ns)

Figure 5 Temperature history of the centre of the bubble around the time of minimum radius(t = 0) in cases I II and III Note the reduction in temperature from case I ( ) to case II( ) as water is allowed to enter the bubble The chemical reactions added in case III (||)further reduce the temperatures

of water vapour (case II) signishy cantly reduces the peak temperature from case IThe decrease in temperature is due mainly to the decrease in the ratio of specishy cheats of the mixture which reduces the compression heating Upon comparison ofcases II and III one observes that the chemical reactions further reduce the peaktemperature due to the endothermic decomposition of the water vapour The centretemperature histories of cases II and III are nearly identical below 4000 K

It is important to understand that the temperature dinoterences in these threecases are not due to subtle dinoterences in bubble dynamics The compression ratio(Rm ax=Rm in ) of the three cases increases from case I to case II to case III whereasthe temperature decreases in this order This decrease in temperature occurs despitethe fact that a higher compression ratio will lead to more compression heating in thebubble collapse all other things being equal In all these cases there are no shocksinside the bubble Shocks will be considered in x 3 e

In shy gure 6 we show the history of the total number of molecules of the dinoterentspecies as a function of time on the same time-scale as in shy gure 5 The lowest curve isthe total amount of water vapour only The next curve is the total amount of watervapour plus the total amount of H2 Each curve represents the addition of one speciesas labelled until the uppermost curve denotes everything but argon We do not labelHO2 and H2O2 as they appear only in trace amounts One observes which speciesare most abundant during the time of the main collapse Note there is a relativelylarge amount of undissociated water considering the elevated temperatures This isprimarily a consequence of the high pressure suppressing the dissociation of H2O

From shy gures 5 and 6 one observes that the onset of signishy cant chemical activity in

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1699

- 4 - 2 0 2 4

109

1010

time relative to minimum radius (ns)

num

ber

of m

olec

ules

H2O

H2

O2

OH

O

H

Figure 6 Cumulative number of molecules of each species in case III The lowest curve is thetotal number of water molecules the next curve is total water + H2 and so on H2 O2 and HO2

are neglected in this plot as they appear in trace amounts

the bubble corresponds to a centre temperature of ca 4000 K Therefore the reducedtemperature of case III shown in shy gure 5 can be attributed to the endothermicchemical reactions Although the bubble dynamics are quite rapid the chemicalreactions keep pace as is shown by the similar time-scales of the changing quantitiesin shy gures 5 and 6 The rapid rate at which the reactions progress is due to the veryhigh density Recall that the rate of progress of each reaction is directly related tothe species concentration (mol cmiexcl3)

(ii) Dynamics and distribution of OH production

In sonochemistry applications the invasion of OH radicals into the liquid isthought to be responsible for much of the complex and useful chemistry that occursTherefore we shall focus on the behaviour of this species in particular In shy gure 7 weplot the total number of OH molecules contained in the bubble on two time-scalesover one acoustic cycle in shy gure 7a and around the time of collapse in shy gure 7b Theinitial concentrations of all species were selected so that the chemical production anddestruction balanced in one acoustic cycle The main collapse is associated with arapid creation of OH once the temperature exceeds 4000 K

Upon rebound the bubble rapidly cools and the OH and other products tend torecombine back into H2O These recombination reactions proceed more slowly thanthe dissociation reactions upon collapse The result is that the bubble is left with asuper-equilibrium amount of OH after it has cooled to ambient temperature Thissuper-equilibrium OH then slowly decays through the rest of the cycle with briefspurts of production and destruction at each of the after-bounces

Proc R Soc Lond A (2000)

1700 B D Storey and A J Szeri

0 10 20 30time (micros)

- 4 - 2 0 2 4

time relative to minimum radius (ns)

106

1010

104

106

108

108

num

ber

of O

H

radi

cals

nu

mbe

r of

OH

ra

dica

ls

Figure 7 Total number of OH molecules contained in the bubble versus time over one acousticcycle (a) and around the time of minimum radius (b) in case III The macrrst sharp peak in (a)is expanded in (b) Subsequent peaks in (a) correspond to local radius minima in the bubbledynamics The initial amounts of OH and all the other species were chosen for no net speciesproduction or destruction over an acoustic cycle

The distribution of OH in the bubble is characterized by a fairly uniform con-centration through much of the bubble interior and a very sharp gradient near thewall This sharp gradient is a result of heat transfer during the collapse Duringcompression conduction between the hot interior and the cool wall sets up a thintemperature boundary layer This boundary layer conshy nes the chemical reactions tothe interior away from the interface The cooler wall leads to a much lower concen-tration of OH at the interface than in the centre at the time of the main collapseIn shy gure 8 we show the molar concentration history of OH at the centre and at theinterface When the bubble is at minimum radius and the OH concentration at thecentre is at its maximum the concentration at the wall is three orders of magnitudelower

Because the chemistry time-scale is much faster than the dinotusion time-scale OHis produced where water vapour was trapped at high enough temperature The liquidinterface is insulated from the high concentration of OH generated deep inside in thebubble The only OH produced at the centre that reaches the bubble interface is theslowly decaying super-equilibrium OH that is left inside the bubble on re-expansion

(c) OH uptake by the liquid does not indeg uence the bubble interior

The inclusion of liquid uptake of OH (case IV) adds further complications wenow try to illuminate some of the main qualitative features We shall argue that OHuptake has little enotect on the gas dynamics and chemical production

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1701

- 4 - 2 0 2 410 - 10

10 - 8

10 - 6

10 - 4

10 - 2

time relative to minimum radius (ns)

conc

entr

atio

n of

OH

(m

ol c

m-3

)

Figure 8 Molar concentration of OH at the centre (||) and the interface ( ) versus timearound the point of minimum radius Note that the concentration at the centre is about threeorders of magnitude greater than at the interface at the point of minimum radius

Due to dinotering ideas on the proper value of the uptake coemacr cient the values areset at pound = 0001 and pound = 001 (cases IV(a) and IV(b)) consistent with the work ofothers to which we made reference earlier The bubble dynamics and temperatureproshy les for cases IV(a) and IV(b) are indistinguishable from those for case III Inshy gure 9 we show the total number of OH molecules in the bubble versus time ontwo dinoterent scales for cases III IV(a) and IV(b) The history for one acoustic cycleis plotted in shy gure 9a and the history concentrated at the main collapse is plottedin shy gure 9b

In the main expansion when the bubble is cool the OH resident in the bubblewill invade the liquid as can be seen by the decrease in the amount of OH at theend of the long expansion as the uptake increases When the bubble collapses thechemical production of OH from recently evaporated water rapidly overwhelms theOH uptake by the liquid As the bubble proceeds into the collapse the curves oftotal OH content for dinoterent uptakes become indistinguishable The total amountof OH in the bubble is dominated by the amount created in the interior away fromthe wall The total amount of OH lost to the liquid during a collapse is only afraction of the amount contained in the bubble Hence the peak amount of OH inthe bubble during the collapse is insensitive to the uptake coe cient of OH Similarstatements apply to uptake of other species

In shy gure 10 we show for the purposes of qualitative discussion the total massof OH exiting the bubble in cases IV(a) and IV(b) While the OH uptake coef-shy cient does not inreguence the total amount of OH contained in the bubble at themain collapse there are large dinoterences in the amount invading the liquid over

Proc R Soc Lond A (2000)

1702 B D Storey and A J Szeri

0 10 20 30

106

1010

102

time (micros)

- 4 - 2 0 2 4104

106

108

time relative to minimum radius (ns)

1010

(b)

(a)nu

mbe

r of

OH

ra

dica

ls

num

ber

of O

H

radi

cals

Figure 9 Comparison of total amount number of OH molecules in the bubble versus time overone acoustic cycle (a) and focused around the time of minimum radius (b) in cases III (||)IV(a) ( ) and IV(b) ( ) The uptake of OH plays a role during the slow expansionswhen the resident amount of OH left in the bubble is well above chemical equilibrium At thecollapse OH uptake does not indeg uence the total amount of OH production

a cycle Much of the OH uptake occurs during the main collapse and the subse-quent re-expansion when the concentration of OH at the wall is at its maximum(see shy gure 8) Further uptake occurs throughout the cycle at each of the after-bounces

(d ) Inclusion of water vapour alters the ereg ect of increased forcing

Next we shall compare the results of cases I II and III with all parameters shy xedexcept for the acoustic pressure amplitude which varies between 10 and 13 barWe that note this is experimentally awkward for SBSL because pressure amplitudeand ambient radius are dependent quantities through the mass exchange equilibriumof rectishy ed dinotusion (Hilgenfeldt et al 1996) The way we present these results isnonetheless useful as it demonstrates the importance of water vapour in the under-standing of sonoluminescence and sonochemistry

Because the bubble dynamics are altered by the inclusion of water vapour weuse the compression ratio as the distinguishing parameter In shy gure 11 we plot themaximum centre temperature versus the compression ratio for cases I II and III Thecalculations in shy gure 11 are devoid of strong shocks a matter to which we devotemore attention below If there are strong shocks then shy gure 11 will look the samein terms of a suitably averaged temperature rather than the centre temperature

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1703

0 10 20 30102

103

104

105

106

num

ber

of O

H

radi

cals

inva

ding

liqu

id

time (micros)

Figure 10 Comparison of the amount of OH absorbed by the liquid versus time for cases IV(a)(|| uptake coeplusmn cient pound = 0001) and IV(b) ( pound = 001) Most of the OH uptakeoccurs at the main collapse and the subsequent after-bounce

In case I the plot shows linear behaviour (on the loglog scale) One observes thatwhen water vapour is not taken into account stronger forcing yields more extremetemperatures

In case II the relationship between compression ratio and maximum temperature isvery dinoterent First we note that the amount of water trapped in the bubble increaseswith compression ratio If we revisit the water-trapping mechanism (x 3 a (iii)) thistrend is readily explained The increase in water-vapour content reduces the ratioof specishy c heats of the gasvapour mixture and hence reduces the compressionheating

At low compression ratios in case II the temperature increases with compressionratio in a manner very similar to case I At low compression ratios the amountof water at collapse is very small the bubble is nearly pure argon As the forcingis increased more water is trapped and the maximum temperature of the bubblebecomes nearly constant In this range of bubble dynamics there is an almost per-fectly compensating enotect (with respect to peak temperature) between the decreasein specishy c-heat ratio and the increase in compression Case III is very similar tocase II except that the plateau in temperature occurs at a lower temperature aconsequence of the endothermic chemical reactions

(e) Inclusion of water vapour promotes the formation of shocks

Water vapour is of great importance in the formation of shocks in very stronglyforced bubbles We shall brieregy consider a 60 m bubble forced with a pressureamplitude of 14 bar at a frequency of 206 kHz (Case A1 in Moss et al (1999))

Proc R Soc Lond A (2000)

1704 B D Storey and A J Szeri

10 1 10 2

10 3

10 4

10 5

Rmax R min

max

imum

tem

pera

ture

(K

)

Figure 11 Change in centre temperature versus compression ratio in cases I ( ) II ( curren )and III (+) In this plot all parameters were held constant except the pressure amplitudewhich was varied between 10 and 13 bar Case I shows a steady increase in temperature withcompression ratio Cases II and III reach a maximum temperature as the forcing is increased

When water vapour is neglected a compression wave is launched into the bubble oncollapse but the wave does not steepen into a shock When water vapour is takeninto account (with or without reactions) the compression wave steepens into a strongshock very near the centre of the bubble Vuong et al (1999) showed that the rateof steepening depends on the amount of compression heating before the compressionwave is launched

In very recent work Lin amp Szeri (2000) used the wavefront expansion technique todetermine the evolution of a compression or rarefaction wavefront that is propagatingspherically into a collapsing or expanding gas of non-uniform entropy This analyticalwork supports the idea that when the speed of sound is higher at the centre (dueto temperature and composition inhomogeneities) shock formation is delayed orprevented Furthermore this work demonstrated that the inward radial velocity alsohinders the steepening of compression waves into shocks

Consistent with the arguments made in Lin amp Szeri (2000) Vuong et al (1999)and Moss et al (1999) our results show that the decrease in the ratio of specishy cheats due to an increase in the water-vapour content can promote shock formationThe point we want to emphasize is that the question of shock formation cannot beundertaken without consideration of water vapour

With respect to total chemical production shocks are of negligible consequenceThe shock inreguences only about the inner 10 of the radius which is only 01 ofthe volume Therefore the increased production of OH at the centre does not anotectthe total amount produced in the bubble

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

1688 B D Storey and A J Szeri

the dinotusivity of mass the Reynolds number is Re sup2 raquo 0R0v0=middot 0 The mean dinotusioncoemacr cient DM

i is a function of temperature pressure and composition It is deshy nedas

DMi =

1 iexcl xiPnj 6= i xj=Dij

where Dij is the binary dinotusion coemacr cient between species i and j The thermal-dinotusion factor is given as

KTi = xi

X

j 6= i

xj not ij

where not ij is the thermal-dinotusion factor between species i and jThe dinotusive mass regux written with the mean dinotusion coemacr cients is an approx-

imation in the interest of computational emacr ciency (Gardiner 1984 Hirschfelder etal 1954) The approximation is valid due to the fact that the other species (besidesargon and water) are present in trace amounts To ensure that the mass reguxes sumto zero as is required for mass conservation the regux of water vapour is computedas minus the sum of the reguxes of all other species rather than in primitive form

The balance of linear momentum written in dimensionless Lagrangian form is

v

t=

jA

wA0

r2

a2

v

aiexcl

wA

wA0

r2

a2

microP

a+ 3

frac12

r

r

a+

frac12

a

para (23)

where v is the mass average velocity and frac12 is the deviatoric stress given as

frac12 =iexcl 4 middot (T P xi)

3Re

sup3v

a

a

riexcl

v

r

acute

The energy equation in these variables is

E

t=

jA

wA0

r2

a2

E

aiexcl

wA

wA0

1

a2

a[r2(q + frac12 v + P v)] (24)

where the radial heat regux is

q = iexcl Cp0T0 raquo 0

P0

k(T P xi)

RePr

T

a

a

r+

nX

i= 1

jiHi

Here Hi is the partial molal enthalpy and Pr sup2 Cp0 middot 0=k0 is the Prandtl numberWe note that

Cp0T0 raquo 0

P0=

reg

reg iexcl 1

for a perfect gas where reg is the ratio of specishy c heats

(b) Transport properties and equation of state

The transport properties ( middot k not ij and Dij) are calculated from equations basedon the ChapmanEnskog theory using a LennardJones 126 potential with correc-tions for high temperatures and densities The individual molecular parameters arecombined in an empirical way to develop enotective mixture parameters for use in thesingle-reguid equations These equations are given in detail in the appendix of SampSbut the essential equations are found in Hirschfelder et al (1954)

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1689

There are a few minor modishy cations to the equations used in SampS In the presentpaper a correction is added in the combination rules for the interaction between polarand non-polar molecules The Eucken correction is used for the thermal conductivityto account for internal degrees of freedom (Hirschfelder et al 1954) Finally thethermal-dinotusion factors are computed using the approximation given by Fristromamp Monchik (1988) for computational emacr ciency We stress that our previous workshows that the results are relatively insensitive to the transport properties as longas the general trends with temperature and pressure are captured

The SoaveRedlichKwong equation of state (EOS) is used with combination rulesfor the empirical constants of the mixture (Reid et al 1987) We use the polynomialdata shy ts for ideal gas specishy c heats at 1 atm from Gardiner (1984) and computedepartures with the EOS While this approach may seem overly simplishy ed comparedwith the EOS used in SampS a similar EOS for water and its reaction products thatallows us to account for shy nite-rate chemistry is not available It was found in SampSand again in the present work that the EOS has no impact on the basic physics ofthe problem and only modishy es the shy nal quantitative answers

(c) Motion and energy transport in the liquid

Rather than solve the NavierStokes equations governing the motion of the liquidwe assume that a RayleighPlesset-type equation for the bubble radius is valid TheRayleighPlesset equation is an ordinary dinoterential equation for the bubble radiuswhich is coupled with the gas dynamics of the interior through the pressure and stressat the bubble wall The form of the equation we use is similar to the one derivedby Prosperetti amp Lezzi (1986) dinotering only in that the present work accounts forthe extra velocity at the interface due to evaporation or condensation The enotect ofevaporationcondensation on the RayleighPlesset equation is very small but theenotect is included for consistency The resulting equation is similar to ones derivedby Yasui (1997a) and by Fujikawa amp Teruaki (1980)

Rather than solve the full energy equation in the liquid we solve the boundary-layer form as described in detail by Fyrillas amp Szeri (1994) The boundary-layerassumption reduces the energy equation to the linear dinotusion equation in Lagrangianboundary-layer coordinates This approximation for the energy equation was used inVuong amp Szeri (1996) and in SampS

(d) Evaporation and condensation

The equation derived from kinetic theory given in Carey (1992) is used to computethe rate of evaporation and condensation at the bubble surface This is written indimensionless variables as

_m00kin =

rMH2 O Z0

2 ordm M0

sup3iexcl PH2 Op

Tin tiexcl P s atp

Tin t

acute

_m00act = frac14 _m00

kin

9gt=

gt(25)

Here _m00kin is the rate at which water molecules hit the interface (from kinetic theory)

frac14 is the accommodation coe cient (sticking probability) _m00act is the rate at which

water molecules hit the interface and undergo phase change PH2 O is the partial

Proc R Soc Lond A (2000)

1690 B D Storey and A J Szeri

pressure of water P s at is the saturation pressure at the temperature of the interfaceTin t M0 is the initial molecular mass of the bubble contents iexcl is a correction forbulk motion to the interface and remains close to one for low mass transfer Weemphasize that this equation is only valid for reasonable temperatures and pressuresand not at the extreme conditions found during a bubble collapse As we argue belowthe evaporation condition is of vanishing consequence during the terminal stages ofbubble collapse when (25) loses validity

There is some discord in measurements of the accommodation coe cient Thevalue we use frac14 = 04 is consistent with that used by Yasui (1997a) (developed frommolecular dynamics simulations) and those recommended by Eames et al (1997)The sensitivity of the results to the accommodation coemacr cient will be consideredlater

(e) Chemical kinetics

Chemical kinetics described by the reaction mechanism determines the produc-tion and destruction of the dinoterent species The reaction mechanism that we use isdescribed in Maas amp Warnatz (1988) The mechanism includes 19 forward and reverseelementary reactions and nine species (Ar H H2 O O2 OH HO2 H2O2) Argononly enters the mechanism as a third body

The net rate of progress of each elementary reaction is the dinoterence in the forwardand reverse rate of progress The forward (reverse) rate of progress is given as theproduct of the reaction rate and the concentrations of all species on the reactant(product) side of the equation The change in concentration of species i with timeis the sum of the net rate of progress of each elementary reaction that containsspecies i The details can be found in several texts (eg Gardiner 1984) In thepresent formulation the chemical kinetics lead to the net chemical production ordestruction term curren i in (22)

The reaction rates are given empirically with the reaction mechanism The Linde-mann form is used to take into account the pressure fall-onot in some of the three-bodyreactions (Gardiner 1984) The high-pressure limit of the reaction rates for availablereactions were taken from Bowman et al (1999)

The elevated pressure in the bubble upon collapse accelerates the rate of progressof the reactions and alters the equilibrium state to which the reactions tend so thatfor a given temperature there is less dissociation at higher pressures This change inequilibrium state with pressure is accounted for in the reaction mechanism becauseforward and reverse reaction rates are related through the equilibrium constantThere is considerable uncertainty in the reaction rates at the high pressures weshall encounter The mechanism and reaction rates we use seem to be the best onespresently available

The reaction mechanism does not include ionization reactions The onset of ion-ization is delayed due to the high pressure The degree of ionization at the conditionswe report is quite low as estimated by the Saha equation Moss et al (1997) putforth the criterion that the onset of ionization in a dense gas is at one-quarter ofthe ionization potential in the present case this temperature is ca 30 000 K Theresults we present are consistent with neglecting ionization based on either of thesecriteria

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1691

(f ) Mass transfer to the liquid

One of the most di cult quantities to determine accurately is the rate of transportof radical species into the liquid While general convectiondinotusionreaction equa-tions can be written for each species dissolved in the liquid this approach involvesmany unknowns For example data on the solubility of OH radicals in liquid watervary by two orders of magnitude (Sander 1998) We know of no solubility data forH or O Moreover there is the possibility of heterogeneous surface reactions (VonSonntag et al 1999) which are poorly understood especially at the time of collapsewhen the liquid very near the bubble interface is supercritical

To avoid these uncertainties we take a very simple approach To determine the rateof radical invasion into the liquid approximately we introduce the uptake coemacr cientpound

pound =Nin iexcl Nou t

Ncoll

(26)

where Nin is the number of molecules invading the liquid Nou t is the number ofmolecules ejected from the liquid and Ncoll is the number of molecules collidingwith the surface (from kinetic theory) This view of mass transfer is oversimplishy edone can easily see that pound is not an intrinsic quantity (Takami et al 1998 Chameides1984)

This approach has nevertheless been used to determine the absorption of radicalspecies onto liquid and solid surfaces The uptake coe cients have been measured forsome of the species in our system over aqueous surfaces (Takami et al 1998 Hansonet al 1992) We shall study the inreguence of uptake coe cient on the results of ourcalculations below

(g) Numerical method

The same Chebyshev collocation method is used that was described in detail inSampS The variables in the gas are approximated by expansions in Chebyshev poly-nomials in the argon Lagrangian coordinate a The temperature in the liquid isapproximated with an expansion in rational Chebyshev polynomials The equationsin the gas evolve with a straightforward explicit predictorcorrector scheme We usestandard operator splitting (Press et al 1988) to solve the chemical kinetics equa-tions separately with a semi-implicit method to avoid time-step restrictions causedby the stinotness of these equations The energy equation in the liquid is integrated intime with a fully implicit method to avoid time-step restrictions associated with theclose spacing of grid points near the bubble wall As an error check global balancesof mass and energy are monitored imbalances always remain below a fraction of aper cent

3 Results and discussion

We begin by presenting detailed results of one particular conshy guration Due to thecomplexity of the problem we shall attempt to present the results in a systematicway in order to demonstrate the importance of certain phenomena The cases to beinvestigated are as follows

Proc R Soc Lond A (2000)

1692 B D Storey and A J Szeri

Table 1 Summary of the direg erent cases investigated in detail

(In all cases the bubble is a 45 m radius argon bubble in water at 298 K acoustically forcedwith a 12 bar pressure amplitude at 265 kHz)

case phase change chemistry OH uptake Rm ax =Rm in ( m) Tm ax (K)

I no no 00 280080 20 900

II yes no 00 313070 9 700

III yes yes 00 317065 7 000

IV(a) yes yes 0001 317065 7 000

IV(b) yes yes 001 317065 7 000

Case I no water vapour allowed in the bubble

Case II water vapour allowed to evaporate and condense but no chemical reactions

Case III water vapour and chemical reactions but no mass transfer across thebubble interface

Cases IV(a) and IV(b) mass transfer of OH across the bubble interface at dif-ferent uptake coe cients

In all cases the bubble is a 45 m-radius argon bubble in 298 K water acousticallyforced with a 12 bar pressure amplitude at 265 kHz These cases are summarized intable 1 With respect to SBSL this bubble was found to be stable to mass exchangeand shape changes though would have minimal light emission (Hilgenfeldt et al 1996 Wu amp Roberts 1998a b) We describe the results for more strongly forcedbubbles below The data we obtain are for steady-state oscillations ie after thedecay of initial transients After presenting the details of these cases we considervariations in the parameters to illustrate interesting trends

(a) Excess water is trapped by the rapid collapse

We shall now argue that signishy cant amounts of water vapour become trapped inthe interior of the bubble when it collapses violently Water is trapped in the interiorbecause the bubble motions become so rapid that the vapour near the centre hasinsu cient time to dinotuse to the bubble wall Non-equilibrium phase change alsoplays a signishy cant role in the amount of water that is trapped All results in thissubsection apply to case II this case allows one to focus on the dynamics of evapora-tion condensation and dinotusion of water inside the bubble without the complicationof chemical reactions

In shy gure 1 we show the radial history of the bubble over one cycle of the acousticforcing in case II The essential features of the bubble dynamics for cases I and II arethe same but the inclusion of water vapour increases the maximum radius and delaysthe collapse The dinoterence is to be expected because case I has no vapour pressureThe compression ratio (Rm ax=Rm in ) for cases I and II is 35 and 44 respectively Thecompression ratio is an approximate way of separating out dinoterences in the resultsdue to dinoterences in the bubble dynamics Typically a higher compression ratio leadsto more compression heating and higher temperatures if all else is equal We returnto this point later

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1693

0 10 20 30

10

20

30

time (micros)

radi

us (

microm

)

Figure 1 Radius of the bubble versus time over one cycle of acoustic forcing in case IIThe essential features for cases I II III and IV are the same

0 10 20 30

10 8

10 9

1010

1011

num

ber

of m

olec

ules

H2O

Ar

time (micros)

Figure 2 Total number of water-vapour molecules contained in the bubble versus time over onecycle of acoustic forcing in case II Note that water comprises ca 14 of the bubble contents atthe end of the main collapse

Proc R Soc Lond A (2000)

1694 B D Storey and A J Szeri

0

4000

8000te

mpe

ratu

re (

K)

1

2

3

time relative to minimum radius (ns)

num

ber

of H

2O m

olec

ules

109

acute-

- 4 - 2 0 2 4

(b)

(a)

Figure 3 Temperature (a) and total number of water molecules contained in the bubble (b) versustime around the point of minimum radius (time = 0) in case II The temperature undergoesorder-of-magnitude changes on this time-scale while the total number of water molecules changesvery little

In shy gure 2 we show the total number of H2O molecules contained within thebubble over the acoustic cycle (case II) A large amount of water evaporates intothe bubble during the main expansion when the pressure is low Water condenses onthe bubble wall during the collapse as the gas pressure increases There is signishy cantwater vapour in the bubble at the main collapse ca 14 (molar basis) of the totalbubble contents is water vapour If the interface were at equilibrium and the speedof mass dinotusion were inshy nite there would be a negligible amount of water in thebubble on collapse

In shy gure 3 we show the temperature history of the centre and the total numberof water molecules in the bubble around the time of minimum radius (time = 0) incase II Over the 10 ns time-scale the total amount of water vapour contained in thebubble is nearly constant whereas the temperature changes by an order of magni-tude This recalls SampS in which it was shown that evolving mixture compositionswere essentially frozen on the time-scale of the temperature peak

The temperature peak may be related to the light emission The equation foroptical power given by Moss et al (1999) with a constant opacity provides an easyestimate for the optical power output as a function of time For case II this estimategives a light emission pulse width (FWHM) of 350 ps which is much shorter than theca 2000 ps temperature peak This optical-pulse-width estimate would be somewhatshortened if the temperature dependence of the opacity were taken into accountThe peak optical power is ca 01 mW although this number is very sensitive to theopacity Typical stable SBSL experiments (at higher forcings than the present case)yield light emission of the order of milliwatts per regash (Gaitan et al 1996 Hiller etal 1992)

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1695

0 5 10 15 20 25

02

04

06

08

10

- 843 ns

- 455 ns

- 299 ns

- 153 ns

- 41 ns

- 15 ns

0 ns - 5 ns

radial position (microm)

mol

ar f

ract

ion

Figure 4 Molar fraction of water vapour versus radius for selected times prior to the moment ofminimum radius (time = 0) in case II Note the development of the non-uniform composition asthe collapse accelerates toward minimum radius In the last 15 ns before minimum radius thecomposition changes by only a few per cent when viewed in the Lagrangian frame instead

There are two mechanisms that determine the excess vapour left behind at thecollapse shy nite-rate mass dinotusion and non-equilibrium phase change We examineeach mechanism and their interaction in turn

(i) Role of direg usion

The shy rst reason for excess water vapour remaining during the collapse is the rapidbubble collapse overwhelming slow mass dinotusion this was extensively discussedin SampS In SampS it was shown that there is a competition between two importanttime-scales the dynamic time-scale of the bubble motions and the mass dinotusiontime-scale These are given explicitly in dimensionless terms as

frac12 d yn = R= _R

frac12 d if = ReScR2

D MH2 O

ordm ReSc1

Rp

T

In the latter we made use of the fact that the dinotusion coe cient of water vapourin the gas mixture D M

H2 O scales as R3pT

The time-scales argument is clarishy ed upon examination of shy gure 4 which showsthe evolving bubble composition as a function of radius at selected times before thetime of minimum radius (case II) In the early part of the collapse the dynamic time-scale is longer than the dinotusion time-scale frac12 d yn frac34 frac12 d if this results in a uniformcomposition of the bubble (upper curves in shy gure 4) As the bubble accelerates into

Proc R Soc Lond A (2000)

1696 B D Storey and A J Szeri

the collapse frac12 d yn sup1 frac12 d if and the decrease in water fraction at the centre begins tolag the decrease in water fraction at the wall (middle curves of shy gure 4) Furtherinto the collapse frac12 d yn frac12 frac12 d if and the water vapour cannot dinotuse to the bubble wallon the time-scale of the rapid collapse This slow dinotusion results in a nearly shy xeddistribution of water vapour (three lowest curves of shy gure 4)

While the three lowest curves in shy gure 4 look quite dinoterent in this Eulerianrepresentation they dinoter by only a few per cent when viewed in the Lagrangianrepresentation (not shown) The curve at iexcl 15 ns is from a time when R = R0 andthe temperature at the centre is ca 750 K The curve at 0 ns is from the time ofminimum radius (015R0) when the temperature at the centre is 9700 K Consistentwith shy gure 3 over times when there are drastic changes in the bubble radius andtemperature the composition of the bubble is almost a material shy eld

(ii) Role of non-equilibrium phase change

Non-equilibrium phase change is the second mechanism by which vapour is trappedin the bubble interior during the collapse By adjusting the accommodation coe -cient one has control over the time-scale for the partial pressure of the water vapourto come to equilibrium with the saturation pressure at the interface The dimension-less condensation time-scale is

frac12 con d =R

frac14

r2 ordm MH2 O

9M0Tin t

When frac12 d yn frac34 frac12 con d the condensation is in quasi-equilibrium with respect to thebubble motions If frac12 d yn frac12 frac12 con d then the bubble collapse is so rapid that no masscan escape the bubble on the time-scale of the collapse

(iii) Interaction of the two mechanisms

It is the competition between the time-scales for mass dinotusion condensation andbubble dynamics that determines which mechanism traps water vapour in the bubbleOnce frac12 d yn frac12 frac12 d if or frac12 d yn frac12 frac12 con d the total amount of water vapour in the bubbleis shy xed Only one of these conditions needs to be met for vapour to be trapped inthe bubble Both mechanisms can however participate to determine the quantity oftrapped vapour

During the bubble collapse the condition frac12 d yn frac12 frac12 d if is reached well before frac12 d yn frac12frac12 con d in case II Therefore in this case (as well as the others presented herein) it isthe slow mass dinotusion that is responsible for trapping vapour in the bubble

For cases where slow mass dinotusion traps the vapour non-equilibrium phasechange helps to determine the amount trapped The accommodation coe cient rep-resents a resistance to condensation at the interface during collapse of the bubble Alower value of accommodation coe cient provides more resistance to phase changeclearly increasing the mass of water retained in the bubble at any time during thecollapse The importance of the non-equilibrium phase change can be estimated bycomparing frac12 d yn with frac12 con d at the time that frac12 d yn = frac12 d if The closer frac12 d yn is to frac12 con d the more important non-equilibrium phase change is in determining the amount ofwater trapped

Colussi et al (1998) developed a model including chemical reactions in a collapsingbubble using an accommodation coe cient of frac14 = 0001 but did not take into

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1697

account the shy nite rate of mass dinotusion The low value of frac14 which works to trapsignishy cant water vapour at collapse was necessary for their model to be consistentwith experimental sonochemistry data When such a low value for frac14 is used thecondition frac12 d yn frac12 frac12 con d is reached well before frac12 d yn frac12 frac12 d if (the opposite of our case)Such a low accommodation coe cient works to trap signishy cant amounts of watervapour in the bubble but in this case the hindered regux at the interface eliminatesmass transfer out of the bubble while the distribution of water vapour within isessentially uniform In the present work we argue that the signishy cant amounts ofwater vapour trapped in the bubble are a consequence of slow dinotusion as well asnon-equilibrium phase change

The temperature of the interface exceeds the critical point for a very brief time(from iexcl 340 ps to 1800 ps relative to the time of minimum radius) For the conditionswe explored the maximum thickness of the supercritical region is quite thin less than1 of the bubble radius ie ca 10 nm We neglect any changes in the liquid equationof state and transport properties when the interface exceeds the critical temperatureThis assumption is justishy ed based on the thinness of the region and the fact that thecritical temperature is reached very late in the collapse Any disturbances caused bythe supercritical interface have insumacr cient time to propagate into the bubble interioron the main collapse Although the evaporation equation is invalid for these shorttimes the total mass added to the bubble is negligible regardless of the magnitudeof evaporative mass regux The issue of the super-critical interface may require futureimprovements but is beyond the scope of the present work

As a shy nal note both thermal and pressure dinotusion also play a role in determiningthe distribution of water vapour during the collapse In the argonwater system thethermal-dinotusion coemacr cient is small and the molecules are of similar size hence pres-sure dinotusion is quite weak Little dinoterence is seen in calculations with and withoutthese other forms of mass dinotusion In a helium or xenon bubble in water thermaland pressure dinotusion are more pronounced and can cause a 1015 dinoterence inthe amount of trapped water vapour In the interest of brevity we will not discussthese results further here

(b) Gas-phase chemical reactions break apart H2O and reduce the temperature

Now we turn to an investigation of what transpires when chemical reactions aretaken into account (case III) We shall show that there is signishy cant time for reactionsto occur ie the time-scale for chemical reaction is similar to that of the bubblecollapse The primarily endothermic reactions absorb a signishy cant amount of energyand further reduce the peak temperatures from cases I and II The distribution ofspecies produced in the bubble is determined by dinotusive transport of water vapourprior to the collapse and heat transfer out of the bubble during the collapse Thesteep temperature gradient at the wall conshy nes the bulk of the chemical reactions tothe bubble interior

(i) Consequences of chemistry

The overall bubble dynamics of cases II and III are essentially the same with acompression ratio of 49 in case III (slightly higher than case II) A convenient pointof departure is to compare the temperature histories of the bubble centre aroundthe main collapse for the three cases we have considered (see shy gure 5) The inclusion

Proc R Soc Lond A (2000)

1698 B D Storey and A J Szeri

- 4 - 2 0 2 410

3

104

tem

pera

ture

(K

)

time relative to minimum radius (ns)

Figure 5 Temperature history of the centre of the bubble around the time of minimum radius(t = 0) in cases I II and III Note the reduction in temperature from case I ( ) to case II( ) as water is allowed to enter the bubble The chemical reactions added in case III (||)further reduce the temperatures

of water vapour (case II) signishy cantly reduces the peak temperature from case IThe decrease in temperature is due mainly to the decrease in the ratio of specishy cheats of the mixture which reduces the compression heating Upon comparison ofcases II and III one observes that the chemical reactions further reduce the peaktemperature due to the endothermic decomposition of the water vapour The centretemperature histories of cases II and III are nearly identical below 4000 K

It is important to understand that the temperature dinoterences in these threecases are not due to subtle dinoterences in bubble dynamics The compression ratio(Rm ax=Rm in ) of the three cases increases from case I to case II to case III whereasthe temperature decreases in this order This decrease in temperature occurs despitethe fact that a higher compression ratio will lead to more compression heating in thebubble collapse all other things being equal In all these cases there are no shocksinside the bubble Shocks will be considered in x 3 e

In shy gure 6 we show the history of the total number of molecules of the dinoterentspecies as a function of time on the same time-scale as in shy gure 5 The lowest curve isthe total amount of water vapour only The next curve is the total amount of watervapour plus the total amount of H2 Each curve represents the addition of one speciesas labelled until the uppermost curve denotes everything but argon We do not labelHO2 and H2O2 as they appear only in trace amounts One observes which speciesare most abundant during the time of the main collapse Note there is a relativelylarge amount of undissociated water considering the elevated temperatures This isprimarily a consequence of the high pressure suppressing the dissociation of H2O

From shy gures 5 and 6 one observes that the onset of signishy cant chemical activity in

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1699

- 4 - 2 0 2 4

109

1010

time relative to minimum radius (ns)

num

ber

of m

olec

ules

H2O

H2

O2

OH

O

H

Figure 6 Cumulative number of molecules of each species in case III The lowest curve is thetotal number of water molecules the next curve is total water + H2 and so on H2 O2 and HO2

are neglected in this plot as they appear in trace amounts

the bubble corresponds to a centre temperature of ca 4000 K Therefore the reducedtemperature of case III shown in shy gure 5 can be attributed to the endothermicchemical reactions Although the bubble dynamics are quite rapid the chemicalreactions keep pace as is shown by the similar time-scales of the changing quantitiesin shy gures 5 and 6 The rapid rate at which the reactions progress is due to the veryhigh density Recall that the rate of progress of each reaction is directly related tothe species concentration (mol cmiexcl3)

(ii) Dynamics and distribution of OH production

In sonochemistry applications the invasion of OH radicals into the liquid isthought to be responsible for much of the complex and useful chemistry that occursTherefore we shall focus on the behaviour of this species in particular In shy gure 7 weplot the total number of OH molecules contained in the bubble on two time-scalesover one acoustic cycle in shy gure 7a and around the time of collapse in shy gure 7b Theinitial concentrations of all species were selected so that the chemical production anddestruction balanced in one acoustic cycle The main collapse is associated with arapid creation of OH once the temperature exceeds 4000 K

Upon rebound the bubble rapidly cools and the OH and other products tend torecombine back into H2O These recombination reactions proceed more slowly thanthe dissociation reactions upon collapse The result is that the bubble is left with asuper-equilibrium amount of OH after it has cooled to ambient temperature Thissuper-equilibrium OH then slowly decays through the rest of the cycle with briefspurts of production and destruction at each of the after-bounces

Proc R Soc Lond A (2000)

1700 B D Storey and A J Szeri

0 10 20 30time (micros)

- 4 - 2 0 2 4

time relative to minimum radius (ns)

106

1010

104

106

108

108

num

ber

of O

H

radi

cals

nu

mbe

r of

OH

ra

dica

ls

Figure 7 Total number of OH molecules contained in the bubble versus time over one acousticcycle (a) and around the time of minimum radius (b) in case III The macrrst sharp peak in (a)is expanded in (b) Subsequent peaks in (a) correspond to local radius minima in the bubbledynamics The initial amounts of OH and all the other species were chosen for no net speciesproduction or destruction over an acoustic cycle

The distribution of OH in the bubble is characterized by a fairly uniform con-centration through much of the bubble interior and a very sharp gradient near thewall This sharp gradient is a result of heat transfer during the collapse Duringcompression conduction between the hot interior and the cool wall sets up a thintemperature boundary layer This boundary layer conshy nes the chemical reactions tothe interior away from the interface The cooler wall leads to a much lower concen-tration of OH at the interface than in the centre at the time of the main collapseIn shy gure 8 we show the molar concentration history of OH at the centre and at theinterface When the bubble is at minimum radius and the OH concentration at thecentre is at its maximum the concentration at the wall is three orders of magnitudelower

Because the chemistry time-scale is much faster than the dinotusion time-scale OHis produced where water vapour was trapped at high enough temperature The liquidinterface is insulated from the high concentration of OH generated deep inside in thebubble The only OH produced at the centre that reaches the bubble interface is theslowly decaying super-equilibrium OH that is left inside the bubble on re-expansion

(c) OH uptake by the liquid does not indeg uence the bubble interior

The inclusion of liquid uptake of OH (case IV) adds further complications wenow try to illuminate some of the main qualitative features We shall argue that OHuptake has little enotect on the gas dynamics and chemical production

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1701

- 4 - 2 0 2 410 - 10

10 - 8

10 - 6

10 - 4

10 - 2

time relative to minimum radius (ns)

conc

entr

atio

n of

OH

(m

ol c

m-3

)

Figure 8 Molar concentration of OH at the centre (||) and the interface ( ) versus timearound the point of minimum radius Note that the concentration at the centre is about threeorders of magnitude greater than at the interface at the point of minimum radius

Due to dinotering ideas on the proper value of the uptake coemacr cient the values areset at pound = 0001 and pound = 001 (cases IV(a) and IV(b)) consistent with the work ofothers to which we made reference earlier The bubble dynamics and temperatureproshy les for cases IV(a) and IV(b) are indistinguishable from those for case III Inshy gure 9 we show the total number of OH molecules in the bubble versus time ontwo dinoterent scales for cases III IV(a) and IV(b) The history for one acoustic cycleis plotted in shy gure 9a and the history concentrated at the main collapse is plottedin shy gure 9b

In the main expansion when the bubble is cool the OH resident in the bubblewill invade the liquid as can be seen by the decrease in the amount of OH at theend of the long expansion as the uptake increases When the bubble collapses thechemical production of OH from recently evaporated water rapidly overwhelms theOH uptake by the liquid As the bubble proceeds into the collapse the curves oftotal OH content for dinoterent uptakes become indistinguishable The total amountof OH in the bubble is dominated by the amount created in the interior away fromthe wall The total amount of OH lost to the liquid during a collapse is only afraction of the amount contained in the bubble Hence the peak amount of OH inthe bubble during the collapse is insensitive to the uptake coe cient of OH Similarstatements apply to uptake of other species

In shy gure 10 we show for the purposes of qualitative discussion the total massof OH exiting the bubble in cases IV(a) and IV(b) While the OH uptake coef-shy cient does not inreguence the total amount of OH contained in the bubble at themain collapse there are large dinoterences in the amount invading the liquid over

Proc R Soc Lond A (2000)

1702 B D Storey and A J Szeri

0 10 20 30

106

1010

102

time (micros)

- 4 - 2 0 2 4104

106

108

time relative to minimum radius (ns)

1010

(b)

(a)nu

mbe

r of

OH

ra

dica

ls

num

ber

of O

H

radi

cals

Figure 9 Comparison of total amount number of OH molecules in the bubble versus time overone acoustic cycle (a) and focused around the time of minimum radius (b) in cases III (||)IV(a) ( ) and IV(b) ( ) The uptake of OH plays a role during the slow expansionswhen the resident amount of OH left in the bubble is well above chemical equilibrium At thecollapse OH uptake does not indeg uence the total amount of OH production

a cycle Much of the OH uptake occurs during the main collapse and the subse-quent re-expansion when the concentration of OH at the wall is at its maximum(see shy gure 8) Further uptake occurs throughout the cycle at each of the after-bounces

(d ) Inclusion of water vapour alters the ereg ect of increased forcing

Next we shall compare the results of cases I II and III with all parameters shy xedexcept for the acoustic pressure amplitude which varies between 10 and 13 barWe that note this is experimentally awkward for SBSL because pressure amplitudeand ambient radius are dependent quantities through the mass exchange equilibriumof rectishy ed dinotusion (Hilgenfeldt et al 1996) The way we present these results isnonetheless useful as it demonstrates the importance of water vapour in the under-standing of sonoluminescence and sonochemistry

Because the bubble dynamics are altered by the inclusion of water vapour weuse the compression ratio as the distinguishing parameter In shy gure 11 we plot themaximum centre temperature versus the compression ratio for cases I II and III Thecalculations in shy gure 11 are devoid of strong shocks a matter to which we devotemore attention below If there are strong shocks then shy gure 11 will look the samein terms of a suitably averaged temperature rather than the centre temperature

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1703

0 10 20 30102

103

104

105

106

num

ber

of O

H

radi

cals

inva

ding

liqu

id

time (micros)

Figure 10 Comparison of the amount of OH absorbed by the liquid versus time for cases IV(a)(|| uptake coeplusmn cient pound = 0001) and IV(b) ( pound = 001) Most of the OH uptakeoccurs at the main collapse and the subsequent after-bounce

In case I the plot shows linear behaviour (on the loglog scale) One observes thatwhen water vapour is not taken into account stronger forcing yields more extremetemperatures

In case II the relationship between compression ratio and maximum temperature isvery dinoterent First we note that the amount of water trapped in the bubble increaseswith compression ratio If we revisit the water-trapping mechanism (x 3 a (iii)) thistrend is readily explained The increase in water-vapour content reduces the ratioof specishy c heats of the gasvapour mixture and hence reduces the compressionheating

At low compression ratios in case II the temperature increases with compressionratio in a manner very similar to case I At low compression ratios the amountof water at collapse is very small the bubble is nearly pure argon As the forcingis increased more water is trapped and the maximum temperature of the bubblebecomes nearly constant In this range of bubble dynamics there is an almost per-fectly compensating enotect (with respect to peak temperature) between the decreasein specishy c-heat ratio and the increase in compression Case III is very similar tocase II except that the plateau in temperature occurs at a lower temperature aconsequence of the endothermic chemical reactions

(e) Inclusion of water vapour promotes the formation of shocks

Water vapour is of great importance in the formation of shocks in very stronglyforced bubbles We shall brieregy consider a 60 m bubble forced with a pressureamplitude of 14 bar at a frequency of 206 kHz (Case A1 in Moss et al (1999))

Proc R Soc Lond A (2000)

1704 B D Storey and A J Szeri

10 1 10 2

10 3

10 4

10 5

Rmax R min

max

imum

tem

pera

ture

(K

)

Figure 11 Change in centre temperature versus compression ratio in cases I ( ) II ( curren )and III (+) In this plot all parameters were held constant except the pressure amplitudewhich was varied between 10 and 13 bar Case I shows a steady increase in temperature withcompression ratio Cases II and III reach a maximum temperature as the forcing is increased

When water vapour is neglected a compression wave is launched into the bubble oncollapse but the wave does not steepen into a shock When water vapour is takeninto account (with or without reactions) the compression wave steepens into a strongshock very near the centre of the bubble Vuong et al (1999) showed that the rateof steepening depends on the amount of compression heating before the compressionwave is launched

In very recent work Lin amp Szeri (2000) used the wavefront expansion technique todetermine the evolution of a compression or rarefaction wavefront that is propagatingspherically into a collapsing or expanding gas of non-uniform entropy This analyticalwork supports the idea that when the speed of sound is higher at the centre (dueto temperature and composition inhomogeneities) shock formation is delayed orprevented Furthermore this work demonstrated that the inward radial velocity alsohinders the steepening of compression waves into shocks

Consistent with the arguments made in Lin amp Szeri (2000) Vuong et al (1999)and Moss et al (1999) our results show that the decrease in the ratio of specishy cheats due to an increase in the water-vapour content can promote shock formationThe point we want to emphasize is that the question of shock formation cannot beundertaken without consideration of water vapour

With respect to total chemical production shocks are of negligible consequenceThe shock inreguences only about the inner 10 of the radius which is only 01 ofthe volume Therefore the increased production of OH at the centre does not anotectthe total amount produced in the bubble

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1689

There are a few minor modishy cations to the equations used in SampS In the presentpaper a correction is added in the combination rules for the interaction between polarand non-polar molecules The Eucken correction is used for the thermal conductivityto account for internal degrees of freedom (Hirschfelder et al 1954) Finally thethermal-dinotusion factors are computed using the approximation given by Fristromamp Monchik (1988) for computational emacr ciency We stress that our previous workshows that the results are relatively insensitive to the transport properties as longas the general trends with temperature and pressure are captured

The SoaveRedlichKwong equation of state (EOS) is used with combination rulesfor the empirical constants of the mixture (Reid et al 1987) We use the polynomialdata shy ts for ideal gas specishy c heats at 1 atm from Gardiner (1984) and computedepartures with the EOS While this approach may seem overly simplishy ed comparedwith the EOS used in SampS a similar EOS for water and its reaction products thatallows us to account for shy nite-rate chemistry is not available It was found in SampSand again in the present work that the EOS has no impact on the basic physics ofthe problem and only modishy es the shy nal quantitative answers

(c) Motion and energy transport in the liquid

Rather than solve the NavierStokes equations governing the motion of the liquidwe assume that a RayleighPlesset-type equation for the bubble radius is valid TheRayleighPlesset equation is an ordinary dinoterential equation for the bubble radiuswhich is coupled with the gas dynamics of the interior through the pressure and stressat the bubble wall The form of the equation we use is similar to the one derivedby Prosperetti amp Lezzi (1986) dinotering only in that the present work accounts forthe extra velocity at the interface due to evaporation or condensation The enotect ofevaporationcondensation on the RayleighPlesset equation is very small but theenotect is included for consistency The resulting equation is similar to ones derivedby Yasui (1997a) and by Fujikawa amp Teruaki (1980)

Rather than solve the full energy equation in the liquid we solve the boundary-layer form as described in detail by Fyrillas amp Szeri (1994) The boundary-layerassumption reduces the energy equation to the linear dinotusion equation in Lagrangianboundary-layer coordinates This approximation for the energy equation was used inVuong amp Szeri (1996) and in SampS

(d) Evaporation and condensation

The equation derived from kinetic theory given in Carey (1992) is used to computethe rate of evaporation and condensation at the bubble surface This is written indimensionless variables as

_m00kin =

rMH2 O Z0

2 ordm M0

sup3iexcl PH2 Op

Tin tiexcl P s atp

Tin t

acute

_m00act = frac14 _m00

kin

9gt=

gt(25)

Here _m00kin is the rate at which water molecules hit the interface (from kinetic theory)

frac14 is the accommodation coe cient (sticking probability) _m00act is the rate at which

water molecules hit the interface and undergo phase change PH2 O is the partial

Proc R Soc Lond A (2000)

1690 B D Storey and A J Szeri

pressure of water P s at is the saturation pressure at the temperature of the interfaceTin t M0 is the initial molecular mass of the bubble contents iexcl is a correction forbulk motion to the interface and remains close to one for low mass transfer Weemphasize that this equation is only valid for reasonable temperatures and pressuresand not at the extreme conditions found during a bubble collapse As we argue belowthe evaporation condition is of vanishing consequence during the terminal stages ofbubble collapse when (25) loses validity

There is some discord in measurements of the accommodation coe cient Thevalue we use frac14 = 04 is consistent with that used by Yasui (1997a) (developed frommolecular dynamics simulations) and those recommended by Eames et al (1997)The sensitivity of the results to the accommodation coemacr cient will be consideredlater

(e) Chemical kinetics

Chemical kinetics described by the reaction mechanism determines the produc-tion and destruction of the dinoterent species The reaction mechanism that we use isdescribed in Maas amp Warnatz (1988) The mechanism includes 19 forward and reverseelementary reactions and nine species (Ar H H2 O O2 OH HO2 H2O2) Argononly enters the mechanism as a third body

The net rate of progress of each elementary reaction is the dinoterence in the forwardand reverse rate of progress The forward (reverse) rate of progress is given as theproduct of the reaction rate and the concentrations of all species on the reactant(product) side of the equation The change in concentration of species i with timeis the sum of the net rate of progress of each elementary reaction that containsspecies i The details can be found in several texts (eg Gardiner 1984) In thepresent formulation the chemical kinetics lead to the net chemical production ordestruction term curren i in (22)

The reaction rates are given empirically with the reaction mechanism The Linde-mann form is used to take into account the pressure fall-onot in some of the three-bodyreactions (Gardiner 1984) The high-pressure limit of the reaction rates for availablereactions were taken from Bowman et al (1999)

The elevated pressure in the bubble upon collapse accelerates the rate of progressof the reactions and alters the equilibrium state to which the reactions tend so thatfor a given temperature there is less dissociation at higher pressures This change inequilibrium state with pressure is accounted for in the reaction mechanism becauseforward and reverse reaction rates are related through the equilibrium constantThere is considerable uncertainty in the reaction rates at the high pressures weshall encounter The mechanism and reaction rates we use seem to be the best onespresently available

The reaction mechanism does not include ionization reactions The onset of ion-ization is delayed due to the high pressure The degree of ionization at the conditionswe report is quite low as estimated by the Saha equation Moss et al (1997) putforth the criterion that the onset of ionization in a dense gas is at one-quarter ofthe ionization potential in the present case this temperature is ca 30 000 K Theresults we present are consistent with neglecting ionization based on either of thesecriteria

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1691

(f ) Mass transfer to the liquid

One of the most di cult quantities to determine accurately is the rate of transportof radical species into the liquid While general convectiondinotusionreaction equa-tions can be written for each species dissolved in the liquid this approach involvesmany unknowns For example data on the solubility of OH radicals in liquid watervary by two orders of magnitude (Sander 1998) We know of no solubility data forH or O Moreover there is the possibility of heterogeneous surface reactions (VonSonntag et al 1999) which are poorly understood especially at the time of collapsewhen the liquid very near the bubble interface is supercritical

To avoid these uncertainties we take a very simple approach To determine the rateof radical invasion into the liquid approximately we introduce the uptake coemacr cientpound

pound =Nin iexcl Nou t

Ncoll

(26)

where Nin is the number of molecules invading the liquid Nou t is the number ofmolecules ejected from the liquid and Ncoll is the number of molecules collidingwith the surface (from kinetic theory) This view of mass transfer is oversimplishy edone can easily see that pound is not an intrinsic quantity (Takami et al 1998 Chameides1984)

This approach has nevertheless been used to determine the absorption of radicalspecies onto liquid and solid surfaces The uptake coe cients have been measured forsome of the species in our system over aqueous surfaces (Takami et al 1998 Hansonet al 1992) We shall study the inreguence of uptake coe cient on the results of ourcalculations below

(g) Numerical method

The same Chebyshev collocation method is used that was described in detail inSampS The variables in the gas are approximated by expansions in Chebyshev poly-nomials in the argon Lagrangian coordinate a The temperature in the liquid isapproximated with an expansion in rational Chebyshev polynomials The equationsin the gas evolve with a straightforward explicit predictorcorrector scheme We usestandard operator splitting (Press et al 1988) to solve the chemical kinetics equa-tions separately with a semi-implicit method to avoid time-step restrictions causedby the stinotness of these equations The energy equation in the liquid is integrated intime with a fully implicit method to avoid time-step restrictions associated with theclose spacing of grid points near the bubble wall As an error check global balancesof mass and energy are monitored imbalances always remain below a fraction of aper cent

3 Results and discussion

We begin by presenting detailed results of one particular conshy guration Due to thecomplexity of the problem we shall attempt to present the results in a systematicway in order to demonstrate the importance of certain phenomena The cases to beinvestigated are as follows

Proc R Soc Lond A (2000)

1692 B D Storey and A J Szeri

Table 1 Summary of the direg erent cases investigated in detail

(In all cases the bubble is a 45 m radius argon bubble in water at 298 K acoustically forcedwith a 12 bar pressure amplitude at 265 kHz)

case phase change chemistry OH uptake Rm ax =Rm in ( m) Tm ax (K)

I no no 00 280080 20 900

II yes no 00 313070 9 700

III yes yes 00 317065 7 000

IV(a) yes yes 0001 317065 7 000

IV(b) yes yes 001 317065 7 000

Case I no water vapour allowed in the bubble

Case II water vapour allowed to evaporate and condense but no chemical reactions

Case III water vapour and chemical reactions but no mass transfer across thebubble interface

Cases IV(a) and IV(b) mass transfer of OH across the bubble interface at dif-ferent uptake coe cients

In all cases the bubble is a 45 m-radius argon bubble in 298 K water acousticallyforced with a 12 bar pressure amplitude at 265 kHz These cases are summarized intable 1 With respect to SBSL this bubble was found to be stable to mass exchangeand shape changes though would have minimal light emission (Hilgenfeldt et al 1996 Wu amp Roberts 1998a b) We describe the results for more strongly forcedbubbles below The data we obtain are for steady-state oscillations ie after thedecay of initial transients After presenting the details of these cases we considervariations in the parameters to illustrate interesting trends

(a) Excess water is trapped by the rapid collapse

We shall now argue that signishy cant amounts of water vapour become trapped inthe interior of the bubble when it collapses violently Water is trapped in the interiorbecause the bubble motions become so rapid that the vapour near the centre hasinsu cient time to dinotuse to the bubble wall Non-equilibrium phase change alsoplays a signishy cant role in the amount of water that is trapped All results in thissubsection apply to case II this case allows one to focus on the dynamics of evapora-tion condensation and dinotusion of water inside the bubble without the complicationof chemical reactions

In shy gure 1 we show the radial history of the bubble over one cycle of the acousticforcing in case II The essential features of the bubble dynamics for cases I and II arethe same but the inclusion of water vapour increases the maximum radius and delaysthe collapse The dinoterence is to be expected because case I has no vapour pressureThe compression ratio (Rm ax=Rm in ) for cases I and II is 35 and 44 respectively Thecompression ratio is an approximate way of separating out dinoterences in the resultsdue to dinoterences in the bubble dynamics Typically a higher compression ratio leadsto more compression heating and higher temperatures if all else is equal We returnto this point later

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1693

0 10 20 30

10

20

30

time (micros)

radi

us (

microm

)

Figure 1 Radius of the bubble versus time over one cycle of acoustic forcing in case IIThe essential features for cases I II III and IV are the same

0 10 20 30

10 8

10 9

1010

1011

num

ber

of m

olec

ules

H2O

Ar

time (micros)

Figure 2 Total number of water-vapour molecules contained in the bubble versus time over onecycle of acoustic forcing in case II Note that water comprises ca 14 of the bubble contents atthe end of the main collapse

Proc R Soc Lond A (2000)

1694 B D Storey and A J Szeri

0

4000

8000te

mpe

ratu

re (

K)

1

2

3

time relative to minimum radius (ns)

num

ber

of H

2O m

olec

ules

109

acute-

- 4 - 2 0 2 4

(b)

(a)

Figure 3 Temperature (a) and total number of water molecules contained in the bubble (b) versustime around the point of minimum radius (time = 0) in case II The temperature undergoesorder-of-magnitude changes on this time-scale while the total number of water molecules changesvery little

In shy gure 2 we show the total number of H2O molecules contained within thebubble over the acoustic cycle (case II) A large amount of water evaporates intothe bubble during the main expansion when the pressure is low Water condenses onthe bubble wall during the collapse as the gas pressure increases There is signishy cantwater vapour in the bubble at the main collapse ca 14 (molar basis) of the totalbubble contents is water vapour If the interface were at equilibrium and the speedof mass dinotusion were inshy nite there would be a negligible amount of water in thebubble on collapse

In shy gure 3 we show the temperature history of the centre and the total numberof water molecules in the bubble around the time of minimum radius (time = 0) incase II Over the 10 ns time-scale the total amount of water vapour contained in thebubble is nearly constant whereas the temperature changes by an order of magni-tude This recalls SampS in which it was shown that evolving mixture compositionswere essentially frozen on the time-scale of the temperature peak

The temperature peak may be related to the light emission The equation foroptical power given by Moss et al (1999) with a constant opacity provides an easyestimate for the optical power output as a function of time For case II this estimategives a light emission pulse width (FWHM) of 350 ps which is much shorter than theca 2000 ps temperature peak This optical-pulse-width estimate would be somewhatshortened if the temperature dependence of the opacity were taken into accountThe peak optical power is ca 01 mW although this number is very sensitive to theopacity Typical stable SBSL experiments (at higher forcings than the present case)yield light emission of the order of milliwatts per regash (Gaitan et al 1996 Hiller etal 1992)

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1695

0 5 10 15 20 25

02

04

06

08

10

- 843 ns

- 455 ns

- 299 ns

- 153 ns

- 41 ns

- 15 ns

0 ns - 5 ns

radial position (microm)

mol

ar f

ract

ion

Figure 4 Molar fraction of water vapour versus radius for selected times prior to the moment ofminimum radius (time = 0) in case II Note the development of the non-uniform composition asthe collapse accelerates toward minimum radius In the last 15 ns before minimum radius thecomposition changes by only a few per cent when viewed in the Lagrangian frame instead

There are two mechanisms that determine the excess vapour left behind at thecollapse shy nite-rate mass dinotusion and non-equilibrium phase change We examineeach mechanism and their interaction in turn

(i) Role of direg usion

The shy rst reason for excess water vapour remaining during the collapse is the rapidbubble collapse overwhelming slow mass dinotusion this was extensively discussedin SampS In SampS it was shown that there is a competition between two importanttime-scales the dynamic time-scale of the bubble motions and the mass dinotusiontime-scale These are given explicitly in dimensionless terms as

frac12 d yn = R= _R

frac12 d if = ReScR2

D MH2 O

ordm ReSc1

Rp

T

In the latter we made use of the fact that the dinotusion coe cient of water vapourin the gas mixture D M

H2 O scales as R3pT

The time-scales argument is clarishy ed upon examination of shy gure 4 which showsthe evolving bubble composition as a function of radius at selected times before thetime of minimum radius (case II) In the early part of the collapse the dynamic time-scale is longer than the dinotusion time-scale frac12 d yn frac34 frac12 d if this results in a uniformcomposition of the bubble (upper curves in shy gure 4) As the bubble accelerates into

Proc R Soc Lond A (2000)

1696 B D Storey and A J Szeri

the collapse frac12 d yn sup1 frac12 d if and the decrease in water fraction at the centre begins tolag the decrease in water fraction at the wall (middle curves of shy gure 4) Furtherinto the collapse frac12 d yn frac12 frac12 d if and the water vapour cannot dinotuse to the bubble wallon the time-scale of the rapid collapse This slow dinotusion results in a nearly shy xeddistribution of water vapour (three lowest curves of shy gure 4)

While the three lowest curves in shy gure 4 look quite dinoterent in this Eulerianrepresentation they dinoter by only a few per cent when viewed in the Lagrangianrepresentation (not shown) The curve at iexcl 15 ns is from a time when R = R0 andthe temperature at the centre is ca 750 K The curve at 0 ns is from the time ofminimum radius (015R0) when the temperature at the centre is 9700 K Consistentwith shy gure 3 over times when there are drastic changes in the bubble radius andtemperature the composition of the bubble is almost a material shy eld

(ii) Role of non-equilibrium phase change

Non-equilibrium phase change is the second mechanism by which vapour is trappedin the bubble interior during the collapse By adjusting the accommodation coe -cient one has control over the time-scale for the partial pressure of the water vapourto come to equilibrium with the saturation pressure at the interface The dimension-less condensation time-scale is

frac12 con d =R

frac14

r2 ordm MH2 O

9M0Tin t

When frac12 d yn frac34 frac12 con d the condensation is in quasi-equilibrium with respect to thebubble motions If frac12 d yn frac12 frac12 con d then the bubble collapse is so rapid that no masscan escape the bubble on the time-scale of the collapse

(iii) Interaction of the two mechanisms

It is the competition between the time-scales for mass dinotusion condensation andbubble dynamics that determines which mechanism traps water vapour in the bubbleOnce frac12 d yn frac12 frac12 d if or frac12 d yn frac12 frac12 con d the total amount of water vapour in the bubbleis shy xed Only one of these conditions needs to be met for vapour to be trapped inthe bubble Both mechanisms can however participate to determine the quantity oftrapped vapour

During the bubble collapse the condition frac12 d yn frac12 frac12 d if is reached well before frac12 d yn frac12frac12 con d in case II Therefore in this case (as well as the others presented herein) it isthe slow mass dinotusion that is responsible for trapping vapour in the bubble

For cases where slow mass dinotusion traps the vapour non-equilibrium phasechange helps to determine the amount trapped The accommodation coe cient rep-resents a resistance to condensation at the interface during collapse of the bubble Alower value of accommodation coe cient provides more resistance to phase changeclearly increasing the mass of water retained in the bubble at any time during thecollapse The importance of the non-equilibrium phase change can be estimated bycomparing frac12 d yn with frac12 con d at the time that frac12 d yn = frac12 d if The closer frac12 d yn is to frac12 con d the more important non-equilibrium phase change is in determining the amount ofwater trapped

Colussi et al (1998) developed a model including chemical reactions in a collapsingbubble using an accommodation coe cient of frac14 = 0001 but did not take into

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1697

account the shy nite rate of mass dinotusion The low value of frac14 which works to trapsignishy cant water vapour at collapse was necessary for their model to be consistentwith experimental sonochemistry data When such a low value for frac14 is used thecondition frac12 d yn frac12 frac12 con d is reached well before frac12 d yn frac12 frac12 d if (the opposite of our case)Such a low accommodation coe cient works to trap signishy cant amounts of watervapour in the bubble but in this case the hindered regux at the interface eliminatesmass transfer out of the bubble while the distribution of water vapour within isessentially uniform In the present work we argue that the signishy cant amounts ofwater vapour trapped in the bubble are a consequence of slow dinotusion as well asnon-equilibrium phase change

The temperature of the interface exceeds the critical point for a very brief time(from iexcl 340 ps to 1800 ps relative to the time of minimum radius) For the conditionswe explored the maximum thickness of the supercritical region is quite thin less than1 of the bubble radius ie ca 10 nm We neglect any changes in the liquid equationof state and transport properties when the interface exceeds the critical temperatureThis assumption is justishy ed based on the thinness of the region and the fact that thecritical temperature is reached very late in the collapse Any disturbances caused bythe supercritical interface have insumacr cient time to propagate into the bubble interioron the main collapse Although the evaporation equation is invalid for these shorttimes the total mass added to the bubble is negligible regardless of the magnitudeof evaporative mass regux The issue of the super-critical interface may require futureimprovements but is beyond the scope of the present work

As a shy nal note both thermal and pressure dinotusion also play a role in determiningthe distribution of water vapour during the collapse In the argonwater system thethermal-dinotusion coemacr cient is small and the molecules are of similar size hence pres-sure dinotusion is quite weak Little dinoterence is seen in calculations with and withoutthese other forms of mass dinotusion In a helium or xenon bubble in water thermaland pressure dinotusion are more pronounced and can cause a 1015 dinoterence inthe amount of trapped water vapour In the interest of brevity we will not discussthese results further here

(b) Gas-phase chemical reactions break apart H2O and reduce the temperature

Now we turn to an investigation of what transpires when chemical reactions aretaken into account (case III) We shall show that there is signishy cant time for reactionsto occur ie the time-scale for chemical reaction is similar to that of the bubblecollapse The primarily endothermic reactions absorb a signishy cant amount of energyand further reduce the peak temperatures from cases I and II The distribution ofspecies produced in the bubble is determined by dinotusive transport of water vapourprior to the collapse and heat transfer out of the bubble during the collapse Thesteep temperature gradient at the wall conshy nes the bulk of the chemical reactions tothe bubble interior

(i) Consequences of chemistry

The overall bubble dynamics of cases II and III are essentially the same with acompression ratio of 49 in case III (slightly higher than case II) A convenient pointof departure is to compare the temperature histories of the bubble centre aroundthe main collapse for the three cases we have considered (see shy gure 5) The inclusion

Proc R Soc Lond A (2000)

1698 B D Storey and A J Szeri

- 4 - 2 0 2 410

3

104

tem

pera

ture

(K

)

time relative to minimum radius (ns)

Figure 5 Temperature history of the centre of the bubble around the time of minimum radius(t = 0) in cases I II and III Note the reduction in temperature from case I ( ) to case II( ) as water is allowed to enter the bubble The chemical reactions added in case III (||)further reduce the temperatures

of water vapour (case II) signishy cantly reduces the peak temperature from case IThe decrease in temperature is due mainly to the decrease in the ratio of specishy cheats of the mixture which reduces the compression heating Upon comparison ofcases II and III one observes that the chemical reactions further reduce the peaktemperature due to the endothermic decomposition of the water vapour The centretemperature histories of cases II and III are nearly identical below 4000 K

It is important to understand that the temperature dinoterences in these threecases are not due to subtle dinoterences in bubble dynamics The compression ratio(Rm ax=Rm in ) of the three cases increases from case I to case II to case III whereasthe temperature decreases in this order This decrease in temperature occurs despitethe fact that a higher compression ratio will lead to more compression heating in thebubble collapse all other things being equal In all these cases there are no shocksinside the bubble Shocks will be considered in x 3 e

In shy gure 6 we show the history of the total number of molecules of the dinoterentspecies as a function of time on the same time-scale as in shy gure 5 The lowest curve isthe total amount of water vapour only The next curve is the total amount of watervapour plus the total amount of H2 Each curve represents the addition of one speciesas labelled until the uppermost curve denotes everything but argon We do not labelHO2 and H2O2 as they appear only in trace amounts One observes which speciesare most abundant during the time of the main collapse Note there is a relativelylarge amount of undissociated water considering the elevated temperatures This isprimarily a consequence of the high pressure suppressing the dissociation of H2O

From shy gures 5 and 6 one observes that the onset of signishy cant chemical activity in

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1699

- 4 - 2 0 2 4

109

1010

time relative to minimum radius (ns)

num

ber

of m

olec

ules

H2O

H2

O2

OH

O

H

Figure 6 Cumulative number of molecules of each species in case III The lowest curve is thetotal number of water molecules the next curve is total water + H2 and so on H2 O2 and HO2

are neglected in this plot as they appear in trace amounts

the bubble corresponds to a centre temperature of ca 4000 K Therefore the reducedtemperature of case III shown in shy gure 5 can be attributed to the endothermicchemical reactions Although the bubble dynamics are quite rapid the chemicalreactions keep pace as is shown by the similar time-scales of the changing quantitiesin shy gures 5 and 6 The rapid rate at which the reactions progress is due to the veryhigh density Recall that the rate of progress of each reaction is directly related tothe species concentration (mol cmiexcl3)

(ii) Dynamics and distribution of OH production

In sonochemistry applications the invasion of OH radicals into the liquid isthought to be responsible for much of the complex and useful chemistry that occursTherefore we shall focus on the behaviour of this species in particular In shy gure 7 weplot the total number of OH molecules contained in the bubble on two time-scalesover one acoustic cycle in shy gure 7a and around the time of collapse in shy gure 7b Theinitial concentrations of all species were selected so that the chemical production anddestruction balanced in one acoustic cycle The main collapse is associated with arapid creation of OH once the temperature exceeds 4000 K

Upon rebound the bubble rapidly cools and the OH and other products tend torecombine back into H2O These recombination reactions proceed more slowly thanthe dissociation reactions upon collapse The result is that the bubble is left with asuper-equilibrium amount of OH after it has cooled to ambient temperature Thissuper-equilibrium OH then slowly decays through the rest of the cycle with briefspurts of production and destruction at each of the after-bounces

Proc R Soc Lond A (2000)

1700 B D Storey and A J Szeri

0 10 20 30time (micros)

- 4 - 2 0 2 4

time relative to minimum radius (ns)

106

1010

104

106

108

108

num

ber

of O

H

radi

cals

nu

mbe

r of

OH

ra

dica

ls

Figure 7 Total number of OH molecules contained in the bubble versus time over one acousticcycle (a) and around the time of minimum radius (b) in case III The macrrst sharp peak in (a)is expanded in (b) Subsequent peaks in (a) correspond to local radius minima in the bubbledynamics The initial amounts of OH and all the other species were chosen for no net speciesproduction or destruction over an acoustic cycle

The distribution of OH in the bubble is characterized by a fairly uniform con-centration through much of the bubble interior and a very sharp gradient near thewall This sharp gradient is a result of heat transfer during the collapse Duringcompression conduction between the hot interior and the cool wall sets up a thintemperature boundary layer This boundary layer conshy nes the chemical reactions tothe interior away from the interface The cooler wall leads to a much lower concen-tration of OH at the interface than in the centre at the time of the main collapseIn shy gure 8 we show the molar concentration history of OH at the centre and at theinterface When the bubble is at minimum radius and the OH concentration at thecentre is at its maximum the concentration at the wall is three orders of magnitudelower

Because the chemistry time-scale is much faster than the dinotusion time-scale OHis produced where water vapour was trapped at high enough temperature The liquidinterface is insulated from the high concentration of OH generated deep inside in thebubble The only OH produced at the centre that reaches the bubble interface is theslowly decaying super-equilibrium OH that is left inside the bubble on re-expansion

(c) OH uptake by the liquid does not indeg uence the bubble interior

The inclusion of liquid uptake of OH (case IV) adds further complications wenow try to illuminate some of the main qualitative features We shall argue that OHuptake has little enotect on the gas dynamics and chemical production

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1701

- 4 - 2 0 2 410 - 10

10 - 8

10 - 6

10 - 4

10 - 2

time relative to minimum radius (ns)

conc

entr

atio

n of

OH

(m

ol c

m-3

)

Figure 8 Molar concentration of OH at the centre (||) and the interface ( ) versus timearound the point of minimum radius Note that the concentration at the centre is about threeorders of magnitude greater than at the interface at the point of minimum radius

Due to dinotering ideas on the proper value of the uptake coemacr cient the values areset at pound = 0001 and pound = 001 (cases IV(a) and IV(b)) consistent with the work ofothers to which we made reference earlier The bubble dynamics and temperatureproshy les for cases IV(a) and IV(b) are indistinguishable from those for case III Inshy gure 9 we show the total number of OH molecules in the bubble versus time ontwo dinoterent scales for cases III IV(a) and IV(b) The history for one acoustic cycleis plotted in shy gure 9a and the history concentrated at the main collapse is plottedin shy gure 9b

In the main expansion when the bubble is cool the OH resident in the bubblewill invade the liquid as can be seen by the decrease in the amount of OH at theend of the long expansion as the uptake increases When the bubble collapses thechemical production of OH from recently evaporated water rapidly overwhelms theOH uptake by the liquid As the bubble proceeds into the collapse the curves oftotal OH content for dinoterent uptakes become indistinguishable The total amountof OH in the bubble is dominated by the amount created in the interior away fromthe wall The total amount of OH lost to the liquid during a collapse is only afraction of the amount contained in the bubble Hence the peak amount of OH inthe bubble during the collapse is insensitive to the uptake coe cient of OH Similarstatements apply to uptake of other species

In shy gure 10 we show for the purposes of qualitative discussion the total massof OH exiting the bubble in cases IV(a) and IV(b) While the OH uptake coef-shy cient does not inreguence the total amount of OH contained in the bubble at themain collapse there are large dinoterences in the amount invading the liquid over

Proc R Soc Lond A (2000)

1702 B D Storey and A J Szeri

0 10 20 30

106

1010

102

time (micros)

- 4 - 2 0 2 4104

106

108

time relative to minimum radius (ns)

1010

(b)

(a)nu

mbe

r of

OH

ra

dica

ls

num

ber

of O

H

radi

cals

Figure 9 Comparison of total amount number of OH molecules in the bubble versus time overone acoustic cycle (a) and focused around the time of minimum radius (b) in cases III (||)IV(a) ( ) and IV(b) ( ) The uptake of OH plays a role during the slow expansionswhen the resident amount of OH left in the bubble is well above chemical equilibrium At thecollapse OH uptake does not indeg uence the total amount of OH production

a cycle Much of the OH uptake occurs during the main collapse and the subse-quent re-expansion when the concentration of OH at the wall is at its maximum(see shy gure 8) Further uptake occurs throughout the cycle at each of the after-bounces

(d ) Inclusion of water vapour alters the ereg ect of increased forcing

Next we shall compare the results of cases I II and III with all parameters shy xedexcept for the acoustic pressure amplitude which varies between 10 and 13 barWe that note this is experimentally awkward for SBSL because pressure amplitudeand ambient radius are dependent quantities through the mass exchange equilibriumof rectishy ed dinotusion (Hilgenfeldt et al 1996) The way we present these results isnonetheless useful as it demonstrates the importance of water vapour in the under-standing of sonoluminescence and sonochemistry

Because the bubble dynamics are altered by the inclusion of water vapour weuse the compression ratio as the distinguishing parameter In shy gure 11 we plot themaximum centre temperature versus the compression ratio for cases I II and III Thecalculations in shy gure 11 are devoid of strong shocks a matter to which we devotemore attention below If there are strong shocks then shy gure 11 will look the samein terms of a suitably averaged temperature rather than the centre temperature

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1703

0 10 20 30102

103

104

105

106

num

ber

of O

H

radi

cals

inva

ding

liqu

id

time (micros)

Figure 10 Comparison of the amount of OH absorbed by the liquid versus time for cases IV(a)(|| uptake coeplusmn cient pound = 0001) and IV(b) ( pound = 001) Most of the OH uptakeoccurs at the main collapse and the subsequent after-bounce

In case I the plot shows linear behaviour (on the loglog scale) One observes thatwhen water vapour is not taken into account stronger forcing yields more extremetemperatures

In case II the relationship between compression ratio and maximum temperature isvery dinoterent First we note that the amount of water trapped in the bubble increaseswith compression ratio If we revisit the water-trapping mechanism (x 3 a (iii)) thistrend is readily explained The increase in water-vapour content reduces the ratioof specishy c heats of the gasvapour mixture and hence reduces the compressionheating

At low compression ratios in case II the temperature increases with compressionratio in a manner very similar to case I At low compression ratios the amountof water at collapse is very small the bubble is nearly pure argon As the forcingis increased more water is trapped and the maximum temperature of the bubblebecomes nearly constant In this range of bubble dynamics there is an almost per-fectly compensating enotect (with respect to peak temperature) between the decreasein specishy c-heat ratio and the increase in compression Case III is very similar tocase II except that the plateau in temperature occurs at a lower temperature aconsequence of the endothermic chemical reactions

(e) Inclusion of water vapour promotes the formation of shocks

Water vapour is of great importance in the formation of shocks in very stronglyforced bubbles We shall brieregy consider a 60 m bubble forced with a pressureamplitude of 14 bar at a frequency of 206 kHz (Case A1 in Moss et al (1999))

Proc R Soc Lond A (2000)

1704 B D Storey and A J Szeri

10 1 10 2

10 3

10 4

10 5

Rmax R min

max

imum

tem

pera

ture

(K

)

Figure 11 Change in centre temperature versus compression ratio in cases I ( ) II ( curren )and III (+) In this plot all parameters were held constant except the pressure amplitudewhich was varied between 10 and 13 bar Case I shows a steady increase in temperature withcompression ratio Cases II and III reach a maximum temperature as the forcing is increased

When water vapour is neglected a compression wave is launched into the bubble oncollapse but the wave does not steepen into a shock When water vapour is takeninto account (with or without reactions) the compression wave steepens into a strongshock very near the centre of the bubble Vuong et al (1999) showed that the rateof steepening depends on the amount of compression heating before the compressionwave is launched

In very recent work Lin amp Szeri (2000) used the wavefront expansion technique todetermine the evolution of a compression or rarefaction wavefront that is propagatingspherically into a collapsing or expanding gas of non-uniform entropy This analyticalwork supports the idea that when the speed of sound is higher at the centre (dueto temperature and composition inhomogeneities) shock formation is delayed orprevented Furthermore this work demonstrated that the inward radial velocity alsohinders the steepening of compression waves into shocks

Consistent with the arguments made in Lin amp Szeri (2000) Vuong et al (1999)and Moss et al (1999) our results show that the decrease in the ratio of specishy cheats due to an increase in the water-vapour content can promote shock formationThe point we want to emphasize is that the question of shock formation cannot beundertaken without consideration of water vapour

With respect to total chemical production shocks are of negligible consequenceThe shock inreguences only about the inner 10 of the radius which is only 01 ofthe volume Therefore the increased production of OH at the centre does not anotectthe total amount produced in the bubble

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

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Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

1690 B D Storey and A J Szeri

pressure of water P s at is the saturation pressure at the temperature of the interfaceTin t M0 is the initial molecular mass of the bubble contents iexcl is a correction forbulk motion to the interface and remains close to one for low mass transfer Weemphasize that this equation is only valid for reasonable temperatures and pressuresand not at the extreme conditions found during a bubble collapse As we argue belowthe evaporation condition is of vanishing consequence during the terminal stages ofbubble collapse when (25) loses validity

There is some discord in measurements of the accommodation coe cient Thevalue we use frac14 = 04 is consistent with that used by Yasui (1997a) (developed frommolecular dynamics simulations) and those recommended by Eames et al (1997)The sensitivity of the results to the accommodation coemacr cient will be consideredlater

(e) Chemical kinetics

Chemical kinetics described by the reaction mechanism determines the produc-tion and destruction of the dinoterent species The reaction mechanism that we use isdescribed in Maas amp Warnatz (1988) The mechanism includes 19 forward and reverseelementary reactions and nine species (Ar H H2 O O2 OH HO2 H2O2) Argononly enters the mechanism as a third body

The net rate of progress of each elementary reaction is the dinoterence in the forwardand reverse rate of progress The forward (reverse) rate of progress is given as theproduct of the reaction rate and the concentrations of all species on the reactant(product) side of the equation The change in concentration of species i with timeis the sum of the net rate of progress of each elementary reaction that containsspecies i The details can be found in several texts (eg Gardiner 1984) In thepresent formulation the chemical kinetics lead to the net chemical production ordestruction term curren i in (22)

The reaction rates are given empirically with the reaction mechanism The Linde-mann form is used to take into account the pressure fall-onot in some of the three-bodyreactions (Gardiner 1984) The high-pressure limit of the reaction rates for availablereactions were taken from Bowman et al (1999)

The elevated pressure in the bubble upon collapse accelerates the rate of progressof the reactions and alters the equilibrium state to which the reactions tend so thatfor a given temperature there is less dissociation at higher pressures This change inequilibrium state with pressure is accounted for in the reaction mechanism becauseforward and reverse reaction rates are related through the equilibrium constantThere is considerable uncertainty in the reaction rates at the high pressures weshall encounter The mechanism and reaction rates we use seem to be the best onespresently available

The reaction mechanism does not include ionization reactions The onset of ion-ization is delayed due to the high pressure The degree of ionization at the conditionswe report is quite low as estimated by the Saha equation Moss et al (1997) putforth the criterion that the onset of ionization in a dense gas is at one-quarter ofthe ionization potential in the present case this temperature is ca 30 000 K Theresults we present are consistent with neglecting ionization based on either of thesecriteria

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1691

(f ) Mass transfer to the liquid

One of the most di cult quantities to determine accurately is the rate of transportof radical species into the liquid While general convectiondinotusionreaction equa-tions can be written for each species dissolved in the liquid this approach involvesmany unknowns For example data on the solubility of OH radicals in liquid watervary by two orders of magnitude (Sander 1998) We know of no solubility data forH or O Moreover there is the possibility of heterogeneous surface reactions (VonSonntag et al 1999) which are poorly understood especially at the time of collapsewhen the liquid very near the bubble interface is supercritical

To avoid these uncertainties we take a very simple approach To determine the rateof radical invasion into the liquid approximately we introduce the uptake coemacr cientpound

pound =Nin iexcl Nou t

Ncoll

(26)

where Nin is the number of molecules invading the liquid Nou t is the number ofmolecules ejected from the liquid and Ncoll is the number of molecules collidingwith the surface (from kinetic theory) This view of mass transfer is oversimplishy edone can easily see that pound is not an intrinsic quantity (Takami et al 1998 Chameides1984)

This approach has nevertheless been used to determine the absorption of radicalspecies onto liquid and solid surfaces The uptake coe cients have been measured forsome of the species in our system over aqueous surfaces (Takami et al 1998 Hansonet al 1992) We shall study the inreguence of uptake coe cient on the results of ourcalculations below

(g) Numerical method

The same Chebyshev collocation method is used that was described in detail inSampS The variables in the gas are approximated by expansions in Chebyshev poly-nomials in the argon Lagrangian coordinate a The temperature in the liquid isapproximated with an expansion in rational Chebyshev polynomials The equationsin the gas evolve with a straightforward explicit predictorcorrector scheme We usestandard operator splitting (Press et al 1988) to solve the chemical kinetics equa-tions separately with a semi-implicit method to avoid time-step restrictions causedby the stinotness of these equations The energy equation in the liquid is integrated intime with a fully implicit method to avoid time-step restrictions associated with theclose spacing of grid points near the bubble wall As an error check global balancesof mass and energy are monitored imbalances always remain below a fraction of aper cent

3 Results and discussion

We begin by presenting detailed results of one particular conshy guration Due to thecomplexity of the problem we shall attempt to present the results in a systematicway in order to demonstrate the importance of certain phenomena The cases to beinvestigated are as follows

Proc R Soc Lond A (2000)

1692 B D Storey and A J Szeri

Table 1 Summary of the direg erent cases investigated in detail

(In all cases the bubble is a 45 m radius argon bubble in water at 298 K acoustically forcedwith a 12 bar pressure amplitude at 265 kHz)

case phase change chemistry OH uptake Rm ax =Rm in ( m) Tm ax (K)

I no no 00 280080 20 900

II yes no 00 313070 9 700

III yes yes 00 317065 7 000

IV(a) yes yes 0001 317065 7 000

IV(b) yes yes 001 317065 7 000

Case I no water vapour allowed in the bubble

Case II water vapour allowed to evaporate and condense but no chemical reactions

Case III water vapour and chemical reactions but no mass transfer across thebubble interface

Cases IV(a) and IV(b) mass transfer of OH across the bubble interface at dif-ferent uptake coe cients

In all cases the bubble is a 45 m-radius argon bubble in 298 K water acousticallyforced with a 12 bar pressure amplitude at 265 kHz These cases are summarized intable 1 With respect to SBSL this bubble was found to be stable to mass exchangeand shape changes though would have minimal light emission (Hilgenfeldt et al 1996 Wu amp Roberts 1998a b) We describe the results for more strongly forcedbubbles below The data we obtain are for steady-state oscillations ie after thedecay of initial transients After presenting the details of these cases we considervariations in the parameters to illustrate interesting trends

(a) Excess water is trapped by the rapid collapse

We shall now argue that signishy cant amounts of water vapour become trapped inthe interior of the bubble when it collapses violently Water is trapped in the interiorbecause the bubble motions become so rapid that the vapour near the centre hasinsu cient time to dinotuse to the bubble wall Non-equilibrium phase change alsoplays a signishy cant role in the amount of water that is trapped All results in thissubsection apply to case II this case allows one to focus on the dynamics of evapora-tion condensation and dinotusion of water inside the bubble without the complicationof chemical reactions

In shy gure 1 we show the radial history of the bubble over one cycle of the acousticforcing in case II The essential features of the bubble dynamics for cases I and II arethe same but the inclusion of water vapour increases the maximum radius and delaysthe collapse The dinoterence is to be expected because case I has no vapour pressureThe compression ratio (Rm ax=Rm in ) for cases I and II is 35 and 44 respectively Thecompression ratio is an approximate way of separating out dinoterences in the resultsdue to dinoterences in the bubble dynamics Typically a higher compression ratio leadsto more compression heating and higher temperatures if all else is equal We returnto this point later

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1693

0 10 20 30

10

20

30

time (micros)

radi

us (

microm

)

Figure 1 Radius of the bubble versus time over one cycle of acoustic forcing in case IIThe essential features for cases I II III and IV are the same

0 10 20 30

10 8

10 9

1010

1011

num

ber

of m

olec

ules

H2O

Ar

time (micros)

Figure 2 Total number of water-vapour molecules contained in the bubble versus time over onecycle of acoustic forcing in case II Note that water comprises ca 14 of the bubble contents atthe end of the main collapse

Proc R Soc Lond A (2000)

1694 B D Storey and A J Szeri

0

4000

8000te

mpe

ratu

re (

K)

1

2

3

time relative to minimum radius (ns)

num

ber

of H

2O m

olec

ules

109

acute-

- 4 - 2 0 2 4

(b)

(a)

Figure 3 Temperature (a) and total number of water molecules contained in the bubble (b) versustime around the point of minimum radius (time = 0) in case II The temperature undergoesorder-of-magnitude changes on this time-scale while the total number of water molecules changesvery little

In shy gure 2 we show the total number of H2O molecules contained within thebubble over the acoustic cycle (case II) A large amount of water evaporates intothe bubble during the main expansion when the pressure is low Water condenses onthe bubble wall during the collapse as the gas pressure increases There is signishy cantwater vapour in the bubble at the main collapse ca 14 (molar basis) of the totalbubble contents is water vapour If the interface were at equilibrium and the speedof mass dinotusion were inshy nite there would be a negligible amount of water in thebubble on collapse

In shy gure 3 we show the temperature history of the centre and the total numberof water molecules in the bubble around the time of minimum radius (time = 0) incase II Over the 10 ns time-scale the total amount of water vapour contained in thebubble is nearly constant whereas the temperature changes by an order of magni-tude This recalls SampS in which it was shown that evolving mixture compositionswere essentially frozen on the time-scale of the temperature peak

The temperature peak may be related to the light emission The equation foroptical power given by Moss et al (1999) with a constant opacity provides an easyestimate for the optical power output as a function of time For case II this estimategives a light emission pulse width (FWHM) of 350 ps which is much shorter than theca 2000 ps temperature peak This optical-pulse-width estimate would be somewhatshortened if the temperature dependence of the opacity were taken into accountThe peak optical power is ca 01 mW although this number is very sensitive to theopacity Typical stable SBSL experiments (at higher forcings than the present case)yield light emission of the order of milliwatts per regash (Gaitan et al 1996 Hiller etal 1992)

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1695

0 5 10 15 20 25

02

04

06

08

10

- 843 ns

- 455 ns

- 299 ns

- 153 ns

- 41 ns

- 15 ns

0 ns - 5 ns

radial position (microm)

mol

ar f

ract

ion

Figure 4 Molar fraction of water vapour versus radius for selected times prior to the moment ofminimum radius (time = 0) in case II Note the development of the non-uniform composition asthe collapse accelerates toward minimum radius In the last 15 ns before minimum radius thecomposition changes by only a few per cent when viewed in the Lagrangian frame instead

There are two mechanisms that determine the excess vapour left behind at thecollapse shy nite-rate mass dinotusion and non-equilibrium phase change We examineeach mechanism and their interaction in turn

(i) Role of direg usion

The shy rst reason for excess water vapour remaining during the collapse is the rapidbubble collapse overwhelming slow mass dinotusion this was extensively discussedin SampS In SampS it was shown that there is a competition between two importanttime-scales the dynamic time-scale of the bubble motions and the mass dinotusiontime-scale These are given explicitly in dimensionless terms as

frac12 d yn = R= _R

frac12 d if = ReScR2

D MH2 O

ordm ReSc1

Rp

T

In the latter we made use of the fact that the dinotusion coe cient of water vapourin the gas mixture D M

H2 O scales as R3pT

The time-scales argument is clarishy ed upon examination of shy gure 4 which showsthe evolving bubble composition as a function of radius at selected times before thetime of minimum radius (case II) In the early part of the collapse the dynamic time-scale is longer than the dinotusion time-scale frac12 d yn frac34 frac12 d if this results in a uniformcomposition of the bubble (upper curves in shy gure 4) As the bubble accelerates into

Proc R Soc Lond A (2000)

1696 B D Storey and A J Szeri

the collapse frac12 d yn sup1 frac12 d if and the decrease in water fraction at the centre begins tolag the decrease in water fraction at the wall (middle curves of shy gure 4) Furtherinto the collapse frac12 d yn frac12 frac12 d if and the water vapour cannot dinotuse to the bubble wallon the time-scale of the rapid collapse This slow dinotusion results in a nearly shy xeddistribution of water vapour (three lowest curves of shy gure 4)

While the three lowest curves in shy gure 4 look quite dinoterent in this Eulerianrepresentation they dinoter by only a few per cent when viewed in the Lagrangianrepresentation (not shown) The curve at iexcl 15 ns is from a time when R = R0 andthe temperature at the centre is ca 750 K The curve at 0 ns is from the time ofminimum radius (015R0) when the temperature at the centre is 9700 K Consistentwith shy gure 3 over times when there are drastic changes in the bubble radius andtemperature the composition of the bubble is almost a material shy eld

(ii) Role of non-equilibrium phase change

Non-equilibrium phase change is the second mechanism by which vapour is trappedin the bubble interior during the collapse By adjusting the accommodation coe -cient one has control over the time-scale for the partial pressure of the water vapourto come to equilibrium with the saturation pressure at the interface The dimension-less condensation time-scale is

frac12 con d =R

frac14

r2 ordm MH2 O

9M0Tin t

When frac12 d yn frac34 frac12 con d the condensation is in quasi-equilibrium with respect to thebubble motions If frac12 d yn frac12 frac12 con d then the bubble collapse is so rapid that no masscan escape the bubble on the time-scale of the collapse

(iii) Interaction of the two mechanisms

It is the competition between the time-scales for mass dinotusion condensation andbubble dynamics that determines which mechanism traps water vapour in the bubbleOnce frac12 d yn frac12 frac12 d if or frac12 d yn frac12 frac12 con d the total amount of water vapour in the bubbleis shy xed Only one of these conditions needs to be met for vapour to be trapped inthe bubble Both mechanisms can however participate to determine the quantity oftrapped vapour

During the bubble collapse the condition frac12 d yn frac12 frac12 d if is reached well before frac12 d yn frac12frac12 con d in case II Therefore in this case (as well as the others presented herein) it isthe slow mass dinotusion that is responsible for trapping vapour in the bubble

For cases where slow mass dinotusion traps the vapour non-equilibrium phasechange helps to determine the amount trapped The accommodation coe cient rep-resents a resistance to condensation at the interface during collapse of the bubble Alower value of accommodation coe cient provides more resistance to phase changeclearly increasing the mass of water retained in the bubble at any time during thecollapse The importance of the non-equilibrium phase change can be estimated bycomparing frac12 d yn with frac12 con d at the time that frac12 d yn = frac12 d if The closer frac12 d yn is to frac12 con d the more important non-equilibrium phase change is in determining the amount ofwater trapped

Colussi et al (1998) developed a model including chemical reactions in a collapsingbubble using an accommodation coe cient of frac14 = 0001 but did not take into

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1697

account the shy nite rate of mass dinotusion The low value of frac14 which works to trapsignishy cant water vapour at collapse was necessary for their model to be consistentwith experimental sonochemistry data When such a low value for frac14 is used thecondition frac12 d yn frac12 frac12 con d is reached well before frac12 d yn frac12 frac12 d if (the opposite of our case)Such a low accommodation coe cient works to trap signishy cant amounts of watervapour in the bubble but in this case the hindered regux at the interface eliminatesmass transfer out of the bubble while the distribution of water vapour within isessentially uniform In the present work we argue that the signishy cant amounts ofwater vapour trapped in the bubble are a consequence of slow dinotusion as well asnon-equilibrium phase change

The temperature of the interface exceeds the critical point for a very brief time(from iexcl 340 ps to 1800 ps relative to the time of minimum radius) For the conditionswe explored the maximum thickness of the supercritical region is quite thin less than1 of the bubble radius ie ca 10 nm We neglect any changes in the liquid equationof state and transport properties when the interface exceeds the critical temperatureThis assumption is justishy ed based on the thinness of the region and the fact that thecritical temperature is reached very late in the collapse Any disturbances caused bythe supercritical interface have insumacr cient time to propagate into the bubble interioron the main collapse Although the evaporation equation is invalid for these shorttimes the total mass added to the bubble is negligible regardless of the magnitudeof evaporative mass regux The issue of the super-critical interface may require futureimprovements but is beyond the scope of the present work

As a shy nal note both thermal and pressure dinotusion also play a role in determiningthe distribution of water vapour during the collapse In the argonwater system thethermal-dinotusion coemacr cient is small and the molecules are of similar size hence pres-sure dinotusion is quite weak Little dinoterence is seen in calculations with and withoutthese other forms of mass dinotusion In a helium or xenon bubble in water thermaland pressure dinotusion are more pronounced and can cause a 1015 dinoterence inthe amount of trapped water vapour In the interest of brevity we will not discussthese results further here

(b) Gas-phase chemical reactions break apart H2O and reduce the temperature

Now we turn to an investigation of what transpires when chemical reactions aretaken into account (case III) We shall show that there is signishy cant time for reactionsto occur ie the time-scale for chemical reaction is similar to that of the bubblecollapse The primarily endothermic reactions absorb a signishy cant amount of energyand further reduce the peak temperatures from cases I and II The distribution ofspecies produced in the bubble is determined by dinotusive transport of water vapourprior to the collapse and heat transfer out of the bubble during the collapse Thesteep temperature gradient at the wall conshy nes the bulk of the chemical reactions tothe bubble interior

(i) Consequences of chemistry

The overall bubble dynamics of cases II and III are essentially the same with acompression ratio of 49 in case III (slightly higher than case II) A convenient pointof departure is to compare the temperature histories of the bubble centre aroundthe main collapse for the three cases we have considered (see shy gure 5) The inclusion

Proc R Soc Lond A (2000)

1698 B D Storey and A J Szeri

- 4 - 2 0 2 410

3

104

tem

pera

ture

(K

)

time relative to minimum radius (ns)

Figure 5 Temperature history of the centre of the bubble around the time of minimum radius(t = 0) in cases I II and III Note the reduction in temperature from case I ( ) to case II( ) as water is allowed to enter the bubble The chemical reactions added in case III (||)further reduce the temperatures

of water vapour (case II) signishy cantly reduces the peak temperature from case IThe decrease in temperature is due mainly to the decrease in the ratio of specishy cheats of the mixture which reduces the compression heating Upon comparison ofcases II and III one observes that the chemical reactions further reduce the peaktemperature due to the endothermic decomposition of the water vapour The centretemperature histories of cases II and III are nearly identical below 4000 K

It is important to understand that the temperature dinoterences in these threecases are not due to subtle dinoterences in bubble dynamics The compression ratio(Rm ax=Rm in ) of the three cases increases from case I to case II to case III whereasthe temperature decreases in this order This decrease in temperature occurs despitethe fact that a higher compression ratio will lead to more compression heating in thebubble collapse all other things being equal In all these cases there are no shocksinside the bubble Shocks will be considered in x 3 e

In shy gure 6 we show the history of the total number of molecules of the dinoterentspecies as a function of time on the same time-scale as in shy gure 5 The lowest curve isthe total amount of water vapour only The next curve is the total amount of watervapour plus the total amount of H2 Each curve represents the addition of one speciesas labelled until the uppermost curve denotes everything but argon We do not labelHO2 and H2O2 as they appear only in trace amounts One observes which speciesare most abundant during the time of the main collapse Note there is a relativelylarge amount of undissociated water considering the elevated temperatures This isprimarily a consequence of the high pressure suppressing the dissociation of H2O

From shy gures 5 and 6 one observes that the onset of signishy cant chemical activity in

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1699

- 4 - 2 0 2 4

109

1010

time relative to minimum radius (ns)

num

ber

of m

olec

ules

H2O

H2

O2

OH

O

H

Figure 6 Cumulative number of molecules of each species in case III The lowest curve is thetotal number of water molecules the next curve is total water + H2 and so on H2 O2 and HO2

are neglected in this plot as they appear in trace amounts

the bubble corresponds to a centre temperature of ca 4000 K Therefore the reducedtemperature of case III shown in shy gure 5 can be attributed to the endothermicchemical reactions Although the bubble dynamics are quite rapid the chemicalreactions keep pace as is shown by the similar time-scales of the changing quantitiesin shy gures 5 and 6 The rapid rate at which the reactions progress is due to the veryhigh density Recall that the rate of progress of each reaction is directly related tothe species concentration (mol cmiexcl3)

(ii) Dynamics and distribution of OH production

In sonochemistry applications the invasion of OH radicals into the liquid isthought to be responsible for much of the complex and useful chemistry that occursTherefore we shall focus on the behaviour of this species in particular In shy gure 7 weplot the total number of OH molecules contained in the bubble on two time-scalesover one acoustic cycle in shy gure 7a and around the time of collapse in shy gure 7b Theinitial concentrations of all species were selected so that the chemical production anddestruction balanced in one acoustic cycle The main collapse is associated with arapid creation of OH once the temperature exceeds 4000 K

Upon rebound the bubble rapidly cools and the OH and other products tend torecombine back into H2O These recombination reactions proceed more slowly thanthe dissociation reactions upon collapse The result is that the bubble is left with asuper-equilibrium amount of OH after it has cooled to ambient temperature Thissuper-equilibrium OH then slowly decays through the rest of the cycle with briefspurts of production and destruction at each of the after-bounces

Proc R Soc Lond A (2000)

1700 B D Storey and A J Szeri

0 10 20 30time (micros)

- 4 - 2 0 2 4

time relative to minimum radius (ns)

106

1010

104

106

108

108

num

ber

of O

H

radi

cals

nu

mbe

r of

OH

ra

dica

ls

Figure 7 Total number of OH molecules contained in the bubble versus time over one acousticcycle (a) and around the time of minimum radius (b) in case III The macrrst sharp peak in (a)is expanded in (b) Subsequent peaks in (a) correspond to local radius minima in the bubbledynamics The initial amounts of OH and all the other species were chosen for no net speciesproduction or destruction over an acoustic cycle

The distribution of OH in the bubble is characterized by a fairly uniform con-centration through much of the bubble interior and a very sharp gradient near thewall This sharp gradient is a result of heat transfer during the collapse Duringcompression conduction between the hot interior and the cool wall sets up a thintemperature boundary layer This boundary layer conshy nes the chemical reactions tothe interior away from the interface The cooler wall leads to a much lower concen-tration of OH at the interface than in the centre at the time of the main collapseIn shy gure 8 we show the molar concentration history of OH at the centre and at theinterface When the bubble is at minimum radius and the OH concentration at thecentre is at its maximum the concentration at the wall is three orders of magnitudelower

Because the chemistry time-scale is much faster than the dinotusion time-scale OHis produced where water vapour was trapped at high enough temperature The liquidinterface is insulated from the high concentration of OH generated deep inside in thebubble The only OH produced at the centre that reaches the bubble interface is theslowly decaying super-equilibrium OH that is left inside the bubble on re-expansion

(c) OH uptake by the liquid does not indeg uence the bubble interior

The inclusion of liquid uptake of OH (case IV) adds further complications wenow try to illuminate some of the main qualitative features We shall argue that OHuptake has little enotect on the gas dynamics and chemical production

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1701

- 4 - 2 0 2 410 - 10

10 - 8

10 - 6

10 - 4

10 - 2

time relative to minimum radius (ns)

conc

entr

atio

n of

OH

(m

ol c

m-3

)

Figure 8 Molar concentration of OH at the centre (||) and the interface ( ) versus timearound the point of minimum radius Note that the concentration at the centre is about threeorders of magnitude greater than at the interface at the point of minimum radius

Due to dinotering ideas on the proper value of the uptake coemacr cient the values areset at pound = 0001 and pound = 001 (cases IV(a) and IV(b)) consistent with the work ofothers to which we made reference earlier The bubble dynamics and temperatureproshy les for cases IV(a) and IV(b) are indistinguishable from those for case III Inshy gure 9 we show the total number of OH molecules in the bubble versus time ontwo dinoterent scales for cases III IV(a) and IV(b) The history for one acoustic cycleis plotted in shy gure 9a and the history concentrated at the main collapse is plottedin shy gure 9b

In the main expansion when the bubble is cool the OH resident in the bubblewill invade the liquid as can be seen by the decrease in the amount of OH at theend of the long expansion as the uptake increases When the bubble collapses thechemical production of OH from recently evaporated water rapidly overwhelms theOH uptake by the liquid As the bubble proceeds into the collapse the curves oftotal OH content for dinoterent uptakes become indistinguishable The total amountof OH in the bubble is dominated by the amount created in the interior away fromthe wall The total amount of OH lost to the liquid during a collapse is only afraction of the amount contained in the bubble Hence the peak amount of OH inthe bubble during the collapse is insensitive to the uptake coe cient of OH Similarstatements apply to uptake of other species

In shy gure 10 we show for the purposes of qualitative discussion the total massof OH exiting the bubble in cases IV(a) and IV(b) While the OH uptake coef-shy cient does not inreguence the total amount of OH contained in the bubble at themain collapse there are large dinoterences in the amount invading the liquid over

Proc R Soc Lond A (2000)

1702 B D Storey and A J Szeri

0 10 20 30

106

1010

102

time (micros)

- 4 - 2 0 2 4104

106

108

time relative to minimum radius (ns)

1010

(b)

(a)nu

mbe

r of

OH

ra

dica

ls

num

ber

of O

H

radi

cals

Figure 9 Comparison of total amount number of OH molecules in the bubble versus time overone acoustic cycle (a) and focused around the time of minimum radius (b) in cases III (||)IV(a) ( ) and IV(b) ( ) The uptake of OH plays a role during the slow expansionswhen the resident amount of OH left in the bubble is well above chemical equilibrium At thecollapse OH uptake does not indeg uence the total amount of OH production

a cycle Much of the OH uptake occurs during the main collapse and the subse-quent re-expansion when the concentration of OH at the wall is at its maximum(see shy gure 8) Further uptake occurs throughout the cycle at each of the after-bounces

(d ) Inclusion of water vapour alters the ereg ect of increased forcing

Next we shall compare the results of cases I II and III with all parameters shy xedexcept for the acoustic pressure amplitude which varies between 10 and 13 barWe that note this is experimentally awkward for SBSL because pressure amplitudeand ambient radius are dependent quantities through the mass exchange equilibriumof rectishy ed dinotusion (Hilgenfeldt et al 1996) The way we present these results isnonetheless useful as it demonstrates the importance of water vapour in the under-standing of sonoluminescence and sonochemistry

Because the bubble dynamics are altered by the inclusion of water vapour weuse the compression ratio as the distinguishing parameter In shy gure 11 we plot themaximum centre temperature versus the compression ratio for cases I II and III Thecalculations in shy gure 11 are devoid of strong shocks a matter to which we devotemore attention below If there are strong shocks then shy gure 11 will look the samein terms of a suitably averaged temperature rather than the centre temperature

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1703

0 10 20 30102

103

104

105

106

num

ber

of O

H

radi

cals

inva

ding

liqu

id

time (micros)

Figure 10 Comparison of the amount of OH absorbed by the liquid versus time for cases IV(a)(|| uptake coeplusmn cient pound = 0001) and IV(b) ( pound = 001) Most of the OH uptakeoccurs at the main collapse and the subsequent after-bounce

In case I the plot shows linear behaviour (on the loglog scale) One observes thatwhen water vapour is not taken into account stronger forcing yields more extremetemperatures

In case II the relationship between compression ratio and maximum temperature isvery dinoterent First we note that the amount of water trapped in the bubble increaseswith compression ratio If we revisit the water-trapping mechanism (x 3 a (iii)) thistrend is readily explained The increase in water-vapour content reduces the ratioof specishy c heats of the gasvapour mixture and hence reduces the compressionheating

At low compression ratios in case II the temperature increases with compressionratio in a manner very similar to case I At low compression ratios the amountof water at collapse is very small the bubble is nearly pure argon As the forcingis increased more water is trapped and the maximum temperature of the bubblebecomes nearly constant In this range of bubble dynamics there is an almost per-fectly compensating enotect (with respect to peak temperature) between the decreasein specishy c-heat ratio and the increase in compression Case III is very similar tocase II except that the plateau in temperature occurs at a lower temperature aconsequence of the endothermic chemical reactions

(e) Inclusion of water vapour promotes the formation of shocks

Water vapour is of great importance in the formation of shocks in very stronglyforced bubbles We shall brieregy consider a 60 m bubble forced with a pressureamplitude of 14 bar at a frequency of 206 kHz (Case A1 in Moss et al (1999))

Proc R Soc Lond A (2000)

1704 B D Storey and A J Szeri

10 1 10 2

10 3

10 4

10 5

Rmax R min

max

imum

tem

pera

ture

(K

)

Figure 11 Change in centre temperature versus compression ratio in cases I ( ) II ( curren )and III (+) In this plot all parameters were held constant except the pressure amplitudewhich was varied between 10 and 13 bar Case I shows a steady increase in temperature withcompression ratio Cases II and III reach a maximum temperature as the forcing is increased

When water vapour is neglected a compression wave is launched into the bubble oncollapse but the wave does not steepen into a shock When water vapour is takeninto account (with or without reactions) the compression wave steepens into a strongshock very near the centre of the bubble Vuong et al (1999) showed that the rateof steepening depends on the amount of compression heating before the compressionwave is launched

In very recent work Lin amp Szeri (2000) used the wavefront expansion technique todetermine the evolution of a compression or rarefaction wavefront that is propagatingspherically into a collapsing or expanding gas of non-uniform entropy This analyticalwork supports the idea that when the speed of sound is higher at the centre (dueto temperature and composition inhomogeneities) shock formation is delayed orprevented Furthermore this work demonstrated that the inward radial velocity alsohinders the steepening of compression waves into shocks

Consistent with the arguments made in Lin amp Szeri (2000) Vuong et al (1999)and Moss et al (1999) our results show that the decrease in the ratio of specishy cheats due to an increase in the water-vapour content can promote shock formationThe point we want to emphasize is that the question of shock formation cannot beundertaken without consideration of water vapour

With respect to total chemical production shocks are of negligible consequenceThe shock inreguences only about the inner 10 of the radius which is only 01 ofthe volume Therefore the increased production of OH at the centre does not anotectthe total amount produced in the bubble

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1691

(f ) Mass transfer to the liquid

One of the most di cult quantities to determine accurately is the rate of transportof radical species into the liquid While general convectiondinotusionreaction equa-tions can be written for each species dissolved in the liquid this approach involvesmany unknowns For example data on the solubility of OH radicals in liquid watervary by two orders of magnitude (Sander 1998) We know of no solubility data forH or O Moreover there is the possibility of heterogeneous surface reactions (VonSonntag et al 1999) which are poorly understood especially at the time of collapsewhen the liquid very near the bubble interface is supercritical

To avoid these uncertainties we take a very simple approach To determine the rateof radical invasion into the liquid approximately we introduce the uptake coemacr cientpound

pound =Nin iexcl Nou t

Ncoll

(26)

where Nin is the number of molecules invading the liquid Nou t is the number ofmolecules ejected from the liquid and Ncoll is the number of molecules collidingwith the surface (from kinetic theory) This view of mass transfer is oversimplishy edone can easily see that pound is not an intrinsic quantity (Takami et al 1998 Chameides1984)

This approach has nevertheless been used to determine the absorption of radicalspecies onto liquid and solid surfaces The uptake coe cients have been measured forsome of the species in our system over aqueous surfaces (Takami et al 1998 Hansonet al 1992) We shall study the inreguence of uptake coe cient on the results of ourcalculations below

(g) Numerical method

The same Chebyshev collocation method is used that was described in detail inSampS The variables in the gas are approximated by expansions in Chebyshev poly-nomials in the argon Lagrangian coordinate a The temperature in the liquid isapproximated with an expansion in rational Chebyshev polynomials The equationsin the gas evolve with a straightforward explicit predictorcorrector scheme We usestandard operator splitting (Press et al 1988) to solve the chemical kinetics equa-tions separately with a semi-implicit method to avoid time-step restrictions causedby the stinotness of these equations The energy equation in the liquid is integrated intime with a fully implicit method to avoid time-step restrictions associated with theclose spacing of grid points near the bubble wall As an error check global balancesof mass and energy are monitored imbalances always remain below a fraction of aper cent

3 Results and discussion

We begin by presenting detailed results of one particular conshy guration Due to thecomplexity of the problem we shall attempt to present the results in a systematicway in order to demonstrate the importance of certain phenomena The cases to beinvestigated are as follows

Proc R Soc Lond A (2000)

1692 B D Storey and A J Szeri

Table 1 Summary of the direg erent cases investigated in detail

(In all cases the bubble is a 45 m radius argon bubble in water at 298 K acoustically forcedwith a 12 bar pressure amplitude at 265 kHz)

case phase change chemistry OH uptake Rm ax =Rm in ( m) Tm ax (K)

I no no 00 280080 20 900

II yes no 00 313070 9 700

III yes yes 00 317065 7 000

IV(a) yes yes 0001 317065 7 000

IV(b) yes yes 001 317065 7 000

Case I no water vapour allowed in the bubble

Case II water vapour allowed to evaporate and condense but no chemical reactions

Case III water vapour and chemical reactions but no mass transfer across thebubble interface

Cases IV(a) and IV(b) mass transfer of OH across the bubble interface at dif-ferent uptake coe cients

In all cases the bubble is a 45 m-radius argon bubble in 298 K water acousticallyforced with a 12 bar pressure amplitude at 265 kHz These cases are summarized intable 1 With respect to SBSL this bubble was found to be stable to mass exchangeand shape changes though would have minimal light emission (Hilgenfeldt et al 1996 Wu amp Roberts 1998a b) We describe the results for more strongly forcedbubbles below The data we obtain are for steady-state oscillations ie after thedecay of initial transients After presenting the details of these cases we considervariations in the parameters to illustrate interesting trends

(a) Excess water is trapped by the rapid collapse

We shall now argue that signishy cant amounts of water vapour become trapped inthe interior of the bubble when it collapses violently Water is trapped in the interiorbecause the bubble motions become so rapid that the vapour near the centre hasinsu cient time to dinotuse to the bubble wall Non-equilibrium phase change alsoplays a signishy cant role in the amount of water that is trapped All results in thissubsection apply to case II this case allows one to focus on the dynamics of evapora-tion condensation and dinotusion of water inside the bubble without the complicationof chemical reactions

In shy gure 1 we show the radial history of the bubble over one cycle of the acousticforcing in case II The essential features of the bubble dynamics for cases I and II arethe same but the inclusion of water vapour increases the maximum radius and delaysthe collapse The dinoterence is to be expected because case I has no vapour pressureThe compression ratio (Rm ax=Rm in ) for cases I and II is 35 and 44 respectively Thecompression ratio is an approximate way of separating out dinoterences in the resultsdue to dinoterences in the bubble dynamics Typically a higher compression ratio leadsto more compression heating and higher temperatures if all else is equal We returnto this point later

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1693

0 10 20 30

10

20

30

time (micros)

radi

us (

microm

)

Figure 1 Radius of the bubble versus time over one cycle of acoustic forcing in case IIThe essential features for cases I II III and IV are the same

0 10 20 30

10 8

10 9

1010

1011

num

ber

of m

olec

ules

H2O

Ar

time (micros)

Figure 2 Total number of water-vapour molecules contained in the bubble versus time over onecycle of acoustic forcing in case II Note that water comprises ca 14 of the bubble contents atthe end of the main collapse

Proc R Soc Lond A (2000)

1694 B D Storey and A J Szeri

0

4000

8000te

mpe

ratu

re (

K)

1

2

3

time relative to minimum radius (ns)

num

ber

of H

2O m

olec

ules

109

acute-

- 4 - 2 0 2 4

(b)

(a)

Figure 3 Temperature (a) and total number of water molecules contained in the bubble (b) versustime around the point of minimum radius (time = 0) in case II The temperature undergoesorder-of-magnitude changes on this time-scale while the total number of water molecules changesvery little

In shy gure 2 we show the total number of H2O molecules contained within thebubble over the acoustic cycle (case II) A large amount of water evaporates intothe bubble during the main expansion when the pressure is low Water condenses onthe bubble wall during the collapse as the gas pressure increases There is signishy cantwater vapour in the bubble at the main collapse ca 14 (molar basis) of the totalbubble contents is water vapour If the interface were at equilibrium and the speedof mass dinotusion were inshy nite there would be a negligible amount of water in thebubble on collapse

In shy gure 3 we show the temperature history of the centre and the total numberof water molecules in the bubble around the time of minimum radius (time = 0) incase II Over the 10 ns time-scale the total amount of water vapour contained in thebubble is nearly constant whereas the temperature changes by an order of magni-tude This recalls SampS in which it was shown that evolving mixture compositionswere essentially frozen on the time-scale of the temperature peak

The temperature peak may be related to the light emission The equation foroptical power given by Moss et al (1999) with a constant opacity provides an easyestimate for the optical power output as a function of time For case II this estimategives a light emission pulse width (FWHM) of 350 ps which is much shorter than theca 2000 ps temperature peak This optical-pulse-width estimate would be somewhatshortened if the temperature dependence of the opacity were taken into accountThe peak optical power is ca 01 mW although this number is very sensitive to theopacity Typical stable SBSL experiments (at higher forcings than the present case)yield light emission of the order of milliwatts per regash (Gaitan et al 1996 Hiller etal 1992)

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1695

0 5 10 15 20 25

02

04

06

08

10

- 843 ns

- 455 ns

- 299 ns

- 153 ns

- 41 ns

- 15 ns

0 ns - 5 ns

radial position (microm)

mol

ar f

ract

ion

Figure 4 Molar fraction of water vapour versus radius for selected times prior to the moment ofminimum radius (time = 0) in case II Note the development of the non-uniform composition asthe collapse accelerates toward minimum radius In the last 15 ns before minimum radius thecomposition changes by only a few per cent when viewed in the Lagrangian frame instead

There are two mechanisms that determine the excess vapour left behind at thecollapse shy nite-rate mass dinotusion and non-equilibrium phase change We examineeach mechanism and their interaction in turn

(i) Role of direg usion

The shy rst reason for excess water vapour remaining during the collapse is the rapidbubble collapse overwhelming slow mass dinotusion this was extensively discussedin SampS In SampS it was shown that there is a competition between two importanttime-scales the dynamic time-scale of the bubble motions and the mass dinotusiontime-scale These are given explicitly in dimensionless terms as

frac12 d yn = R= _R

frac12 d if = ReScR2

D MH2 O

ordm ReSc1

Rp

T

In the latter we made use of the fact that the dinotusion coe cient of water vapourin the gas mixture D M

H2 O scales as R3pT

The time-scales argument is clarishy ed upon examination of shy gure 4 which showsthe evolving bubble composition as a function of radius at selected times before thetime of minimum radius (case II) In the early part of the collapse the dynamic time-scale is longer than the dinotusion time-scale frac12 d yn frac34 frac12 d if this results in a uniformcomposition of the bubble (upper curves in shy gure 4) As the bubble accelerates into

Proc R Soc Lond A (2000)

1696 B D Storey and A J Szeri

the collapse frac12 d yn sup1 frac12 d if and the decrease in water fraction at the centre begins tolag the decrease in water fraction at the wall (middle curves of shy gure 4) Furtherinto the collapse frac12 d yn frac12 frac12 d if and the water vapour cannot dinotuse to the bubble wallon the time-scale of the rapid collapse This slow dinotusion results in a nearly shy xeddistribution of water vapour (three lowest curves of shy gure 4)

While the three lowest curves in shy gure 4 look quite dinoterent in this Eulerianrepresentation they dinoter by only a few per cent when viewed in the Lagrangianrepresentation (not shown) The curve at iexcl 15 ns is from a time when R = R0 andthe temperature at the centre is ca 750 K The curve at 0 ns is from the time ofminimum radius (015R0) when the temperature at the centre is 9700 K Consistentwith shy gure 3 over times when there are drastic changes in the bubble radius andtemperature the composition of the bubble is almost a material shy eld

(ii) Role of non-equilibrium phase change

Non-equilibrium phase change is the second mechanism by which vapour is trappedin the bubble interior during the collapse By adjusting the accommodation coe -cient one has control over the time-scale for the partial pressure of the water vapourto come to equilibrium with the saturation pressure at the interface The dimension-less condensation time-scale is

frac12 con d =R

frac14

r2 ordm MH2 O

9M0Tin t

When frac12 d yn frac34 frac12 con d the condensation is in quasi-equilibrium with respect to thebubble motions If frac12 d yn frac12 frac12 con d then the bubble collapse is so rapid that no masscan escape the bubble on the time-scale of the collapse

(iii) Interaction of the two mechanisms

It is the competition between the time-scales for mass dinotusion condensation andbubble dynamics that determines which mechanism traps water vapour in the bubbleOnce frac12 d yn frac12 frac12 d if or frac12 d yn frac12 frac12 con d the total amount of water vapour in the bubbleis shy xed Only one of these conditions needs to be met for vapour to be trapped inthe bubble Both mechanisms can however participate to determine the quantity oftrapped vapour

During the bubble collapse the condition frac12 d yn frac12 frac12 d if is reached well before frac12 d yn frac12frac12 con d in case II Therefore in this case (as well as the others presented herein) it isthe slow mass dinotusion that is responsible for trapping vapour in the bubble

For cases where slow mass dinotusion traps the vapour non-equilibrium phasechange helps to determine the amount trapped The accommodation coe cient rep-resents a resistance to condensation at the interface during collapse of the bubble Alower value of accommodation coe cient provides more resistance to phase changeclearly increasing the mass of water retained in the bubble at any time during thecollapse The importance of the non-equilibrium phase change can be estimated bycomparing frac12 d yn with frac12 con d at the time that frac12 d yn = frac12 d if The closer frac12 d yn is to frac12 con d the more important non-equilibrium phase change is in determining the amount ofwater trapped

Colussi et al (1998) developed a model including chemical reactions in a collapsingbubble using an accommodation coe cient of frac14 = 0001 but did not take into

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1697

account the shy nite rate of mass dinotusion The low value of frac14 which works to trapsignishy cant water vapour at collapse was necessary for their model to be consistentwith experimental sonochemistry data When such a low value for frac14 is used thecondition frac12 d yn frac12 frac12 con d is reached well before frac12 d yn frac12 frac12 d if (the opposite of our case)Such a low accommodation coe cient works to trap signishy cant amounts of watervapour in the bubble but in this case the hindered regux at the interface eliminatesmass transfer out of the bubble while the distribution of water vapour within isessentially uniform In the present work we argue that the signishy cant amounts ofwater vapour trapped in the bubble are a consequence of slow dinotusion as well asnon-equilibrium phase change

The temperature of the interface exceeds the critical point for a very brief time(from iexcl 340 ps to 1800 ps relative to the time of minimum radius) For the conditionswe explored the maximum thickness of the supercritical region is quite thin less than1 of the bubble radius ie ca 10 nm We neglect any changes in the liquid equationof state and transport properties when the interface exceeds the critical temperatureThis assumption is justishy ed based on the thinness of the region and the fact that thecritical temperature is reached very late in the collapse Any disturbances caused bythe supercritical interface have insumacr cient time to propagate into the bubble interioron the main collapse Although the evaporation equation is invalid for these shorttimes the total mass added to the bubble is negligible regardless of the magnitudeof evaporative mass regux The issue of the super-critical interface may require futureimprovements but is beyond the scope of the present work

As a shy nal note both thermal and pressure dinotusion also play a role in determiningthe distribution of water vapour during the collapse In the argonwater system thethermal-dinotusion coemacr cient is small and the molecules are of similar size hence pres-sure dinotusion is quite weak Little dinoterence is seen in calculations with and withoutthese other forms of mass dinotusion In a helium or xenon bubble in water thermaland pressure dinotusion are more pronounced and can cause a 1015 dinoterence inthe amount of trapped water vapour In the interest of brevity we will not discussthese results further here

(b) Gas-phase chemical reactions break apart H2O and reduce the temperature

Now we turn to an investigation of what transpires when chemical reactions aretaken into account (case III) We shall show that there is signishy cant time for reactionsto occur ie the time-scale for chemical reaction is similar to that of the bubblecollapse The primarily endothermic reactions absorb a signishy cant amount of energyand further reduce the peak temperatures from cases I and II The distribution ofspecies produced in the bubble is determined by dinotusive transport of water vapourprior to the collapse and heat transfer out of the bubble during the collapse Thesteep temperature gradient at the wall conshy nes the bulk of the chemical reactions tothe bubble interior

(i) Consequences of chemistry

The overall bubble dynamics of cases II and III are essentially the same with acompression ratio of 49 in case III (slightly higher than case II) A convenient pointof departure is to compare the temperature histories of the bubble centre aroundthe main collapse for the three cases we have considered (see shy gure 5) The inclusion

Proc R Soc Lond A (2000)

1698 B D Storey and A J Szeri

- 4 - 2 0 2 410

3

104

tem

pera

ture

(K

)

time relative to minimum radius (ns)

Figure 5 Temperature history of the centre of the bubble around the time of minimum radius(t = 0) in cases I II and III Note the reduction in temperature from case I ( ) to case II( ) as water is allowed to enter the bubble The chemical reactions added in case III (||)further reduce the temperatures

of water vapour (case II) signishy cantly reduces the peak temperature from case IThe decrease in temperature is due mainly to the decrease in the ratio of specishy cheats of the mixture which reduces the compression heating Upon comparison ofcases II and III one observes that the chemical reactions further reduce the peaktemperature due to the endothermic decomposition of the water vapour The centretemperature histories of cases II and III are nearly identical below 4000 K

It is important to understand that the temperature dinoterences in these threecases are not due to subtle dinoterences in bubble dynamics The compression ratio(Rm ax=Rm in ) of the three cases increases from case I to case II to case III whereasthe temperature decreases in this order This decrease in temperature occurs despitethe fact that a higher compression ratio will lead to more compression heating in thebubble collapse all other things being equal In all these cases there are no shocksinside the bubble Shocks will be considered in x 3 e

In shy gure 6 we show the history of the total number of molecules of the dinoterentspecies as a function of time on the same time-scale as in shy gure 5 The lowest curve isthe total amount of water vapour only The next curve is the total amount of watervapour plus the total amount of H2 Each curve represents the addition of one speciesas labelled until the uppermost curve denotes everything but argon We do not labelHO2 and H2O2 as they appear only in trace amounts One observes which speciesare most abundant during the time of the main collapse Note there is a relativelylarge amount of undissociated water considering the elevated temperatures This isprimarily a consequence of the high pressure suppressing the dissociation of H2O

From shy gures 5 and 6 one observes that the onset of signishy cant chemical activity in

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1699

- 4 - 2 0 2 4

109

1010

time relative to minimum radius (ns)

num

ber

of m

olec

ules

H2O

H2

O2

OH

O

H

Figure 6 Cumulative number of molecules of each species in case III The lowest curve is thetotal number of water molecules the next curve is total water + H2 and so on H2 O2 and HO2

are neglected in this plot as they appear in trace amounts

the bubble corresponds to a centre temperature of ca 4000 K Therefore the reducedtemperature of case III shown in shy gure 5 can be attributed to the endothermicchemical reactions Although the bubble dynamics are quite rapid the chemicalreactions keep pace as is shown by the similar time-scales of the changing quantitiesin shy gures 5 and 6 The rapid rate at which the reactions progress is due to the veryhigh density Recall that the rate of progress of each reaction is directly related tothe species concentration (mol cmiexcl3)

(ii) Dynamics and distribution of OH production

In sonochemistry applications the invasion of OH radicals into the liquid isthought to be responsible for much of the complex and useful chemistry that occursTherefore we shall focus on the behaviour of this species in particular In shy gure 7 weplot the total number of OH molecules contained in the bubble on two time-scalesover one acoustic cycle in shy gure 7a and around the time of collapse in shy gure 7b Theinitial concentrations of all species were selected so that the chemical production anddestruction balanced in one acoustic cycle The main collapse is associated with arapid creation of OH once the temperature exceeds 4000 K

Upon rebound the bubble rapidly cools and the OH and other products tend torecombine back into H2O These recombination reactions proceed more slowly thanthe dissociation reactions upon collapse The result is that the bubble is left with asuper-equilibrium amount of OH after it has cooled to ambient temperature Thissuper-equilibrium OH then slowly decays through the rest of the cycle with briefspurts of production and destruction at each of the after-bounces

Proc R Soc Lond A (2000)

1700 B D Storey and A J Szeri

0 10 20 30time (micros)

- 4 - 2 0 2 4

time relative to minimum radius (ns)

106

1010

104

106

108

108

num

ber

of O

H

radi

cals

nu

mbe

r of

OH

ra

dica

ls

Figure 7 Total number of OH molecules contained in the bubble versus time over one acousticcycle (a) and around the time of minimum radius (b) in case III The macrrst sharp peak in (a)is expanded in (b) Subsequent peaks in (a) correspond to local radius minima in the bubbledynamics The initial amounts of OH and all the other species were chosen for no net speciesproduction or destruction over an acoustic cycle

The distribution of OH in the bubble is characterized by a fairly uniform con-centration through much of the bubble interior and a very sharp gradient near thewall This sharp gradient is a result of heat transfer during the collapse Duringcompression conduction between the hot interior and the cool wall sets up a thintemperature boundary layer This boundary layer conshy nes the chemical reactions tothe interior away from the interface The cooler wall leads to a much lower concen-tration of OH at the interface than in the centre at the time of the main collapseIn shy gure 8 we show the molar concentration history of OH at the centre and at theinterface When the bubble is at minimum radius and the OH concentration at thecentre is at its maximum the concentration at the wall is three orders of magnitudelower

Because the chemistry time-scale is much faster than the dinotusion time-scale OHis produced where water vapour was trapped at high enough temperature The liquidinterface is insulated from the high concentration of OH generated deep inside in thebubble The only OH produced at the centre that reaches the bubble interface is theslowly decaying super-equilibrium OH that is left inside the bubble on re-expansion

(c) OH uptake by the liquid does not indeg uence the bubble interior

The inclusion of liquid uptake of OH (case IV) adds further complications wenow try to illuminate some of the main qualitative features We shall argue that OHuptake has little enotect on the gas dynamics and chemical production

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1701

- 4 - 2 0 2 410 - 10

10 - 8

10 - 6

10 - 4

10 - 2

time relative to minimum radius (ns)

conc

entr

atio

n of

OH

(m

ol c

m-3

)

Figure 8 Molar concentration of OH at the centre (||) and the interface ( ) versus timearound the point of minimum radius Note that the concentration at the centre is about threeorders of magnitude greater than at the interface at the point of minimum radius

Due to dinotering ideas on the proper value of the uptake coemacr cient the values areset at pound = 0001 and pound = 001 (cases IV(a) and IV(b)) consistent with the work ofothers to which we made reference earlier The bubble dynamics and temperatureproshy les for cases IV(a) and IV(b) are indistinguishable from those for case III Inshy gure 9 we show the total number of OH molecules in the bubble versus time ontwo dinoterent scales for cases III IV(a) and IV(b) The history for one acoustic cycleis plotted in shy gure 9a and the history concentrated at the main collapse is plottedin shy gure 9b

In the main expansion when the bubble is cool the OH resident in the bubblewill invade the liquid as can be seen by the decrease in the amount of OH at theend of the long expansion as the uptake increases When the bubble collapses thechemical production of OH from recently evaporated water rapidly overwhelms theOH uptake by the liquid As the bubble proceeds into the collapse the curves oftotal OH content for dinoterent uptakes become indistinguishable The total amountof OH in the bubble is dominated by the amount created in the interior away fromthe wall The total amount of OH lost to the liquid during a collapse is only afraction of the amount contained in the bubble Hence the peak amount of OH inthe bubble during the collapse is insensitive to the uptake coe cient of OH Similarstatements apply to uptake of other species

In shy gure 10 we show for the purposes of qualitative discussion the total massof OH exiting the bubble in cases IV(a) and IV(b) While the OH uptake coef-shy cient does not inreguence the total amount of OH contained in the bubble at themain collapse there are large dinoterences in the amount invading the liquid over

Proc R Soc Lond A (2000)

1702 B D Storey and A J Szeri

0 10 20 30

106

1010

102

time (micros)

- 4 - 2 0 2 4104

106

108

time relative to minimum radius (ns)

1010

(b)

(a)nu

mbe

r of

OH

ra

dica

ls

num

ber

of O

H

radi

cals

Figure 9 Comparison of total amount number of OH molecules in the bubble versus time overone acoustic cycle (a) and focused around the time of minimum radius (b) in cases III (||)IV(a) ( ) and IV(b) ( ) The uptake of OH plays a role during the slow expansionswhen the resident amount of OH left in the bubble is well above chemical equilibrium At thecollapse OH uptake does not indeg uence the total amount of OH production

a cycle Much of the OH uptake occurs during the main collapse and the subse-quent re-expansion when the concentration of OH at the wall is at its maximum(see shy gure 8) Further uptake occurs throughout the cycle at each of the after-bounces

(d ) Inclusion of water vapour alters the ereg ect of increased forcing

Next we shall compare the results of cases I II and III with all parameters shy xedexcept for the acoustic pressure amplitude which varies between 10 and 13 barWe that note this is experimentally awkward for SBSL because pressure amplitudeand ambient radius are dependent quantities through the mass exchange equilibriumof rectishy ed dinotusion (Hilgenfeldt et al 1996) The way we present these results isnonetheless useful as it demonstrates the importance of water vapour in the under-standing of sonoluminescence and sonochemistry

Because the bubble dynamics are altered by the inclusion of water vapour weuse the compression ratio as the distinguishing parameter In shy gure 11 we plot themaximum centre temperature versus the compression ratio for cases I II and III Thecalculations in shy gure 11 are devoid of strong shocks a matter to which we devotemore attention below If there are strong shocks then shy gure 11 will look the samein terms of a suitably averaged temperature rather than the centre temperature

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1703

0 10 20 30102

103

104

105

106

num

ber

of O

H

radi

cals

inva

ding

liqu

id

time (micros)

Figure 10 Comparison of the amount of OH absorbed by the liquid versus time for cases IV(a)(|| uptake coeplusmn cient pound = 0001) and IV(b) ( pound = 001) Most of the OH uptakeoccurs at the main collapse and the subsequent after-bounce

In case I the plot shows linear behaviour (on the loglog scale) One observes thatwhen water vapour is not taken into account stronger forcing yields more extremetemperatures

In case II the relationship between compression ratio and maximum temperature isvery dinoterent First we note that the amount of water trapped in the bubble increaseswith compression ratio If we revisit the water-trapping mechanism (x 3 a (iii)) thistrend is readily explained The increase in water-vapour content reduces the ratioof specishy c heats of the gasvapour mixture and hence reduces the compressionheating

At low compression ratios in case II the temperature increases with compressionratio in a manner very similar to case I At low compression ratios the amountof water at collapse is very small the bubble is nearly pure argon As the forcingis increased more water is trapped and the maximum temperature of the bubblebecomes nearly constant In this range of bubble dynamics there is an almost per-fectly compensating enotect (with respect to peak temperature) between the decreasein specishy c-heat ratio and the increase in compression Case III is very similar tocase II except that the plateau in temperature occurs at a lower temperature aconsequence of the endothermic chemical reactions

(e) Inclusion of water vapour promotes the formation of shocks

Water vapour is of great importance in the formation of shocks in very stronglyforced bubbles We shall brieregy consider a 60 m bubble forced with a pressureamplitude of 14 bar at a frequency of 206 kHz (Case A1 in Moss et al (1999))

Proc R Soc Lond A (2000)

1704 B D Storey and A J Szeri

10 1 10 2

10 3

10 4

10 5

Rmax R min

max

imum

tem

pera

ture

(K

)

Figure 11 Change in centre temperature versus compression ratio in cases I ( ) II ( curren )and III (+) In this plot all parameters were held constant except the pressure amplitudewhich was varied between 10 and 13 bar Case I shows a steady increase in temperature withcompression ratio Cases II and III reach a maximum temperature as the forcing is increased

When water vapour is neglected a compression wave is launched into the bubble oncollapse but the wave does not steepen into a shock When water vapour is takeninto account (with or without reactions) the compression wave steepens into a strongshock very near the centre of the bubble Vuong et al (1999) showed that the rateof steepening depends on the amount of compression heating before the compressionwave is launched

In very recent work Lin amp Szeri (2000) used the wavefront expansion technique todetermine the evolution of a compression or rarefaction wavefront that is propagatingspherically into a collapsing or expanding gas of non-uniform entropy This analyticalwork supports the idea that when the speed of sound is higher at the centre (dueto temperature and composition inhomogeneities) shock formation is delayed orprevented Furthermore this work demonstrated that the inward radial velocity alsohinders the steepening of compression waves into shocks

Consistent with the arguments made in Lin amp Szeri (2000) Vuong et al (1999)and Moss et al (1999) our results show that the decrease in the ratio of specishy cheats due to an increase in the water-vapour content can promote shock formationThe point we want to emphasize is that the question of shock formation cannot beundertaken without consideration of water vapour

With respect to total chemical production shocks are of negligible consequenceThe shock inreguences only about the inner 10 of the radius which is only 01 ofthe volume Therefore the increased production of OH at the centre does not anotectthe total amount produced in the bubble

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

1692 B D Storey and A J Szeri

Table 1 Summary of the direg erent cases investigated in detail

(In all cases the bubble is a 45 m radius argon bubble in water at 298 K acoustically forcedwith a 12 bar pressure amplitude at 265 kHz)

case phase change chemistry OH uptake Rm ax =Rm in ( m) Tm ax (K)

I no no 00 280080 20 900

II yes no 00 313070 9 700

III yes yes 00 317065 7 000

IV(a) yes yes 0001 317065 7 000

IV(b) yes yes 001 317065 7 000

Case I no water vapour allowed in the bubble

Case II water vapour allowed to evaporate and condense but no chemical reactions

Case III water vapour and chemical reactions but no mass transfer across thebubble interface

Cases IV(a) and IV(b) mass transfer of OH across the bubble interface at dif-ferent uptake coe cients

In all cases the bubble is a 45 m-radius argon bubble in 298 K water acousticallyforced with a 12 bar pressure amplitude at 265 kHz These cases are summarized intable 1 With respect to SBSL this bubble was found to be stable to mass exchangeand shape changes though would have minimal light emission (Hilgenfeldt et al 1996 Wu amp Roberts 1998a b) We describe the results for more strongly forcedbubbles below The data we obtain are for steady-state oscillations ie after thedecay of initial transients After presenting the details of these cases we considervariations in the parameters to illustrate interesting trends

(a) Excess water is trapped by the rapid collapse

We shall now argue that signishy cant amounts of water vapour become trapped inthe interior of the bubble when it collapses violently Water is trapped in the interiorbecause the bubble motions become so rapid that the vapour near the centre hasinsu cient time to dinotuse to the bubble wall Non-equilibrium phase change alsoplays a signishy cant role in the amount of water that is trapped All results in thissubsection apply to case II this case allows one to focus on the dynamics of evapora-tion condensation and dinotusion of water inside the bubble without the complicationof chemical reactions

In shy gure 1 we show the radial history of the bubble over one cycle of the acousticforcing in case II The essential features of the bubble dynamics for cases I and II arethe same but the inclusion of water vapour increases the maximum radius and delaysthe collapse The dinoterence is to be expected because case I has no vapour pressureThe compression ratio (Rm ax=Rm in ) for cases I and II is 35 and 44 respectively Thecompression ratio is an approximate way of separating out dinoterences in the resultsdue to dinoterences in the bubble dynamics Typically a higher compression ratio leadsto more compression heating and higher temperatures if all else is equal We returnto this point later

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1693

0 10 20 30

10

20

30

time (micros)

radi

us (

microm

)

Figure 1 Radius of the bubble versus time over one cycle of acoustic forcing in case IIThe essential features for cases I II III and IV are the same

0 10 20 30

10 8

10 9

1010

1011

num

ber

of m

olec

ules

H2O

Ar

time (micros)

Figure 2 Total number of water-vapour molecules contained in the bubble versus time over onecycle of acoustic forcing in case II Note that water comprises ca 14 of the bubble contents atthe end of the main collapse

Proc R Soc Lond A (2000)

1694 B D Storey and A J Szeri

0

4000

8000te

mpe

ratu

re (

K)

1

2

3

time relative to minimum radius (ns)

num

ber

of H

2O m

olec

ules

109

acute-

- 4 - 2 0 2 4

(b)

(a)

Figure 3 Temperature (a) and total number of water molecules contained in the bubble (b) versustime around the point of minimum radius (time = 0) in case II The temperature undergoesorder-of-magnitude changes on this time-scale while the total number of water molecules changesvery little

In shy gure 2 we show the total number of H2O molecules contained within thebubble over the acoustic cycle (case II) A large amount of water evaporates intothe bubble during the main expansion when the pressure is low Water condenses onthe bubble wall during the collapse as the gas pressure increases There is signishy cantwater vapour in the bubble at the main collapse ca 14 (molar basis) of the totalbubble contents is water vapour If the interface were at equilibrium and the speedof mass dinotusion were inshy nite there would be a negligible amount of water in thebubble on collapse

In shy gure 3 we show the temperature history of the centre and the total numberof water molecules in the bubble around the time of minimum radius (time = 0) incase II Over the 10 ns time-scale the total amount of water vapour contained in thebubble is nearly constant whereas the temperature changes by an order of magni-tude This recalls SampS in which it was shown that evolving mixture compositionswere essentially frozen on the time-scale of the temperature peak

The temperature peak may be related to the light emission The equation foroptical power given by Moss et al (1999) with a constant opacity provides an easyestimate for the optical power output as a function of time For case II this estimategives a light emission pulse width (FWHM) of 350 ps which is much shorter than theca 2000 ps temperature peak This optical-pulse-width estimate would be somewhatshortened if the temperature dependence of the opacity were taken into accountThe peak optical power is ca 01 mW although this number is very sensitive to theopacity Typical stable SBSL experiments (at higher forcings than the present case)yield light emission of the order of milliwatts per regash (Gaitan et al 1996 Hiller etal 1992)

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1695

0 5 10 15 20 25

02

04

06

08

10

- 843 ns

- 455 ns

- 299 ns

- 153 ns

- 41 ns

- 15 ns

0 ns - 5 ns

radial position (microm)

mol

ar f

ract

ion

Figure 4 Molar fraction of water vapour versus radius for selected times prior to the moment ofminimum radius (time = 0) in case II Note the development of the non-uniform composition asthe collapse accelerates toward minimum radius In the last 15 ns before minimum radius thecomposition changes by only a few per cent when viewed in the Lagrangian frame instead

There are two mechanisms that determine the excess vapour left behind at thecollapse shy nite-rate mass dinotusion and non-equilibrium phase change We examineeach mechanism and their interaction in turn

(i) Role of direg usion

The shy rst reason for excess water vapour remaining during the collapse is the rapidbubble collapse overwhelming slow mass dinotusion this was extensively discussedin SampS In SampS it was shown that there is a competition between two importanttime-scales the dynamic time-scale of the bubble motions and the mass dinotusiontime-scale These are given explicitly in dimensionless terms as

frac12 d yn = R= _R

frac12 d if = ReScR2

D MH2 O

ordm ReSc1

Rp

T

In the latter we made use of the fact that the dinotusion coe cient of water vapourin the gas mixture D M

H2 O scales as R3pT

The time-scales argument is clarishy ed upon examination of shy gure 4 which showsthe evolving bubble composition as a function of radius at selected times before thetime of minimum radius (case II) In the early part of the collapse the dynamic time-scale is longer than the dinotusion time-scale frac12 d yn frac34 frac12 d if this results in a uniformcomposition of the bubble (upper curves in shy gure 4) As the bubble accelerates into

Proc R Soc Lond A (2000)

1696 B D Storey and A J Szeri

the collapse frac12 d yn sup1 frac12 d if and the decrease in water fraction at the centre begins tolag the decrease in water fraction at the wall (middle curves of shy gure 4) Furtherinto the collapse frac12 d yn frac12 frac12 d if and the water vapour cannot dinotuse to the bubble wallon the time-scale of the rapid collapse This slow dinotusion results in a nearly shy xeddistribution of water vapour (three lowest curves of shy gure 4)

While the three lowest curves in shy gure 4 look quite dinoterent in this Eulerianrepresentation they dinoter by only a few per cent when viewed in the Lagrangianrepresentation (not shown) The curve at iexcl 15 ns is from a time when R = R0 andthe temperature at the centre is ca 750 K The curve at 0 ns is from the time ofminimum radius (015R0) when the temperature at the centre is 9700 K Consistentwith shy gure 3 over times when there are drastic changes in the bubble radius andtemperature the composition of the bubble is almost a material shy eld

(ii) Role of non-equilibrium phase change

Non-equilibrium phase change is the second mechanism by which vapour is trappedin the bubble interior during the collapse By adjusting the accommodation coe -cient one has control over the time-scale for the partial pressure of the water vapourto come to equilibrium with the saturation pressure at the interface The dimension-less condensation time-scale is

frac12 con d =R

frac14

r2 ordm MH2 O

9M0Tin t

When frac12 d yn frac34 frac12 con d the condensation is in quasi-equilibrium with respect to thebubble motions If frac12 d yn frac12 frac12 con d then the bubble collapse is so rapid that no masscan escape the bubble on the time-scale of the collapse

(iii) Interaction of the two mechanisms

It is the competition between the time-scales for mass dinotusion condensation andbubble dynamics that determines which mechanism traps water vapour in the bubbleOnce frac12 d yn frac12 frac12 d if or frac12 d yn frac12 frac12 con d the total amount of water vapour in the bubbleis shy xed Only one of these conditions needs to be met for vapour to be trapped inthe bubble Both mechanisms can however participate to determine the quantity oftrapped vapour

During the bubble collapse the condition frac12 d yn frac12 frac12 d if is reached well before frac12 d yn frac12frac12 con d in case II Therefore in this case (as well as the others presented herein) it isthe slow mass dinotusion that is responsible for trapping vapour in the bubble

For cases where slow mass dinotusion traps the vapour non-equilibrium phasechange helps to determine the amount trapped The accommodation coe cient rep-resents a resistance to condensation at the interface during collapse of the bubble Alower value of accommodation coe cient provides more resistance to phase changeclearly increasing the mass of water retained in the bubble at any time during thecollapse The importance of the non-equilibrium phase change can be estimated bycomparing frac12 d yn with frac12 con d at the time that frac12 d yn = frac12 d if The closer frac12 d yn is to frac12 con d the more important non-equilibrium phase change is in determining the amount ofwater trapped

Colussi et al (1998) developed a model including chemical reactions in a collapsingbubble using an accommodation coe cient of frac14 = 0001 but did not take into

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1697

account the shy nite rate of mass dinotusion The low value of frac14 which works to trapsignishy cant water vapour at collapse was necessary for their model to be consistentwith experimental sonochemistry data When such a low value for frac14 is used thecondition frac12 d yn frac12 frac12 con d is reached well before frac12 d yn frac12 frac12 d if (the opposite of our case)Such a low accommodation coe cient works to trap signishy cant amounts of watervapour in the bubble but in this case the hindered regux at the interface eliminatesmass transfer out of the bubble while the distribution of water vapour within isessentially uniform In the present work we argue that the signishy cant amounts ofwater vapour trapped in the bubble are a consequence of slow dinotusion as well asnon-equilibrium phase change

The temperature of the interface exceeds the critical point for a very brief time(from iexcl 340 ps to 1800 ps relative to the time of minimum radius) For the conditionswe explored the maximum thickness of the supercritical region is quite thin less than1 of the bubble radius ie ca 10 nm We neglect any changes in the liquid equationof state and transport properties when the interface exceeds the critical temperatureThis assumption is justishy ed based on the thinness of the region and the fact that thecritical temperature is reached very late in the collapse Any disturbances caused bythe supercritical interface have insumacr cient time to propagate into the bubble interioron the main collapse Although the evaporation equation is invalid for these shorttimes the total mass added to the bubble is negligible regardless of the magnitudeof evaporative mass regux The issue of the super-critical interface may require futureimprovements but is beyond the scope of the present work

As a shy nal note both thermal and pressure dinotusion also play a role in determiningthe distribution of water vapour during the collapse In the argonwater system thethermal-dinotusion coemacr cient is small and the molecules are of similar size hence pres-sure dinotusion is quite weak Little dinoterence is seen in calculations with and withoutthese other forms of mass dinotusion In a helium or xenon bubble in water thermaland pressure dinotusion are more pronounced and can cause a 1015 dinoterence inthe amount of trapped water vapour In the interest of brevity we will not discussthese results further here

(b) Gas-phase chemical reactions break apart H2O and reduce the temperature

Now we turn to an investigation of what transpires when chemical reactions aretaken into account (case III) We shall show that there is signishy cant time for reactionsto occur ie the time-scale for chemical reaction is similar to that of the bubblecollapse The primarily endothermic reactions absorb a signishy cant amount of energyand further reduce the peak temperatures from cases I and II The distribution ofspecies produced in the bubble is determined by dinotusive transport of water vapourprior to the collapse and heat transfer out of the bubble during the collapse Thesteep temperature gradient at the wall conshy nes the bulk of the chemical reactions tothe bubble interior

(i) Consequences of chemistry

The overall bubble dynamics of cases II and III are essentially the same with acompression ratio of 49 in case III (slightly higher than case II) A convenient pointof departure is to compare the temperature histories of the bubble centre aroundthe main collapse for the three cases we have considered (see shy gure 5) The inclusion

Proc R Soc Lond A (2000)

1698 B D Storey and A J Szeri

- 4 - 2 0 2 410

3

104

tem

pera

ture

(K

)

time relative to minimum radius (ns)

Figure 5 Temperature history of the centre of the bubble around the time of minimum radius(t = 0) in cases I II and III Note the reduction in temperature from case I ( ) to case II( ) as water is allowed to enter the bubble The chemical reactions added in case III (||)further reduce the temperatures

of water vapour (case II) signishy cantly reduces the peak temperature from case IThe decrease in temperature is due mainly to the decrease in the ratio of specishy cheats of the mixture which reduces the compression heating Upon comparison ofcases II and III one observes that the chemical reactions further reduce the peaktemperature due to the endothermic decomposition of the water vapour The centretemperature histories of cases II and III are nearly identical below 4000 K

It is important to understand that the temperature dinoterences in these threecases are not due to subtle dinoterences in bubble dynamics The compression ratio(Rm ax=Rm in ) of the three cases increases from case I to case II to case III whereasthe temperature decreases in this order This decrease in temperature occurs despitethe fact that a higher compression ratio will lead to more compression heating in thebubble collapse all other things being equal In all these cases there are no shocksinside the bubble Shocks will be considered in x 3 e

In shy gure 6 we show the history of the total number of molecules of the dinoterentspecies as a function of time on the same time-scale as in shy gure 5 The lowest curve isthe total amount of water vapour only The next curve is the total amount of watervapour plus the total amount of H2 Each curve represents the addition of one speciesas labelled until the uppermost curve denotes everything but argon We do not labelHO2 and H2O2 as they appear only in trace amounts One observes which speciesare most abundant during the time of the main collapse Note there is a relativelylarge amount of undissociated water considering the elevated temperatures This isprimarily a consequence of the high pressure suppressing the dissociation of H2O

From shy gures 5 and 6 one observes that the onset of signishy cant chemical activity in

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1699

- 4 - 2 0 2 4

109

1010

time relative to minimum radius (ns)

num

ber

of m

olec

ules

H2O

H2

O2

OH

O

H

Figure 6 Cumulative number of molecules of each species in case III The lowest curve is thetotal number of water molecules the next curve is total water + H2 and so on H2 O2 and HO2

are neglected in this plot as they appear in trace amounts

the bubble corresponds to a centre temperature of ca 4000 K Therefore the reducedtemperature of case III shown in shy gure 5 can be attributed to the endothermicchemical reactions Although the bubble dynamics are quite rapid the chemicalreactions keep pace as is shown by the similar time-scales of the changing quantitiesin shy gures 5 and 6 The rapid rate at which the reactions progress is due to the veryhigh density Recall that the rate of progress of each reaction is directly related tothe species concentration (mol cmiexcl3)

(ii) Dynamics and distribution of OH production

In sonochemistry applications the invasion of OH radicals into the liquid isthought to be responsible for much of the complex and useful chemistry that occursTherefore we shall focus on the behaviour of this species in particular In shy gure 7 weplot the total number of OH molecules contained in the bubble on two time-scalesover one acoustic cycle in shy gure 7a and around the time of collapse in shy gure 7b Theinitial concentrations of all species were selected so that the chemical production anddestruction balanced in one acoustic cycle The main collapse is associated with arapid creation of OH once the temperature exceeds 4000 K

Upon rebound the bubble rapidly cools and the OH and other products tend torecombine back into H2O These recombination reactions proceed more slowly thanthe dissociation reactions upon collapse The result is that the bubble is left with asuper-equilibrium amount of OH after it has cooled to ambient temperature Thissuper-equilibrium OH then slowly decays through the rest of the cycle with briefspurts of production and destruction at each of the after-bounces

Proc R Soc Lond A (2000)

1700 B D Storey and A J Szeri

0 10 20 30time (micros)

- 4 - 2 0 2 4

time relative to minimum radius (ns)

106

1010

104

106

108

108

num

ber

of O

H

radi

cals

nu

mbe

r of

OH

ra

dica

ls

Figure 7 Total number of OH molecules contained in the bubble versus time over one acousticcycle (a) and around the time of minimum radius (b) in case III The macrrst sharp peak in (a)is expanded in (b) Subsequent peaks in (a) correspond to local radius minima in the bubbledynamics The initial amounts of OH and all the other species were chosen for no net speciesproduction or destruction over an acoustic cycle

The distribution of OH in the bubble is characterized by a fairly uniform con-centration through much of the bubble interior and a very sharp gradient near thewall This sharp gradient is a result of heat transfer during the collapse Duringcompression conduction between the hot interior and the cool wall sets up a thintemperature boundary layer This boundary layer conshy nes the chemical reactions tothe interior away from the interface The cooler wall leads to a much lower concen-tration of OH at the interface than in the centre at the time of the main collapseIn shy gure 8 we show the molar concentration history of OH at the centre and at theinterface When the bubble is at minimum radius and the OH concentration at thecentre is at its maximum the concentration at the wall is three orders of magnitudelower

Because the chemistry time-scale is much faster than the dinotusion time-scale OHis produced where water vapour was trapped at high enough temperature The liquidinterface is insulated from the high concentration of OH generated deep inside in thebubble The only OH produced at the centre that reaches the bubble interface is theslowly decaying super-equilibrium OH that is left inside the bubble on re-expansion

(c) OH uptake by the liquid does not indeg uence the bubble interior

The inclusion of liquid uptake of OH (case IV) adds further complications wenow try to illuminate some of the main qualitative features We shall argue that OHuptake has little enotect on the gas dynamics and chemical production

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1701

- 4 - 2 0 2 410 - 10

10 - 8

10 - 6

10 - 4

10 - 2

time relative to minimum radius (ns)

conc

entr

atio

n of

OH

(m

ol c

m-3

)

Figure 8 Molar concentration of OH at the centre (||) and the interface ( ) versus timearound the point of minimum radius Note that the concentration at the centre is about threeorders of magnitude greater than at the interface at the point of minimum radius

Due to dinotering ideas on the proper value of the uptake coemacr cient the values areset at pound = 0001 and pound = 001 (cases IV(a) and IV(b)) consistent with the work ofothers to which we made reference earlier The bubble dynamics and temperatureproshy les for cases IV(a) and IV(b) are indistinguishable from those for case III Inshy gure 9 we show the total number of OH molecules in the bubble versus time ontwo dinoterent scales for cases III IV(a) and IV(b) The history for one acoustic cycleis plotted in shy gure 9a and the history concentrated at the main collapse is plottedin shy gure 9b

In the main expansion when the bubble is cool the OH resident in the bubblewill invade the liquid as can be seen by the decrease in the amount of OH at theend of the long expansion as the uptake increases When the bubble collapses thechemical production of OH from recently evaporated water rapidly overwhelms theOH uptake by the liquid As the bubble proceeds into the collapse the curves oftotal OH content for dinoterent uptakes become indistinguishable The total amountof OH in the bubble is dominated by the amount created in the interior away fromthe wall The total amount of OH lost to the liquid during a collapse is only afraction of the amount contained in the bubble Hence the peak amount of OH inthe bubble during the collapse is insensitive to the uptake coe cient of OH Similarstatements apply to uptake of other species

In shy gure 10 we show for the purposes of qualitative discussion the total massof OH exiting the bubble in cases IV(a) and IV(b) While the OH uptake coef-shy cient does not inreguence the total amount of OH contained in the bubble at themain collapse there are large dinoterences in the amount invading the liquid over

Proc R Soc Lond A (2000)

1702 B D Storey and A J Szeri

0 10 20 30

106

1010

102

time (micros)

- 4 - 2 0 2 4104

106

108

time relative to minimum radius (ns)

1010

(b)

(a)nu

mbe

r of

OH

ra

dica

ls

num

ber

of O

H

radi

cals

Figure 9 Comparison of total amount number of OH molecules in the bubble versus time overone acoustic cycle (a) and focused around the time of minimum radius (b) in cases III (||)IV(a) ( ) and IV(b) ( ) The uptake of OH plays a role during the slow expansionswhen the resident amount of OH left in the bubble is well above chemical equilibrium At thecollapse OH uptake does not indeg uence the total amount of OH production

a cycle Much of the OH uptake occurs during the main collapse and the subse-quent re-expansion when the concentration of OH at the wall is at its maximum(see shy gure 8) Further uptake occurs throughout the cycle at each of the after-bounces

(d ) Inclusion of water vapour alters the ereg ect of increased forcing

Next we shall compare the results of cases I II and III with all parameters shy xedexcept for the acoustic pressure amplitude which varies between 10 and 13 barWe that note this is experimentally awkward for SBSL because pressure amplitudeand ambient radius are dependent quantities through the mass exchange equilibriumof rectishy ed dinotusion (Hilgenfeldt et al 1996) The way we present these results isnonetheless useful as it demonstrates the importance of water vapour in the under-standing of sonoluminescence and sonochemistry

Because the bubble dynamics are altered by the inclusion of water vapour weuse the compression ratio as the distinguishing parameter In shy gure 11 we plot themaximum centre temperature versus the compression ratio for cases I II and III Thecalculations in shy gure 11 are devoid of strong shocks a matter to which we devotemore attention below If there are strong shocks then shy gure 11 will look the samein terms of a suitably averaged temperature rather than the centre temperature

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1703

0 10 20 30102

103

104

105

106

num

ber

of O

H

radi

cals

inva

ding

liqu

id

time (micros)

Figure 10 Comparison of the amount of OH absorbed by the liquid versus time for cases IV(a)(|| uptake coeplusmn cient pound = 0001) and IV(b) ( pound = 001) Most of the OH uptakeoccurs at the main collapse and the subsequent after-bounce

In case I the plot shows linear behaviour (on the loglog scale) One observes thatwhen water vapour is not taken into account stronger forcing yields more extremetemperatures

In case II the relationship between compression ratio and maximum temperature isvery dinoterent First we note that the amount of water trapped in the bubble increaseswith compression ratio If we revisit the water-trapping mechanism (x 3 a (iii)) thistrend is readily explained The increase in water-vapour content reduces the ratioof specishy c heats of the gasvapour mixture and hence reduces the compressionheating

At low compression ratios in case II the temperature increases with compressionratio in a manner very similar to case I At low compression ratios the amountof water at collapse is very small the bubble is nearly pure argon As the forcingis increased more water is trapped and the maximum temperature of the bubblebecomes nearly constant In this range of bubble dynamics there is an almost per-fectly compensating enotect (with respect to peak temperature) between the decreasein specishy c-heat ratio and the increase in compression Case III is very similar tocase II except that the plateau in temperature occurs at a lower temperature aconsequence of the endothermic chemical reactions

(e) Inclusion of water vapour promotes the formation of shocks

Water vapour is of great importance in the formation of shocks in very stronglyforced bubbles We shall brieregy consider a 60 m bubble forced with a pressureamplitude of 14 bar at a frequency of 206 kHz (Case A1 in Moss et al (1999))

Proc R Soc Lond A (2000)

1704 B D Storey and A J Szeri

10 1 10 2

10 3

10 4

10 5

Rmax R min

max

imum

tem

pera

ture

(K

)

Figure 11 Change in centre temperature versus compression ratio in cases I ( ) II ( curren )and III (+) In this plot all parameters were held constant except the pressure amplitudewhich was varied between 10 and 13 bar Case I shows a steady increase in temperature withcompression ratio Cases II and III reach a maximum temperature as the forcing is increased

When water vapour is neglected a compression wave is launched into the bubble oncollapse but the wave does not steepen into a shock When water vapour is takeninto account (with or without reactions) the compression wave steepens into a strongshock very near the centre of the bubble Vuong et al (1999) showed that the rateof steepening depends on the amount of compression heating before the compressionwave is launched

In very recent work Lin amp Szeri (2000) used the wavefront expansion technique todetermine the evolution of a compression or rarefaction wavefront that is propagatingspherically into a collapsing or expanding gas of non-uniform entropy This analyticalwork supports the idea that when the speed of sound is higher at the centre (dueto temperature and composition inhomogeneities) shock formation is delayed orprevented Furthermore this work demonstrated that the inward radial velocity alsohinders the steepening of compression waves into shocks

Consistent with the arguments made in Lin amp Szeri (2000) Vuong et al (1999)and Moss et al (1999) our results show that the decrease in the ratio of specishy cheats due to an increase in the water-vapour content can promote shock formationThe point we want to emphasize is that the question of shock formation cannot beundertaken without consideration of water vapour

With respect to total chemical production shocks are of negligible consequenceThe shock inreguences only about the inner 10 of the radius which is only 01 ofthe volume Therefore the increased production of OH at the centre does not anotectthe total amount produced in the bubble

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1693

0 10 20 30

10

20

30

time (micros)

radi

us (

microm

)

Figure 1 Radius of the bubble versus time over one cycle of acoustic forcing in case IIThe essential features for cases I II III and IV are the same

0 10 20 30

10 8

10 9

1010

1011

num

ber

of m

olec

ules

H2O

Ar

time (micros)

Figure 2 Total number of water-vapour molecules contained in the bubble versus time over onecycle of acoustic forcing in case II Note that water comprises ca 14 of the bubble contents atthe end of the main collapse

Proc R Soc Lond A (2000)

1694 B D Storey and A J Szeri

0

4000

8000te

mpe

ratu

re (

K)

1

2

3

time relative to minimum radius (ns)

num

ber

of H

2O m

olec

ules

109

acute-

- 4 - 2 0 2 4

(b)

(a)

Figure 3 Temperature (a) and total number of water molecules contained in the bubble (b) versustime around the point of minimum radius (time = 0) in case II The temperature undergoesorder-of-magnitude changes on this time-scale while the total number of water molecules changesvery little

In shy gure 2 we show the total number of H2O molecules contained within thebubble over the acoustic cycle (case II) A large amount of water evaporates intothe bubble during the main expansion when the pressure is low Water condenses onthe bubble wall during the collapse as the gas pressure increases There is signishy cantwater vapour in the bubble at the main collapse ca 14 (molar basis) of the totalbubble contents is water vapour If the interface were at equilibrium and the speedof mass dinotusion were inshy nite there would be a negligible amount of water in thebubble on collapse

In shy gure 3 we show the temperature history of the centre and the total numberof water molecules in the bubble around the time of minimum radius (time = 0) incase II Over the 10 ns time-scale the total amount of water vapour contained in thebubble is nearly constant whereas the temperature changes by an order of magni-tude This recalls SampS in which it was shown that evolving mixture compositionswere essentially frozen on the time-scale of the temperature peak

The temperature peak may be related to the light emission The equation foroptical power given by Moss et al (1999) with a constant opacity provides an easyestimate for the optical power output as a function of time For case II this estimategives a light emission pulse width (FWHM) of 350 ps which is much shorter than theca 2000 ps temperature peak This optical-pulse-width estimate would be somewhatshortened if the temperature dependence of the opacity were taken into accountThe peak optical power is ca 01 mW although this number is very sensitive to theopacity Typical stable SBSL experiments (at higher forcings than the present case)yield light emission of the order of milliwatts per regash (Gaitan et al 1996 Hiller etal 1992)

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1695

0 5 10 15 20 25

02

04

06

08

10

- 843 ns

- 455 ns

- 299 ns

- 153 ns

- 41 ns

- 15 ns

0 ns - 5 ns

radial position (microm)

mol

ar f

ract

ion

Figure 4 Molar fraction of water vapour versus radius for selected times prior to the moment ofminimum radius (time = 0) in case II Note the development of the non-uniform composition asthe collapse accelerates toward minimum radius In the last 15 ns before minimum radius thecomposition changes by only a few per cent when viewed in the Lagrangian frame instead

There are two mechanisms that determine the excess vapour left behind at thecollapse shy nite-rate mass dinotusion and non-equilibrium phase change We examineeach mechanism and their interaction in turn

(i) Role of direg usion

The shy rst reason for excess water vapour remaining during the collapse is the rapidbubble collapse overwhelming slow mass dinotusion this was extensively discussedin SampS In SampS it was shown that there is a competition between two importanttime-scales the dynamic time-scale of the bubble motions and the mass dinotusiontime-scale These are given explicitly in dimensionless terms as

frac12 d yn = R= _R

frac12 d if = ReScR2

D MH2 O

ordm ReSc1

Rp

T

In the latter we made use of the fact that the dinotusion coe cient of water vapourin the gas mixture D M

H2 O scales as R3pT

The time-scales argument is clarishy ed upon examination of shy gure 4 which showsthe evolving bubble composition as a function of radius at selected times before thetime of minimum radius (case II) In the early part of the collapse the dynamic time-scale is longer than the dinotusion time-scale frac12 d yn frac34 frac12 d if this results in a uniformcomposition of the bubble (upper curves in shy gure 4) As the bubble accelerates into

Proc R Soc Lond A (2000)

1696 B D Storey and A J Szeri

the collapse frac12 d yn sup1 frac12 d if and the decrease in water fraction at the centre begins tolag the decrease in water fraction at the wall (middle curves of shy gure 4) Furtherinto the collapse frac12 d yn frac12 frac12 d if and the water vapour cannot dinotuse to the bubble wallon the time-scale of the rapid collapse This slow dinotusion results in a nearly shy xeddistribution of water vapour (three lowest curves of shy gure 4)

While the three lowest curves in shy gure 4 look quite dinoterent in this Eulerianrepresentation they dinoter by only a few per cent when viewed in the Lagrangianrepresentation (not shown) The curve at iexcl 15 ns is from a time when R = R0 andthe temperature at the centre is ca 750 K The curve at 0 ns is from the time ofminimum radius (015R0) when the temperature at the centre is 9700 K Consistentwith shy gure 3 over times when there are drastic changes in the bubble radius andtemperature the composition of the bubble is almost a material shy eld

(ii) Role of non-equilibrium phase change

Non-equilibrium phase change is the second mechanism by which vapour is trappedin the bubble interior during the collapse By adjusting the accommodation coe -cient one has control over the time-scale for the partial pressure of the water vapourto come to equilibrium with the saturation pressure at the interface The dimension-less condensation time-scale is

frac12 con d =R

frac14

r2 ordm MH2 O

9M0Tin t

When frac12 d yn frac34 frac12 con d the condensation is in quasi-equilibrium with respect to thebubble motions If frac12 d yn frac12 frac12 con d then the bubble collapse is so rapid that no masscan escape the bubble on the time-scale of the collapse

(iii) Interaction of the two mechanisms

It is the competition between the time-scales for mass dinotusion condensation andbubble dynamics that determines which mechanism traps water vapour in the bubbleOnce frac12 d yn frac12 frac12 d if or frac12 d yn frac12 frac12 con d the total amount of water vapour in the bubbleis shy xed Only one of these conditions needs to be met for vapour to be trapped inthe bubble Both mechanisms can however participate to determine the quantity oftrapped vapour

During the bubble collapse the condition frac12 d yn frac12 frac12 d if is reached well before frac12 d yn frac12frac12 con d in case II Therefore in this case (as well as the others presented herein) it isthe slow mass dinotusion that is responsible for trapping vapour in the bubble

For cases where slow mass dinotusion traps the vapour non-equilibrium phasechange helps to determine the amount trapped The accommodation coe cient rep-resents a resistance to condensation at the interface during collapse of the bubble Alower value of accommodation coe cient provides more resistance to phase changeclearly increasing the mass of water retained in the bubble at any time during thecollapse The importance of the non-equilibrium phase change can be estimated bycomparing frac12 d yn with frac12 con d at the time that frac12 d yn = frac12 d if The closer frac12 d yn is to frac12 con d the more important non-equilibrium phase change is in determining the amount ofwater trapped

Colussi et al (1998) developed a model including chemical reactions in a collapsingbubble using an accommodation coe cient of frac14 = 0001 but did not take into

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1697

account the shy nite rate of mass dinotusion The low value of frac14 which works to trapsignishy cant water vapour at collapse was necessary for their model to be consistentwith experimental sonochemistry data When such a low value for frac14 is used thecondition frac12 d yn frac12 frac12 con d is reached well before frac12 d yn frac12 frac12 d if (the opposite of our case)Such a low accommodation coe cient works to trap signishy cant amounts of watervapour in the bubble but in this case the hindered regux at the interface eliminatesmass transfer out of the bubble while the distribution of water vapour within isessentially uniform In the present work we argue that the signishy cant amounts ofwater vapour trapped in the bubble are a consequence of slow dinotusion as well asnon-equilibrium phase change

The temperature of the interface exceeds the critical point for a very brief time(from iexcl 340 ps to 1800 ps relative to the time of minimum radius) For the conditionswe explored the maximum thickness of the supercritical region is quite thin less than1 of the bubble radius ie ca 10 nm We neglect any changes in the liquid equationof state and transport properties when the interface exceeds the critical temperatureThis assumption is justishy ed based on the thinness of the region and the fact that thecritical temperature is reached very late in the collapse Any disturbances caused bythe supercritical interface have insumacr cient time to propagate into the bubble interioron the main collapse Although the evaporation equation is invalid for these shorttimes the total mass added to the bubble is negligible regardless of the magnitudeof evaporative mass regux The issue of the super-critical interface may require futureimprovements but is beyond the scope of the present work

As a shy nal note both thermal and pressure dinotusion also play a role in determiningthe distribution of water vapour during the collapse In the argonwater system thethermal-dinotusion coemacr cient is small and the molecules are of similar size hence pres-sure dinotusion is quite weak Little dinoterence is seen in calculations with and withoutthese other forms of mass dinotusion In a helium or xenon bubble in water thermaland pressure dinotusion are more pronounced and can cause a 1015 dinoterence inthe amount of trapped water vapour In the interest of brevity we will not discussthese results further here

(b) Gas-phase chemical reactions break apart H2O and reduce the temperature

Now we turn to an investigation of what transpires when chemical reactions aretaken into account (case III) We shall show that there is signishy cant time for reactionsto occur ie the time-scale for chemical reaction is similar to that of the bubblecollapse The primarily endothermic reactions absorb a signishy cant amount of energyand further reduce the peak temperatures from cases I and II The distribution ofspecies produced in the bubble is determined by dinotusive transport of water vapourprior to the collapse and heat transfer out of the bubble during the collapse Thesteep temperature gradient at the wall conshy nes the bulk of the chemical reactions tothe bubble interior

(i) Consequences of chemistry

The overall bubble dynamics of cases II and III are essentially the same with acompression ratio of 49 in case III (slightly higher than case II) A convenient pointof departure is to compare the temperature histories of the bubble centre aroundthe main collapse for the three cases we have considered (see shy gure 5) The inclusion

Proc R Soc Lond A (2000)

1698 B D Storey and A J Szeri

- 4 - 2 0 2 410

3

104

tem

pera

ture

(K

)

time relative to minimum radius (ns)

Figure 5 Temperature history of the centre of the bubble around the time of minimum radius(t = 0) in cases I II and III Note the reduction in temperature from case I ( ) to case II( ) as water is allowed to enter the bubble The chemical reactions added in case III (||)further reduce the temperatures

of water vapour (case II) signishy cantly reduces the peak temperature from case IThe decrease in temperature is due mainly to the decrease in the ratio of specishy cheats of the mixture which reduces the compression heating Upon comparison ofcases II and III one observes that the chemical reactions further reduce the peaktemperature due to the endothermic decomposition of the water vapour The centretemperature histories of cases II and III are nearly identical below 4000 K

It is important to understand that the temperature dinoterences in these threecases are not due to subtle dinoterences in bubble dynamics The compression ratio(Rm ax=Rm in ) of the three cases increases from case I to case II to case III whereasthe temperature decreases in this order This decrease in temperature occurs despitethe fact that a higher compression ratio will lead to more compression heating in thebubble collapse all other things being equal In all these cases there are no shocksinside the bubble Shocks will be considered in x 3 e

In shy gure 6 we show the history of the total number of molecules of the dinoterentspecies as a function of time on the same time-scale as in shy gure 5 The lowest curve isthe total amount of water vapour only The next curve is the total amount of watervapour plus the total amount of H2 Each curve represents the addition of one speciesas labelled until the uppermost curve denotes everything but argon We do not labelHO2 and H2O2 as they appear only in trace amounts One observes which speciesare most abundant during the time of the main collapse Note there is a relativelylarge amount of undissociated water considering the elevated temperatures This isprimarily a consequence of the high pressure suppressing the dissociation of H2O

From shy gures 5 and 6 one observes that the onset of signishy cant chemical activity in

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1699

- 4 - 2 0 2 4

109

1010

time relative to minimum radius (ns)

num

ber

of m

olec

ules

H2O

H2

O2

OH

O

H

Figure 6 Cumulative number of molecules of each species in case III The lowest curve is thetotal number of water molecules the next curve is total water + H2 and so on H2 O2 and HO2

are neglected in this plot as they appear in trace amounts

the bubble corresponds to a centre temperature of ca 4000 K Therefore the reducedtemperature of case III shown in shy gure 5 can be attributed to the endothermicchemical reactions Although the bubble dynamics are quite rapid the chemicalreactions keep pace as is shown by the similar time-scales of the changing quantitiesin shy gures 5 and 6 The rapid rate at which the reactions progress is due to the veryhigh density Recall that the rate of progress of each reaction is directly related tothe species concentration (mol cmiexcl3)

(ii) Dynamics and distribution of OH production

In sonochemistry applications the invasion of OH radicals into the liquid isthought to be responsible for much of the complex and useful chemistry that occursTherefore we shall focus on the behaviour of this species in particular In shy gure 7 weplot the total number of OH molecules contained in the bubble on two time-scalesover one acoustic cycle in shy gure 7a and around the time of collapse in shy gure 7b Theinitial concentrations of all species were selected so that the chemical production anddestruction balanced in one acoustic cycle The main collapse is associated with arapid creation of OH once the temperature exceeds 4000 K

Upon rebound the bubble rapidly cools and the OH and other products tend torecombine back into H2O These recombination reactions proceed more slowly thanthe dissociation reactions upon collapse The result is that the bubble is left with asuper-equilibrium amount of OH after it has cooled to ambient temperature Thissuper-equilibrium OH then slowly decays through the rest of the cycle with briefspurts of production and destruction at each of the after-bounces

Proc R Soc Lond A (2000)

1700 B D Storey and A J Szeri

0 10 20 30time (micros)

- 4 - 2 0 2 4

time relative to minimum radius (ns)

106

1010

104

106

108

108

num

ber

of O

H

radi

cals

nu

mbe

r of

OH

ra

dica

ls

Figure 7 Total number of OH molecules contained in the bubble versus time over one acousticcycle (a) and around the time of minimum radius (b) in case III The macrrst sharp peak in (a)is expanded in (b) Subsequent peaks in (a) correspond to local radius minima in the bubbledynamics The initial amounts of OH and all the other species were chosen for no net speciesproduction or destruction over an acoustic cycle

The distribution of OH in the bubble is characterized by a fairly uniform con-centration through much of the bubble interior and a very sharp gradient near thewall This sharp gradient is a result of heat transfer during the collapse Duringcompression conduction between the hot interior and the cool wall sets up a thintemperature boundary layer This boundary layer conshy nes the chemical reactions tothe interior away from the interface The cooler wall leads to a much lower concen-tration of OH at the interface than in the centre at the time of the main collapseIn shy gure 8 we show the molar concentration history of OH at the centre and at theinterface When the bubble is at minimum radius and the OH concentration at thecentre is at its maximum the concentration at the wall is three orders of magnitudelower

Because the chemistry time-scale is much faster than the dinotusion time-scale OHis produced where water vapour was trapped at high enough temperature The liquidinterface is insulated from the high concentration of OH generated deep inside in thebubble The only OH produced at the centre that reaches the bubble interface is theslowly decaying super-equilibrium OH that is left inside the bubble on re-expansion

(c) OH uptake by the liquid does not indeg uence the bubble interior

The inclusion of liquid uptake of OH (case IV) adds further complications wenow try to illuminate some of the main qualitative features We shall argue that OHuptake has little enotect on the gas dynamics and chemical production

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1701

- 4 - 2 0 2 410 - 10

10 - 8

10 - 6

10 - 4

10 - 2

time relative to minimum radius (ns)

conc

entr

atio

n of

OH

(m

ol c

m-3

)

Figure 8 Molar concentration of OH at the centre (||) and the interface ( ) versus timearound the point of minimum radius Note that the concentration at the centre is about threeorders of magnitude greater than at the interface at the point of minimum radius

Due to dinotering ideas on the proper value of the uptake coemacr cient the values areset at pound = 0001 and pound = 001 (cases IV(a) and IV(b)) consistent with the work ofothers to which we made reference earlier The bubble dynamics and temperatureproshy les for cases IV(a) and IV(b) are indistinguishable from those for case III Inshy gure 9 we show the total number of OH molecules in the bubble versus time ontwo dinoterent scales for cases III IV(a) and IV(b) The history for one acoustic cycleis plotted in shy gure 9a and the history concentrated at the main collapse is plottedin shy gure 9b

In the main expansion when the bubble is cool the OH resident in the bubblewill invade the liquid as can be seen by the decrease in the amount of OH at theend of the long expansion as the uptake increases When the bubble collapses thechemical production of OH from recently evaporated water rapidly overwhelms theOH uptake by the liquid As the bubble proceeds into the collapse the curves oftotal OH content for dinoterent uptakes become indistinguishable The total amountof OH in the bubble is dominated by the amount created in the interior away fromthe wall The total amount of OH lost to the liquid during a collapse is only afraction of the amount contained in the bubble Hence the peak amount of OH inthe bubble during the collapse is insensitive to the uptake coe cient of OH Similarstatements apply to uptake of other species

In shy gure 10 we show for the purposes of qualitative discussion the total massof OH exiting the bubble in cases IV(a) and IV(b) While the OH uptake coef-shy cient does not inreguence the total amount of OH contained in the bubble at themain collapse there are large dinoterences in the amount invading the liquid over

Proc R Soc Lond A (2000)

1702 B D Storey and A J Szeri

0 10 20 30

106

1010

102

time (micros)

- 4 - 2 0 2 4104

106

108

time relative to minimum radius (ns)

1010

(b)

(a)nu

mbe

r of

OH

ra

dica

ls

num

ber

of O

H

radi

cals

Figure 9 Comparison of total amount number of OH molecules in the bubble versus time overone acoustic cycle (a) and focused around the time of minimum radius (b) in cases III (||)IV(a) ( ) and IV(b) ( ) The uptake of OH plays a role during the slow expansionswhen the resident amount of OH left in the bubble is well above chemical equilibrium At thecollapse OH uptake does not indeg uence the total amount of OH production

a cycle Much of the OH uptake occurs during the main collapse and the subse-quent re-expansion when the concentration of OH at the wall is at its maximum(see shy gure 8) Further uptake occurs throughout the cycle at each of the after-bounces

(d ) Inclusion of water vapour alters the ereg ect of increased forcing

Next we shall compare the results of cases I II and III with all parameters shy xedexcept for the acoustic pressure amplitude which varies between 10 and 13 barWe that note this is experimentally awkward for SBSL because pressure amplitudeand ambient radius are dependent quantities through the mass exchange equilibriumof rectishy ed dinotusion (Hilgenfeldt et al 1996) The way we present these results isnonetheless useful as it demonstrates the importance of water vapour in the under-standing of sonoluminescence and sonochemistry

Because the bubble dynamics are altered by the inclusion of water vapour weuse the compression ratio as the distinguishing parameter In shy gure 11 we plot themaximum centre temperature versus the compression ratio for cases I II and III Thecalculations in shy gure 11 are devoid of strong shocks a matter to which we devotemore attention below If there are strong shocks then shy gure 11 will look the samein terms of a suitably averaged temperature rather than the centre temperature

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1703

0 10 20 30102

103

104

105

106

num

ber

of O

H

radi

cals

inva

ding

liqu

id

time (micros)

Figure 10 Comparison of the amount of OH absorbed by the liquid versus time for cases IV(a)(|| uptake coeplusmn cient pound = 0001) and IV(b) ( pound = 001) Most of the OH uptakeoccurs at the main collapse and the subsequent after-bounce

In case I the plot shows linear behaviour (on the loglog scale) One observes thatwhen water vapour is not taken into account stronger forcing yields more extremetemperatures

In case II the relationship between compression ratio and maximum temperature isvery dinoterent First we note that the amount of water trapped in the bubble increaseswith compression ratio If we revisit the water-trapping mechanism (x 3 a (iii)) thistrend is readily explained The increase in water-vapour content reduces the ratioof specishy c heats of the gasvapour mixture and hence reduces the compressionheating

At low compression ratios in case II the temperature increases with compressionratio in a manner very similar to case I At low compression ratios the amountof water at collapse is very small the bubble is nearly pure argon As the forcingis increased more water is trapped and the maximum temperature of the bubblebecomes nearly constant In this range of bubble dynamics there is an almost per-fectly compensating enotect (with respect to peak temperature) between the decreasein specishy c-heat ratio and the increase in compression Case III is very similar tocase II except that the plateau in temperature occurs at a lower temperature aconsequence of the endothermic chemical reactions

(e) Inclusion of water vapour promotes the formation of shocks

Water vapour is of great importance in the formation of shocks in very stronglyforced bubbles We shall brieregy consider a 60 m bubble forced with a pressureamplitude of 14 bar at a frequency of 206 kHz (Case A1 in Moss et al (1999))

Proc R Soc Lond A (2000)

1704 B D Storey and A J Szeri

10 1 10 2

10 3

10 4

10 5

Rmax R min

max

imum

tem

pera

ture

(K

)

Figure 11 Change in centre temperature versus compression ratio in cases I ( ) II ( curren )and III (+) In this plot all parameters were held constant except the pressure amplitudewhich was varied between 10 and 13 bar Case I shows a steady increase in temperature withcompression ratio Cases II and III reach a maximum temperature as the forcing is increased

When water vapour is neglected a compression wave is launched into the bubble oncollapse but the wave does not steepen into a shock When water vapour is takeninto account (with or without reactions) the compression wave steepens into a strongshock very near the centre of the bubble Vuong et al (1999) showed that the rateof steepening depends on the amount of compression heating before the compressionwave is launched

In very recent work Lin amp Szeri (2000) used the wavefront expansion technique todetermine the evolution of a compression or rarefaction wavefront that is propagatingspherically into a collapsing or expanding gas of non-uniform entropy This analyticalwork supports the idea that when the speed of sound is higher at the centre (dueto temperature and composition inhomogeneities) shock formation is delayed orprevented Furthermore this work demonstrated that the inward radial velocity alsohinders the steepening of compression waves into shocks

Consistent with the arguments made in Lin amp Szeri (2000) Vuong et al (1999)and Moss et al (1999) our results show that the decrease in the ratio of specishy cheats due to an increase in the water-vapour content can promote shock formationThe point we want to emphasize is that the question of shock formation cannot beundertaken without consideration of water vapour

With respect to total chemical production shocks are of negligible consequenceThe shock inreguences only about the inner 10 of the radius which is only 01 ofthe volume Therefore the increased production of OH at the centre does not anotectthe total amount produced in the bubble

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

1694 B D Storey and A J Szeri

0

4000

8000te

mpe

ratu

re (

K)

1

2

3

time relative to minimum radius (ns)

num

ber

of H

2O m

olec

ules

109

acute-

- 4 - 2 0 2 4

(b)

(a)

Figure 3 Temperature (a) and total number of water molecules contained in the bubble (b) versustime around the point of minimum radius (time = 0) in case II The temperature undergoesorder-of-magnitude changes on this time-scale while the total number of water molecules changesvery little

In shy gure 2 we show the total number of H2O molecules contained within thebubble over the acoustic cycle (case II) A large amount of water evaporates intothe bubble during the main expansion when the pressure is low Water condenses onthe bubble wall during the collapse as the gas pressure increases There is signishy cantwater vapour in the bubble at the main collapse ca 14 (molar basis) of the totalbubble contents is water vapour If the interface were at equilibrium and the speedof mass dinotusion were inshy nite there would be a negligible amount of water in thebubble on collapse

In shy gure 3 we show the temperature history of the centre and the total numberof water molecules in the bubble around the time of minimum radius (time = 0) incase II Over the 10 ns time-scale the total amount of water vapour contained in thebubble is nearly constant whereas the temperature changes by an order of magni-tude This recalls SampS in which it was shown that evolving mixture compositionswere essentially frozen on the time-scale of the temperature peak

The temperature peak may be related to the light emission The equation foroptical power given by Moss et al (1999) with a constant opacity provides an easyestimate for the optical power output as a function of time For case II this estimategives a light emission pulse width (FWHM) of 350 ps which is much shorter than theca 2000 ps temperature peak This optical-pulse-width estimate would be somewhatshortened if the temperature dependence of the opacity were taken into accountThe peak optical power is ca 01 mW although this number is very sensitive to theopacity Typical stable SBSL experiments (at higher forcings than the present case)yield light emission of the order of milliwatts per regash (Gaitan et al 1996 Hiller etal 1992)

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1695

0 5 10 15 20 25

02

04

06

08

10

- 843 ns

- 455 ns

- 299 ns

- 153 ns

- 41 ns

- 15 ns

0 ns - 5 ns

radial position (microm)

mol

ar f

ract

ion

Figure 4 Molar fraction of water vapour versus radius for selected times prior to the moment ofminimum radius (time = 0) in case II Note the development of the non-uniform composition asthe collapse accelerates toward minimum radius In the last 15 ns before minimum radius thecomposition changes by only a few per cent when viewed in the Lagrangian frame instead

There are two mechanisms that determine the excess vapour left behind at thecollapse shy nite-rate mass dinotusion and non-equilibrium phase change We examineeach mechanism and their interaction in turn

(i) Role of direg usion

The shy rst reason for excess water vapour remaining during the collapse is the rapidbubble collapse overwhelming slow mass dinotusion this was extensively discussedin SampS In SampS it was shown that there is a competition between two importanttime-scales the dynamic time-scale of the bubble motions and the mass dinotusiontime-scale These are given explicitly in dimensionless terms as

frac12 d yn = R= _R

frac12 d if = ReScR2

D MH2 O

ordm ReSc1

Rp

T

In the latter we made use of the fact that the dinotusion coe cient of water vapourin the gas mixture D M

H2 O scales as R3pT

The time-scales argument is clarishy ed upon examination of shy gure 4 which showsthe evolving bubble composition as a function of radius at selected times before thetime of minimum radius (case II) In the early part of the collapse the dynamic time-scale is longer than the dinotusion time-scale frac12 d yn frac34 frac12 d if this results in a uniformcomposition of the bubble (upper curves in shy gure 4) As the bubble accelerates into

Proc R Soc Lond A (2000)

1696 B D Storey and A J Szeri

the collapse frac12 d yn sup1 frac12 d if and the decrease in water fraction at the centre begins tolag the decrease in water fraction at the wall (middle curves of shy gure 4) Furtherinto the collapse frac12 d yn frac12 frac12 d if and the water vapour cannot dinotuse to the bubble wallon the time-scale of the rapid collapse This slow dinotusion results in a nearly shy xeddistribution of water vapour (three lowest curves of shy gure 4)

While the three lowest curves in shy gure 4 look quite dinoterent in this Eulerianrepresentation they dinoter by only a few per cent when viewed in the Lagrangianrepresentation (not shown) The curve at iexcl 15 ns is from a time when R = R0 andthe temperature at the centre is ca 750 K The curve at 0 ns is from the time ofminimum radius (015R0) when the temperature at the centre is 9700 K Consistentwith shy gure 3 over times when there are drastic changes in the bubble radius andtemperature the composition of the bubble is almost a material shy eld

(ii) Role of non-equilibrium phase change

Non-equilibrium phase change is the second mechanism by which vapour is trappedin the bubble interior during the collapse By adjusting the accommodation coe -cient one has control over the time-scale for the partial pressure of the water vapourto come to equilibrium with the saturation pressure at the interface The dimension-less condensation time-scale is

frac12 con d =R

frac14

r2 ordm MH2 O

9M0Tin t

When frac12 d yn frac34 frac12 con d the condensation is in quasi-equilibrium with respect to thebubble motions If frac12 d yn frac12 frac12 con d then the bubble collapse is so rapid that no masscan escape the bubble on the time-scale of the collapse

(iii) Interaction of the two mechanisms

It is the competition between the time-scales for mass dinotusion condensation andbubble dynamics that determines which mechanism traps water vapour in the bubbleOnce frac12 d yn frac12 frac12 d if or frac12 d yn frac12 frac12 con d the total amount of water vapour in the bubbleis shy xed Only one of these conditions needs to be met for vapour to be trapped inthe bubble Both mechanisms can however participate to determine the quantity oftrapped vapour

During the bubble collapse the condition frac12 d yn frac12 frac12 d if is reached well before frac12 d yn frac12frac12 con d in case II Therefore in this case (as well as the others presented herein) it isthe slow mass dinotusion that is responsible for trapping vapour in the bubble

For cases where slow mass dinotusion traps the vapour non-equilibrium phasechange helps to determine the amount trapped The accommodation coe cient rep-resents a resistance to condensation at the interface during collapse of the bubble Alower value of accommodation coe cient provides more resistance to phase changeclearly increasing the mass of water retained in the bubble at any time during thecollapse The importance of the non-equilibrium phase change can be estimated bycomparing frac12 d yn with frac12 con d at the time that frac12 d yn = frac12 d if The closer frac12 d yn is to frac12 con d the more important non-equilibrium phase change is in determining the amount ofwater trapped

Colussi et al (1998) developed a model including chemical reactions in a collapsingbubble using an accommodation coe cient of frac14 = 0001 but did not take into

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1697

account the shy nite rate of mass dinotusion The low value of frac14 which works to trapsignishy cant water vapour at collapse was necessary for their model to be consistentwith experimental sonochemistry data When such a low value for frac14 is used thecondition frac12 d yn frac12 frac12 con d is reached well before frac12 d yn frac12 frac12 d if (the opposite of our case)Such a low accommodation coe cient works to trap signishy cant amounts of watervapour in the bubble but in this case the hindered regux at the interface eliminatesmass transfer out of the bubble while the distribution of water vapour within isessentially uniform In the present work we argue that the signishy cant amounts ofwater vapour trapped in the bubble are a consequence of slow dinotusion as well asnon-equilibrium phase change

The temperature of the interface exceeds the critical point for a very brief time(from iexcl 340 ps to 1800 ps relative to the time of minimum radius) For the conditionswe explored the maximum thickness of the supercritical region is quite thin less than1 of the bubble radius ie ca 10 nm We neglect any changes in the liquid equationof state and transport properties when the interface exceeds the critical temperatureThis assumption is justishy ed based on the thinness of the region and the fact that thecritical temperature is reached very late in the collapse Any disturbances caused bythe supercritical interface have insumacr cient time to propagate into the bubble interioron the main collapse Although the evaporation equation is invalid for these shorttimes the total mass added to the bubble is negligible regardless of the magnitudeof evaporative mass regux The issue of the super-critical interface may require futureimprovements but is beyond the scope of the present work

As a shy nal note both thermal and pressure dinotusion also play a role in determiningthe distribution of water vapour during the collapse In the argonwater system thethermal-dinotusion coemacr cient is small and the molecules are of similar size hence pres-sure dinotusion is quite weak Little dinoterence is seen in calculations with and withoutthese other forms of mass dinotusion In a helium or xenon bubble in water thermaland pressure dinotusion are more pronounced and can cause a 1015 dinoterence inthe amount of trapped water vapour In the interest of brevity we will not discussthese results further here

(b) Gas-phase chemical reactions break apart H2O and reduce the temperature

Now we turn to an investigation of what transpires when chemical reactions aretaken into account (case III) We shall show that there is signishy cant time for reactionsto occur ie the time-scale for chemical reaction is similar to that of the bubblecollapse The primarily endothermic reactions absorb a signishy cant amount of energyand further reduce the peak temperatures from cases I and II The distribution ofspecies produced in the bubble is determined by dinotusive transport of water vapourprior to the collapse and heat transfer out of the bubble during the collapse Thesteep temperature gradient at the wall conshy nes the bulk of the chemical reactions tothe bubble interior

(i) Consequences of chemistry

The overall bubble dynamics of cases II and III are essentially the same with acompression ratio of 49 in case III (slightly higher than case II) A convenient pointof departure is to compare the temperature histories of the bubble centre aroundthe main collapse for the three cases we have considered (see shy gure 5) The inclusion

Proc R Soc Lond A (2000)

1698 B D Storey and A J Szeri

- 4 - 2 0 2 410

3

104

tem

pera

ture

(K

)

time relative to minimum radius (ns)

Figure 5 Temperature history of the centre of the bubble around the time of minimum radius(t = 0) in cases I II and III Note the reduction in temperature from case I ( ) to case II( ) as water is allowed to enter the bubble The chemical reactions added in case III (||)further reduce the temperatures

of water vapour (case II) signishy cantly reduces the peak temperature from case IThe decrease in temperature is due mainly to the decrease in the ratio of specishy cheats of the mixture which reduces the compression heating Upon comparison ofcases II and III one observes that the chemical reactions further reduce the peaktemperature due to the endothermic decomposition of the water vapour The centretemperature histories of cases II and III are nearly identical below 4000 K

It is important to understand that the temperature dinoterences in these threecases are not due to subtle dinoterences in bubble dynamics The compression ratio(Rm ax=Rm in ) of the three cases increases from case I to case II to case III whereasthe temperature decreases in this order This decrease in temperature occurs despitethe fact that a higher compression ratio will lead to more compression heating in thebubble collapse all other things being equal In all these cases there are no shocksinside the bubble Shocks will be considered in x 3 e

In shy gure 6 we show the history of the total number of molecules of the dinoterentspecies as a function of time on the same time-scale as in shy gure 5 The lowest curve isthe total amount of water vapour only The next curve is the total amount of watervapour plus the total amount of H2 Each curve represents the addition of one speciesas labelled until the uppermost curve denotes everything but argon We do not labelHO2 and H2O2 as they appear only in trace amounts One observes which speciesare most abundant during the time of the main collapse Note there is a relativelylarge amount of undissociated water considering the elevated temperatures This isprimarily a consequence of the high pressure suppressing the dissociation of H2O

From shy gures 5 and 6 one observes that the onset of signishy cant chemical activity in

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1699

- 4 - 2 0 2 4

109

1010

time relative to minimum radius (ns)

num

ber

of m

olec

ules

H2O

H2

O2

OH

O

H

Figure 6 Cumulative number of molecules of each species in case III The lowest curve is thetotal number of water molecules the next curve is total water + H2 and so on H2 O2 and HO2

are neglected in this plot as they appear in trace amounts

the bubble corresponds to a centre temperature of ca 4000 K Therefore the reducedtemperature of case III shown in shy gure 5 can be attributed to the endothermicchemical reactions Although the bubble dynamics are quite rapid the chemicalreactions keep pace as is shown by the similar time-scales of the changing quantitiesin shy gures 5 and 6 The rapid rate at which the reactions progress is due to the veryhigh density Recall that the rate of progress of each reaction is directly related tothe species concentration (mol cmiexcl3)

(ii) Dynamics and distribution of OH production

In sonochemistry applications the invasion of OH radicals into the liquid isthought to be responsible for much of the complex and useful chemistry that occursTherefore we shall focus on the behaviour of this species in particular In shy gure 7 weplot the total number of OH molecules contained in the bubble on two time-scalesover one acoustic cycle in shy gure 7a and around the time of collapse in shy gure 7b Theinitial concentrations of all species were selected so that the chemical production anddestruction balanced in one acoustic cycle The main collapse is associated with arapid creation of OH once the temperature exceeds 4000 K

Upon rebound the bubble rapidly cools and the OH and other products tend torecombine back into H2O These recombination reactions proceed more slowly thanthe dissociation reactions upon collapse The result is that the bubble is left with asuper-equilibrium amount of OH after it has cooled to ambient temperature Thissuper-equilibrium OH then slowly decays through the rest of the cycle with briefspurts of production and destruction at each of the after-bounces

Proc R Soc Lond A (2000)

1700 B D Storey and A J Szeri

0 10 20 30time (micros)

- 4 - 2 0 2 4

time relative to minimum radius (ns)

106

1010

104

106

108

108

num

ber

of O

H

radi

cals

nu

mbe

r of

OH

ra

dica

ls

Figure 7 Total number of OH molecules contained in the bubble versus time over one acousticcycle (a) and around the time of minimum radius (b) in case III The macrrst sharp peak in (a)is expanded in (b) Subsequent peaks in (a) correspond to local radius minima in the bubbledynamics The initial amounts of OH and all the other species were chosen for no net speciesproduction or destruction over an acoustic cycle

The distribution of OH in the bubble is characterized by a fairly uniform con-centration through much of the bubble interior and a very sharp gradient near thewall This sharp gradient is a result of heat transfer during the collapse Duringcompression conduction between the hot interior and the cool wall sets up a thintemperature boundary layer This boundary layer conshy nes the chemical reactions tothe interior away from the interface The cooler wall leads to a much lower concen-tration of OH at the interface than in the centre at the time of the main collapseIn shy gure 8 we show the molar concentration history of OH at the centre and at theinterface When the bubble is at minimum radius and the OH concentration at thecentre is at its maximum the concentration at the wall is three orders of magnitudelower

Because the chemistry time-scale is much faster than the dinotusion time-scale OHis produced where water vapour was trapped at high enough temperature The liquidinterface is insulated from the high concentration of OH generated deep inside in thebubble The only OH produced at the centre that reaches the bubble interface is theslowly decaying super-equilibrium OH that is left inside the bubble on re-expansion

(c) OH uptake by the liquid does not indeg uence the bubble interior

The inclusion of liquid uptake of OH (case IV) adds further complications wenow try to illuminate some of the main qualitative features We shall argue that OHuptake has little enotect on the gas dynamics and chemical production

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1701

- 4 - 2 0 2 410 - 10

10 - 8

10 - 6

10 - 4

10 - 2

time relative to minimum radius (ns)

conc

entr

atio

n of

OH

(m

ol c

m-3

)

Figure 8 Molar concentration of OH at the centre (||) and the interface ( ) versus timearound the point of minimum radius Note that the concentration at the centre is about threeorders of magnitude greater than at the interface at the point of minimum radius

Due to dinotering ideas on the proper value of the uptake coemacr cient the values areset at pound = 0001 and pound = 001 (cases IV(a) and IV(b)) consistent with the work ofothers to which we made reference earlier The bubble dynamics and temperatureproshy les for cases IV(a) and IV(b) are indistinguishable from those for case III Inshy gure 9 we show the total number of OH molecules in the bubble versus time ontwo dinoterent scales for cases III IV(a) and IV(b) The history for one acoustic cycleis plotted in shy gure 9a and the history concentrated at the main collapse is plottedin shy gure 9b

In the main expansion when the bubble is cool the OH resident in the bubblewill invade the liquid as can be seen by the decrease in the amount of OH at theend of the long expansion as the uptake increases When the bubble collapses thechemical production of OH from recently evaporated water rapidly overwhelms theOH uptake by the liquid As the bubble proceeds into the collapse the curves oftotal OH content for dinoterent uptakes become indistinguishable The total amountof OH in the bubble is dominated by the amount created in the interior away fromthe wall The total amount of OH lost to the liquid during a collapse is only afraction of the amount contained in the bubble Hence the peak amount of OH inthe bubble during the collapse is insensitive to the uptake coe cient of OH Similarstatements apply to uptake of other species

In shy gure 10 we show for the purposes of qualitative discussion the total massof OH exiting the bubble in cases IV(a) and IV(b) While the OH uptake coef-shy cient does not inreguence the total amount of OH contained in the bubble at themain collapse there are large dinoterences in the amount invading the liquid over

Proc R Soc Lond A (2000)

1702 B D Storey and A J Szeri

0 10 20 30

106

1010

102

time (micros)

- 4 - 2 0 2 4104

106

108

time relative to minimum radius (ns)

1010

(b)

(a)nu

mbe

r of

OH

ra

dica

ls

num

ber

of O

H

radi

cals

Figure 9 Comparison of total amount number of OH molecules in the bubble versus time overone acoustic cycle (a) and focused around the time of minimum radius (b) in cases III (||)IV(a) ( ) and IV(b) ( ) The uptake of OH plays a role during the slow expansionswhen the resident amount of OH left in the bubble is well above chemical equilibrium At thecollapse OH uptake does not indeg uence the total amount of OH production

a cycle Much of the OH uptake occurs during the main collapse and the subse-quent re-expansion when the concentration of OH at the wall is at its maximum(see shy gure 8) Further uptake occurs throughout the cycle at each of the after-bounces

(d ) Inclusion of water vapour alters the ereg ect of increased forcing

Next we shall compare the results of cases I II and III with all parameters shy xedexcept for the acoustic pressure amplitude which varies between 10 and 13 barWe that note this is experimentally awkward for SBSL because pressure amplitudeand ambient radius are dependent quantities through the mass exchange equilibriumof rectishy ed dinotusion (Hilgenfeldt et al 1996) The way we present these results isnonetheless useful as it demonstrates the importance of water vapour in the under-standing of sonoluminescence and sonochemistry

Because the bubble dynamics are altered by the inclusion of water vapour weuse the compression ratio as the distinguishing parameter In shy gure 11 we plot themaximum centre temperature versus the compression ratio for cases I II and III Thecalculations in shy gure 11 are devoid of strong shocks a matter to which we devotemore attention below If there are strong shocks then shy gure 11 will look the samein terms of a suitably averaged temperature rather than the centre temperature

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1703

0 10 20 30102

103

104

105

106

num

ber

of O

H

radi

cals

inva

ding

liqu

id

time (micros)

Figure 10 Comparison of the amount of OH absorbed by the liquid versus time for cases IV(a)(|| uptake coeplusmn cient pound = 0001) and IV(b) ( pound = 001) Most of the OH uptakeoccurs at the main collapse and the subsequent after-bounce

In case I the plot shows linear behaviour (on the loglog scale) One observes thatwhen water vapour is not taken into account stronger forcing yields more extremetemperatures

In case II the relationship between compression ratio and maximum temperature isvery dinoterent First we note that the amount of water trapped in the bubble increaseswith compression ratio If we revisit the water-trapping mechanism (x 3 a (iii)) thistrend is readily explained The increase in water-vapour content reduces the ratioof specishy c heats of the gasvapour mixture and hence reduces the compressionheating

At low compression ratios in case II the temperature increases with compressionratio in a manner very similar to case I At low compression ratios the amountof water at collapse is very small the bubble is nearly pure argon As the forcingis increased more water is trapped and the maximum temperature of the bubblebecomes nearly constant In this range of bubble dynamics there is an almost per-fectly compensating enotect (with respect to peak temperature) between the decreasein specishy c-heat ratio and the increase in compression Case III is very similar tocase II except that the plateau in temperature occurs at a lower temperature aconsequence of the endothermic chemical reactions

(e) Inclusion of water vapour promotes the formation of shocks

Water vapour is of great importance in the formation of shocks in very stronglyforced bubbles We shall brieregy consider a 60 m bubble forced with a pressureamplitude of 14 bar at a frequency of 206 kHz (Case A1 in Moss et al (1999))

Proc R Soc Lond A (2000)

1704 B D Storey and A J Szeri

10 1 10 2

10 3

10 4

10 5

Rmax R min

max

imum

tem

pera

ture

(K

)

Figure 11 Change in centre temperature versus compression ratio in cases I ( ) II ( curren )and III (+) In this plot all parameters were held constant except the pressure amplitudewhich was varied between 10 and 13 bar Case I shows a steady increase in temperature withcompression ratio Cases II and III reach a maximum temperature as the forcing is increased

When water vapour is neglected a compression wave is launched into the bubble oncollapse but the wave does not steepen into a shock When water vapour is takeninto account (with or without reactions) the compression wave steepens into a strongshock very near the centre of the bubble Vuong et al (1999) showed that the rateof steepening depends on the amount of compression heating before the compressionwave is launched

In very recent work Lin amp Szeri (2000) used the wavefront expansion technique todetermine the evolution of a compression or rarefaction wavefront that is propagatingspherically into a collapsing or expanding gas of non-uniform entropy This analyticalwork supports the idea that when the speed of sound is higher at the centre (dueto temperature and composition inhomogeneities) shock formation is delayed orprevented Furthermore this work demonstrated that the inward radial velocity alsohinders the steepening of compression waves into shocks

Consistent with the arguments made in Lin amp Szeri (2000) Vuong et al (1999)and Moss et al (1999) our results show that the decrease in the ratio of specishy cheats due to an increase in the water-vapour content can promote shock formationThe point we want to emphasize is that the question of shock formation cannot beundertaken without consideration of water vapour

With respect to total chemical production shocks are of negligible consequenceThe shock inreguences only about the inner 10 of the radius which is only 01 ofthe volume Therefore the increased production of OH at the centre does not anotectthe total amount produced in the bubble

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1695

0 5 10 15 20 25

02

04

06

08

10

- 843 ns

- 455 ns

- 299 ns

- 153 ns

- 41 ns

- 15 ns

0 ns - 5 ns

radial position (microm)

mol

ar f

ract

ion

Figure 4 Molar fraction of water vapour versus radius for selected times prior to the moment ofminimum radius (time = 0) in case II Note the development of the non-uniform composition asthe collapse accelerates toward minimum radius In the last 15 ns before minimum radius thecomposition changes by only a few per cent when viewed in the Lagrangian frame instead

There are two mechanisms that determine the excess vapour left behind at thecollapse shy nite-rate mass dinotusion and non-equilibrium phase change We examineeach mechanism and their interaction in turn

(i) Role of direg usion

The shy rst reason for excess water vapour remaining during the collapse is the rapidbubble collapse overwhelming slow mass dinotusion this was extensively discussedin SampS In SampS it was shown that there is a competition between two importanttime-scales the dynamic time-scale of the bubble motions and the mass dinotusiontime-scale These are given explicitly in dimensionless terms as

frac12 d yn = R= _R

frac12 d if = ReScR2

D MH2 O

ordm ReSc1

Rp

T

In the latter we made use of the fact that the dinotusion coe cient of water vapourin the gas mixture D M

H2 O scales as R3pT

The time-scales argument is clarishy ed upon examination of shy gure 4 which showsthe evolving bubble composition as a function of radius at selected times before thetime of minimum radius (case II) In the early part of the collapse the dynamic time-scale is longer than the dinotusion time-scale frac12 d yn frac34 frac12 d if this results in a uniformcomposition of the bubble (upper curves in shy gure 4) As the bubble accelerates into

Proc R Soc Lond A (2000)

1696 B D Storey and A J Szeri

the collapse frac12 d yn sup1 frac12 d if and the decrease in water fraction at the centre begins tolag the decrease in water fraction at the wall (middle curves of shy gure 4) Furtherinto the collapse frac12 d yn frac12 frac12 d if and the water vapour cannot dinotuse to the bubble wallon the time-scale of the rapid collapse This slow dinotusion results in a nearly shy xeddistribution of water vapour (three lowest curves of shy gure 4)

While the three lowest curves in shy gure 4 look quite dinoterent in this Eulerianrepresentation they dinoter by only a few per cent when viewed in the Lagrangianrepresentation (not shown) The curve at iexcl 15 ns is from a time when R = R0 andthe temperature at the centre is ca 750 K The curve at 0 ns is from the time ofminimum radius (015R0) when the temperature at the centre is 9700 K Consistentwith shy gure 3 over times when there are drastic changes in the bubble radius andtemperature the composition of the bubble is almost a material shy eld

(ii) Role of non-equilibrium phase change

Non-equilibrium phase change is the second mechanism by which vapour is trappedin the bubble interior during the collapse By adjusting the accommodation coe -cient one has control over the time-scale for the partial pressure of the water vapourto come to equilibrium with the saturation pressure at the interface The dimension-less condensation time-scale is

frac12 con d =R

frac14

r2 ordm MH2 O

9M0Tin t

When frac12 d yn frac34 frac12 con d the condensation is in quasi-equilibrium with respect to thebubble motions If frac12 d yn frac12 frac12 con d then the bubble collapse is so rapid that no masscan escape the bubble on the time-scale of the collapse

(iii) Interaction of the two mechanisms

It is the competition between the time-scales for mass dinotusion condensation andbubble dynamics that determines which mechanism traps water vapour in the bubbleOnce frac12 d yn frac12 frac12 d if or frac12 d yn frac12 frac12 con d the total amount of water vapour in the bubbleis shy xed Only one of these conditions needs to be met for vapour to be trapped inthe bubble Both mechanisms can however participate to determine the quantity oftrapped vapour

During the bubble collapse the condition frac12 d yn frac12 frac12 d if is reached well before frac12 d yn frac12frac12 con d in case II Therefore in this case (as well as the others presented herein) it isthe slow mass dinotusion that is responsible for trapping vapour in the bubble

For cases where slow mass dinotusion traps the vapour non-equilibrium phasechange helps to determine the amount trapped The accommodation coe cient rep-resents a resistance to condensation at the interface during collapse of the bubble Alower value of accommodation coe cient provides more resistance to phase changeclearly increasing the mass of water retained in the bubble at any time during thecollapse The importance of the non-equilibrium phase change can be estimated bycomparing frac12 d yn with frac12 con d at the time that frac12 d yn = frac12 d if The closer frac12 d yn is to frac12 con d the more important non-equilibrium phase change is in determining the amount ofwater trapped

Colussi et al (1998) developed a model including chemical reactions in a collapsingbubble using an accommodation coe cient of frac14 = 0001 but did not take into

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1697

account the shy nite rate of mass dinotusion The low value of frac14 which works to trapsignishy cant water vapour at collapse was necessary for their model to be consistentwith experimental sonochemistry data When such a low value for frac14 is used thecondition frac12 d yn frac12 frac12 con d is reached well before frac12 d yn frac12 frac12 d if (the opposite of our case)Such a low accommodation coe cient works to trap signishy cant amounts of watervapour in the bubble but in this case the hindered regux at the interface eliminatesmass transfer out of the bubble while the distribution of water vapour within isessentially uniform In the present work we argue that the signishy cant amounts ofwater vapour trapped in the bubble are a consequence of slow dinotusion as well asnon-equilibrium phase change

The temperature of the interface exceeds the critical point for a very brief time(from iexcl 340 ps to 1800 ps relative to the time of minimum radius) For the conditionswe explored the maximum thickness of the supercritical region is quite thin less than1 of the bubble radius ie ca 10 nm We neglect any changes in the liquid equationof state and transport properties when the interface exceeds the critical temperatureThis assumption is justishy ed based on the thinness of the region and the fact that thecritical temperature is reached very late in the collapse Any disturbances caused bythe supercritical interface have insumacr cient time to propagate into the bubble interioron the main collapse Although the evaporation equation is invalid for these shorttimes the total mass added to the bubble is negligible regardless of the magnitudeof evaporative mass regux The issue of the super-critical interface may require futureimprovements but is beyond the scope of the present work

As a shy nal note both thermal and pressure dinotusion also play a role in determiningthe distribution of water vapour during the collapse In the argonwater system thethermal-dinotusion coemacr cient is small and the molecules are of similar size hence pres-sure dinotusion is quite weak Little dinoterence is seen in calculations with and withoutthese other forms of mass dinotusion In a helium or xenon bubble in water thermaland pressure dinotusion are more pronounced and can cause a 1015 dinoterence inthe amount of trapped water vapour In the interest of brevity we will not discussthese results further here

(b) Gas-phase chemical reactions break apart H2O and reduce the temperature

Now we turn to an investigation of what transpires when chemical reactions aretaken into account (case III) We shall show that there is signishy cant time for reactionsto occur ie the time-scale for chemical reaction is similar to that of the bubblecollapse The primarily endothermic reactions absorb a signishy cant amount of energyand further reduce the peak temperatures from cases I and II The distribution ofspecies produced in the bubble is determined by dinotusive transport of water vapourprior to the collapse and heat transfer out of the bubble during the collapse Thesteep temperature gradient at the wall conshy nes the bulk of the chemical reactions tothe bubble interior

(i) Consequences of chemistry

The overall bubble dynamics of cases II and III are essentially the same with acompression ratio of 49 in case III (slightly higher than case II) A convenient pointof departure is to compare the temperature histories of the bubble centre aroundthe main collapse for the three cases we have considered (see shy gure 5) The inclusion

Proc R Soc Lond A (2000)

1698 B D Storey and A J Szeri

- 4 - 2 0 2 410

3

104

tem

pera

ture

(K

)

time relative to minimum radius (ns)

Figure 5 Temperature history of the centre of the bubble around the time of minimum radius(t = 0) in cases I II and III Note the reduction in temperature from case I ( ) to case II( ) as water is allowed to enter the bubble The chemical reactions added in case III (||)further reduce the temperatures

of water vapour (case II) signishy cantly reduces the peak temperature from case IThe decrease in temperature is due mainly to the decrease in the ratio of specishy cheats of the mixture which reduces the compression heating Upon comparison ofcases II and III one observes that the chemical reactions further reduce the peaktemperature due to the endothermic decomposition of the water vapour The centretemperature histories of cases II and III are nearly identical below 4000 K

It is important to understand that the temperature dinoterences in these threecases are not due to subtle dinoterences in bubble dynamics The compression ratio(Rm ax=Rm in ) of the three cases increases from case I to case II to case III whereasthe temperature decreases in this order This decrease in temperature occurs despitethe fact that a higher compression ratio will lead to more compression heating in thebubble collapse all other things being equal In all these cases there are no shocksinside the bubble Shocks will be considered in x 3 e

In shy gure 6 we show the history of the total number of molecules of the dinoterentspecies as a function of time on the same time-scale as in shy gure 5 The lowest curve isthe total amount of water vapour only The next curve is the total amount of watervapour plus the total amount of H2 Each curve represents the addition of one speciesas labelled until the uppermost curve denotes everything but argon We do not labelHO2 and H2O2 as they appear only in trace amounts One observes which speciesare most abundant during the time of the main collapse Note there is a relativelylarge amount of undissociated water considering the elevated temperatures This isprimarily a consequence of the high pressure suppressing the dissociation of H2O

From shy gures 5 and 6 one observes that the onset of signishy cant chemical activity in

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1699

- 4 - 2 0 2 4

109

1010

time relative to minimum radius (ns)

num

ber

of m

olec

ules

H2O

H2

O2

OH

O

H

Figure 6 Cumulative number of molecules of each species in case III The lowest curve is thetotal number of water molecules the next curve is total water + H2 and so on H2 O2 and HO2

are neglected in this plot as they appear in trace amounts

the bubble corresponds to a centre temperature of ca 4000 K Therefore the reducedtemperature of case III shown in shy gure 5 can be attributed to the endothermicchemical reactions Although the bubble dynamics are quite rapid the chemicalreactions keep pace as is shown by the similar time-scales of the changing quantitiesin shy gures 5 and 6 The rapid rate at which the reactions progress is due to the veryhigh density Recall that the rate of progress of each reaction is directly related tothe species concentration (mol cmiexcl3)

(ii) Dynamics and distribution of OH production

In sonochemistry applications the invasion of OH radicals into the liquid isthought to be responsible for much of the complex and useful chemistry that occursTherefore we shall focus on the behaviour of this species in particular In shy gure 7 weplot the total number of OH molecules contained in the bubble on two time-scalesover one acoustic cycle in shy gure 7a and around the time of collapse in shy gure 7b Theinitial concentrations of all species were selected so that the chemical production anddestruction balanced in one acoustic cycle The main collapse is associated with arapid creation of OH once the temperature exceeds 4000 K

Upon rebound the bubble rapidly cools and the OH and other products tend torecombine back into H2O These recombination reactions proceed more slowly thanthe dissociation reactions upon collapse The result is that the bubble is left with asuper-equilibrium amount of OH after it has cooled to ambient temperature Thissuper-equilibrium OH then slowly decays through the rest of the cycle with briefspurts of production and destruction at each of the after-bounces

Proc R Soc Lond A (2000)

1700 B D Storey and A J Szeri

0 10 20 30time (micros)

- 4 - 2 0 2 4

time relative to minimum radius (ns)

106

1010

104

106

108

108

num

ber

of O

H

radi

cals

nu

mbe

r of

OH

ra

dica

ls

Figure 7 Total number of OH molecules contained in the bubble versus time over one acousticcycle (a) and around the time of minimum radius (b) in case III The macrrst sharp peak in (a)is expanded in (b) Subsequent peaks in (a) correspond to local radius minima in the bubbledynamics The initial amounts of OH and all the other species were chosen for no net speciesproduction or destruction over an acoustic cycle

The distribution of OH in the bubble is characterized by a fairly uniform con-centration through much of the bubble interior and a very sharp gradient near thewall This sharp gradient is a result of heat transfer during the collapse Duringcompression conduction between the hot interior and the cool wall sets up a thintemperature boundary layer This boundary layer conshy nes the chemical reactions tothe interior away from the interface The cooler wall leads to a much lower concen-tration of OH at the interface than in the centre at the time of the main collapseIn shy gure 8 we show the molar concentration history of OH at the centre and at theinterface When the bubble is at minimum radius and the OH concentration at thecentre is at its maximum the concentration at the wall is three orders of magnitudelower

Because the chemistry time-scale is much faster than the dinotusion time-scale OHis produced where water vapour was trapped at high enough temperature The liquidinterface is insulated from the high concentration of OH generated deep inside in thebubble The only OH produced at the centre that reaches the bubble interface is theslowly decaying super-equilibrium OH that is left inside the bubble on re-expansion

(c) OH uptake by the liquid does not indeg uence the bubble interior

The inclusion of liquid uptake of OH (case IV) adds further complications wenow try to illuminate some of the main qualitative features We shall argue that OHuptake has little enotect on the gas dynamics and chemical production

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1701

- 4 - 2 0 2 410 - 10

10 - 8

10 - 6

10 - 4

10 - 2

time relative to minimum radius (ns)

conc

entr

atio

n of

OH

(m

ol c

m-3

)

Figure 8 Molar concentration of OH at the centre (||) and the interface ( ) versus timearound the point of minimum radius Note that the concentration at the centre is about threeorders of magnitude greater than at the interface at the point of minimum radius

Due to dinotering ideas on the proper value of the uptake coemacr cient the values areset at pound = 0001 and pound = 001 (cases IV(a) and IV(b)) consistent with the work ofothers to which we made reference earlier The bubble dynamics and temperatureproshy les for cases IV(a) and IV(b) are indistinguishable from those for case III Inshy gure 9 we show the total number of OH molecules in the bubble versus time ontwo dinoterent scales for cases III IV(a) and IV(b) The history for one acoustic cycleis plotted in shy gure 9a and the history concentrated at the main collapse is plottedin shy gure 9b

In the main expansion when the bubble is cool the OH resident in the bubblewill invade the liquid as can be seen by the decrease in the amount of OH at theend of the long expansion as the uptake increases When the bubble collapses thechemical production of OH from recently evaporated water rapidly overwhelms theOH uptake by the liquid As the bubble proceeds into the collapse the curves oftotal OH content for dinoterent uptakes become indistinguishable The total amountof OH in the bubble is dominated by the amount created in the interior away fromthe wall The total amount of OH lost to the liquid during a collapse is only afraction of the amount contained in the bubble Hence the peak amount of OH inthe bubble during the collapse is insensitive to the uptake coe cient of OH Similarstatements apply to uptake of other species

In shy gure 10 we show for the purposes of qualitative discussion the total massof OH exiting the bubble in cases IV(a) and IV(b) While the OH uptake coef-shy cient does not inreguence the total amount of OH contained in the bubble at themain collapse there are large dinoterences in the amount invading the liquid over

Proc R Soc Lond A (2000)

1702 B D Storey and A J Szeri

0 10 20 30

106

1010

102

time (micros)

- 4 - 2 0 2 4104

106

108

time relative to minimum radius (ns)

1010

(b)

(a)nu

mbe

r of

OH

ra

dica

ls

num

ber

of O

H

radi

cals

Figure 9 Comparison of total amount number of OH molecules in the bubble versus time overone acoustic cycle (a) and focused around the time of minimum radius (b) in cases III (||)IV(a) ( ) and IV(b) ( ) The uptake of OH plays a role during the slow expansionswhen the resident amount of OH left in the bubble is well above chemical equilibrium At thecollapse OH uptake does not indeg uence the total amount of OH production

a cycle Much of the OH uptake occurs during the main collapse and the subse-quent re-expansion when the concentration of OH at the wall is at its maximum(see shy gure 8) Further uptake occurs throughout the cycle at each of the after-bounces

(d ) Inclusion of water vapour alters the ereg ect of increased forcing

Next we shall compare the results of cases I II and III with all parameters shy xedexcept for the acoustic pressure amplitude which varies between 10 and 13 barWe that note this is experimentally awkward for SBSL because pressure amplitudeand ambient radius are dependent quantities through the mass exchange equilibriumof rectishy ed dinotusion (Hilgenfeldt et al 1996) The way we present these results isnonetheless useful as it demonstrates the importance of water vapour in the under-standing of sonoluminescence and sonochemistry

Because the bubble dynamics are altered by the inclusion of water vapour weuse the compression ratio as the distinguishing parameter In shy gure 11 we plot themaximum centre temperature versus the compression ratio for cases I II and III Thecalculations in shy gure 11 are devoid of strong shocks a matter to which we devotemore attention below If there are strong shocks then shy gure 11 will look the samein terms of a suitably averaged temperature rather than the centre temperature

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1703

0 10 20 30102

103

104

105

106

num

ber

of O

H

radi

cals

inva

ding

liqu

id

time (micros)

Figure 10 Comparison of the amount of OH absorbed by the liquid versus time for cases IV(a)(|| uptake coeplusmn cient pound = 0001) and IV(b) ( pound = 001) Most of the OH uptakeoccurs at the main collapse and the subsequent after-bounce

In case I the plot shows linear behaviour (on the loglog scale) One observes thatwhen water vapour is not taken into account stronger forcing yields more extremetemperatures

In case II the relationship between compression ratio and maximum temperature isvery dinoterent First we note that the amount of water trapped in the bubble increaseswith compression ratio If we revisit the water-trapping mechanism (x 3 a (iii)) thistrend is readily explained The increase in water-vapour content reduces the ratioof specishy c heats of the gasvapour mixture and hence reduces the compressionheating

At low compression ratios in case II the temperature increases with compressionratio in a manner very similar to case I At low compression ratios the amountof water at collapse is very small the bubble is nearly pure argon As the forcingis increased more water is trapped and the maximum temperature of the bubblebecomes nearly constant In this range of bubble dynamics there is an almost per-fectly compensating enotect (with respect to peak temperature) between the decreasein specishy c-heat ratio and the increase in compression Case III is very similar tocase II except that the plateau in temperature occurs at a lower temperature aconsequence of the endothermic chemical reactions

(e) Inclusion of water vapour promotes the formation of shocks

Water vapour is of great importance in the formation of shocks in very stronglyforced bubbles We shall brieregy consider a 60 m bubble forced with a pressureamplitude of 14 bar at a frequency of 206 kHz (Case A1 in Moss et al (1999))

Proc R Soc Lond A (2000)

1704 B D Storey and A J Szeri

10 1 10 2

10 3

10 4

10 5

Rmax R min

max

imum

tem

pera

ture

(K

)

Figure 11 Change in centre temperature versus compression ratio in cases I ( ) II ( curren )and III (+) In this plot all parameters were held constant except the pressure amplitudewhich was varied between 10 and 13 bar Case I shows a steady increase in temperature withcompression ratio Cases II and III reach a maximum temperature as the forcing is increased

When water vapour is neglected a compression wave is launched into the bubble oncollapse but the wave does not steepen into a shock When water vapour is takeninto account (with or without reactions) the compression wave steepens into a strongshock very near the centre of the bubble Vuong et al (1999) showed that the rateof steepening depends on the amount of compression heating before the compressionwave is launched

In very recent work Lin amp Szeri (2000) used the wavefront expansion technique todetermine the evolution of a compression or rarefaction wavefront that is propagatingspherically into a collapsing or expanding gas of non-uniform entropy This analyticalwork supports the idea that when the speed of sound is higher at the centre (dueto temperature and composition inhomogeneities) shock formation is delayed orprevented Furthermore this work demonstrated that the inward radial velocity alsohinders the steepening of compression waves into shocks

Consistent with the arguments made in Lin amp Szeri (2000) Vuong et al (1999)and Moss et al (1999) our results show that the decrease in the ratio of specishy cheats due to an increase in the water-vapour content can promote shock formationThe point we want to emphasize is that the question of shock formation cannot beundertaken without consideration of water vapour

With respect to total chemical production shocks are of negligible consequenceThe shock inreguences only about the inner 10 of the radius which is only 01 ofthe volume Therefore the increased production of OH at the centre does not anotectthe total amount produced in the bubble

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

1696 B D Storey and A J Szeri

the collapse frac12 d yn sup1 frac12 d if and the decrease in water fraction at the centre begins tolag the decrease in water fraction at the wall (middle curves of shy gure 4) Furtherinto the collapse frac12 d yn frac12 frac12 d if and the water vapour cannot dinotuse to the bubble wallon the time-scale of the rapid collapse This slow dinotusion results in a nearly shy xeddistribution of water vapour (three lowest curves of shy gure 4)

While the three lowest curves in shy gure 4 look quite dinoterent in this Eulerianrepresentation they dinoter by only a few per cent when viewed in the Lagrangianrepresentation (not shown) The curve at iexcl 15 ns is from a time when R = R0 andthe temperature at the centre is ca 750 K The curve at 0 ns is from the time ofminimum radius (015R0) when the temperature at the centre is 9700 K Consistentwith shy gure 3 over times when there are drastic changes in the bubble radius andtemperature the composition of the bubble is almost a material shy eld

(ii) Role of non-equilibrium phase change

Non-equilibrium phase change is the second mechanism by which vapour is trappedin the bubble interior during the collapse By adjusting the accommodation coe -cient one has control over the time-scale for the partial pressure of the water vapourto come to equilibrium with the saturation pressure at the interface The dimension-less condensation time-scale is

frac12 con d =R

frac14

r2 ordm MH2 O

9M0Tin t

When frac12 d yn frac34 frac12 con d the condensation is in quasi-equilibrium with respect to thebubble motions If frac12 d yn frac12 frac12 con d then the bubble collapse is so rapid that no masscan escape the bubble on the time-scale of the collapse

(iii) Interaction of the two mechanisms

It is the competition between the time-scales for mass dinotusion condensation andbubble dynamics that determines which mechanism traps water vapour in the bubbleOnce frac12 d yn frac12 frac12 d if or frac12 d yn frac12 frac12 con d the total amount of water vapour in the bubbleis shy xed Only one of these conditions needs to be met for vapour to be trapped inthe bubble Both mechanisms can however participate to determine the quantity oftrapped vapour

During the bubble collapse the condition frac12 d yn frac12 frac12 d if is reached well before frac12 d yn frac12frac12 con d in case II Therefore in this case (as well as the others presented herein) it isthe slow mass dinotusion that is responsible for trapping vapour in the bubble

For cases where slow mass dinotusion traps the vapour non-equilibrium phasechange helps to determine the amount trapped The accommodation coe cient rep-resents a resistance to condensation at the interface during collapse of the bubble Alower value of accommodation coe cient provides more resistance to phase changeclearly increasing the mass of water retained in the bubble at any time during thecollapse The importance of the non-equilibrium phase change can be estimated bycomparing frac12 d yn with frac12 con d at the time that frac12 d yn = frac12 d if The closer frac12 d yn is to frac12 con d the more important non-equilibrium phase change is in determining the amount ofwater trapped

Colussi et al (1998) developed a model including chemical reactions in a collapsingbubble using an accommodation coe cient of frac14 = 0001 but did not take into

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1697

account the shy nite rate of mass dinotusion The low value of frac14 which works to trapsignishy cant water vapour at collapse was necessary for their model to be consistentwith experimental sonochemistry data When such a low value for frac14 is used thecondition frac12 d yn frac12 frac12 con d is reached well before frac12 d yn frac12 frac12 d if (the opposite of our case)Such a low accommodation coe cient works to trap signishy cant amounts of watervapour in the bubble but in this case the hindered regux at the interface eliminatesmass transfer out of the bubble while the distribution of water vapour within isessentially uniform In the present work we argue that the signishy cant amounts ofwater vapour trapped in the bubble are a consequence of slow dinotusion as well asnon-equilibrium phase change

The temperature of the interface exceeds the critical point for a very brief time(from iexcl 340 ps to 1800 ps relative to the time of minimum radius) For the conditionswe explored the maximum thickness of the supercritical region is quite thin less than1 of the bubble radius ie ca 10 nm We neglect any changes in the liquid equationof state and transport properties when the interface exceeds the critical temperatureThis assumption is justishy ed based on the thinness of the region and the fact that thecritical temperature is reached very late in the collapse Any disturbances caused bythe supercritical interface have insumacr cient time to propagate into the bubble interioron the main collapse Although the evaporation equation is invalid for these shorttimes the total mass added to the bubble is negligible regardless of the magnitudeof evaporative mass regux The issue of the super-critical interface may require futureimprovements but is beyond the scope of the present work

As a shy nal note both thermal and pressure dinotusion also play a role in determiningthe distribution of water vapour during the collapse In the argonwater system thethermal-dinotusion coemacr cient is small and the molecules are of similar size hence pres-sure dinotusion is quite weak Little dinoterence is seen in calculations with and withoutthese other forms of mass dinotusion In a helium or xenon bubble in water thermaland pressure dinotusion are more pronounced and can cause a 1015 dinoterence inthe amount of trapped water vapour In the interest of brevity we will not discussthese results further here

(b) Gas-phase chemical reactions break apart H2O and reduce the temperature

Now we turn to an investigation of what transpires when chemical reactions aretaken into account (case III) We shall show that there is signishy cant time for reactionsto occur ie the time-scale for chemical reaction is similar to that of the bubblecollapse The primarily endothermic reactions absorb a signishy cant amount of energyand further reduce the peak temperatures from cases I and II The distribution ofspecies produced in the bubble is determined by dinotusive transport of water vapourprior to the collapse and heat transfer out of the bubble during the collapse Thesteep temperature gradient at the wall conshy nes the bulk of the chemical reactions tothe bubble interior

(i) Consequences of chemistry

The overall bubble dynamics of cases II and III are essentially the same with acompression ratio of 49 in case III (slightly higher than case II) A convenient pointof departure is to compare the temperature histories of the bubble centre aroundthe main collapse for the three cases we have considered (see shy gure 5) The inclusion

Proc R Soc Lond A (2000)

1698 B D Storey and A J Szeri

- 4 - 2 0 2 410

3

104

tem

pera

ture

(K

)

time relative to minimum radius (ns)

Figure 5 Temperature history of the centre of the bubble around the time of minimum radius(t = 0) in cases I II and III Note the reduction in temperature from case I ( ) to case II( ) as water is allowed to enter the bubble The chemical reactions added in case III (||)further reduce the temperatures

of water vapour (case II) signishy cantly reduces the peak temperature from case IThe decrease in temperature is due mainly to the decrease in the ratio of specishy cheats of the mixture which reduces the compression heating Upon comparison ofcases II and III one observes that the chemical reactions further reduce the peaktemperature due to the endothermic decomposition of the water vapour The centretemperature histories of cases II and III are nearly identical below 4000 K

It is important to understand that the temperature dinoterences in these threecases are not due to subtle dinoterences in bubble dynamics The compression ratio(Rm ax=Rm in ) of the three cases increases from case I to case II to case III whereasthe temperature decreases in this order This decrease in temperature occurs despitethe fact that a higher compression ratio will lead to more compression heating in thebubble collapse all other things being equal In all these cases there are no shocksinside the bubble Shocks will be considered in x 3 e

In shy gure 6 we show the history of the total number of molecules of the dinoterentspecies as a function of time on the same time-scale as in shy gure 5 The lowest curve isthe total amount of water vapour only The next curve is the total amount of watervapour plus the total amount of H2 Each curve represents the addition of one speciesas labelled until the uppermost curve denotes everything but argon We do not labelHO2 and H2O2 as they appear only in trace amounts One observes which speciesare most abundant during the time of the main collapse Note there is a relativelylarge amount of undissociated water considering the elevated temperatures This isprimarily a consequence of the high pressure suppressing the dissociation of H2O

From shy gures 5 and 6 one observes that the onset of signishy cant chemical activity in

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1699

- 4 - 2 0 2 4

109

1010

time relative to minimum radius (ns)

num

ber

of m

olec

ules

H2O

H2

O2

OH

O

H

Figure 6 Cumulative number of molecules of each species in case III The lowest curve is thetotal number of water molecules the next curve is total water + H2 and so on H2 O2 and HO2

are neglected in this plot as they appear in trace amounts

the bubble corresponds to a centre temperature of ca 4000 K Therefore the reducedtemperature of case III shown in shy gure 5 can be attributed to the endothermicchemical reactions Although the bubble dynamics are quite rapid the chemicalreactions keep pace as is shown by the similar time-scales of the changing quantitiesin shy gures 5 and 6 The rapid rate at which the reactions progress is due to the veryhigh density Recall that the rate of progress of each reaction is directly related tothe species concentration (mol cmiexcl3)

(ii) Dynamics and distribution of OH production

In sonochemistry applications the invasion of OH radicals into the liquid isthought to be responsible for much of the complex and useful chemistry that occursTherefore we shall focus on the behaviour of this species in particular In shy gure 7 weplot the total number of OH molecules contained in the bubble on two time-scalesover one acoustic cycle in shy gure 7a and around the time of collapse in shy gure 7b Theinitial concentrations of all species were selected so that the chemical production anddestruction balanced in one acoustic cycle The main collapse is associated with arapid creation of OH once the temperature exceeds 4000 K

Upon rebound the bubble rapidly cools and the OH and other products tend torecombine back into H2O These recombination reactions proceed more slowly thanthe dissociation reactions upon collapse The result is that the bubble is left with asuper-equilibrium amount of OH after it has cooled to ambient temperature Thissuper-equilibrium OH then slowly decays through the rest of the cycle with briefspurts of production and destruction at each of the after-bounces

Proc R Soc Lond A (2000)

1700 B D Storey and A J Szeri

0 10 20 30time (micros)

- 4 - 2 0 2 4

time relative to minimum radius (ns)

106

1010

104

106

108

108

num

ber

of O

H

radi

cals

nu

mbe

r of

OH

ra

dica

ls

Figure 7 Total number of OH molecules contained in the bubble versus time over one acousticcycle (a) and around the time of minimum radius (b) in case III The macrrst sharp peak in (a)is expanded in (b) Subsequent peaks in (a) correspond to local radius minima in the bubbledynamics The initial amounts of OH and all the other species were chosen for no net speciesproduction or destruction over an acoustic cycle

The distribution of OH in the bubble is characterized by a fairly uniform con-centration through much of the bubble interior and a very sharp gradient near thewall This sharp gradient is a result of heat transfer during the collapse Duringcompression conduction between the hot interior and the cool wall sets up a thintemperature boundary layer This boundary layer conshy nes the chemical reactions tothe interior away from the interface The cooler wall leads to a much lower concen-tration of OH at the interface than in the centre at the time of the main collapseIn shy gure 8 we show the molar concentration history of OH at the centre and at theinterface When the bubble is at minimum radius and the OH concentration at thecentre is at its maximum the concentration at the wall is three orders of magnitudelower

Because the chemistry time-scale is much faster than the dinotusion time-scale OHis produced where water vapour was trapped at high enough temperature The liquidinterface is insulated from the high concentration of OH generated deep inside in thebubble The only OH produced at the centre that reaches the bubble interface is theslowly decaying super-equilibrium OH that is left inside the bubble on re-expansion

(c) OH uptake by the liquid does not indeg uence the bubble interior

The inclusion of liquid uptake of OH (case IV) adds further complications wenow try to illuminate some of the main qualitative features We shall argue that OHuptake has little enotect on the gas dynamics and chemical production

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1701

- 4 - 2 0 2 410 - 10

10 - 8

10 - 6

10 - 4

10 - 2

time relative to minimum radius (ns)

conc

entr

atio

n of

OH

(m

ol c

m-3

)

Figure 8 Molar concentration of OH at the centre (||) and the interface ( ) versus timearound the point of minimum radius Note that the concentration at the centre is about threeorders of magnitude greater than at the interface at the point of minimum radius

Due to dinotering ideas on the proper value of the uptake coemacr cient the values areset at pound = 0001 and pound = 001 (cases IV(a) and IV(b)) consistent with the work ofothers to which we made reference earlier The bubble dynamics and temperatureproshy les for cases IV(a) and IV(b) are indistinguishable from those for case III Inshy gure 9 we show the total number of OH molecules in the bubble versus time ontwo dinoterent scales for cases III IV(a) and IV(b) The history for one acoustic cycleis plotted in shy gure 9a and the history concentrated at the main collapse is plottedin shy gure 9b

In the main expansion when the bubble is cool the OH resident in the bubblewill invade the liquid as can be seen by the decrease in the amount of OH at theend of the long expansion as the uptake increases When the bubble collapses thechemical production of OH from recently evaporated water rapidly overwhelms theOH uptake by the liquid As the bubble proceeds into the collapse the curves oftotal OH content for dinoterent uptakes become indistinguishable The total amountof OH in the bubble is dominated by the amount created in the interior away fromthe wall The total amount of OH lost to the liquid during a collapse is only afraction of the amount contained in the bubble Hence the peak amount of OH inthe bubble during the collapse is insensitive to the uptake coe cient of OH Similarstatements apply to uptake of other species

In shy gure 10 we show for the purposes of qualitative discussion the total massof OH exiting the bubble in cases IV(a) and IV(b) While the OH uptake coef-shy cient does not inreguence the total amount of OH contained in the bubble at themain collapse there are large dinoterences in the amount invading the liquid over

Proc R Soc Lond A (2000)

1702 B D Storey and A J Szeri

0 10 20 30

106

1010

102

time (micros)

- 4 - 2 0 2 4104

106

108

time relative to minimum radius (ns)

1010

(b)

(a)nu

mbe

r of

OH

ra

dica

ls

num

ber

of O

H

radi

cals

Figure 9 Comparison of total amount number of OH molecules in the bubble versus time overone acoustic cycle (a) and focused around the time of minimum radius (b) in cases III (||)IV(a) ( ) and IV(b) ( ) The uptake of OH plays a role during the slow expansionswhen the resident amount of OH left in the bubble is well above chemical equilibrium At thecollapse OH uptake does not indeg uence the total amount of OH production

a cycle Much of the OH uptake occurs during the main collapse and the subse-quent re-expansion when the concentration of OH at the wall is at its maximum(see shy gure 8) Further uptake occurs throughout the cycle at each of the after-bounces

(d ) Inclusion of water vapour alters the ereg ect of increased forcing

Next we shall compare the results of cases I II and III with all parameters shy xedexcept for the acoustic pressure amplitude which varies between 10 and 13 barWe that note this is experimentally awkward for SBSL because pressure amplitudeand ambient radius are dependent quantities through the mass exchange equilibriumof rectishy ed dinotusion (Hilgenfeldt et al 1996) The way we present these results isnonetheless useful as it demonstrates the importance of water vapour in the under-standing of sonoluminescence and sonochemistry

Because the bubble dynamics are altered by the inclusion of water vapour weuse the compression ratio as the distinguishing parameter In shy gure 11 we plot themaximum centre temperature versus the compression ratio for cases I II and III Thecalculations in shy gure 11 are devoid of strong shocks a matter to which we devotemore attention below If there are strong shocks then shy gure 11 will look the samein terms of a suitably averaged temperature rather than the centre temperature

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1703

0 10 20 30102

103

104

105

106

num

ber

of O

H

radi

cals

inva

ding

liqu

id

time (micros)

Figure 10 Comparison of the amount of OH absorbed by the liquid versus time for cases IV(a)(|| uptake coeplusmn cient pound = 0001) and IV(b) ( pound = 001) Most of the OH uptakeoccurs at the main collapse and the subsequent after-bounce

In case I the plot shows linear behaviour (on the loglog scale) One observes thatwhen water vapour is not taken into account stronger forcing yields more extremetemperatures

In case II the relationship between compression ratio and maximum temperature isvery dinoterent First we note that the amount of water trapped in the bubble increaseswith compression ratio If we revisit the water-trapping mechanism (x 3 a (iii)) thistrend is readily explained The increase in water-vapour content reduces the ratioof specishy c heats of the gasvapour mixture and hence reduces the compressionheating

At low compression ratios in case II the temperature increases with compressionratio in a manner very similar to case I At low compression ratios the amountof water at collapse is very small the bubble is nearly pure argon As the forcingis increased more water is trapped and the maximum temperature of the bubblebecomes nearly constant In this range of bubble dynamics there is an almost per-fectly compensating enotect (with respect to peak temperature) between the decreasein specishy c-heat ratio and the increase in compression Case III is very similar tocase II except that the plateau in temperature occurs at a lower temperature aconsequence of the endothermic chemical reactions

(e) Inclusion of water vapour promotes the formation of shocks

Water vapour is of great importance in the formation of shocks in very stronglyforced bubbles We shall brieregy consider a 60 m bubble forced with a pressureamplitude of 14 bar at a frequency of 206 kHz (Case A1 in Moss et al (1999))

Proc R Soc Lond A (2000)

1704 B D Storey and A J Szeri

10 1 10 2

10 3

10 4

10 5

Rmax R min

max

imum

tem

pera

ture

(K

)

Figure 11 Change in centre temperature versus compression ratio in cases I ( ) II ( curren )and III (+) In this plot all parameters were held constant except the pressure amplitudewhich was varied between 10 and 13 bar Case I shows a steady increase in temperature withcompression ratio Cases II and III reach a maximum temperature as the forcing is increased

When water vapour is neglected a compression wave is launched into the bubble oncollapse but the wave does not steepen into a shock When water vapour is takeninto account (with or without reactions) the compression wave steepens into a strongshock very near the centre of the bubble Vuong et al (1999) showed that the rateof steepening depends on the amount of compression heating before the compressionwave is launched

In very recent work Lin amp Szeri (2000) used the wavefront expansion technique todetermine the evolution of a compression or rarefaction wavefront that is propagatingspherically into a collapsing or expanding gas of non-uniform entropy This analyticalwork supports the idea that when the speed of sound is higher at the centre (dueto temperature and composition inhomogeneities) shock formation is delayed orprevented Furthermore this work demonstrated that the inward radial velocity alsohinders the steepening of compression waves into shocks

Consistent with the arguments made in Lin amp Szeri (2000) Vuong et al (1999)and Moss et al (1999) our results show that the decrease in the ratio of specishy cheats due to an increase in the water-vapour content can promote shock formationThe point we want to emphasize is that the question of shock formation cannot beundertaken without consideration of water vapour

With respect to total chemical production shocks are of negligible consequenceThe shock inreguences only about the inner 10 of the radius which is only 01 ofthe volume Therefore the increased production of OH at the centre does not anotectthe total amount produced in the bubble

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1697

account the shy nite rate of mass dinotusion The low value of frac14 which works to trapsignishy cant water vapour at collapse was necessary for their model to be consistentwith experimental sonochemistry data When such a low value for frac14 is used thecondition frac12 d yn frac12 frac12 con d is reached well before frac12 d yn frac12 frac12 d if (the opposite of our case)Such a low accommodation coe cient works to trap signishy cant amounts of watervapour in the bubble but in this case the hindered regux at the interface eliminatesmass transfer out of the bubble while the distribution of water vapour within isessentially uniform In the present work we argue that the signishy cant amounts ofwater vapour trapped in the bubble are a consequence of slow dinotusion as well asnon-equilibrium phase change

The temperature of the interface exceeds the critical point for a very brief time(from iexcl 340 ps to 1800 ps relative to the time of minimum radius) For the conditionswe explored the maximum thickness of the supercritical region is quite thin less than1 of the bubble radius ie ca 10 nm We neglect any changes in the liquid equationof state and transport properties when the interface exceeds the critical temperatureThis assumption is justishy ed based on the thinness of the region and the fact that thecritical temperature is reached very late in the collapse Any disturbances caused bythe supercritical interface have insumacr cient time to propagate into the bubble interioron the main collapse Although the evaporation equation is invalid for these shorttimes the total mass added to the bubble is negligible regardless of the magnitudeof evaporative mass regux The issue of the super-critical interface may require futureimprovements but is beyond the scope of the present work

As a shy nal note both thermal and pressure dinotusion also play a role in determiningthe distribution of water vapour during the collapse In the argonwater system thethermal-dinotusion coemacr cient is small and the molecules are of similar size hence pres-sure dinotusion is quite weak Little dinoterence is seen in calculations with and withoutthese other forms of mass dinotusion In a helium or xenon bubble in water thermaland pressure dinotusion are more pronounced and can cause a 1015 dinoterence inthe amount of trapped water vapour In the interest of brevity we will not discussthese results further here

(b) Gas-phase chemical reactions break apart H2O and reduce the temperature

Now we turn to an investigation of what transpires when chemical reactions aretaken into account (case III) We shall show that there is signishy cant time for reactionsto occur ie the time-scale for chemical reaction is similar to that of the bubblecollapse The primarily endothermic reactions absorb a signishy cant amount of energyand further reduce the peak temperatures from cases I and II The distribution ofspecies produced in the bubble is determined by dinotusive transport of water vapourprior to the collapse and heat transfer out of the bubble during the collapse Thesteep temperature gradient at the wall conshy nes the bulk of the chemical reactions tothe bubble interior

(i) Consequences of chemistry

The overall bubble dynamics of cases II and III are essentially the same with acompression ratio of 49 in case III (slightly higher than case II) A convenient pointof departure is to compare the temperature histories of the bubble centre aroundthe main collapse for the three cases we have considered (see shy gure 5) The inclusion

Proc R Soc Lond A (2000)

1698 B D Storey and A J Szeri

- 4 - 2 0 2 410

3

104

tem

pera

ture

(K

)

time relative to minimum radius (ns)

Figure 5 Temperature history of the centre of the bubble around the time of minimum radius(t = 0) in cases I II and III Note the reduction in temperature from case I ( ) to case II( ) as water is allowed to enter the bubble The chemical reactions added in case III (||)further reduce the temperatures

of water vapour (case II) signishy cantly reduces the peak temperature from case IThe decrease in temperature is due mainly to the decrease in the ratio of specishy cheats of the mixture which reduces the compression heating Upon comparison ofcases II and III one observes that the chemical reactions further reduce the peaktemperature due to the endothermic decomposition of the water vapour The centretemperature histories of cases II and III are nearly identical below 4000 K

It is important to understand that the temperature dinoterences in these threecases are not due to subtle dinoterences in bubble dynamics The compression ratio(Rm ax=Rm in ) of the three cases increases from case I to case II to case III whereasthe temperature decreases in this order This decrease in temperature occurs despitethe fact that a higher compression ratio will lead to more compression heating in thebubble collapse all other things being equal In all these cases there are no shocksinside the bubble Shocks will be considered in x 3 e

In shy gure 6 we show the history of the total number of molecules of the dinoterentspecies as a function of time on the same time-scale as in shy gure 5 The lowest curve isthe total amount of water vapour only The next curve is the total amount of watervapour plus the total amount of H2 Each curve represents the addition of one speciesas labelled until the uppermost curve denotes everything but argon We do not labelHO2 and H2O2 as they appear only in trace amounts One observes which speciesare most abundant during the time of the main collapse Note there is a relativelylarge amount of undissociated water considering the elevated temperatures This isprimarily a consequence of the high pressure suppressing the dissociation of H2O

From shy gures 5 and 6 one observes that the onset of signishy cant chemical activity in

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1699

- 4 - 2 0 2 4

109

1010

time relative to minimum radius (ns)

num

ber

of m

olec

ules

H2O

H2

O2

OH

O

H

Figure 6 Cumulative number of molecules of each species in case III The lowest curve is thetotal number of water molecules the next curve is total water + H2 and so on H2 O2 and HO2

are neglected in this plot as they appear in trace amounts

the bubble corresponds to a centre temperature of ca 4000 K Therefore the reducedtemperature of case III shown in shy gure 5 can be attributed to the endothermicchemical reactions Although the bubble dynamics are quite rapid the chemicalreactions keep pace as is shown by the similar time-scales of the changing quantitiesin shy gures 5 and 6 The rapid rate at which the reactions progress is due to the veryhigh density Recall that the rate of progress of each reaction is directly related tothe species concentration (mol cmiexcl3)

(ii) Dynamics and distribution of OH production

In sonochemistry applications the invasion of OH radicals into the liquid isthought to be responsible for much of the complex and useful chemistry that occursTherefore we shall focus on the behaviour of this species in particular In shy gure 7 weplot the total number of OH molecules contained in the bubble on two time-scalesover one acoustic cycle in shy gure 7a and around the time of collapse in shy gure 7b Theinitial concentrations of all species were selected so that the chemical production anddestruction balanced in one acoustic cycle The main collapse is associated with arapid creation of OH once the temperature exceeds 4000 K

Upon rebound the bubble rapidly cools and the OH and other products tend torecombine back into H2O These recombination reactions proceed more slowly thanthe dissociation reactions upon collapse The result is that the bubble is left with asuper-equilibrium amount of OH after it has cooled to ambient temperature Thissuper-equilibrium OH then slowly decays through the rest of the cycle with briefspurts of production and destruction at each of the after-bounces

Proc R Soc Lond A (2000)

1700 B D Storey and A J Szeri

0 10 20 30time (micros)

- 4 - 2 0 2 4

time relative to minimum radius (ns)

106

1010

104

106

108

108

num

ber

of O

H

radi

cals

nu

mbe

r of

OH

ra

dica

ls

Figure 7 Total number of OH molecules contained in the bubble versus time over one acousticcycle (a) and around the time of minimum radius (b) in case III The macrrst sharp peak in (a)is expanded in (b) Subsequent peaks in (a) correspond to local radius minima in the bubbledynamics The initial amounts of OH and all the other species were chosen for no net speciesproduction or destruction over an acoustic cycle

The distribution of OH in the bubble is characterized by a fairly uniform con-centration through much of the bubble interior and a very sharp gradient near thewall This sharp gradient is a result of heat transfer during the collapse Duringcompression conduction between the hot interior and the cool wall sets up a thintemperature boundary layer This boundary layer conshy nes the chemical reactions tothe interior away from the interface The cooler wall leads to a much lower concen-tration of OH at the interface than in the centre at the time of the main collapseIn shy gure 8 we show the molar concentration history of OH at the centre and at theinterface When the bubble is at minimum radius and the OH concentration at thecentre is at its maximum the concentration at the wall is three orders of magnitudelower

Because the chemistry time-scale is much faster than the dinotusion time-scale OHis produced where water vapour was trapped at high enough temperature The liquidinterface is insulated from the high concentration of OH generated deep inside in thebubble The only OH produced at the centre that reaches the bubble interface is theslowly decaying super-equilibrium OH that is left inside the bubble on re-expansion

(c) OH uptake by the liquid does not indeg uence the bubble interior

The inclusion of liquid uptake of OH (case IV) adds further complications wenow try to illuminate some of the main qualitative features We shall argue that OHuptake has little enotect on the gas dynamics and chemical production

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1701

- 4 - 2 0 2 410 - 10

10 - 8

10 - 6

10 - 4

10 - 2

time relative to minimum radius (ns)

conc

entr

atio

n of

OH

(m

ol c

m-3

)

Figure 8 Molar concentration of OH at the centre (||) and the interface ( ) versus timearound the point of minimum radius Note that the concentration at the centre is about threeorders of magnitude greater than at the interface at the point of minimum radius

Due to dinotering ideas on the proper value of the uptake coemacr cient the values areset at pound = 0001 and pound = 001 (cases IV(a) and IV(b)) consistent with the work ofothers to which we made reference earlier The bubble dynamics and temperatureproshy les for cases IV(a) and IV(b) are indistinguishable from those for case III Inshy gure 9 we show the total number of OH molecules in the bubble versus time ontwo dinoterent scales for cases III IV(a) and IV(b) The history for one acoustic cycleis plotted in shy gure 9a and the history concentrated at the main collapse is plottedin shy gure 9b

In the main expansion when the bubble is cool the OH resident in the bubblewill invade the liquid as can be seen by the decrease in the amount of OH at theend of the long expansion as the uptake increases When the bubble collapses thechemical production of OH from recently evaporated water rapidly overwhelms theOH uptake by the liquid As the bubble proceeds into the collapse the curves oftotal OH content for dinoterent uptakes become indistinguishable The total amountof OH in the bubble is dominated by the amount created in the interior away fromthe wall The total amount of OH lost to the liquid during a collapse is only afraction of the amount contained in the bubble Hence the peak amount of OH inthe bubble during the collapse is insensitive to the uptake coe cient of OH Similarstatements apply to uptake of other species

In shy gure 10 we show for the purposes of qualitative discussion the total massof OH exiting the bubble in cases IV(a) and IV(b) While the OH uptake coef-shy cient does not inreguence the total amount of OH contained in the bubble at themain collapse there are large dinoterences in the amount invading the liquid over

Proc R Soc Lond A (2000)

1702 B D Storey and A J Szeri

0 10 20 30

106

1010

102

time (micros)

- 4 - 2 0 2 4104

106

108

time relative to minimum radius (ns)

1010

(b)

(a)nu

mbe

r of

OH

ra

dica

ls

num

ber

of O

H

radi

cals

Figure 9 Comparison of total amount number of OH molecules in the bubble versus time overone acoustic cycle (a) and focused around the time of minimum radius (b) in cases III (||)IV(a) ( ) and IV(b) ( ) The uptake of OH plays a role during the slow expansionswhen the resident amount of OH left in the bubble is well above chemical equilibrium At thecollapse OH uptake does not indeg uence the total amount of OH production

a cycle Much of the OH uptake occurs during the main collapse and the subse-quent re-expansion when the concentration of OH at the wall is at its maximum(see shy gure 8) Further uptake occurs throughout the cycle at each of the after-bounces

(d ) Inclusion of water vapour alters the ereg ect of increased forcing

Next we shall compare the results of cases I II and III with all parameters shy xedexcept for the acoustic pressure amplitude which varies between 10 and 13 barWe that note this is experimentally awkward for SBSL because pressure amplitudeand ambient radius are dependent quantities through the mass exchange equilibriumof rectishy ed dinotusion (Hilgenfeldt et al 1996) The way we present these results isnonetheless useful as it demonstrates the importance of water vapour in the under-standing of sonoluminescence and sonochemistry

Because the bubble dynamics are altered by the inclusion of water vapour weuse the compression ratio as the distinguishing parameter In shy gure 11 we plot themaximum centre temperature versus the compression ratio for cases I II and III Thecalculations in shy gure 11 are devoid of strong shocks a matter to which we devotemore attention below If there are strong shocks then shy gure 11 will look the samein terms of a suitably averaged temperature rather than the centre temperature

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1703

0 10 20 30102

103

104

105

106

num

ber

of O

H

radi

cals

inva

ding

liqu

id

time (micros)

Figure 10 Comparison of the amount of OH absorbed by the liquid versus time for cases IV(a)(|| uptake coeplusmn cient pound = 0001) and IV(b) ( pound = 001) Most of the OH uptakeoccurs at the main collapse and the subsequent after-bounce

In case I the plot shows linear behaviour (on the loglog scale) One observes thatwhen water vapour is not taken into account stronger forcing yields more extremetemperatures

In case II the relationship between compression ratio and maximum temperature isvery dinoterent First we note that the amount of water trapped in the bubble increaseswith compression ratio If we revisit the water-trapping mechanism (x 3 a (iii)) thistrend is readily explained The increase in water-vapour content reduces the ratioof specishy c heats of the gasvapour mixture and hence reduces the compressionheating

At low compression ratios in case II the temperature increases with compressionratio in a manner very similar to case I At low compression ratios the amountof water at collapse is very small the bubble is nearly pure argon As the forcingis increased more water is trapped and the maximum temperature of the bubblebecomes nearly constant In this range of bubble dynamics there is an almost per-fectly compensating enotect (with respect to peak temperature) between the decreasein specishy c-heat ratio and the increase in compression Case III is very similar tocase II except that the plateau in temperature occurs at a lower temperature aconsequence of the endothermic chemical reactions

(e) Inclusion of water vapour promotes the formation of shocks

Water vapour is of great importance in the formation of shocks in very stronglyforced bubbles We shall brieregy consider a 60 m bubble forced with a pressureamplitude of 14 bar at a frequency of 206 kHz (Case A1 in Moss et al (1999))

Proc R Soc Lond A (2000)

1704 B D Storey and A J Szeri

10 1 10 2

10 3

10 4

10 5

Rmax R min

max

imum

tem

pera

ture

(K

)

Figure 11 Change in centre temperature versus compression ratio in cases I ( ) II ( curren )and III (+) In this plot all parameters were held constant except the pressure amplitudewhich was varied between 10 and 13 bar Case I shows a steady increase in temperature withcompression ratio Cases II and III reach a maximum temperature as the forcing is increased

When water vapour is neglected a compression wave is launched into the bubble oncollapse but the wave does not steepen into a shock When water vapour is takeninto account (with or without reactions) the compression wave steepens into a strongshock very near the centre of the bubble Vuong et al (1999) showed that the rateof steepening depends on the amount of compression heating before the compressionwave is launched

In very recent work Lin amp Szeri (2000) used the wavefront expansion technique todetermine the evolution of a compression or rarefaction wavefront that is propagatingspherically into a collapsing or expanding gas of non-uniform entropy This analyticalwork supports the idea that when the speed of sound is higher at the centre (dueto temperature and composition inhomogeneities) shock formation is delayed orprevented Furthermore this work demonstrated that the inward radial velocity alsohinders the steepening of compression waves into shocks

Consistent with the arguments made in Lin amp Szeri (2000) Vuong et al (1999)and Moss et al (1999) our results show that the decrease in the ratio of specishy cheats due to an increase in the water-vapour content can promote shock formationThe point we want to emphasize is that the question of shock formation cannot beundertaken without consideration of water vapour

With respect to total chemical production shocks are of negligible consequenceThe shock inreguences only about the inner 10 of the radius which is only 01 ofthe volume Therefore the increased production of OH at the centre does not anotectthe total amount produced in the bubble

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

1698 B D Storey and A J Szeri

- 4 - 2 0 2 410

3

104

tem

pera

ture

(K

)

time relative to minimum radius (ns)

Figure 5 Temperature history of the centre of the bubble around the time of minimum radius(t = 0) in cases I II and III Note the reduction in temperature from case I ( ) to case II( ) as water is allowed to enter the bubble The chemical reactions added in case III (||)further reduce the temperatures

of water vapour (case II) signishy cantly reduces the peak temperature from case IThe decrease in temperature is due mainly to the decrease in the ratio of specishy cheats of the mixture which reduces the compression heating Upon comparison ofcases II and III one observes that the chemical reactions further reduce the peaktemperature due to the endothermic decomposition of the water vapour The centretemperature histories of cases II and III are nearly identical below 4000 K

It is important to understand that the temperature dinoterences in these threecases are not due to subtle dinoterences in bubble dynamics The compression ratio(Rm ax=Rm in ) of the three cases increases from case I to case II to case III whereasthe temperature decreases in this order This decrease in temperature occurs despitethe fact that a higher compression ratio will lead to more compression heating in thebubble collapse all other things being equal In all these cases there are no shocksinside the bubble Shocks will be considered in x 3 e

In shy gure 6 we show the history of the total number of molecules of the dinoterentspecies as a function of time on the same time-scale as in shy gure 5 The lowest curve isthe total amount of water vapour only The next curve is the total amount of watervapour plus the total amount of H2 Each curve represents the addition of one speciesas labelled until the uppermost curve denotes everything but argon We do not labelHO2 and H2O2 as they appear only in trace amounts One observes which speciesare most abundant during the time of the main collapse Note there is a relativelylarge amount of undissociated water considering the elevated temperatures This isprimarily a consequence of the high pressure suppressing the dissociation of H2O

From shy gures 5 and 6 one observes that the onset of signishy cant chemical activity in

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1699

- 4 - 2 0 2 4

109

1010

time relative to minimum radius (ns)

num

ber

of m

olec

ules

H2O

H2

O2

OH

O

H

Figure 6 Cumulative number of molecules of each species in case III The lowest curve is thetotal number of water molecules the next curve is total water + H2 and so on H2 O2 and HO2

are neglected in this plot as they appear in trace amounts

the bubble corresponds to a centre temperature of ca 4000 K Therefore the reducedtemperature of case III shown in shy gure 5 can be attributed to the endothermicchemical reactions Although the bubble dynamics are quite rapid the chemicalreactions keep pace as is shown by the similar time-scales of the changing quantitiesin shy gures 5 and 6 The rapid rate at which the reactions progress is due to the veryhigh density Recall that the rate of progress of each reaction is directly related tothe species concentration (mol cmiexcl3)

(ii) Dynamics and distribution of OH production

In sonochemistry applications the invasion of OH radicals into the liquid isthought to be responsible for much of the complex and useful chemistry that occursTherefore we shall focus on the behaviour of this species in particular In shy gure 7 weplot the total number of OH molecules contained in the bubble on two time-scalesover one acoustic cycle in shy gure 7a and around the time of collapse in shy gure 7b Theinitial concentrations of all species were selected so that the chemical production anddestruction balanced in one acoustic cycle The main collapse is associated with arapid creation of OH once the temperature exceeds 4000 K

Upon rebound the bubble rapidly cools and the OH and other products tend torecombine back into H2O These recombination reactions proceed more slowly thanthe dissociation reactions upon collapse The result is that the bubble is left with asuper-equilibrium amount of OH after it has cooled to ambient temperature Thissuper-equilibrium OH then slowly decays through the rest of the cycle with briefspurts of production and destruction at each of the after-bounces

Proc R Soc Lond A (2000)

1700 B D Storey and A J Szeri

0 10 20 30time (micros)

- 4 - 2 0 2 4

time relative to minimum radius (ns)

106

1010

104

106

108

108

num

ber

of O

H

radi

cals

nu

mbe

r of

OH

ra

dica

ls

Figure 7 Total number of OH molecules contained in the bubble versus time over one acousticcycle (a) and around the time of minimum radius (b) in case III The macrrst sharp peak in (a)is expanded in (b) Subsequent peaks in (a) correspond to local radius minima in the bubbledynamics The initial amounts of OH and all the other species were chosen for no net speciesproduction or destruction over an acoustic cycle

The distribution of OH in the bubble is characterized by a fairly uniform con-centration through much of the bubble interior and a very sharp gradient near thewall This sharp gradient is a result of heat transfer during the collapse Duringcompression conduction between the hot interior and the cool wall sets up a thintemperature boundary layer This boundary layer conshy nes the chemical reactions tothe interior away from the interface The cooler wall leads to a much lower concen-tration of OH at the interface than in the centre at the time of the main collapseIn shy gure 8 we show the molar concentration history of OH at the centre and at theinterface When the bubble is at minimum radius and the OH concentration at thecentre is at its maximum the concentration at the wall is three orders of magnitudelower

Because the chemistry time-scale is much faster than the dinotusion time-scale OHis produced where water vapour was trapped at high enough temperature The liquidinterface is insulated from the high concentration of OH generated deep inside in thebubble The only OH produced at the centre that reaches the bubble interface is theslowly decaying super-equilibrium OH that is left inside the bubble on re-expansion

(c) OH uptake by the liquid does not indeg uence the bubble interior

The inclusion of liquid uptake of OH (case IV) adds further complications wenow try to illuminate some of the main qualitative features We shall argue that OHuptake has little enotect on the gas dynamics and chemical production

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1701

- 4 - 2 0 2 410 - 10

10 - 8

10 - 6

10 - 4

10 - 2

time relative to minimum radius (ns)

conc

entr

atio

n of

OH

(m

ol c

m-3

)

Figure 8 Molar concentration of OH at the centre (||) and the interface ( ) versus timearound the point of minimum radius Note that the concentration at the centre is about threeorders of magnitude greater than at the interface at the point of minimum radius

Due to dinotering ideas on the proper value of the uptake coemacr cient the values areset at pound = 0001 and pound = 001 (cases IV(a) and IV(b)) consistent with the work ofothers to which we made reference earlier The bubble dynamics and temperatureproshy les for cases IV(a) and IV(b) are indistinguishable from those for case III Inshy gure 9 we show the total number of OH molecules in the bubble versus time ontwo dinoterent scales for cases III IV(a) and IV(b) The history for one acoustic cycleis plotted in shy gure 9a and the history concentrated at the main collapse is plottedin shy gure 9b

In the main expansion when the bubble is cool the OH resident in the bubblewill invade the liquid as can be seen by the decrease in the amount of OH at theend of the long expansion as the uptake increases When the bubble collapses thechemical production of OH from recently evaporated water rapidly overwhelms theOH uptake by the liquid As the bubble proceeds into the collapse the curves oftotal OH content for dinoterent uptakes become indistinguishable The total amountof OH in the bubble is dominated by the amount created in the interior away fromthe wall The total amount of OH lost to the liquid during a collapse is only afraction of the amount contained in the bubble Hence the peak amount of OH inthe bubble during the collapse is insensitive to the uptake coe cient of OH Similarstatements apply to uptake of other species

In shy gure 10 we show for the purposes of qualitative discussion the total massof OH exiting the bubble in cases IV(a) and IV(b) While the OH uptake coef-shy cient does not inreguence the total amount of OH contained in the bubble at themain collapse there are large dinoterences in the amount invading the liquid over

Proc R Soc Lond A (2000)

1702 B D Storey and A J Szeri

0 10 20 30

106

1010

102

time (micros)

- 4 - 2 0 2 4104

106

108

time relative to minimum radius (ns)

1010

(b)

(a)nu

mbe

r of

OH

ra

dica

ls

num

ber

of O

H

radi

cals

Figure 9 Comparison of total amount number of OH molecules in the bubble versus time overone acoustic cycle (a) and focused around the time of minimum radius (b) in cases III (||)IV(a) ( ) and IV(b) ( ) The uptake of OH plays a role during the slow expansionswhen the resident amount of OH left in the bubble is well above chemical equilibrium At thecollapse OH uptake does not indeg uence the total amount of OH production

a cycle Much of the OH uptake occurs during the main collapse and the subse-quent re-expansion when the concentration of OH at the wall is at its maximum(see shy gure 8) Further uptake occurs throughout the cycle at each of the after-bounces

(d ) Inclusion of water vapour alters the ereg ect of increased forcing

Next we shall compare the results of cases I II and III with all parameters shy xedexcept for the acoustic pressure amplitude which varies between 10 and 13 barWe that note this is experimentally awkward for SBSL because pressure amplitudeand ambient radius are dependent quantities through the mass exchange equilibriumof rectishy ed dinotusion (Hilgenfeldt et al 1996) The way we present these results isnonetheless useful as it demonstrates the importance of water vapour in the under-standing of sonoluminescence and sonochemistry

Because the bubble dynamics are altered by the inclusion of water vapour weuse the compression ratio as the distinguishing parameter In shy gure 11 we plot themaximum centre temperature versus the compression ratio for cases I II and III Thecalculations in shy gure 11 are devoid of strong shocks a matter to which we devotemore attention below If there are strong shocks then shy gure 11 will look the samein terms of a suitably averaged temperature rather than the centre temperature

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1703

0 10 20 30102

103

104

105

106

num

ber

of O

H

radi

cals

inva

ding

liqu

id

time (micros)

Figure 10 Comparison of the amount of OH absorbed by the liquid versus time for cases IV(a)(|| uptake coeplusmn cient pound = 0001) and IV(b) ( pound = 001) Most of the OH uptakeoccurs at the main collapse and the subsequent after-bounce

In case I the plot shows linear behaviour (on the loglog scale) One observes thatwhen water vapour is not taken into account stronger forcing yields more extremetemperatures

In case II the relationship between compression ratio and maximum temperature isvery dinoterent First we note that the amount of water trapped in the bubble increaseswith compression ratio If we revisit the water-trapping mechanism (x 3 a (iii)) thistrend is readily explained The increase in water-vapour content reduces the ratioof specishy c heats of the gasvapour mixture and hence reduces the compressionheating

At low compression ratios in case II the temperature increases with compressionratio in a manner very similar to case I At low compression ratios the amountof water at collapse is very small the bubble is nearly pure argon As the forcingis increased more water is trapped and the maximum temperature of the bubblebecomes nearly constant In this range of bubble dynamics there is an almost per-fectly compensating enotect (with respect to peak temperature) between the decreasein specishy c-heat ratio and the increase in compression Case III is very similar tocase II except that the plateau in temperature occurs at a lower temperature aconsequence of the endothermic chemical reactions

(e) Inclusion of water vapour promotes the formation of shocks

Water vapour is of great importance in the formation of shocks in very stronglyforced bubbles We shall brieregy consider a 60 m bubble forced with a pressureamplitude of 14 bar at a frequency of 206 kHz (Case A1 in Moss et al (1999))

Proc R Soc Lond A (2000)

1704 B D Storey and A J Szeri

10 1 10 2

10 3

10 4

10 5

Rmax R min

max

imum

tem

pera

ture

(K

)

Figure 11 Change in centre temperature versus compression ratio in cases I ( ) II ( curren )and III (+) In this plot all parameters were held constant except the pressure amplitudewhich was varied between 10 and 13 bar Case I shows a steady increase in temperature withcompression ratio Cases II and III reach a maximum temperature as the forcing is increased

When water vapour is neglected a compression wave is launched into the bubble oncollapse but the wave does not steepen into a shock When water vapour is takeninto account (with or without reactions) the compression wave steepens into a strongshock very near the centre of the bubble Vuong et al (1999) showed that the rateof steepening depends on the amount of compression heating before the compressionwave is launched

In very recent work Lin amp Szeri (2000) used the wavefront expansion technique todetermine the evolution of a compression or rarefaction wavefront that is propagatingspherically into a collapsing or expanding gas of non-uniform entropy This analyticalwork supports the idea that when the speed of sound is higher at the centre (dueto temperature and composition inhomogeneities) shock formation is delayed orprevented Furthermore this work demonstrated that the inward radial velocity alsohinders the steepening of compression waves into shocks

Consistent with the arguments made in Lin amp Szeri (2000) Vuong et al (1999)and Moss et al (1999) our results show that the decrease in the ratio of specishy cheats due to an increase in the water-vapour content can promote shock formationThe point we want to emphasize is that the question of shock formation cannot beundertaken without consideration of water vapour

With respect to total chemical production shocks are of negligible consequenceThe shock inreguences only about the inner 10 of the radius which is only 01 ofthe volume Therefore the increased production of OH at the centre does not anotectthe total amount produced in the bubble

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1699

- 4 - 2 0 2 4

109

1010

time relative to minimum radius (ns)

num

ber

of m

olec

ules

H2O

H2

O2

OH

O

H

Figure 6 Cumulative number of molecules of each species in case III The lowest curve is thetotal number of water molecules the next curve is total water + H2 and so on H2 O2 and HO2

are neglected in this plot as they appear in trace amounts

the bubble corresponds to a centre temperature of ca 4000 K Therefore the reducedtemperature of case III shown in shy gure 5 can be attributed to the endothermicchemical reactions Although the bubble dynamics are quite rapid the chemicalreactions keep pace as is shown by the similar time-scales of the changing quantitiesin shy gures 5 and 6 The rapid rate at which the reactions progress is due to the veryhigh density Recall that the rate of progress of each reaction is directly related tothe species concentration (mol cmiexcl3)

(ii) Dynamics and distribution of OH production

In sonochemistry applications the invasion of OH radicals into the liquid isthought to be responsible for much of the complex and useful chemistry that occursTherefore we shall focus on the behaviour of this species in particular In shy gure 7 weplot the total number of OH molecules contained in the bubble on two time-scalesover one acoustic cycle in shy gure 7a and around the time of collapse in shy gure 7b Theinitial concentrations of all species were selected so that the chemical production anddestruction balanced in one acoustic cycle The main collapse is associated with arapid creation of OH once the temperature exceeds 4000 K

Upon rebound the bubble rapidly cools and the OH and other products tend torecombine back into H2O These recombination reactions proceed more slowly thanthe dissociation reactions upon collapse The result is that the bubble is left with asuper-equilibrium amount of OH after it has cooled to ambient temperature Thissuper-equilibrium OH then slowly decays through the rest of the cycle with briefspurts of production and destruction at each of the after-bounces

Proc R Soc Lond A (2000)

1700 B D Storey and A J Szeri

0 10 20 30time (micros)

- 4 - 2 0 2 4

time relative to minimum radius (ns)

106

1010

104

106

108

108

num

ber

of O

H

radi

cals

nu

mbe

r of

OH

ra

dica

ls

Figure 7 Total number of OH molecules contained in the bubble versus time over one acousticcycle (a) and around the time of minimum radius (b) in case III The macrrst sharp peak in (a)is expanded in (b) Subsequent peaks in (a) correspond to local radius minima in the bubbledynamics The initial amounts of OH and all the other species were chosen for no net speciesproduction or destruction over an acoustic cycle

The distribution of OH in the bubble is characterized by a fairly uniform con-centration through much of the bubble interior and a very sharp gradient near thewall This sharp gradient is a result of heat transfer during the collapse Duringcompression conduction between the hot interior and the cool wall sets up a thintemperature boundary layer This boundary layer conshy nes the chemical reactions tothe interior away from the interface The cooler wall leads to a much lower concen-tration of OH at the interface than in the centre at the time of the main collapseIn shy gure 8 we show the molar concentration history of OH at the centre and at theinterface When the bubble is at minimum radius and the OH concentration at thecentre is at its maximum the concentration at the wall is three orders of magnitudelower

Because the chemistry time-scale is much faster than the dinotusion time-scale OHis produced where water vapour was trapped at high enough temperature The liquidinterface is insulated from the high concentration of OH generated deep inside in thebubble The only OH produced at the centre that reaches the bubble interface is theslowly decaying super-equilibrium OH that is left inside the bubble on re-expansion

(c) OH uptake by the liquid does not indeg uence the bubble interior

The inclusion of liquid uptake of OH (case IV) adds further complications wenow try to illuminate some of the main qualitative features We shall argue that OHuptake has little enotect on the gas dynamics and chemical production

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1701

- 4 - 2 0 2 410 - 10

10 - 8

10 - 6

10 - 4

10 - 2

time relative to minimum radius (ns)

conc

entr

atio

n of

OH

(m

ol c

m-3

)

Figure 8 Molar concentration of OH at the centre (||) and the interface ( ) versus timearound the point of minimum radius Note that the concentration at the centre is about threeorders of magnitude greater than at the interface at the point of minimum radius

Due to dinotering ideas on the proper value of the uptake coemacr cient the values areset at pound = 0001 and pound = 001 (cases IV(a) and IV(b)) consistent with the work ofothers to which we made reference earlier The bubble dynamics and temperatureproshy les for cases IV(a) and IV(b) are indistinguishable from those for case III Inshy gure 9 we show the total number of OH molecules in the bubble versus time ontwo dinoterent scales for cases III IV(a) and IV(b) The history for one acoustic cycleis plotted in shy gure 9a and the history concentrated at the main collapse is plottedin shy gure 9b

In the main expansion when the bubble is cool the OH resident in the bubblewill invade the liquid as can be seen by the decrease in the amount of OH at theend of the long expansion as the uptake increases When the bubble collapses thechemical production of OH from recently evaporated water rapidly overwhelms theOH uptake by the liquid As the bubble proceeds into the collapse the curves oftotal OH content for dinoterent uptakes become indistinguishable The total amountof OH in the bubble is dominated by the amount created in the interior away fromthe wall The total amount of OH lost to the liquid during a collapse is only afraction of the amount contained in the bubble Hence the peak amount of OH inthe bubble during the collapse is insensitive to the uptake coe cient of OH Similarstatements apply to uptake of other species

In shy gure 10 we show for the purposes of qualitative discussion the total massof OH exiting the bubble in cases IV(a) and IV(b) While the OH uptake coef-shy cient does not inreguence the total amount of OH contained in the bubble at themain collapse there are large dinoterences in the amount invading the liquid over

Proc R Soc Lond A (2000)

1702 B D Storey and A J Szeri

0 10 20 30

106

1010

102

time (micros)

- 4 - 2 0 2 4104

106

108

time relative to minimum radius (ns)

1010

(b)

(a)nu

mbe

r of

OH

ra

dica

ls

num

ber

of O

H

radi

cals

Figure 9 Comparison of total amount number of OH molecules in the bubble versus time overone acoustic cycle (a) and focused around the time of minimum radius (b) in cases III (||)IV(a) ( ) and IV(b) ( ) The uptake of OH plays a role during the slow expansionswhen the resident amount of OH left in the bubble is well above chemical equilibrium At thecollapse OH uptake does not indeg uence the total amount of OH production

a cycle Much of the OH uptake occurs during the main collapse and the subse-quent re-expansion when the concentration of OH at the wall is at its maximum(see shy gure 8) Further uptake occurs throughout the cycle at each of the after-bounces

(d ) Inclusion of water vapour alters the ereg ect of increased forcing

Next we shall compare the results of cases I II and III with all parameters shy xedexcept for the acoustic pressure amplitude which varies between 10 and 13 barWe that note this is experimentally awkward for SBSL because pressure amplitudeand ambient radius are dependent quantities through the mass exchange equilibriumof rectishy ed dinotusion (Hilgenfeldt et al 1996) The way we present these results isnonetheless useful as it demonstrates the importance of water vapour in the under-standing of sonoluminescence and sonochemistry

Because the bubble dynamics are altered by the inclusion of water vapour weuse the compression ratio as the distinguishing parameter In shy gure 11 we plot themaximum centre temperature versus the compression ratio for cases I II and III Thecalculations in shy gure 11 are devoid of strong shocks a matter to which we devotemore attention below If there are strong shocks then shy gure 11 will look the samein terms of a suitably averaged temperature rather than the centre temperature

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1703

0 10 20 30102

103

104

105

106

num

ber

of O

H

radi

cals

inva

ding

liqu

id

time (micros)

Figure 10 Comparison of the amount of OH absorbed by the liquid versus time for cases IV(a)(|| uptake coeplusmn cient pound = 0001) and IV(b) ( pound = 001) Most of the OH uptakeoccurs at the main collapse and the subsequent after-bounce

In case I the plot shows linear behaviour (on the loglog scale) One observes thatwhen water vapour is not taken into account stronger forcing yields more extremetemperatures

In case II the relationship between compression ratio and maximum temperature isvery dinoterent First we note that the amount of water trapped in the bubble increaseswith compression ratio If we revisit the water-trapping mechanism (x 3 a (iii)) thistrend is readily explained The increase in water-vapour content reduces the ratioof specishy c heats of the gasvapour mixture and hence reduces the compressionheating

At low compression ratios in case II the temperature increases with compressionratio in a manner very similar to case I At low compression ratios the amountof water at collapse is very small the bubble is nearly pure argon As the forcingis increased more water is trapped and the maximum temperature of the bubblebecomes nearly constant In this range of bubble dynamics there is an almost per-fectly compensating enotect (with respect to peak temperature) between the decreasein specishy c-heat ratio and the increase in compression Case III is very similar tocase II except that the plateau in temperature occurs at a lower temperature aconsequence of the endothermic chemical reactions

(e) Inclusion of water vapour promotes the formation of shocks

Water vapour is of great importance in the formation of shocks in very stronglyforced bubbles We shall brieregy consider a 60 m bubble forced with a pressureamplitude of 14 bar at a frequency of 206 kHz (Case A1 in Moss et al (1999))

Proc R Soc Lond A (2000)

1704 B D Storey and A J Szeri

10 1 10 2

10 3

10 4

10 5

Rmax R min

max

imum

tem

pera

ture

(K

)

Figure 11 Change in centre temperature versus compression ratio in cases I ( ) II ( curren )and III (+) In this plot all parameters were held constant except the pressure amplitudewhich was varied between 10 and 13 bar Case I shows a steady increase in temperature withcompression ratio Cases II and III reach a maximum temperature as the forcing is increased

When water vapour is neglected a compression wave is launched into the bubble oncollapse but the wave does not steepen into a shock When water vapour is takeninto account (with or without reactions) the compression wave steepens into a strongshock very near the centre of the bubble Vuong et al (1999) showed that the rateof steepening depends on the amount of compression heating before the compressionwave is launched

In very recent work Lin amp Szeri (2000) used the wavefront expansion technique todetermine the evolution of a compression or rarefaction wavefront that is propagatingspherically into a collapsing or expanding gas of non-uniform entropy This analyticalwork supports the idea that when the speed of sound is higher at the centre (dueto temperature and composition inhomogeneities) shock formation is delayed orprevented Furthermore this work demonstrated that the inward radial velocity alsohinders the steepening of compression waves into shocks

Consistent with the arguments made in Lin amp Szeri (2000) Vuong et al (1999)and Moss et al (1999) our results show that the decrease in the ratio of specishy cheats due to an increase in the water-vapour content can promote shock formationThe point we want to emphasize is that the question of shock formation cannot beundertaken without consideration of water vapour

With respect to total chemical production shocks are of negligible consequenceThe shock inreguences only about the inner 10 of the radius which is only 01 ofthe volume Therefore the increased production of OH at the centre does not anotectthe total amount produced in the bubble

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

1700 B D Storey and A J Szeri

0 10 20 30time (micros)

- 4 - 2 0 2 4

time relative to minimum radius (ns)

106

1010

104

106

108

108

num

ber

of O

H

radi

cals

nu

mbe

r of

OH

ra

dica

ls

Figure 7 Total number of OH molecules contained in the bubble versus time over one acousticcycle (a) and around the time of minimum radius (b) in case III The macrrst sharp peak in (a)is expanded in (b) Subsequent peaks in (a) correspond to local radius minima in the bubbledynamics The initial amounts of OH and all the other species were chosen for no net speciesproduction or destruction over an acoustic cycle

The distribution of OH in the bubble is characterized by a fairly uniform con-centration through much of the bubble interior and a very sharp gradient near thewall This sharp gradient is a result of heat transfer during the collapse Duringcompression conduction between the hot interior and the cool wall sets up a thintemperature boundary layer This boundary layer conshy nes the chemical reactions tothe interior away from the interface The cooler wall leads to a much lower concen-tration of OH at the interface than in the centre at the time of the main collapseIn shy gure 8 we show the molar concentration history of OH at the centre and at theinterface When the bubble is at minimum radius and the OH concentration at thecentre is at its maximum the concentration at the wall is three orders of magnitudelower

Because the chemistry time-scale is much faster than the dinotusion time-scale OHis produced where water vapour was trapped at high enough temperature The liquidinterface is insulated from the high concentration of OH generated deep inside in thebubble The only OH produced at the centre that reaches the bubble interface is theslowly decaying super-equilibrium OH that is left inside the bubble on re-expansion

(c) OH uptake by the liquid does not indeg uence the bubble interior

The inclusion of liquid uptake of OH (case IV) adds further complications wenow try to illuminate some of the main qualitative features We shall argue that OHuptake has little enotect on the gas dynamics and chemical production

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1701

- 4 - 2 0 2 410 - 10

10 - 8

10 - 6

10 - 4

10 - 2

time relative to minimum radius (ns)

conc

entr

atio

n of

OH

(m

ol c

m-3

)

Figure 8 Molar concentration of OH at the centre (||) and the interface ( ) versus timearound the point of minimum radius Note that the concentration at the centre is about threeorders of magnitude greater than at the interface at the point of minimum radius

Due to dinotering ideas on the proper value of the uptake coemacr cient the values areset at pound = 0001 and pound = 001 (cases IV(a) and IV(b)) consistent with the work ofothers to which we made reference earlier The bubble dynamics and temperatureproshy les for cases IV(a) and IV(b) are indistinguishable from those for case III Inshy gure 9 we show the total number of OH molecules in the bubble versus time ontwo dinoterent scales for cases III IV(a) and IV(b) The history for one acoustic cycleis plotted in shy gure 9a and the history concentrated at the main collapse is plottedin shy gure 9b

In the main expansion when the bubble is cool the OH resident in the bubblewill invade the liquid as can be seen by the decrease in the amount of OH at theend of the long expansion as the uptake increases When the bubble collapses thechemical production of OH from recently evaporated water rapidly overwhelms theOH uptake by the liquid As the bubble proceeds into the collapse the curves oftotal OH content for dinoterent uptakes become indistinguishable The total amountof OH in the bubble is dominated by the amount created in the interior away fromthe wall The total amount of OH lost to the liquid during a collapse is only afraction of the amount contained in the bubble Hence the peak amount of OH inthe bubble during the collapse is insensitive to the uptake coe cient of OH Similarstatements apply to uptake of other species

In shy gure 10 we show for the purposes of qualitative discussion the total massof OH exiting the bubble in cases IV(a) and IV(b) While the OH uptake coef-shy cient does not inreguence the total amount of OH contained in the bubble at themain collapse there are large dinoterences in the amount invading the liquid over

Proc R Soc Lond A (2000)

1702 B D Storey and A J Szeri

0 10 20 30

106

1010

102

time (micros)

- 4 - 2 0 2 4104

106

108

time relative to minimum radius (ns)

1010

(b)

(a)nu

mbe

r of

OH

ra

dica

ls

num

ber

of O

H

radi

cals

Figure 9 Comparison of total amount number of OH molecules in the bubble versus time overone acoustic cycle (a) and focused around the time of minimum radius (b) in cases III (||)IV(a) ( ) and IV(b) ( ) The uptake of OH plays a role during the slow expansionswhen the resident amount of OH left in the bubble is well above chemical equilibrium At thecollapse OH uptake does not indeg uence the total amount of OH production

a cycle Much of the OH uptake occurs during the main collapse and the subse-quent re-expansion when the concentration of OH at the wall is at its maximum(see shy gure 8) Further uptake occurs throughout the cycle at each of the after-bounces

(d ) Inclusion of water vapour alters the ereg ect of increased forcing

Next we shall compare the results of cases I II and III with all parameters shy xedexcept for the acoustic pressure amplitude which varies between 10 and 13 barWe that note this is experimentally awkward for SBSL because pressure amplitudeand ambient radius are dependent quantities through the mass exchange equilibriumof rectishy ed dinotusion (Hilgenfeldt et al 1996) The way we present these results isnonetheless useful as it demonstrates the importance of water vapour in the under-standing of sonoluminescence and sonochemistry

Because the bubble dynamics are altered by the inclusion of water vapour weuse the compression ratio as the distinguishing parameter In shy gure 11 we plot themaximum centre temperature versus the compression ratio for cases I II and III Thecalculations in shy gure 11 are devoid of strong shocks a matter to which we devotemore attention below If there are strong shocks then shy gure 11 will look the samein terms of a suitably averaged temperature rather than the centre temperature

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1703

0 10 20 30102

103

104

105

106

num

ber

of O

H

radi

cals

inva

ding

liqu

id

time (micros)

Figure 10 Comparison of the amount of OH absorbed by the liquid versus time for cases IV(a)(|| uptake coeplusmn cient pound = 0001) and IV(b) ( pound = 001) Most of the OH uptakeoccurs at the main collapse and the subsequent after-bounce

In case I the plot shows linear behaviour (on the loglog scale) One observes thatwhen water vapour is not taken into account stronger forcing yields more extremetemperatures

In case II the relationship between compression ratio and maximum temperature isvery dinoterent First we note that the amount of water trapped in the bubble increaseswith compression ratio If we revisit the water-trapping mechanism (x 3 a (iii)) thistrend is readily explained The increase in water-vapour content reduces the ratioof specishy c heats of the gasvapour mixture and hence reduces the compressionheating

At low compression ratios in case II the temperature increases with compressionratio in a manner very similar to case I At low compression ratios the amountof water at collapse is very small the bubble is nearly pure argon As the forcingis increased more water is trapped and the maximum temperature of the bubblebecomes nearly constant In this range of bubble dynamics there is an almost per-fectly compensating enotect (with respect to peak temperature) between the decreasein specishy c-heat ratio and the increase in compression Case III is very similar tocase II except that the plateau in temperature occurs at a lower temperature aconsequence of the endothermic chemical reactions

(e) Inclusion of water vapour promotes the formation of shocks

Water vapour is of great importance in the formation of shocks in very stronglyforced bubbles We shall brieregy consider a 60 m bubble forced with a pressureamplitude of 14 bar at a frequency of 206 kHz (Case A1 in Moss et al (1999))

Proc R Soc Lond A (2000)

1704 B D Storey and A J Szeri

10 1 10 2

10 3

10 4

10 5

Rmax R min

max

imum

tem

pera

ture

(K

)

Figure 11 Change in centre temperature versus compression ratio in cases I ( ) II ( curren )and III (+) In this plot all parameters were held constant except the pressure amplitudewhich was varied between 10 and 13 bar Case I shows a steady increase in temperature withcompression ratio Cases II and III reach a maximum temperature as the forcing is increased

When water vapour is neglected a compression wave is launched into the bubble oncollapse but the wave does not steepen into a shock When water vapour is takeninto account (with or without reactions) the compression wave steepens into a strongshock very near the centre of the bubble Vuong et al (1999) showed that the rateof steepening depends on the amount of compression heating before the compressionwave is launched

In very recent work Lin amp Szeri (2000) used the wavefront expansion technique todetermine the evolution of a compression or rarefaction wavefront that is propagatingspherically into a collapsing or expanding gas of non-uniform entropy This analyticalwork supports the idea that when the speed of sound is higher at the centre (dueto temperature and composition inhomogeneities) shock formation is delayed orprevented Furthermore this work demonstrated that the inward radial velocity alsohinders the steepening of compression waves into shocks

Consistent with the arguments made in Lin amp Szeri (2000) Vuong et al (1999)and Moss et al (1999) our results show that the decrease in the ratio of specishy cheats due to an increase in the water-vapour content can promote shock formationThe point we want to emphasize is that the question of shock formation cannot beundertaken without consideration of water vapour

With respect to total chemical production shocks are of negligible consequenceThe shock inreguences only about the inner 10 of the radius which is only 01 ofthe volume Therefore the increased production of OH at the centre does not anotectthe total amount produced in the bubble

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1701

- 4 - 2 0 2 410 - 10

10 - 8

10 - 6

10 - 4

10 - 2

time relative to minimum radius (ns)

conc

entr

atio

n of

OH

(m

ol c

m-3

)

Figure 8 Molar concentration of OH at the centre (||) and the interface ( ) versus timearound the point of minimum radius Note that the concentration at the centre is about threeorders of magnitude greater than at the interface at the point of minimum radius

Due to dinotering ideas on the proper value of the uptake coemacr cient the values areset at pound = 0001 and pound = 001 (cases IV(a) and IV(b)) consistent with the work ofothers to which we made reference earlier The bubble dynamics and temperatureproshy les for cases IV(a) and IV(b) are indistinguishable from those for case III Inshy gure 9 we show the total number of OH molecules in the bubble versus time ontwo dinoterent scales for cases III IV(a) and IV(b) The history for one acoustic cycleis plotted in shy gure 9a and the history concentrated at the main collapse is plottedin shy gure 9b

In the main expansion when the bubble is cool the OH resident in the bubblewill invade the liquid as can be seen by the decrease in the amount of OH at theend of the long expansion as the uptake increases When the bubble collapses thechemical production of OH from recently evaporated water rapidly overwhelms theOH uptake by the liquid As the bubble proceeds into the collapse the curves oftotal OH content for dinoterent uptakes become indistinguishable The total amountof OH in the bubble is dominated by the amount created in the interior away fromthe wall The total amount of OH lost to the liquid during a collapse is only afraction of the amount contained in the bubble Hence the peak amount of OH inthe bubble during the collapse is insensitive to the uptake coe cient of OH Similarstatements apply to uptake of other species

In shy gure 10 we show for the purposes of qualitative discussion the total massof OH exiting the bubble in cases IV(a) and IV(b) While the OH uptake coef-shy cient does not inreguence the total amount of OH contained in the bubble at themain collapse there are large dinoterences in the amount invading the liquid over

Proc R Soc Lond A (2000)

1702 B D Storey and A J Szeri

0 10 20 30

106

1010

102

time (micros)

- 4 - 2 0 2 4104

106

108

time relative to minimum radius (ns)

1010

(b)

(a)nu

mbe

r of

OH

ra

dica

ls

num

ber

of O

H

radi

cals

Figure 9 Comparison of total amount number of OH molecules in the bubble versus time overone acoustic cycle (a) and focused around the time of minimum radius (b) in cases III (||)IV(a) ( ) and IV(b) ( ) The uptake of OH plays a role during the slow expansionswhen the resident amount of OH left in the bubble is well above chemical equilibrium At thecollapse OH uptake does not indeg uence the total amount of OH production

a cycle Much of the OH uptake occurs during the main collapse and the subse-quent re-expansion when the concentration of OH at the wall is at its maximum(see shy gure 8) Further uptake occurs throughout the cycle at each of the after-bounces

(d ) Inclusion of water vapour alters the ereg ect of increased forcing

Next we shall compare the results of cases I II and III with all parameters shy xedexcept for the acoustic pressure amplitude which varies between 10 and 13 barWe that note this is experimentally awkward for SBSL because pressure amplitudeand ambient radius are dependent quantities through the mass exchange equilibriumof rectishy ed dinotusion (Hilgenfeldt et al 1996) The way we present these results isnonetheless useful as it demonstrates the importance of water vapour in the under-standing of sonoluminescence and sonochemistry

Because the bubble dynamics are altered by the inclusion of water vapour weuse the compression ratio as the distinguishing parameter In shy gure 11 we plot themaximum centre temperature versus the compression ratio for cases I II and III Thecalculations in shy gure 11 are devoid of strong shocks a matter to which we devotemore attention below If there are strong shocks then shy gure 11 will look the samein terms of a suitably averaged temperature rather than the centre temperature

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1703

0 10 20 30102

103

104

105

106

num

ber

of O

H

radi

cals

inva

ding

liqu

id

time (micros)

Figure 10 Comparison of the amount of OH absorbed by the liquid versus time for cases IV(a)(|| uptake coeplusmn cient pound = 0001) and IV(b) ( pound = 001) Most of the OH uptakeoccurs at the main collapse and the subsequent after-bounce

In case I the plot shows linear behaviour (on the loglog scale) One observes thatwhen water vapour is not taken into account stronger forcing yields more extremetemperatures

In case II the relationship between compression ratio and maximum temperature isvery dinoterent First we note that the amount of water trapped in the bubble increaseswith compression ratio If we revisit the water-trapping mechanism (x 3 a (iii)) thistrend is readily explained The increase in water-vapour content reduces the ratioof specishy c heats of the gasvapour mixture and hence reduces the compressionheating

At low compression ratios in case II the temperature increases with compressionratio in a manner very similar to case I At low compression ratios the amountof water at collapse is very small the bubble is nearly pure argon As the forcingis increased more water is trapped and the maximum temperature of the bubblebecomes nearly constant In this range of bubble dynamics there is an almost per-fectly compensating enotect (with respect to peak temperature) between the decreasein specishy c-heat ratio and the increase in compression Case III is very similar tocase II except that the plateau in temperature occurs at a lower temperature aconsequence of the endothermic chemical reactions

(e) Inclusion of water vapour promotes the formation of shocks

Water vapour is of great importance in the formation of shocks in very stronglyforced bubbles We shall brieregy consider a 60 m bubble forced with a pressureamplitude of 14 bar at a frequency of 206 kHz (Case A1 in Moss et al (1999))

Proc R Soc Lond A (2000)

1704 B D Storey and A J Szeri

10 1 10 2

10 3

10 4

10 5

Rmax R min

max

imum

tem

pera

ture

(K

)

Figure 11 Change in centre temperature versus compression ratio in cases I ( ) II ( curren )and III (+) In this plot all parameters were held constant except the pressure amplitudewhich was varied between 10 and 13 bar Case I shows a steady increase in temperature withcompression ratio Cases II and III reach a maximum temperature as the forcing is increased

When water vapour is neglected a compression wave is launched into the bubble oncollapse but the wave does not steepen into a shock When water vapour is takeninto account (with or without reactions) the compression wave steepens into a strongshock very near the centre of the bubble Vuong et al (1999) showed that the rateof steepening depends on the amount of compression heating before the compressionwave is launched

In very recent work Lin amp Szeri (2000) used the wavefront expansion technique todetermine the evolution of a compression or rarefaction wavefront that is propagatingspherically into a collapsing or expanding gas of non-uniform entropy This analyticalwork supports the idea that when the speed of sound is higher at the centre (dueto temperature and composition inhomogeneities) shock formation is delayed orprevented Furthermore this work demonstrated that the inward radial velocity alsohinders the steepening of compression waves into shocks

Consistent with the arguments made in Lin amp Szeri (2000) Vuong et al (1999)and Moss et al (1999) our results show that the decrease in the ratio of specishy cheats due to an increase in the water-vapour content can promote shock formationThe point we want to emphasize is that the question of shock formation cannot beundertaken without consideration of water vapour

With respect to total chemical production shocks are of negligible consequenceThe shock inreguences only about the inner 10 of the radius which is only 01 ofthe volume Therefore the increased production of OH at the centre does not anotectthe total amount produced in the bubble

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

1702 B D Storey and A J Szeri

0 10 20 30

106

1010

102

time (micros)

- 4 - 2 0 2 4104

106

108

time relative to minimum radius (ns)

1010

(b)

(a)nu

mbe

r of

OH

ra

dica

ls

num

ber

of O

H

radi

cals

Figure 9 Comparison of total amount number of OH molecules in the bubble versus time overone acoustic cycle (a) and focused around the time of minimum radius (b) in cases III (||)IV(a) ( ) and IV(b) ( ) The uptake of OH plays a role during the slow expansionswhen the resident amount of OH left in the bubble is well above chemical equilibrium At thecollapse OH uptake does not indeg uence the total amount of OH production

a cycle Much of the OH uptake occurs during the main collapse and the subse-quent re-expansion when the concentration of OH at the wall is at its maximum(see shy gure 8) Further uptake occurs throughout the cycle at each of the after-bounces

(d ) Inclusion of water vapour alters the ereg ect of increased forcing

Next we shall compare the results of cases I II and III with all parameters shy xedexcept for the acoustic pressure amplitude which varies between 10 and 13 barWe that note this is experimentally awkward for SBSL because pressure amplitudeand ambient radius are dependent quantities through the mass exchange equilibriumof rectishy ed dinotusion (Hilgenfeldt et al 1996) The way we present these results isnonetheless useful as it demonstrates the importance of water vapour in the under-standing of sonoluminescence and sonochemistry

Because the bubble dynamics are altered by the inclusion of water vapour weuse the compression ratio as the distinguishing parameter In shy gure 11 we plot themaximum centre temperature versus the compression ratio for cases I II and III Thecalculations in shy gure 11 are devoid of strong shocks a matter to which we devotemore attention below If there are strong shocks then shy gure 11 will look the samein terms of a suitably averaged temperature rather than the centre temperature

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1703

0 10 20 30102

103

104

105

106

num

ber

of O

H

radi

cals

inva

ding

liqu

id

time (micros)

Figure 10 Comparison of the amount of OH absorbed by the liquid versus time for cases IV(a)(|| uptake coeplusmn cient pound = 0001) and IV(b) ( pound = 001) Most of the OH uptakeoccurs at the main collapse and the subsequent after-bounce

In case I the plot shows linear behaviour (on the loglog scale) One observes thatwhen water vapour is not taken into account stronger forcing yields more extremetemperatures

In case II the relationship between compression ratio and maximum temperature isvery dinoterent First we note that the amount of water trapped in the bubble increaseswith compression ratio If we revisit the water-trapping mechanism (x 3 a (iii)) thistrend is readily explained The increase in water-vapour content reduces the ratioof specishy c heats of the gasvapour mixture and hence reduces the compressionheating

At low compression ratios in case II the temperature increases with compressionratio in a manner very similar to case I At low compression ratios the amountof water at collapse is very small the bubble is nearly pure argon As the forcingis increased more water is trapped and the maximum temperature of the bubblebecomes nearly constant In this range of bubble dynamics there is an almost per-fectly compensating enotect (with respect to peak temperature) between the decreasein specishy c-heat ratio and the increase in compression Case III is very similar tocase II except that the plateau in temperature occurs at a lower temperature aconsequence of the endothermic chemical reactions

(e) Inclusion of water vapour promotes the formation of shocks

Water vapour is of great importance in the formation of shocks in very stronglyforced bubbles We shall brieregy consider a 60 m bubble forced with a pressureamplitude of 14 bar at a frequency of 206 kHz (Case A1 in Moss et al (1999))

Proc R Soc Lond A (2000)

1704 B D Storey and A J Szeri

10 1 10 2

10 3

10 4

10 5

Rmax R min

max

imum

tem

pera

ture

(K

)

Figure 11 Change in centre temperature versus compression ratio in cases I ( ) II ( curren )and III (+) In this plot all parameters were held constant except the pressure amplitudewhich was varied between 10 and 13 bar Case I shows a steady increase in temperature withcompression ratio Cases II and III reach a maximum temperature as the forcing is increased

When water vapour is neglected a compression wave is launched into the bubble oncollapse but the wave does not steepen into a shock When water vapour is takeninto account (with or without reactions) the compression wave steepens into a strongshock very near the centre of the bubble Vuong et al (1999) showed that the rateof steepening depends on the amount of compression heating before the compressionwave is launched

In very recent work Lin amp Szeri (2000) used the wavefront expansion technique todetermine the evolution of a compression or rarefaction wavefront that is propagatingspherically into a collapsing or expanding gas of non-uniform entropy This analyticalwork supports the idea that when the speed of sound is higher at the centre (dueto temperature and composition inhomogeneities) shock formation is delayed orprevented Furthermore this work demonstrated that the inward radial velocity alsohinders the steepening of compression waves into shocks

Consistent with the arguments made in Lin amp Szeri (2000) Vuong et al (1999)and Moss et al (1999) our results show that the decrease in the ratio of specishy cheats due to an increase in the water-vapour content can promote shock formationThe point we want to emphasize is that the question of shock formation cannot beundertaken without consideration of water vapour

With respect to total chemical production shocks are of negligible consequenceThe shock inreguences only about the inner 10 of the radius which is only 01 ofthe volume Therefore the increased production of OH at the centre does not anotectthe total amount produced in the bubble

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1703

0 10 20 30102

103

104

105

106

num

ber

of O

H

radi

cals

inva

ding

liqu

id

time (micros)

Figure 10 Comparison of the amount of OH absorbed by the liquid versus time for cases IV(a)(|| uptake coeplusmn cient pound = 0001) and IV(b) ( pound = 001) Most of the OH uptakeoccurs at the main collapse and the subsequent after-bounce

In case I the plot shows linear behaviour (on the loglog scale) One observes thatwhen water vapour is not taken into account stronger forcing yields more extremetemperatures

In case II the relationship between compression ratio and maximum temperature isvery dinoterent First we note that the amount of water trapped in the bubble increaseswith compression ratio If we revisit the water-trapping mechanism (x 3 a (iii)) thistrend is readily explained The increase in water-vapour content reduces the ratioof specishy c heats of the gasvapour mixture and hence reduces the compressionheating

At low compression ratios in case II the temperature increases with compressionratio in a manner very similar to case I At low compression ratios the amountof water at collapse is very small the bubble is nearly pure argon As the forcingis increased more water is trapped and the maximum temperature of the bubblebecomes nearly constant In this range of bubble dynamics there is an almost per-fectly compensating enotect (with respect to peak temperature) between the decreasein specishy c-heat ratio and the increase in compression Case III is very similar tocase II except that the plateau in temperature occurs at a lower temperature aconsequence of the endothermic chemical reactions

(e) Inclusion of water vapour promotes the formation of shocks

Water vapour is of great importance in the formation of shocks in very stronglyforced bubbles We shall brieregy consider a 60 m bubble forced with a pressureamplitude of 14 bar at a frequency of 206 kHz (Case A1 in Moss et al (1999))

Proc R Soc Lond A (2000)

1704 B D Storey and A J Szeri

10 1 10 2

10 3

10 4

10 5

Rmax R min

max

imum

tem

pera

ture

(K

)

Figure 11 Change in centre temperature versus compression ratio in cases I ( ) II ( curren )and III (+) In this plot all parameters were held constant except the pressure amplitudewhich was varied between 10 and 13 bar Case I shows a steady increase in temperature withcompression ratio Cases II and III reach a maximum temperature as the forcing is increased

When water vapour is neglected a compression wave is launched into the bubble oncollapse but the wave does not steepen into a shock When water vapour is takeninto account (with or without reactions) the compression wave steepens into a strongshock very near the centre of the bubble Vuong et al (1999) showed that the rateof steepening depends on the amount of compression heating before the compressionwave is launched

In very recent work Lin amp Szeri (2000) used the wavefront expansion technique todetermine the evolution of a compression or rarefaction wavefront that is propagatingspherically into a collapsing or expanding gas of non-uniform entropy This analyticalwork supports the idea that when the speed of sound is higher at the centre (dueto temperature and composition inhomogeneities) shock formation is delayed orprevented Furthermore this work demonstrated that the inward radial velocity alsohinders the steepening of compression waves into shocks

Consistent with the arguments made in Lin amp Szeri (2000) Vuong et al (1999)and Moss et al (1999) our results show that the decrease in the ratio of specishy cheats due to an increase in the water-vapour content can promote shock formationThe point we want to emphasize is that the question of shock formation cannot beundertaken without consideration of water vapour

With respect to total chemical production shocks are of negligible consequenceThe shock inreguences only about the inner 10 of the radius which is only 01 ofthe volume Therefore the increased production of OH at the centre does not anotectthe total amount produced in the bubble

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

1704 B D Storey and A J Szeri

10 1 10 2

10 3

10 4

10 5

Rmax R min

max

imum

tem

pera

ture

(K

)

Figure 11 Change in centre temperature versus compression ratio in cases I ( ) II ( curren )and III (+) In this plot all parameters were held constant except the pressure amplitudewhich was varied between 10 and 13 bar Case I shows a steady increase in temperature withcompression ratio Cases II and III reach a maximum temperature as the forcing is increased

When water vapour is neglected a compression wave is launched into the bubble oncollapse but the wave does not steepen into a shock When water vapour is takeninto account (with or without reactions) the compression wave steepens into a strongshock very near the centre of the bubble Vuong et al (1999) showed that the rateof steepening depends on the amount of compression heating before the compressionwave is launched

In very recent work Lin amp Szeri (2000) used the wavefront expansion technique todetermine the evolution of a compression or rarefaction wavefront that is propagatingspherically into a collapsing or expanding gas of non-uniform entropy This analyticalwork supports the idea that when the speed of sound is higher at the centre (dueto temperature and composition inhomogeneities) shock formation is delayed orprevented Furthermore this work demonstrated that the inward radial velocity alsohinders the steepening of compression waves into shocks

Consistent with the arguments made in Lin amp Szeri (2000) Vuong et al (1999)and Moss et al (1999) our results show that the decrease in the ratio of specishy cheats due to an increase in the water-vapour content can promote shock formationThe point we want to emphasize is that the question of shock formation cannot beundertaken without consideration of water vapour

With respect to total chemical production shocks are of negligible consequenceThe shock inreguences only about the inner 10 of the radius which is only 01 ofthe volume Therefore the increased production of OH at the centre does not anotectthe total amount produced in the bubble

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1705

4 Relevance to sonochemistry

In sonochemistry one interpretation of how much OH is available to react in theliquid is to assume that in multi-bubble situations the bubbles will be unstable oncollapse and break apart in some way dispersing their total OH content into theliquid This view was promoted by Colussi et al (1998) If we take this view wemight regard the peak amount of OH in shy gure 7 to be the available sonochemicalOH We remind the reader that peak amounts of total OH at the collapses arecompletely insensitive to the uptake coe cient

An alternative view is that the bubbles are stable and the OH radicals enterthe liquid by uptake across the interface If we take this view we might regard theamount of OH in shy gure 10 to be the available sonochemical OH for each cycle Thetotal mass of OH leaving the bubble depends on the concentration at the wall andon the uptake coe cient as we have shown

We expect that in reality the OH transport to the liquid would be some com-bination of these two proposed mechanisms the important mechanism will dependupon the application The total amount of OH available in the bubble is a few ordersof magnitude greater than the amount taken up by the liquid in one cycle Thusthe unstable bubble mechanism may be the important one in many applicationsBubbles would have to be stable for thousands of cycles for the mass-transfer mecha-nism to compete in terms of quantity of OH donated to the liquid The OH -uptakemechanism may be important in other applications

The total amount of OH in the bubble is insensitive to the uptake by the liquidbecause the evaporation of water on every expansion provides the `fuelrsquo to producemany more OH radicals at the collapse than are lost via uptake to the liquid (assum-ing a stable bubble) Reactions involving O and H chemistry should be contrastedwith the chemistry of other species One example is the dissociation hypothesis of airbubbles (Lohse et al 1997) The hypothesis is that on the collapse of an air bubblethe nitrogen reacts with water vapour producing NH3 a very soluble species TheNH3 will tend to leave the bubble at a fast rate (ie the associated uptake coemacr cientis large) While small amounts of dissolved N2 will come into the bubble on thenext expansion when the bubble is larger than the ambient radius this rate is slowcompared with the uptake of NH3 The bubble is soon devoid of nitrogen due to theease with which NH3 leaves the bubble compared with the dimacr culty with which N2

enters itHence for understanding the composition of stable bubbles OH uptake by the

liquid is not important due to rapid evaporation of H2O on the expansion NH3

uptake is important due to the slow rate that N2 enters the bubble on the expansionOne should keep these facts in mind when thinking of how the present results onwater-vapour chemistry may apply to chemical systems more complicated than amonatomic gas and water

Finally on the basis of the shy ndings presented in this paper one might be temptedto think that simply by suppressing evaporation of water into the bubble eg byadding salt or reducing the bath temperature one has direct control over conditionsat the collapse The situation is however more complicated because adding salt(Wall et al 1999) or manipulating the liquid temperature can drastically change thegas solubility The change in solubility can anotect the ambient bubble radius leadingto dinoterent bubble dynamics

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

1706 B D Storey and A J Szeri

5 Conclusions

In this paper we attempted to illuminate the important role that water vapour playsin the physics of strongly collapsing bubbles Water vapour is trapped inside the bub-ble during the violent collapse The amount and distribution of water vapour in thebubble is frozen on the time-scale of the peak temperature The trapping mechanismcan be understood by a relative-time-scales argument the bubble-dynamics time-scale becomes much faster than the mass-dinotusion time-scale at some point in thecollapse The amount of water left behind depends sensitively on the non-equilibriumphase change

The trapped water vapour undergoes chemical reactions on the time-scale of thebubble dynamics The products of reaction inside the bubble are distributed accord-ing to where chemical reactions occur dinotusion enotects are small in this regard Dueto the large temperature gradient at the bubble wall most of the chemical activityoccurs within the bubble interior This temperature gradient keeps the concentrationof OH about three orders of magnitude lower at the wall than at the centre Watervapour and its destruction in endothermic chemical reactions signishy cantly reduce thepeak temperatures achieved during the rapid collapse

The phenomenon of OH uptake into the liquid had only a small enotect on theoverall properties of the bubble (ie mass temperature composition) This insensi-tivity to OH uptake into the liquid is due to the fact that the large amounts of OHproduced by the bubble remain in the centre and the OH concentration at the wallremains relatively low

Through variations of parameters we explored some interesting trends when watervapour and reactions are taken into account When water vapour is neglected thepeak temperature achieved in the collapse increases with forcing when all otherparameters are shy xed When water vapour is included the simultaneous increasein water content and compression ratio with increased forcing nearly compensateone another the consequence is that the peak temperature of the bubble is nearlyconstant with increased forcing This is an example of an important trend thatchanges if water is not taken into account

We also investigated the case of more strongly forced bubbles We showed thatneglect of water vapour can prevent shocks from forming When water vapour isallowed into the bubble a shock can form near the centre in very strongly forcedcases Physical phenomena can be overlooked if water vapour is neglected

We discussed the relevance of this work to sonochemistry Two ideas have beenadvanced for OH dispersal into the liquid The shy rst mechanism assumes that thebubble bursts on collapse and that the OH is somehow mixed with the liquid Thesecond mechanism assumes that the bubble is stable and that the OH is taken inby the liquid throughout the bubble oscillations The second mechanism is far lessproductive than the shy rst

Finally we shy nish with a connection to light emission that provides a measure ofconshy dence in the present results The amount of water vapour that Moss et al (1999)used was determined by matching their calculated results with the experimental datafor photon yields and pulse widths measured by Gaitan amp Holt (1999) and Hiller etal (1998) respectively In table 2 we compare the uniform molar fraction of watervapour that Moss et al (1999) used with the amount we determined with the presentmethod Because the composition is non-uniform we give the fraction of vapour at

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1707

Table 2 Comparison of the amount of water vapour trapped in the bubble at collapsedetermined by the present work versus the amount determined by Moss et al (1999)

(The vapour in Moss et al (1999) was assumed to be uniform throughout the bubble The molefraction of vapour at the centre and based on the total contents determined by the presentmethod is compared with the (uniform) fraction determined by Moss et al (1999) to yieldphoton counts and pulse widths in concert with experiments (see text) The case number refersto table 1 in Moss et al (1999))

R0 PA vapour vapour vapour

case ( m) (bar) Moss () centre () total ()

A1 60 140 40 56 33

B1 40 132 36 48 27

C1 21 129 30 41 22

the centre and based on the total bubble contents One observes not only that theactual numbers are in excellent agreement but that the trend is captured as well

This research was supported by the National Science Foundation The authors thank W CMoss D A Young J-Y Chen M Frenklach J Reisse K Bartik H-P Schuchmann andC Von Sonntag for useful discussions and help with references

References

Barber B P Weninger K amp Putterman S J 1997 Sonoluminescence Phil Trans R SocLond A 355 641644

Bird R B Stewart W E amp Lightfoot E N 1960 Transport phenomena Wiley

Bowman C T Hanson R K Davidson D F Gardiner W C Lissianski V SmithG P Golden D M Frenklach M amp Goldenberg M 1999 httpwwwmeberkeleyedugri mech

Carey V P 1992 Liquidvapor phase change phenomena London Taylor amp Francis

Chameides W L 1984 The photochemistry of remote marine stratiform cloud J GeophysRes 89 47394755

Colussi A J Weavers L K amp Horeg man M R 1998 Chemical bubble dynamics and quanti-tative sonochemistry J Phys Chem A 102 69276934

Eames I W Marr N J amp Sabir H 1997 The evaporation coeplusmn cient of water a review IntJ Heat Mass Transfer 40 29632973

Fristrom R M amp Monchik L 1988 Two simple approximations to the thermal-direg usion factorand their applications to deg ame studies Comb Flame 71 8999

Fujikawa S amp Teruaki A 1980 Ereg ects of the non-equilibrium condensation of vapour on thepressure wave produced by the collapse of a bubble in a liquid J Fluid Mech 97 481512

Fyrillas M M amp Szeri A J 1994 Dissolution or growth of soluble spherical oscillating bubblesJ Fluid Mech 277 381407

Gaitan D F amp Holt R G 1999 Experimental observations of bubble response and lightintensity near the threshold for single bubble sonoluminescence in an airwater system PhysRev E 59 54955502

Gaitan D F Atchley A A Lewia S D Carlson J T Maruyama X K Moran M ampSweider D 1996 Spectra of sonoluminescence in water and glycerinwater mixtures PhysRev E 54 525528

Gardiner W C 1984 Combustion chemistry Springer

Proc R Soc Lond A (2000)

1708 B D Storey and A J Szeri

Gong C amp Hart D P 1998 Ultrasound induced cavitation and sonochemical yields J AcoustSoc Am 104 26752682

Hanson D R Burkholder J B Howard C J Ravishankara A R 1992 Measurement of OHand HO2 radical uptake coeplusmn cients on water and sulfuric acid surfaces J Phys Chem 9649794985

Hilgenfeldt S Lohse D amp Brenner M P 1996 Phase diagrams for sonoluminescing bubblesPhys Fluids 8 28082826

Hiller R Putterman S J amp Barber B P 1992 Spectrum of synchronous picosecond sonolu-minescence Phys Rev Lett 69 11821184

Hiller R A Putterman S J amp Weninger K R 1998 Time-resolved spectra of sonolumines-cence Phys Rev Lett 80 10901093

Hirschfelder J O Curtiss C F amp Bird R B 1954 Molecular theory of gases and liquidsWiley

Kamath V Prosperetti A amp Egolfopoulos F N 1993 A theoretical study of sonoluminescenceJ Acoust Soc Am 94 248260

Lin H amp Szeri A J 2000 Shock formation in the presence of entropy gradients J Fluid Mech(Submitted)

Lohse D Brenner M P Dupont T F Hilgenfeldt S amp Johnston B 1997 Sonoluminescingair bubbles rectify argon Phys Rev Lett 78 13591362

Maas U amp Warnatz J 1988 Ignition processes in hydrogenoxygen mixtures Combust Flame74 5369

Mason T J 1999 Sonochemistry current uses and future prospects in the chemical and pro-cessing industries Phil Trans R Soc Lond A 357 355369

Moss W C Clarke D B amp Young D A 1997 Calculated pulse widths and spectra of a singlesonoluminescence bubble Science 276 13981401

Moss W C Young D A Harte J A Levatin J L Rozsnyai B F Zimmerman G B ampZimmerman I H 1999 Computed optical emissions from a sonoluminescing bubble PhysRev E 59 29862992

Press W H Teukolsky S A Vetterling W T amp Flannery B P 1988 Numerical recipes inC Cambridge University Press

Prosperetti A amp Lezzi A 1986 Bubble dynamics in a compressible liquid Part 1 First-ordertheory J Fluid Mech 168 457477

Reid R C Prausnitz J M amp Poling B E 1987 Properties of gases and liquids McGraw-Hill

Sander R 1998 Henryrsquos Law constants In NIST Chemistry WebBook (ed W G Mallard ampP J Linstrom) NIST Standard Reference Database no 69 Gaithersburg MD NationalInstitute of Standards and Technology httpwebbooknistgov

Sochard S Wilhelm A M amp Delmas H 1997 Modeling of free radicals production in acollapsing gas-vapour bubble Ultrason Sonochem 4 7784

Storey B D amp Szeri A J 1999 Mixture segregation within sonoluminescence bubbles J FluidMech 396 203221

Suslick K S Didenko Y Fang M F Hyeon T Kolbeck K J McNamara W B MdleleniM M amp Wong M 1999 Acoustic cavitation and its chemical consequences Phil Trans RSoc Lond A 357 335353

Takami A Kato S Shimono A amp Seichiro K 1998 Uptake coeplusmn cient of OH radical onaqueous surface Chem Phys 231 215227

Von Sonntag C Mark G Tauber A amp Schuchmann H-P 1999 OH radical formation anddosimetry in the sonolysis of aqueous solutions Adv Sonochem 5 109145

Vuong V Q amp Szeri A J 1996 Sonoluminescence and direg usive transport Phys Fluids 823542364

Vuong V Q Szeri A J amp Young D A 1999 Shock formation within sonoluminescencebubbles Phys Fluids 11 1017

Proc R Soc Lond A (2000)

Water vapour sonoluminescence and sonochemistry 1709

Wall M Ashokkumar M Tronson R amp Grieser F 1999 Multibubble sonoluminescence inaqueous salt solutions Ultrason Sonochem 6 714

Wu C C amp Roberts P H 1998a Bubble shape instability and sonoluminescence Phys LettA 250 131136

Wu C C amp Roberts P H 1998b On rectimacred direg usion and sonoluminescence Theor ComputFluid Dynam 10 357372

Yasui K 1997a Alternative model of single-bubble sonoluminescence Phys Rev E 56 67506760

Yasui K 1997b Chemical reactions in a sonoluminescing bubble J Phys Soc Japan 66 29112920

Proc R Soc Lond A (2000)