a conceptual model for acoustic microcavitation

A conceptual model for acoustic microcavitation Sameer I. Madanshetty Aerospace and Mechanical Engineering, Boston University, Boston, Massachusetts 02215

(Received 25 May 1994; revised 8 May 1995; accepted 8 May 1995)

A conceptual model is developed to explain microcavitation events at low megahertz frequencies and low duty cycles, facilitated by smooth, spherical, submicrometer sized particles in clean water. The extremely high levels of accelerations associated with high-frequency acoustic fields, the active detector field, can, according to this model, "coax" such smooth particles to nucleate cavitation events at reduced thresholds. This conceptual model should help further experimentation exploring the phenomenon of "acoustic coaxing" for microcavitation. ̧ 1995 Acoustical Society of America.

PACS numbers: 43.35.Ei

INTRODUCTION

One always thinks of cavitation in terms of a preexisting gaseous presence either as a bubble or a stabilized pocket of gas trapped in a suitable container crevice. Cavitation can occur if a sound wave imposes sufficiently strong tensile fields. The oscillating pressures of a sound wave imply re- strictions on the time duration of the available tensile field

within a period of oscillation. At high frequencies it becomes increasingly difficult to bring about cavitation because insuf- ficient time is available for bubble growth, which precedes the cavitational implosion. Consequently the cavitation thresholds of bubbles become greater at higher frequencies, and for practical purposes, it becomes very difficult to bring about cavitation (in water) at frequencies in excess of 10 MHz (cf .... in excess of 3 MHz, Neppiras•). Therefore, it appears paradoxical that a high-frequency field may actually help lower the thresholds and facilitate cavitation. Such lowering of thresholds in the presence of high-frequency fields has been reported by Madanshetty. 2

To explore the mechanism of how high-frequency acoustic fields "coax" smooth, spherical particles to nucleate cavitation events at reduced thresholds, we have selected for

discussion several experiments from earlier work. 2-5 This paper is arranged as follows: We start by giving a brief review of cavitation and then describe the particular experiment that highlighted the paradox. The central thesis of the conceptual model is then presented, and we elaborate upon the constructs that build the model. It is difficult to be fully rigorous at this stage; we therefore seek only supporting evidence and arguments, and present some more experiments that corroborate the thesis, at least in principle.

I. BACKGROUND

A. Brief review of cavitation

Cavitation is a fascinating phenomenon. It can erode metallic surfaces, help shatter kidney stones, accelerate chemical reactions, and even lead to light production-- sonoluminescence, in the case of acoustic cavitation. Acous-

tic cavitation has been exhaustively reviewed by Flynn, 6 Neppiras, • Apfel, 7 and Prosperetti. 8 We will not dwell on the

details of bubble dynamics, but we will recollect in simple terms the concepts of rectified diffusion, resonance, and transient cavitation.

Consider a free bubble in the path of a sound wave. The bubble expands and contracts in response to the pressure alternations of the sound wave; the energy stored during the expansion being returned concentrated during the possibly implosive collapse. Should a bubble grow to about two and a half times its original size during the negative excursion of the acoustic pressure, then during the following positive half- cycle its speed of collapse could become supersonic. 9 Such almost single cycle violent events, called transient cavitation, may explain the energetic manifestations of cavitation.

Unlike the dramatic bubble growth within a single acoustic cycle as seen in transient (inertial) cavitation, there exists a more gradual process, termed rectified diffusion. Un- der favorable conditions, a small bubble exposed to a con- tinuous sound wave tends to grow in size if rectified diffusion is dominant. According to Henry's law, for a gas soluble in liquid, the equilibrium concentration of the dissolved gas in the liquid is directly proportional to the partial pressure of the gas above the liquid surface, the constant of proportion- ality being a function of temperature only. When a bubble expands, the pressure in the bubble interior falls and gas diffuses into the bubble from the surrounding liquid. When the bubble contracts, the pressure in the interior increases and the gas diffuses into solution in the surrounding liquid. However, the area available for diffusion is larger in the expansion mode than in the contraction mode. Consequently, there is a net diffusion of the gas into the bubble from the surrounding liquid over a complete cycle and thus the bubble grows due to rectified diffusion.

However, a bubble can grow only up to a critical size-- the resonance radius determined by the frequency of the impressed sound wave. For small amplitude oscillations a bubble acts like a simple linear oscillator of mass equal to the virtual mass of the pulsating sphere, which is 3x the mass of displaced fluid, and stiffness primarily given by the internal pressure of the bubble times the ratio of specific heats. (Surface tension effects are significant for small bubbles.) Following Minnaert, •ø ignoring surface tension, there is a simple relation for the resonance radius of air bubbles in water:

2681 J. Acoust. Soc. Am. 98 (5), Pt. 1, November 1995 0001-4966/95/98(5)/2681/9/$6.00 ¸ 1995 Acoustical Society of America 2681

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(Resonance radius in /•m)

X (insonification frequency in MHz)= 3.2.

This relation is valid to within 5% even for bubbles of radius

10/•m. The bubble response becomes increasingly vigorous at resonance radius, and is limited by the damping mecha- nisms in the bubble environment--viscous damping, acoustic radiation damping, and thermal damping. A postresonance bubble may exhibit nonlinear modes of oscillations or become transient if the forcing acoustic pressure amplitude is adequately high.

The above discussion presupposes the presence of a free bubble in the path of a sound wave. Free bubbles, however, do not last long in a standing body of water. The larger ones are rapidly removed due to buoyancy and the smaller ones dissolve even in slightly undersaturated water. While a 10-/•m air bubble rises in water at a terminal speed of 300 /•m/s, it can survive for about 5 s before dissolving completely. The dissolution of bubbles is driven essentially by the excess pressure inside the bubble due to the surface tension.

It is very difficult to cavitate clean liquids. • A pure liquid purged of all particulate impurities and stored in a perfectly smooth container can attain its theoretical tensile strength before undergoing cavitation or fracture. Under ideal conditions water can be as strong as aluminum; the tensile strength of water based on the homogeneous nucleation theory exceeds 1000 bars. (In cavitation studies tensile strengths are often quoted in terms of negative pressures, and threshold is understood as the pressure amplitude at which the first occurrence of cavitation is detected.) The observed strengths (thresholds) in practice, however, are very much lower, rarely exceeding a few bars for reasonably clean liquids. This is because there exist gas pockets within the liquid which provide the necessary seeding for cavitation. A gas site is often stabilized in a crevice, •2 either in the container wall or on a fluidborne particle. Incomplete wetting traps gas at the root of a sharp crevice, stabilizing it against dissolution. Unlike a free bubble, surface tension in this case acts on

a meniscus which is concave toward the liquid. Overpressur- ing the liquid for sufficient duration prior to insonification can force the meniscus further into the crevice and may cause full wetting of the crevice, which then gives rise to increased thresholds.

II. THE EXPERIMENT

While researching acoustic microcavitation 2 in water primarily at 0.75-MHz frequency and 1% duty cycle--as a precursor to the work to follow at more diagnostically rel- evant frequencies--two kinds of acoustic detectors were used to detect cavitation (see schematic of the test cell, Fig. 1). The first one was an unfocused, untuned 1-MHz transducer which served as a "passive detector." The second was a focused 30-MHz transducer used in a pulse-echo mode as the "active detector." Cavitation itself was brought about by a focused PZT-8, 0.75-MHz transducer driven in pulse mode. Focused systems have the advantage that the zone of cavitation is localized within the bulk of the liquid, away from the walls of the container and the rough transducer face. The

rho-c TM

•bsorption wall

acoustically transpaten DIAPHRAGM

APPROXIMATE

I ACTIVE

DETECTOR

•i•i•ii•i::::::::::::::'"'"'"'"'":i:•:E:•:•:•:i:i:!:E::

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I BEAM

'""D P 'ETESIc•R

FIG. 1. Schematic of the test cell. The test cell is divided into two compartments so that the rho-c TM impedance matching rubber wall immersed in water provides a nonreflecting boundary while remaining isolated from the cavitation chamber. An acoustically transparent stainless shim stock forms a watertight seal between the two compartments. To detect cavitation two kinds of acoustic detectors are used. The first one is an unfocused, untuned 1-MHz receiver transducer which serves as a passive detector. The other one is a focused 30-MHz transducer which is used in pulse-echo mode and is called the active detector. Cavitation itself is brought about by a focused 0.75-MHz PZT-8 crystal driven in pulse mode (tone bursts), typically 10- /•s-long pulses at 1-kHz PRF. The active detector is arranged confocally with respect to the cavitation transducer. Both the interrogating pulse and the cavitation pulse arrive simultaneously at the common focus, which is the region of cavitation (see Ref. 2).

active detector was arranged to be confocal with respect to the cavitation transducer. Both the interrogating pulse and the cavitation pulse were arranged to arrive simultaneously at the common focus, the region of cavitation. With the test chamber filled with clean water (distilled, de-ionized, and extensively filtered) no cavitation was observed, even when the cavitation transducer was driven to give its peak output in excess of 22 bar peak negative.

When the test chamber with polystyrene latex microparticles suspended in a clean water host was irradiated with short (10/•s long) acoustic pulses from the focused cavitation transducer, the cavitation thresholds indicated by the passive detector were around 15 bar peak negative. When the 30-MHz focused active transducer was switched on to operate in the pulse-echo mode, it measured thresholds of 5 bar peak negative. This observation in itself may amply speak for the superior sensitivity of the active detector. However, when the thresholds were measured with the passive transducer, while the active transducer field was left switched on, the passive thresholds dropped to around 7 bar peak negative. This difference in the measured passive thresholds, 15

2682 J. Acoust. Soc. Am., Vol. 98, No. 5, Pt. 1, November 1995 Sameer I. Madanshetty: Model for acoustic microcavitation 2682


bar peak negative with the active detector off, and 7 bar peak negative with the active detector on, suggests that the cavitation environment was being influenced by the detection field of the active detector. In the experimental setup, the active detector focal pressure amplitudes were invariably less than 0.5 bar peak negative. Even with the crudest superposi- tion, this cannot account for the lowering of passive detector thresholds from 15 to 7 bar peak negative. In fact, it becomes increasingly difficult to initiate cavitation at increased acoustic frequencies, and it was found that the active detector did not cause cavitation on its own.

Note that the polystyrene latex microparticles of 0.984-/zm mean diameter added to the host water were monodispersed, spherical, and smooth. Extensive observation of the particles under the scanning electron microscope did not reveal any significant surface flaws or crevices down to about 500 •, the resolution limit of the device used. The specific area based on adsorption studies--information from manufacturer's test reports--was also close to the geometric surface area calculated from the assumption of sphericity. These observations of the smoothness of the latex micro-

spheres in the context of the crevice model raise the following questions: How can smooth spherical particles help initiate cavitation? How does the acoustic field of the active detector facilitate the process?

III. CONCEPTUAL MODEL FOR ACOUSTIC MICROCAVITATION

In this section we present a conceptual model to answer these questions. Significant statements used in the descrip- tion of this model will be justified in later sections. See Fig. 2--the scenario.

Although the surfaces of the polystyrene latex particles are smooth down to 500 •, they could have nanoscale gas pockets on their surfaces. This gas could be caused by im- perfect wetting of the pores, or by adsorbed nonpolar constituents desorbing when the particle enters the reduced pressure region of the main cavitation transducer. If a single nano-gas-dot of diameter 50 nm (500 •) were to cavitate alone in the tensile environment of the cavitation transducer,

the estimated threshold would be around 60 bar peak negative. Consequently, the observed lower thresholds in the presence of the active detector field are possible if the nano- gas-dots aggregate to form sufficiently larger gas patches. In water, the wavelengths at 30 and 0.75 MHz are 50/zm and 2 mm, respectively, while the largest particles we have used are less than 1 /zm in diameter. In these circumstances a particle feels essentially a uniform pressure over its surface. Also, in the focal zone of a focused transducer one may assume that the waves are nearly plane. In water, the active detector operating at 30 MHz at a modest pressure level setting of 0.5 bar peak negative can give rise to a particle (fluid parcel) acceleration of about 6.47x 10 6 rn/s 2 or, equivalently, 6.5 X 105 gs. At any given point a particle denser than water will inertially lag the acceleration, while a less dense particle will move in the direction of acceleration. In the present case we are considering polystyrene spheres whose density is 1.05 g/cc, and are reasonably density matched with water. Air density, on the other hand, is about 1.2x 10 -3 g/cc. Coupled

The Scenario:

1. A plane wave of 30 MHz at a pressure amplitude of 0.5 bar

gives rise to an acceleration of 6.5 x l0 s g units. 2. Air is 830 times lighter than water.

3. Formation of lens shaped gas caps on the oscillating particle

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!:i:! :i:i:i:i:!:i :!:i :!:i:!:i:i:i :i: !: !: i:!:i:i:i: !:!:!:i:!:i:10:i :.i ß 2.2-2-2-2-2.2-2• 2-2 ø 2.•ß 2-2.2-2-2-5-2-2-2-2-2-2.2-292.2.2-2.2-2-2.•. 2-i-2ø2 ø 2-2-2-2ø 2-2-2-2-2.•. 2-2-2-2-2-2-•-2-2-•-2-5'2-2-5-2.2-2-2-2-2-2ø 2-2-2-2-•-2-2-2-2ø2 -

4. For cavitation to occur, the negative pressure in the tensile

environment should overcome the surface tension force.

o red = p(rcd 2/4)

FIG. 2. The scenario. The nano-gas-dots on the particle surface tend to agglomerate toward the fore and aft regions on the oscillating particle to form lens shaped gas caps which give rise to cavitation events in suitable tensile environments. Nano-gas-dots anywhere on the particle surface, at regions A, A', B, and B', experience kinetic buoyancy; only those near B, B' experience rectified kinetic buoyancy, while those at A and A' mostly oscillate in the sound field.

with high acceleration fields, this density contrast, a factor of 830, will enhance the kinetic buoyancy and urge the nano-gas-dots toward the fore and aft regions on the sphere where they could agglomerate and form gas caps. One can visualize these gas caps as lenslike regions located at the extremities of an oscillating (to and fro) spherical particle. A cavitation event can then occur when the force due to surface

tension on the perimeter of the gas cap is overcome by the tensile forces effective on the gas region. It is to be noted that the use of macroscopic concepts such as surface tension down to near molecular levels is not wholly unjustified. 8

In the case of the polystyrene spheres of mean diameter 0.984/zm, the measured passive detector threshold was 7 bar peak negative 2 in the presence of the active detector field. The surface tension (0.073 N/m) force acting on the perimeter of a circle of diameter 0.984/zm can be overcome by a tensile force due to a pressure of just 3 bar peak negative acting on the corresponding projected area. Here we have assumed a gas patch of the same diameter as the particle. In reality, the gas patch would be smaller than the particle di- mensions. This would, as observed, invariably give rise to thresholds higher than the lower limit corresponding to a gas patch as large as the particle diameter, or a bubble exactly encapsulating the particle. Recall that for a single nano-gas- dot of size 50 nm, to cavitate alone in the tensile environment of the cavitation transducer the estimated threshold

would be around 60 bar peak negative. While the above mechanism of gas collection due to

kinetic buoyancy effects helps us explain cavitation pro- moted by smooth particles and the reduced thresholds in the



presence of the active detector field--acoustic coaxingrowe are not in a position to predict the exact threshold values for a given particle/host context, and more detailed experiments may help.

IV. DETAILED JUSTIFICATION

A number of statements need to be justified for the above model to be viable. (1) The particles considered are small enough that there are no gradients of pressure or velocity across their extent, and that the particles ideally are entrained in the flow. (2) The particles are smooth, spherical, and monodispersed. The suspensions are dilute enough that there are no agglomerations, and particles stay separated from each other. The particles are smooth in that there are no significant, large crevices to nucleate cavitation directly. However, imperfectly wet sites of several nanometers are envisaged to be present on the particle surface. Though the particles are spherical, the model does not heavily depend on such sphericity. (3) High acceleration fields due to the active detector somehow educe gas/vapor into lens shaped regions on the particle surface, which eventually give rise to cavitation.

In the sections to follow we seek to explain the present understanding of acoustic coaxing as rigorously as possible.

A. Information from crevice model

Cavitation experiments routinely report thresholds considerably lower than those corresponding to homogeneous nucleation. These low observed thresholds are due to hetero-

geneous nucleation requiring bubblelike presence in the liquid being cavitated. As mentioned earlier free bubbles do not survive for long in a standing body of water. To explain the presence of stabilized gaseous regions Harvey •2 introduced the crevice model. Over the years the crevice model has been considerably refined, and a recent paper by Atchley and Prosperetti •3 discusses in detail the various possibilities of dynamic, crevice-trapped gas cavities. It is informative to note that a crevice-trapped gas pocket can last indefinitely even in an unsaturated liquid environment. The liquid surface has to be convex toward the trapped gas and the crevice should be sharp enough to permit sufficiently large interface curvature to preclude any dissolution of the trapped gas in the unsaturated liquid surroundings. In fact, extremely narrow crevices can equilibrate the meniscus so that the crevice root has no gas in it and yet it remains unwet. Further, Atch- ley and Prosperetti •3 posit that the mechanism responsible for contact angle hysteresis may be operative even on submicrometer length scales. In the absence of any hysteresis in contact angle only crevices with specific geometries can survive in saturated liquids, and this would make cavitation a rare phenomenon.

While following the movement of the interface in crev- icelike geometries it is useful to note that for a fixed dissolved gas content of the liquid, an increase in static pressure makes the liquid behave as if it is undersaturated. Conversely a decrease in the static pressure of the liquid makes the liquid appear effectively supersaturated.

B. Incomplete wetting of nearly smooth spherical particles

Very few solids have geometrically smooth surfaces. Solids in a glass state may have smooth surfaces. Also, care- fully cleaved crystal surfaces can be smooth initially. From a practical point of view, the notion of smoothness for real surfaces implies that one ignores roughness below a certain length scale. If we decide not to be blind to the nature of things as they are even below such a fine length scale, then roughness must persist. Adsorption studies on various pow- der surfaces have revealed the presence of fractal surfaces even to the molecular levels. TM Consequently, at some scale of fineness there must be unwet sites on the solid surface, as liquids tend to minimize the free surface.

Our observations of the particles under the scanning electron microscope reveal that they are smooth down to the resolution limits of 500 • (or 50 nm). Hence we postulate that any roughness present must be at the nanometer length scales. These nanogas dots could be comprised of gas re- leased from the dissolved state, vapor molecules from the host liquid, and under favorable circumstances desorption of the adsorbed nonpolar constituents. How these unwet sites, the nano-gas-dots, can be activated for the purposes of cavitation will be seen in the following section.

C. Response of waterborne particle and nano-gas- dots to high-frequency sound fields

Batchelor •5 has analyzed the problem of a sphere set in motion relative to a fluid by the passage of a sound wave through the fluid. That analysis is repeated below in the context of this paper. We are considering waterborne spherical particles small compared to the wavelength of the sound wave. Let V be the velocity of the fluid everywhere in the neighborhood and at the location of the spherical particle, in the absence of that particle. The acceleration of this fluid, in the absence of the sphere, is approximately I•' (where the dot denotes differentiation with respect to time), the contribution V.VV being negligible for a sound wave (in the long- wavelength limit). In a reference frame moving with a velocity V and acceleration I•', there will appear in the equation of motion of the sphere an effective force -I7 per unit mass, which leads to a buoyancy force MoI•' on the body (M 0 being the mass of the fluid displaced by the sphere). Pro- vided the fluid moves irrotationally, the equation of motion, with no force applied directly to the sphere and with the neglect of gravity, is

M l) - - «M o ( l) - I• ) + M o I•, (1) where U is the sphere velocity relative to the unaccelerated axes, and M is its mass. Integrating the above equation and setting the constant of integration to zero, with the assumption that the sphere does not drift through the fluid, we have

U=[-}Mo/(M+ «Mo)]. (2)

This relation is applicable to the oscillations of a small sphere suspended in a fluid through which a sound wave is passing, provided that the frequency is large enough to make the thickness of the vorticity boundary layer small. The same



relation holds between the accelerations of the particle and the fluid. Since the model only considers a smooth particle whose surface is sparsely populated with nano-gas-dots the above analysis is adequate for the present purpose.

Vorticity arises wholly from the boundaries and, if the relative fluid motion at the boundaries is purely periodic (with frequency to in radians/second), the rate of generation of vorticity there is alternately positive and negative. It is reasonable to assume that no net vorticity is generated in one cycle and that the vorticity is zero outside a narrow region near the boundary within which the alternate layers of positive and negative vorticity are diffusing and canceling. The time available for diffusion of vorticity of one sign from the boundary is 2•rlw (0.03/as at 30 MHz), and the thickness of the layer of nonzero vorticity, 6, is of order (v/w) m, v being the kinematic fluid viscosity. At 30 MHz the thickness of the boundary layer, 6, for a particle in water is around 70 nm. This is an oscillating boundary layer with an exponentially decaying envelope extending into the surrounding fluid. The decaying velocity amplitude of this boundary layer is the relative velocity between the particle and the fluid. In the case considered, the density contrast between the polystyrene particles and the host fluid, water, is very slight. Conse- quently the boundary layer does not present strong velocity contrast. For a perfectly density matched particle the boundary layer vanishes altogether. The fact that the nano-gas-dots reside within the boundary layer does not preclude them from experiencing the acceleration field of the passing sound wave; regardless of where they are in the acceleration field, they experience the kinetic buoyancy effects.

Equation (2) indicates that a sphere heavier than the host fluid oscillates with an amplitude smaller than the surrounding fluid, and a sphere lighter than its surroundings oscillates with a larger amplitude than the surrounding fluid. The polystyrene spheres of density 1.05 g/cc are closely matched to the host fluid, water, whereas any air bubbles, being less dense than water by a factor of 830, tend to have an amplitude of oscillation 3 x greater than that of surrounding water. For the nano-gas-dots on the particle surface, such oscillations are of significance in that a nano-gas-dot may bridge with a neighboring dot. Even though the high-density contrast yields a limiting maximum relative amplitude ratio of

ß

three, the relative acceleration given by U-V can be very high, indeed. Additionally, because a more dense particle always lags the host liquid, and the nano-gas-dots lead the host liquid, the movement of the nano-gas-dots on the particle surface is reinforced.

One may extend the above ideas to understand how gas bubbles in liquids approach one another and coalesce when the bubbles undergo volume oscillations in phase. Each oscillating bubble produces an accelerating radial motion in the surrounding liquid and two neighboring bubbles are able to influence each other's motion. One needs to use the general form of Eq. (1) to include variation of displaced mass M0 due to volume oscillations. This reads

dU 1 d{Mo( U- V)} dV M dt 2 dt + Mø dt ' (3)

For a gas bubble in liquid, M<•M o , so that

• 3 Q-(U- V)Mo/M o. (4)

Now if V is periodic with zero mean and M 0 is periodic with a relatively small fluctuating part, the fluctuating part of U is approximately equal to 3 V and the average value of the bubble acceleration, (t)), over one cycle is

( t,))• - 2(VMo)/(Mo), (5)

which may be nonzero. Consider two spherical bubbles which are a distance r apart and displace masses p(r• + r[ sin tot), p(r2 + r• sin tot), respectively, of liquid density p. Here r• >> r[ and r2 >> r•. The first bubble produces at the position of the second an approximately uniform velocity of magnitude to r• cos wt/4•rr 2, and so the average acceleration of the second bubble along the line joining the bubbles becomes - 2 'r•/4 •r 2 the negative sign indicating at- O) 1•' 1 1•'2, traction toward the first bubble. This is the attraction between

two bubbles, or between one bubble and a plane boundary. This attraction ultimately leads to a steady drift velocity of each bubble, since viscous forces resist migration. The at- tractive force is usually small, but ultrasonic vibrations of a liquid are known to clear it of gas bubbles (this discussion has been reproduced almost exactly from Batchelor•5).

The above analysis guides us in thinking about the nano- gas-dots on a particle surface. The initial formation of the gas caps through the cooperative aggregation of nano-gas- dots may be seen as follows:

(1) Particle density does not differ significantly from the host liquid; consequently the particle follows the liquid faith- fully. The nano-gas-dots, however, have an amplitude of oscillation almost 3 X that of the surrounding liquid. This may lead to patch linking as at A, A' and patch distortion and possibly nonuniform enlargement as at B, B' (Fig. 2).

(2) Two pulsating hemispherical nano-gas-dots closely separated on a planermore specifically, closely separated 50-nm gas dots on a 1-/xm-diameter sphere surface-•can coalesce as a result of mutual attraction (for favorable conditions at the contact line, possibility of slipping or appropri- ately receding contact angle). This can progressively enlarge the gas cap.

(3) Additionally at B, as the particle accelerates to the left, indicated by the nonsolid (gray) arrow, the liquid in proximity should appear supersaturated (with dissolved gas) as a consequence of high accelerations, and may facilitate transfer of gas into the gas cap. (It is often observed that centrifuged samples of liquid indicate an increasing concentration of dissolved gases toward the center of rotation fre- quently giving rise to a cavitationlike severance of the liquid column from the center of rotation.) This may also be how the active detector field expropriates the dissolved gas from the host liquid into creating and growing nano-gas-dots.

(4) When the acceleration reverses, the gas patch developed at B tends to be squashed, flattened out against the barrier posed by the sphere surface, thus enlarging the area of the gas film patch on the particle. The gas patch, as at A, is free to oscillate to and fro on the top of the sphere. In the case of a nano-gas-dot midway between A and B (or, say, at 30 ø latitude with BB' as equator), during acceleration indicated by the solid arrow, it can move toward B easily. The



same nano-gas-dot during the reversed acceleration phase encounters the material barrier posed by the spherical particle, which hinders its reverse motion. Under these circumstances, the nano-gas-dot can be thought to be experiencing a rectified kinetic buoyancy, urging it to move only in one direction, toward B in the present case. This promotes the agglomeration of nano-gas-dots. A nano-gas-dot, as a rough estimate, experiences high uniform acceleration over every half a cycle in the active tone burst. It can then cover a distance of 1/2XaccelerationX (half-period) 2, which is around 10 • per (half) cycle. Now at 30 MHz there are 300 cycles in a 10-•s pulse, and the nano-gas-dot can move an estimated distance of 0.3/xm on the particle surface. A nano- gas-dot subjected to sharp pulses of acceleration combined with the rectified kinetic buoyancy can be thought of as experiencing "ratcheted motion." The relative velocity buildup in such sharp pulses is small enough for Stokes drag to be inconsequential, hence permitting the nano-gas-dot to traverse across the particle.

(5) Should a substantial gas patch develop at A, the particle could rotate under the unbalanced radiation torque so that A takes up the position at B or B'. We are postulating that the largest patch tends to be at B, B', fore and aft, along the axis of oscillation.

(6) The tensile field around the particle ensures that the gas patch is crescent shaped; the excess pressure for spherical surface formation in the presence of surface tension is provided by the tensile environment. Though the gas cap is likely to be partly vaporous, the original nano-gas-dots are unlikely to contain significant amounts of vapor. Through the gas caps we are only envisaging a discontinuity, the separa- tion between the liquid in contact with the solid sphere. Along the perimeter where the gas cap contacts the sphere, surface tension can effectively resist the tensile wrenching effective on the gas cap, only up to a critical gas cap size. The critical gas cap size is discussed in the next section. However, it should be noted that the sum total of the gas initially present in all the nano-gas-dots on the particle that gives rise to the gas caps need not equal the volume of the gas caps. What is crucial to the model is that surface tension finds an adequate perimeter--three phase contact line--to act on, to resist the tensile field.

Before discussing the cavitation event let us briefly note the effect of viscosity. Following Skudrzyk, 16 Eq. (1) can be written as follows:

•+ «40) (0- •)=0. (6)

Since no external force is applied to the liquidborne spherical particle as the sound wave passes its location, this equation expresses the motion of that liquidborne particle, as an equi-

.

librium of two forces: (M-M0)V, the force needed to drive the sphere in exactly the same manner as the fluid particles in the incident wave, and (M+ «M0)(•-l?), the additional force needed to provide for the relative velocity of the particle with respect to the fluid. Viscous effects in terms of a Stokes drag on a particle of radius R can be included in the statement of equilibrium as

(M- M0) 12+ 6 rr•R( U- V) + (M + «M0) (/_)- 12)=0. (7)

Introducing harmonic time dependence of 6o= 2 rrf, where f is the frequency of the acoustic wave, we have

(M-Mo)jtoV+6rrixR( U- V)+(M+ «Mo)ito( U- V)=0. (8)

Defining the parameters f0 and/5 as

6 rr/xR f

fø:2rr(M0+ «M0)' /•:•00' (9) we can express the ratio of the particle velocity to the fluid velocity by

U 1 _j•-i . (10) 1 V [(M+•Mo)/(Mo+«Mo)]_j•-•

f0, the characteristic frequency, can also be written in terms of the radius of the spherical particle as fo=(3•/p)/2rrR 2. For f<fo viscous effects dominate and the spherical particle is almost completely entrained, while for f>fo inertial effects are dominant and the gas cap formation is likely to be vigorous. The characteristic frequencies for particles sizes of 0.984, 0.481, and 0.245/xm are 2.16, 9, and 34 MHz, respectively. Since the active detector frequency is 30 MHz it ren- ders the larger two of the three particle sizes inertially dominated. For particles of diameter smaller than 0.245 •m in the presence of the active detector field of 30 MHz, the gas cap formation is likely to be sluggish compared to the larger particles. Amongst all the particles smaller than the wavelength, this viscous analysis actually helps us understand the significance of the smallness of such small particles as re- gards the efficacy of particle entrainment in the sound wave.

D. The cavitation event

When the tensile pressure acting on the gas cap can overcome the surface tension resistance along the perimeter of contact, fracture occurs. It is this fracturing event and the consequent bubble activity that is picked up by the detector transducers. The peak negative pressure amplitude which causes this fracture is the cavitation threshold. While in the

context of Fig. 2 the equation

p rrr2 - 2 rrr rr (11)

establishes the equilibrium condition, it does not explicitly state the nature of the equilibrium. Here 2 r = d is the gas cap diameter, rr is the surface tension for water, and p is the pressure (tensile) in the cavitation field. The stability of a free bubble has been formulated in terms of the Blake thresh-

old, and for our purposes we may extend the Blake threshold arguments to nano-gas-dots by noting that for a half-bubble (or a partial bubble) on a plane, the same analysis applies with little modification in both the extreme conditions of the

contact line motion: one permitting a freely slipping contact line on a solid surface, and the other where the contact line is completely anchored in the initial configuration. This partial bubble may be the interface coming out of the crevice mouth. One can then think of such half-bubbles or nano-gas-



dots as being equivalent to free bubbles of the same radius of curvature for interpretation in terms of the Blake threshold. Typically Blake thresholds of 55, 11.8, 6.2, and 1.4 bar peak negative correspond to air bubbles of radii 10, 50, 100, and 1000 nm, respectively, or the same Blake threshold values correspond to half-bubbles of the same given sizes. In the case of a half-bubble on a plane, the radius of curvature is same as the gas cap size. However, as suggested by the scenario, the radius of curvature may be comparable to the particle size. Then, for the three sizes of particles used, with 0.984-, 0.481-, and 0.245-/.tm diameters, the Blake thresholds are 1.96, 3.1, and 5.3 bar peak negative, respectively. So, even apart from the fact that high-frequency fields do not afford enough time for significant bubble growth, the maximum pressure amplitude available in the active detector field, 0.5 bar peak negative, is well below the Blake threshold value to cause unbounded growth in the gas caps. It certainly is unable to wrench off the gas cap; Eq. (11) would necessitate a gas cap size much larger than the particle size. Also, the observed thresholds for the various particle sizes, as mentioned later in Sec. V A, are always greater than the corresponding Blake thresholds, and are also greater than the tensile pressures required to form gas bubbles encapsulating the spherical particles.

Summarizing, the acoustic field of the active detector only preprocesses the particles for cavitation by engendering the formation of a gas cap larger than the nano-gas-dots envisaged earlier. As the particle is convected into the focal zone of the main cavitation field, it experiences stronger tensile environment and more pronounced motion of the gas caps, which causes them to reach critical size, whence the tensile rupture ensues, at threshold values.

If we assume that the surface tension is fixed in a given experiment, then for a given threshold pressure [cf. Eq. (11)] smaller gas caps will be surface tension dominated and are stable against rupture. They are stable against being completely shut by surface tension because of the gas content of the agglomerated nano-gas-dots, which ultimately resist the surface tension forces at some shrunk size. Similarly larger gas caps are pressure dominated and should rupture immedi- ately at threshold pressures. The gas cap structure is supported by the tensile field around the particle, and a cap is mainly an empty cavity. Its gas content and vapor content are not needed to support its shape; the force needed to balance the surface tension comes from the tensile environment sur-

rounding the particle. This equilibrium of the gas cap, where the surface tension forces balance the tensile forces, is an unstable one. The acceleration effects of the active detector

field commence the gas cap formation on the particle, grow it to the critical equilibrium radius in the tensile environment of the cavitation field, and eventually lead to rupture at threshold conditions.

A question naturally arises: Suppose there is no active detector field, and the main cavitation transducer is acting alone. Should its field not give rise to the acceleration effects like the active detector and affect the thresholds? This question is explored in the following section.

I ' I ' I ' I ' I I

ß 0.245 gm spheres

ß 0.481 gm spheres

ß 0.984 gm spheres

0.0 0.2 0.4 0.6 0.8 1.0

Normalized Active Detector Field

FIG. 3. Particle size effect on passive thresholds. The monodispersed polystyrene latex microspheres of three different sizes were Used: 0.254, 0.481, and 0.984/xm. In the plot they are represented by solid squares, triangles, and circles, respectively. The particle number density was maintained the same--l.9X108 particles/cc. Passive thresholds decrease with increase in particle size, and in all the cases thresholds invariably decrease with stronger active detector field. With the active detector switched off, the passive threshold for different particles has almost the same value (see Ref. 2). Normal active field corresponds to 30-MHz tone burst, 10-/xs pulse, 1-kHz PRF, confocal setting. Test cell: water (70% dissolved air saturation).

V. ADDITIONAL EXPERIMENTS

A. Passive thresholds for various active detector

settings and various particle sizes

Figure 3 shows how the passive thresholds vary in the presence of the active detector field. At normal setting the active detector is operated at 10-/.rs pulse width, 1-kHz PRF, and a peak pressure amplitude of 0.5 bars. Zero on the ab- scissa corresponds to zero active detector field: active detector switched off. Essentially monodispersed, polystyrene particles of three different sizes, having mean diameters of 0.245, 0.481, and 0.984/.tm, were tested. 2 The particle number density was maintained the same at 1.9 x 108 particles/cc. The dissolved air saturation in the test cell was held around

70%.

When the active detector is switched off the thresholds

are solely due to the main cavitation field because the active detector cannot participate in the cavitation process. The observed passive thresholds are around 15 bar peak' negative with the smaller particles giving a slightly greater value (Fig. 3). It should be noted that if the main cavitation transducer were to act alone, in order to give rise to acceleration levels as high as those due to the active detector operating at 30 MHz at a pressure amplitude of 0.5 bar peak negative, the



pressure amplitude of the main cavitation transducer ought to be greater by a factor of 30 so as to keep the "frequency- pressure" product almost the same. At similar acceleration levels it is reasonable to expect that the particles will be processed for gas cap formation similarly. In the case of the main cavitation transducer acting alone to produce comparable levels of acceleration,' the cavitation event can take place readily as the main cavitation field is already stronger than the threshold values when the active detector field is

present. At the intermediate levels of the active detector settings the observed passive detector thresholds assume intermediate values. While there is nothing unusual about this trend, understanding the cooperative mechanics between the active detector field and the main cavitation field at these

settings will be crucial to quantifying the frequency effects in acoustic coaxing induced microcavitation.

Note that experiments with increased number density of particles in the test cell failed to produce any saturation effects; the passive thresholds in the presence of the active field remained lower than those in the absence of the active

field.

B. Effect of the active detector field on silica particles

In an attempt to seek a more conclusive evidence for the effect of active detector field on acoustic microcavitation, the

following experiment is useful. If the nano-gas-dots on the particle surface are responsible for starting the gas caps, then can we test particles for cavitation before and after purging the nano-gas-dots?

Silica microspheres, developed for packing columns for liquid chromatography, seem to be a suitable candidate. These particles are grown by a polymerization process, called the "sol-gel" process. We have used 0.75-/zm- diameter particles, suspended in distilled water and completely surfactant free. The particles were smooth, spherical, and monodispersed. The particle material was unfused quartz of density 2.1 g/cc. In the unfused state surface porosity is conducive to the formation of nano-gas-dots. However, on sintering, due to surface fusion and glass formation, the particle surface is rendered almost free of any nano-gas-dots. A way to test for the degree of surface fusion is to test the silica particles for alkali tolerance: Unfused silica is strongly at- tacked by strong alkalis like NaOH. Strong alkalis are regu- larly stored in glass bottles, while unfused silica readily vanishes in such alkalis. Before measuring the thresholds both the sintered and unsintered silica particles were qualified for smoothness by SEM examination.

The experiment was conducted at a particle number density of 3.64X108 particles/cc. The dissolved air saturation in the test cell was around 80%. The observed passive thresholds in the presence of the normal active detector field for unsintered silica were around 6 bar peak negative while those for sintered silica particles, under same test conditions, were around 13 bar peak negative. 2 This supports the hy- pothesis that the presence of the nano-gas-dots is essential for initiating the gas cap formation under the dictates of the active detector field. The stronger density contrast between the silica/water combination compared to polystyrene/water

combination did not seem to affect the observed cavitation

thresholds strongly.

Vl. CONCLUSIONS AND PROPOSED EXPERIMENTS

We have attempted to conceptualize a mechanism by which smooth, spherical particles may nucleate cavitation events. Ordinarily when exposed to a strong sound field (the main cavitation field), liquidborne microparticles may cause little cavitation at threshold conditions. But, if in addition to the sound field there exists a weak, high-frequency auxiliary field, such as the active detector field, cavitation by the microparticles is readily facilitated and cavitation thresholds are significantly reduced. However, these reduced thresholds are still greater than the corresponding Blake thresholds and even those based on a bubble formation encapsulating the particle. The mechanism of acoustic coaxing conceptualized here explains how the observed thresholds might come about. While we have attempted to establish the reasonable- ness of such a process, detailed experiments are needed to quantify the coaxing effect. A brief list of some of the experiments to be performed is as follows:

(1) The direct experimental demonstration of the coaxing effect is possible through real-time scattering studies of a single leviated particle or a tagged particle. How the weak, high-frequency field helps the oscillating liquidborne micro- particle to expropriate dissolved gas from the host liquid onto the nano-unwet sites on the particle surface can be observed by monitoring the scattering signature throughout the evolution of the gas caps.

(2) Attempt to establish the acceleration level below which no significant acoustic coaxing occurs. Adjust the active detector parameters so that it no longer reduces the thresholds. Attempt to improve its sensitivity if necessary. With the active detector operating only below these cutoff parameters, try other high-frequency fields. Test how the acceleration, effective in acoustic coaxing, depends on field parameters. Adjust the operating pressure amplitude and the frequency of the auxiliary acoustic field (the coaxing field) to obtain a fixed acceleration level, especially at various frequencies, and check if the threshold reduction is consistent. This should lend credence to the argument: agglomeration of nanogas dots through acceleration effects. Initially, instead of trying to operate the active detector below the cutoff levels, use a passive detector and seek coaxing through different frequencies of the auxiliary field. Investigate acoustic coaxing in terms of the time exposure of the particles being processed for gas cap formation.

(3) Keeping the active detector the same, measure thresholds for different frequencies of the main-cavitation field. It is important to do this test with the particles suspended in the test tank and also by running a sheath-flow jet of particles through the common focus of the cavitation and the detection fields. In the latter case, since the particles are exposed to the active detector field only for the short time that it takes the jet to convect them, the particles may not be processed effectively for the gas cap formation, and hence there may be little reduction in the observed thresholds.

(4) Keeping the active detector the same, measure thresholds for different frequencies of the main-cavitation



field for a fixed length of the tone burst. As remarked earlier in Sec. V A the statement--similar levels of acceleration

may process the particles for gas cap formation similarly.-- needs to be quantified. Even though the frequency-pressure product determines the effective acceleration level, the manner in which the acceleration pulses are effected is expected to influence the movement of the nano-gas-dots on the particle surface.

(5) Observe the coaxing effect on prepressurized samples of particle suspensions. Prepressurization is expected to denucleate particles by possibly closing off a few nanocrevices, and hence impede gas cap formation.

(6) One needs to study the cavitation activity above threshold conditions in the presence and absence of the active field.

ACKNOWLEDGMENTS

I am grateful to professors R. E. Apfel, B. T. Chu, A. Nadim, and A. A. Atchley for the discussions during the writing of this paper. This work was supported by NSF Grant No. CTS-9110920.

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a conceptual model for acoustic microcavitation

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