characteristic times for pressure and electrostatic force driven thin film drainage

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IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE

Copyright copy 2008 American Scientific PublishersAll rights reservedPrinted in the United States of America

Journal ofComputational and Theoretical Nanoscience

Vol 5 2060ndash2066 2008

Characteristic Times for Pressure andElectrostatic Force Driven Thin Film Drainage

David G Crawford Charles R Koch and Subir Bhattacharjeelowast

Department of Mechanical Engineering 4-9 Mechanical Engineering Building University of AlbertaEdmonton AB T6G 2G8 Canada

The drainage of thin liquid films confined between two electrodes was studied employing the longwave approximation of the pertinent hydrodynamic equations A general mathematical formulationwas used for the case of electric field driven drainage of the film by solving the Laplace equationin conjunction with the approximate flow equations The drainage of small periodic segments of thefilm was then studied employing the model under conditions that emulate purely pressure baseddrainage as well as drainage in presence of an applied electric field The final film morphologiesand the drainage times in presence of electric fields were found to be considerably different fromthose observed for pressure based drainage

Keywords Thin Liquid Films Electrostatic Drainage Long Wave Approximation FiniteDifference van der Waals Interaction

1 INTRODUCTION

Thin macroscopic films occur in a wide range of biolog-ical physical and engineering systems When the filmsare subjected to mechanical thermal chemical or elec-trical perturbations their drainage can occur due to adiverse range of dynamics such as wave propagationwave steepening and chaotic responses The films mayor may not rupture holes or dry spots may form andother complicated structures may result The phenomenathat govern thin film drainage are different than those intypical bulk fluids and present unique challenges1 In thinfilms with thickness lt100 nm fluid molecules interactnot only with surrounding fluid molecules (as in a typi-cal bulk environment) but also the materials that boundthe film23 The strength and range of the intermolecu-lar forces depend on the physical and chemical proper-ties of the system and are predominantly classified as theLifshitz van der-Waals (LVDW) electrostatic and polarinteractions3 The excess intermolecular interactions resultin a conjoiningdisjoining pressure which causes filmdrainage

Modification of the drainage behavior of films by apply-ing an electric field has immense importance in a vari-ety of applications including formation of well-controlledpatterns leading to novel nanostructures of relevance tooptoelectronic devices sensors biochips and membranes4

lowastAuthor to whom correspondence should be addressed

Considerable theoretical attention has been devoted to elu-cidate pattern formation in thin polymeric films by appli-cation of electric fields5ndash9 Most of these studies primarilydevote their attention to the relationship between the per-tinent length scales and the range of the interactions thatdetermine the final structure of the film after drainageIn this context a comparison of the time scales betweenconventional pressure based drainage and electric fieldinduced drainage has not been a major focus of these stud-ies Some of our recent experiments with electric fieldinduced drainage of free films lead to the notion that elec-tric fields result in a much more rapid drainage of the filmsand the ensuing patterns can be more complex compared topressure induced drainage10 To address this observationthis paper theoretically explores the electric field induceddrainage of thin films supported by a solid substrate andcompares the drainage times and final patterns with thoseobtained during solely pressure based drainage

2 MATHEMATICAL MODEL

When a thin film is perturbed by an infinitesimal distur-bance the perturbations may grow or be damped out Ifthe film is thicker then a critical thickness the perturba-tions will grow in a periodic manner creating waves on thefilm surface211 The perturbations grow with a character-istic wavelength () The wavelengths of the disturbanceare much larger than the mean film thickness The differ-ence in length scale between the film thickness and the

2060 J Comput Theor Nanosci 2008 Vol 5 No 10 1546-1955200852060007 doi101166jctn20081014


IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE

Crawford et al Characteristic Times for Pressure and Electrostatic Force Driven Thin Film Drainage

z = 0

z = h0

z = d

λ

z = h(x y t)

Upper electrode

Bounding fluid ε2 micro2

Lower electrode

x

z

Liquid film ε1 micro1

ψ = ψlow

ψ = ψup

Fig 1 Schematic representation of the thin film system The liquid filmunder consideration is supported by the lower electrode The boundingfluid is modeled as an inert fluid (similar to air) with 2 1 Forelectrostatic force induced drainage a potential difference of low minusup

is applied between the two electrodes with up typically set as groundThe initial height of the unperturbed film is h0 and the vertical distancebetween the two electrodes is d

waves that form on the film interface allow the simplifica-tion of the governing equations A lsquolong-waversquo asymptoticanalysis can be used to simplify the governing equations1

The thin film system studied is shown schematically inFigure 1 It depicts a liquid film with a bounding fluidon top sandwiched between two electrodes As shown inthe figure d is the electrode spacing hx y t is the filmthickness at spatial coordinates (x y) and time (t) h0 isthe average initial film thickness subscript i indicates thefluid phase under consideration with i= 1 for the film andi = 2 for the bounding fluid i (i = 12) indicates thepermittivity of fluid i i is the viscosity of fluid (i= 12) is the wavelength of disturbances on the film interfaceup is an applied voltage at upper electrode (z = d) andlow is the applied voltage at the lower electrode (z= 0)

21 Governing Equations

The fluids are modeled using a continuum assumption andthe motion is defined by the Navier Stokes (NS) equationsand a continuity relation

i

( ui

t+ ui middot ui

)

=minuspi + middot i ui + uiT + fi (1a)

middot i ui= 0 (1b)

The term ui is the velocity vector (u= u vw) pi is thepressure i is the density is the cartesian operator =x y z and fi = minusi represents the bodyforces where i is the conjoiningdisjoining pressure Theconjoiningdisjoining pressure is an excess energy per unitvolume that describes forces that exist on the film TheNavier Stokes system shown in Eq (1) is applied for boththe film and bounding fluid (i = 12)

The general formulation for thin film drainage is com-pleted by the definition of boundary conditions At the

upper and lower electrodes no penetration and no slipboundary conditions are applied

u1 = 0 at z= 0

u2 = 0 at z= d(2)

At the interface between the fluids continuity of velocityand a stress balance exist The two fluids are modeled asimmiscible so no fluid mixing or mass transfer occursat their interface This ensures that a sharp interface ismaintained and that velocity is continuous across it

u1 = u2 at z= hx y t (3)

The velocity of the fluids at the interface is related to thefilm thickness by a kinematic condition11 The kinematiccondition is obtained by taking the partial derivative ofthe film thickness z= hx y t with respect to time Thechain rule and velocity definitions can then be used torelate the film thickness to the interfacial velocity

w = h

t+u

h

x+v

h

yat z= hx y t (4)

The kinematic condition in Eq (4) is used in place of thecontinuity of velocity (Eq 3)

At the interface a stress balance between the two fluidsis performed by taking the normal and tangential projec-tions of the stresses at the fluid interface

n middot macr1 middot nminus macr2 middot n = + fe middot n (5)

tj middot macr1 middot nminus macr2 middot n= fe middot tj (6)

where tj = tjx tjy tjz is the jth unit tangent vector tothe fluid interface where (j = 12) n is the unit normalvector to the fluid interface which points from the film tothe bounding fluid with components n = nxny nz isthe mean interfacial curvature is the interfacial tensionfe is the external surface force vector and macri is the stresstensor for the ith fluid phase (i = 12)

The solution of the NS equation system presented abovewould be difficult since the interface between the fluidsis a free boundary Fortunately the problem scaling andphysics allow the system to be reduced to a single evolu-tion equation for the film thickness The lsquolong waversquo orlsquosmall slopersquo scaling argument uses the fact that dominantwavelength of disturbances which dictates the lengthscale of the problem is much larger than the film thick-ness h171112 For simplicity a scaling factor k is definedto indicate the difference in lengths and aid in the simpli-fication of the hydrodynamic equations

k = h0

1 (7)

where h0 is the initial mean film thickness The lengths arenon-dimensionalized by using the mean film thickness h0

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RESEARCHARTICLE

Characteristic Times for Pressure and Electrostatic Force Driven Thin Film Drainage Crawford et al

for z and the fastest growing disturbance for x and yThe scaled spatial dimensions and the spatial derivativesare explicitly given by (the uppercase variables representthe scaled quantities)

x = X y = Y z= h0Z (8a)

x= 1

X

y= 1

Y

z= 1

h0

Z(8b)

Equation (8) are simplified using Eq (7) and can be usedto determine the scaling required for the other problemparameters If the fluid velocity components in the lat-eral directions (x y) will have characteristic values u0v0 = O1 then the non-dimensional velocities in thesedirections can be written as

U = u

u0

V = v

u0

(9)

The continuity Eq (1) using relations (8) and (9)becomes

u0

U

X+ u0

V

Y+ w

z= 0 (10)

which suggests a scaling of the fluid velocity in z direc-tion as

W =

u0h0

w = w

u0k(11)

The time is re-scaled to reflect the long length scalesconsidered defining a new slow time (T ) based on thecharacteristic velocity (u0) and long length scale ()

T = ku0

h0

t (12)

Following the work of Oron et al1 the flow within thefilm is locally parallel so that px sim uzz and py sim vzzThis suggests scaling for the pressure (P ) and body forces() as

P= kh0

u0

p (13)

Re-writing the NS system with the scalings introducedby Eqs (8ndash13) gives

kRe UT +UUX +VUY +WUZ

=minusP +X +k2UXX +UYY +UZZ (14a)

kRe VT +UVX +VVY +WVZ

=minusP +Y +k2VXX +VYY +VZZ (14b)

k3Re WT +UWX +VWY +WWZ

=minusP +Z +k2k2WXX +WYY +WZZ (14c)

0 = UX +VY +WZ (14d)

where Eqs (14andash14d) apply for both fluids The subscriptsin the above equations represent derivatives with respect tothe scaled variable The boundary conditions at the upperand lower surface are not changed by the scaling

The focus of this study is confined films where a viscos-ity ratio of the film to bounding fluid is large (12 1)Therefore in the fluid films studied the bounding fluidacts like an inert gas and eliminates the requirement forsolving the evolution equations for both fluids Since thefluid film is thin (h0 lt 100 nm) the inertial effects willbe negligible (Re = u0h0 1) These simplificationsalong with consideration of the leading order terms inEqs (14andash14d) yield

minusPX +X+UZZ = 0 (15a)

minusPY +Y +VZZ = 0 (15b)

PZ +Z = 0 (15c)

UX +VY +WZ = 0 (15d)

The fluid properties are assumed constant in space andtime in the derivation of Eqs (15andash15d) Equations (15)form the basis for the derivation of the interface motionequation The boundary conditions used are

U = 0 at z= 0 (16a)

UZ = VZ = 0 at z= hx y t (16b)

P minusPext = HXX +HYY at z= hx y t (16c)

where Pext is the external reference pressure (set to zero)The kinematic condition at the fluid interface z= hx y tis also used to relate the temporal changes of the interfaceto the fluid velocity

HT +UHX +VHY =W (17)

The interface motion equation is derived using the sys-tem of equations shown in (15) with the boundary condi-tions from (16) and the kinematic condition (17)

Extracting the relationships between the scaled veloc-ity components U V and W with the scaled film thick-ness H and the pressure P from the above equationsleads to the final evolution equation for the film

H

t+

X

[H 3 0

X

]+

Y

[H 3 0

Y

]= 0 (18)

where

0 = 2H

X2+ 2H

Y 2minus (19)

The function represents the overall body forces actingon the film which results from intermolecular and electro-static interactions

The intermolecular forces considered in the presentwork consists of a long range Lifshitz-van der Waals(LVDW) force and a short ranged repulsion similar toBorn repulsion The energy per unit volume or conjoin-ingdisjoining pressure in the film due to the LVDW forcescan be written as213

LV DW = AL

63h3(20)

2062 J Comput Theor Nanosci 5 2060ndash2066 2008


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RESEARCHARTICLE


where AL is the effective Hamaker constant for interactionsbetween the upper (free) surface of the film and lower solidthrough the fluid film The above expression diverges whenthe film dries out at the lower solid substrate (h= 0) Thisdivergence is prevented by considering a repulsive forcethat maintains a minimum equilibrium liquid thickness onthe substrate as suggested by Mitlin14 In this work anabsorbed layer of thickness l0 = 1 nm is maintained oneither solid surface A 1 nm thick absorbed layer doesnot significantly influence the film drainage while ensuringthe continuity assumption remains valid The close rangerepulsive term takes the form

BL=minus8BL

h9(21)

The close range repulsive constant BL is found by set-ting the conjoiningdisjoining pressure equal to zero at thelower surface (z= l0= 0)

The electrostatics driven drainage of the film is causedby the Maxwell stress at the fluid interface This stressis computed by solving for the electric potential distribu-tion between the two electrodes represented by the Laplaceequation

2i = 0 i = 12 (22)

where i is the potential in fluid i = 12 The boundaryconditions for the electrostatic problem are defined as

1 = low at z= 0

1 = 2 at z= hx y t

1

1

z= 2

2

zat z= hx y t

2 = 0 at z= d

(23)

The above system of equations can be solved analyti-cally for a thin film trapped between two homogeneouselectrodes The potential energy per unit area due to theelectrostatic interaction and the electric field at the film-bounding fluid interface respectively are given by4

EL =minus0650pp minus1E2p (24)

and

Ep =low

pdminus p minus1h(25)

Here p = 12 is the ratio of relative permittivity of thefilm liquid to the bounding fluid

The contributions of electrostatics and the intermolec-ular body forces are added to yield the total conjoin-ingdisjoining pressure

= LV DW +BL +EL (26)

22 Scaling

The problem scaling is derived from a linear stability for-mulation for the thin film equation The general form forthe time (tlowast) and length (xlowast) scales as given by Vermaet al13 from the linear stability analysis are

tlowast = 12

h3h2xlowast =

radicminus832

h(27)

In the case of pressure dominated flow where only LVDWforces are present the time and length scales can befound using Eq (27) Following the work of Khanna andSharma12 the time (tlowast) and length (xlowast) scaling as well asthe dominant length scale (c) for a pressure dominatedflow are

tlowast = 1232h50

A2L

xlowast = h20

radic23

AL

(28)

cLV DW = 23h20

radic23

AL

(29)

The scaling for an electrostatically dominated flow is

tlowast = 3h30

0650pp minus12low

2

xlowast = 23

radich3

0

0650pp minus12low

(30)

cEL = 23

radic2b

0pp minus12Eminus32

p (31)

3 NUMERICAL METHOD

The solution methodology employed was the discretiza-tion of spatial derivatives using finite differences to reducethe partial differential equation system to a differential-algebraic equation (DAE) system in time which wasthen solved employing a differential algebraic solver(DASSL)15 In the simulations performed a uniform carte-sian grid was used The thin film evolution equation wasdiscretized using central differences in a pseudo stag-gered grid approach The simulations were performed ona film cross-section of dimensions 777 m times 7677 mand grid size used to discretize the spatial domain (XY )was 60times60 The time steps were modified adaptively bythe DAE solver

The discretization scheme was formulated to avoidso-called lsquochecker boardingrsquo in the solution This formula-tion requires mid-point values for the film thickness whichare used in the non-linear terms The mid-point valuescan either be approximated based on calculated values orobtained by solving additional equations The latter wouldrequire an additional set of nx minus1ny minus1 equations and



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RESEARCHARTICLE


the storage of accompanying values Since significantlyincreasing the number of equations was not desirable themid-point values were approximated using a third orderEverettrsquos method16

In the problems considered in this work periodic bound-ary conditions were employed In this case the assumptionwas that the portion of the domain modeled was repeatedas periodic images over the entire system This means thatany flow out of a boundary will result in inflow on theopposite boundary Following11 the boundary conditionimplies that the first and last points on a row or column ofmesh are equal This means that a continuity of derivativesoccurs over the boundaries

The initial condition for the problem was assignedas a random small-amplitude perturbation of theheight H around the initial equilibrium film thickness(H0 = hh0 = 1) The maximum amplitude of the pertur-bation was ca 1 of the initial scaled height

A fortran program was developed to implement the solu-tion algorithm The program was run on a personal com-puter with an Intel Pentium IV 2 GHz processor and 2 GBRAM A typical dynamic simulation required about twohours of run time for completion

4 RESULTS AND DISCUSSION

In this section we compare the evolved film morpholo-gies and the time scales observed during pressure basedand electrostatically driven drainage of thin films Thecomparison of film morphologies pertain to qualitativeobservations of the film structures and length scales ofdominant features The comparison of time scales is con-ducted by studying the times required to attain the finalpseudo-stationary states starting from the initial uniformthickness All simulations were performed for a homoge-neous liquid film formed on a chemically homogeneoussubstrate

41 Pressure Based Drainage

Here we examine pressure based drainage on a homoge-neous substrate In pressure based drainage only LVDWinteractions influence the film drainage The base lineparameters for pressure induced drainage on a homoge-neous substrate are shown in Table I The values in Table Iwere used for all simulations that follow unless explicitlystated otherwise

Figure 2 shows the time evolution of the pressure baseddrainage for an initially 20 nm thick film A square domainof side L = 2c where c is given by Eq (29) wasused in these simulations The domain was discretized intoa 60times 60 cartesian grid The gray-scale levels representthe non-dimensional film thickness The film drains froma small initial random perturbation eventually forming acoherent structure of depressions and bumps Some of the

Table I Base line parameters employed in the simulations for thin filmdrainage

Parameter Value

Viscosity (film ) 1 Pa sInterfacial tension () 0038 NmRelative permittivity ratio (p) 25 [minus]Electrode spacing (d) 100 nmInitial film thickness (h0) 20 nmFilm dimension (square) (L) 777 mEffective Hamaker constant (AL) 1times10minus19 JShort-range repulsion constant (BL) 6663times10minus76 Jm6

depressions randomly turn into the fastest draining partsof the film and eventually completely drain to form cir-cular holes The number of holes then grows and in thefinal stages of the drainage the spacing between two adja-cent holes (shown in the final image of Fig 2) is 1638cLinear theory predicts that hole spacing in pressure baseddrainage should have a spacing of

radic2c Thus our numer-

ical simulations seem to be in reasonable agreement withlinear theory12

The time evolution of the non-dimensional film thick-ness along a section shown by the dashed line inFigure 2(f) is depicted in Figure 3 The 2D section viewsshow the non-dimensional film thickness corresponding tothe slice at Y = 560256 The section cut was chosen sothat it intersected the first hole that forms in the domainThe figure shows that ridges or lips form around the edgesof the hole It also demonstrates the filmrsquos behavior awayfrom the hole As holes or dry spots in the film form theremainder of the film thickens and assumes a new equi-librium height but is otherwise not influenced by the for-mation of the hole Pressure driven film drainage results

(a) (b) (c)

(d) (e) (f)

X

Y

Fig 2 Evolution of a 20 nm thick film taken as gray scale images ofthe film thickness The times corresponding to the snapshots are (a) 058(b) 2140 (c) 3082 (d) 3298 (e) 3514 and (f) 3572 seconds A sixlevel linear gray scale was used in all images shown The maximum is2596 nm (white) and the minimum is 1 nm (black) All other parameterswere taken as base line values (Table (I)) The mean thickness of thefilm in (a) is 20 nm with a lt1 amplitude random perturbation Thescaled dimensions in the X and Y directions in each panel range fromminus23 to 23 The origin is located at the center of each panel (shown inpanel (a))



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ndash6 ndash4 ndash2 0 2 4 60

02

04

06

08

1

Non-dimensional X [ndash]

Non

-dim

ensi

onal

film

thic

knes

s (h

h0)

t = 057 s t = 3082 st = 3298 st = 3514 st = 3572 s

Fig 3 Section views of the non-dimensional film thickness at differenttimes The section views are taken at Y = 560256 (dashed line in panel (f)of Fig 2) All other parameters were taken as base line values (Table I)

in the formation a series of circular holes and surround-ing ridges The drainage is very slow at the initial stageswhile the final morphology is attained very rapidly overthe last few seconds

42 Electrostatic Drainage

Figure 4 shows a series of gray scale images that representthe final film structures formed by drainage of an initially20 nm thick film subjected to increasing initial appliedelectric fields cf Eq (25) on a uniform domain size ofL = 7677 m As the magnitude of the applied electricfield is increased the maximum height that the film attainsincreases Comparing panels (a) and (b) of Figure 4 it isapparent that the final film morphology observed for solelypressure based drainage (E = 0 panel (a)) is quite similar

Fig 4 Comparison of final film structures for a 20 nm film subjectedto different magnitudes of electric field The magnitude of the initiallyapplied electric fields at the interface are (from panel a to panel (f)respectively) Ep = 0 045 450 1136 2727 and 4545 MVm A lineargray scale is shown in all images with a maximum value of 3961 nm(white) to a minimum value of 1 nm (black) All other parameters are asstated in Table I

to the structure observed in presence of an applied electricfield of 045 MVm (panel (b)) When the applied fieldis increased another order of magnitude to 450 MVm(panel (c)) the film morphology changes noticeably Thehole growth becomes more uniform with the size of theholes that form becoming almost equal The applicationof small electric fields even as low as 450 MVm adduniformity to the structure that is formed This behavioris also observed in panel (d) corresponding to an appliedfield of 1136 MVm When the applied field was increasedto 2727 MVm (panel (e)) the film morphology againchanges significantly The size of the holes that form isreduced and localized elevated patches or columns can beseen As the magnitude of the applied field is increasedto 4545 MVm (panel (f)) the size of holes and columnsthat form becomes even smaller

The morphology of the drained film clearly indicateshow increase in the applied field above a threshold valueof ca 5 MVm changes the nature of drainage At higherapplied fields the film morphology develops subtle dif-ferences from those obtained under solely pressure drivendrainage With the application of an electric field ofmagnitude gt 25 MVm the film drainage becomes dom-inated by electrostatic effects as evidenced by formationof columnar structures separating smaller holes as well asan increase in the density of features observed over thesimulated region The application of an electric field willalso increase the maximum observed film thickness andin the case where large fields are applied (Ep gt 25 MVm)columns will form

Figure 5 shows the evolution of the minimum non-dimensional film thickness with time The minimum

0 10 20 30 40 500

01

02

03

04

05

06

07

08

09

1

Time (t) [s]

ψlow = 00 Vψlow = 01 Vψlow = 10 Vψlow = 25 Vψlow = 50 Vψlow = 100 V

Min

imum

dim

ensi

onle

ss fi

lm th

icne

ss (

Hm

in)

[ndash]

Fig 5 Minimum non-dimensional film thickness for an initially 20 nmthick film for various applied potentials on the lower electrode lowkeeping the upper electrode grounded The initial electric fields atthe interface (Ep) corresponding to these potentials are shown inFigures 4(a)ndash(f) respectively The minimum thickness corresponds to thefastest draining location on the film All other parameters are as statedin Table I



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RESEARCHARTICLE


thickness was found by locating the fastest growing holeand tracking its drainage The drainage time decreasesas the magnitude of the applied voltage on the lowerelectrode low and therefore the initial electric field atthe interface cf Eq (25) is increased Note that theelectric field changes with time as the film morphologyevolves in each simulation although the applied poten-tial low remains fixed For applied voltages of up to01 V no perceptible change in the drainage behavior com-pared to pressure based drainage is observed At higherapplied potentials the drainage becomes quicker The ini-tial decrease in height is almost independent of the appliedpotential following which a rapid change in height withtime is observed As the magnitude of the applied elec-tric field is increased the time taken to attain the finalheight is considerably reduced Finally at applied voltageof 10 V the drainage is virtually instantaneous comparedto the time taken for drainage of a film under the influenceof purely van der Waals interactions

5 CONCLUDING REMARKS

Numerical simulations of thin film drainage employingthe long wave approximation were conducted to comparethe drainage behavior under influence of pressure and anapplied electric field across the film It is noted that apartfrom morphological differences in the final film structuresobtained under these conditions there is also a significantdifference in time scales of the drainage with electro-static drainage becoming almost instantaneous comparedto pressure based drainage at high applied fields Theseobservations indicate that drainage of films by application

of electric field follows considerably different dynamicscompared to pressure based drainage

Acknowledgment Financial support from NSERC isgratefully acknowledged

References

1 A Oron S H Davis and S G Bankoff Rev Mod Phys 69 931(1997)

2 E Ruckenstein and R K Jain J Chem Soc Faraday Trans II 70132 (1974)

3 A Sharma and A T Jameel J Colloid Interface Sci 161 190 (1993)4 R Verma A Sharma K Kargupta and J Bhaumik Langmuir 21

3710 (2005)5 E Schaffer T Thurn-Albrecht T P Russell and U Steiner Nature

403 874 (2000)6 L F Pease and W B Russel J Non-Newtonian Fluid Mech 102

233 (2002)7 V Shankar and A Sharma J Colloid Interface Sci 274 294 (2004)8 N Wu L F Pease and W B Russel Langmuir 21 12290

(2005)9 N Wu L F Pease and W B Russel Adv Funct Mat 16 1992

(2006)10 F Mostowfi K Khristov J Czarnecki J Masliyah and

S Bhattacharjee Appl Phys Lett 90 184102 (2007)11 M B Williams and S H Davis J Colloid Interface Sci 90 220

(1982)12 R Khanna and A Sharma J Colloid Interface Sci 195 42

(1997)13 R Verma A Sharma I Banerjee and K Kargupta J Colloid Inter-

face Sci 296 220 (2006)14 V S Mitlin J Colloid Interface Sci 156 491 (1993)15 R S Maier L R Petzold and W Rath Proc Scalable Parallel

Libraries Conference (1993) pp 174ndash18216 F S Acton Numerical Methods that Work Harper and Row

New York (1970)

Received 12 October 2007 Accepted 21 December 2007



IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE


z = 0

z = h0

z = d

λ

z = h(x y t)

Upper electrode

Bounding fluid ε2 micro2

Lower electrode

x

z

Liquid film ε1 micro1

ψ = ψlow

ψ = ψup

Fig 1 Schematic representation of the thin film system The liquid filmunder consideration is supported by the lower electrode The boundingfluid is modeled as an inert fluid (similar to air) with 2 1 Forelectrostatic force induced drainage a potential difference of low minusup

is applied between the two electrodes with up typically set as groundThe initial height of the unperturbed film is h0 and the vertical distancebetween the two electrodes is d

waves that form on the film interface allow the simplifica-tion of the governing equations A lsquolong-waversquo asymptoticanalysis can be used to simplify the governing equations1

The thin film system studied is shown schematically inFigure 1 It depicts a liquid film with a bounding fluidon top sandwiched between two electrodes As shown inthe figure d is the electrode spacing hx y t is the filmthickness at spatial coordinates (x y) and time (t) h0 isthe average initial film thickness subscript i indicates thefluid phase under consideration with i= 1 for the film andi = 2 for the bounding fluid i (i = 12) indicates thepermittivity of fluid i i is the viscosity of fluid (i= 12) is the wavelength of disturbances on the film interfaceup is an applied voltage at upper electrode (z = d) andlow is the applied voltage at the lower electrode (z= 0)

21 Governing Equations

The fluids are modeled using a continuum assumption andthe motion is defined by the Navier Stokes (NS) equationsand a continuity relation

i

( ui

t+ ui middot ui

)

=minuspi + middot i ui + uiT + fi (1a)

middot i ui= 0 (1b)

The term ui is the velocity vector (u= u vw) pi is thepressure i is the density is the cartesian operator =x y z and fi = minusi represents the bodyforces where i is the conjoiningdisjoining pressure Theconjoiningdisjoining pressure is an excess energy per unitvolume that describes forces that exist on the film TheNavier Stokes system shown in Eq (1) is applied for boththe film and bounding fluid (i = 12)

The general formulation for thin film drainage is com-pleted by the definition of boundary conditions At the

upper and lower electrodes no penetration and no slipboundary conditions are applied

u1 = 0 at z= 0

u2 = 0 at z= d(2)

At the interface between the fluids continuity of velocityand a stress balance exist The two fluids are modeled asimmiscible so no fluid mixing or mass transfer occursat their interface This ensures that a sharp interface ismaintained and that velocity is continuous across it

u1 = u2 at z= hx y t (3)

The velocity of the fluids at the interface is related to thefilm thickness by a kinematic condition11 The kinematiccondition is obtained by taking the partial derivative ofthe film thickness z= hx y t with respect to time Thechain rule and velocity definitions can then be used torelate the film thickness to the interfacial velocity

w = h

t+u

h

x+v

h

yat z= hx y t (4)

The kinematic condition in Eq (4) is used in place of thecontinuity of velocity (Eq 3)

At the interface a stress balance between the two fluidsis performed by taking the normal and tangential projec-tions of the stresses at the fluid interface

n middot macr1 middot nminus macr2 middot n = + fe middot n (5)

tj middot macr1 middot nminus macr2 middot n= fe middot tj (6)

where tj = tjx tjy tjz is the jth unit tangent vector tothe fluid interface where (j = 12) n is the unit normalvector to the fluid interface which points from the film tothe bounding fluid with components n = nxny nz isthe mean interfacial curvature is the interfacial tensionfe is the external surface force vector and macri is the stresstensor for the ith fluid phase (i = 12)

The solution of the NS equation system presented abovewould be difficult since the interface between the fluidsis a free boundary Fortunately the problem scaling andphysics allow the system to be reduced to a single evolu-tion equation for the film thickness The lsquolong waversquo orlsquosmall slopersquo scaling argument uses the fact that dominantwavelength of disturbances which dictates the lengthscale of the problem is much larger than the film thick-ness h171112 For simplicity a scaling factor k is definedto indicate the difference in lengths and aid in the simpli-fication of the hydrodynamic equations

k = h0

1 (7)

where h0 is the initial mean film thickness The lengths arenon-dimensionalized by using the mean film thickness h0



IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE



x = X y = Y z= h0Z (8a)

x= 1

X

y= 1

Y

z= 1

h0

Z(8b)


U = u

u0

V = v

u0

(9)


u0

U

X+ u0

V

Y+ w

z= 0 (10)


W =

u0h0

w = w

u0k(11)


T = ku0

h0

t (12)


P= kh0

u0

p (13)








0 = UX +VY +WZ (14d)





PZ +Z = 0 (15c)

UX +VY +WZ = 0 (15d)


U = 0 at z= 0 (16a)







H

t+

X

[H 3 0

X

]+

Y

[H 3 0

Y

]= 0 (18)

where

0 = 2H

X2+ 2H

Y 2minus (19)



LV DW = AL

63h3(20)



IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE



BL=minus8BL

h9(21)



2i = 0 i = 12 (22)


1 = low at z= 0

1 = 2 at z= hx y t

1

1

z= 2

2

zat z= hx y t

2 = 0 at z= d

(23)



and

Ep =low





22 Scaling


tlowast = 12

h3h2xlowast =

radicminus832

h(27)


tlowast = 1232h50

A2L

xlowast = h20

radic23

AL

(28)

cLV DW = 23h20

radic23

AL

(29)


tlowast = 3h30

0650pp minus12low

2

xlowast = 23

radich3

0

0650pp minus12low

(30)

cEL = 23

radic2b

0pp minus12Eminus32

p (31)

3 NUMERICAL METHOD





IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE












Parameter Value






(a) (b) (c)

(d) (e) (f)

X

Y




IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE



02

04

06

08

1


Non

-dim

ensi

onal

film

thic

knes

s (h

h0)

t = 057 s t = 3082 st = 3298 st = 3514 st = 3572 s









0 10 20 30 40 500

01

02

03

04

05

06

07

08

09

1

Time (t) [s]


Min

imum

dim

ensi

onle

ss fi

lm th

icne

ss (

Hm

in)

[ndash]




IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE







References














New York (1970)




IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE



x = X y = Y z= h0Z (8a)

x= 1

X

y= 1

Y

z= 1

h0

Z(8b)


U = u

u0

V = v

u0

(9)


u0

U

X+ u0

V

Y+ w

z= 0 (10)


W =

u0h0

w = w

u0k(11)


T = ku0

h0

t (12)


P= kh0

u0

p (13)








0 = UX +VY +WZ (14d)





PZ +Z = 0 (15c)

UX +VY +WZ = 0 (15d)


U = 0 at z= 0 (16a)







H

t+

X

[H 3 0

X

]+

Y

[H 3 0

Y

]= 0 (18)

where

0 = 2H

X2+ 2H

Y 2minus (19)



LV DW = AL

63h3(20)



IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE



BL=minus8BL

h9(21)



2i = 0 i = 12 (22)


1 = low at z= 0

1 = 2 at z= hx y t

1

1

z= 2

2

zat z= hx y t

2 = 0 at z= d

(23)



and

Ep =low





22 Scaling


tlowast = 12

h3h2xlowast =

radicminus832

h(27)


tlowast = 1232h50

A2L

xlowast = h20

radic23

AL

(28)

cLV DW = 23h20

radic23

AL

(29)


tlowast = 3h30

0650pp minus12low

2

xlowast = 23

radich3

0

0650pp minus12low

(30)

cEL = 23

radic2b

0pp minus12Eminus32

p (31)

3 NUMERICAL METHOD





IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE












Parameter Value






(a) (b) (c)

(d) (e) (f)

X

Y




IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE



02

04

06

08

1


Non

-dim

ensi

onal

film

thic

knes

s (h

h0)

t = 057 s t = 3082 st = 3298 st = 3514 st = 3572 s









0 10 20 30 40 500

01

02

03

04

05

06

07

08

09

1

Time (t) [s]


Min

imum

dim

ensi

onle

ss fi

lm th

icne

ss (

Hm

in)

[ndash]




IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE







References














New York (1970)




IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE



BL=minus8BL

h9(21)



2i = 0 i = 12 (22)


1 = low at z= 0

1 = 2 at z= hx y t

1

1

z= 2

2

zat z= hx y t

2 = 0 at z= d

(23)



and

Ep =low





22 Scaling


tlowast = 12

h3h2xlowast =

radicminus832

h(27)


tlowast = 1232h50

A2L

xlowast = h20

radic23

AL

(28)

cLV DW = 23h20

radic23

AL

(29)


tlowast = 3h30

0650pp minus12low

2

xlowast = 23

radich3

0

0650pp minus12low

(30)

cEL = 23

radic2b

0pp minus12Eminus32

p (31)

3 NUMERICAL METHOD





IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE












Parameter Value






(a) (b) (c)

(d) (e) (f)

X

Y




IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE



02

04

06

08

1


Non

-dim

ensi

onal

film

thic

knes

s (h

h0)

t = 057 s t = 3082 st = 3298 st = 3514 st = 3572 s









0 10 20 30 40 500

01

02

03

04

05

06

07

08

09

1

Time (t) [s]


Min

imum

dim

ensi

onle

ss fi

lm th

icne

ss (

Hm

in)

[ndash]




IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE







References














New York (1970)




IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE












Parameter Value






(a) (b) (c)

(d) (e) (f)

X

Y




IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE



02

04

06

08

1


Non

-dim

ensi

onal

film

thic

knes

s (h

h0)

t = 057 s t = 3082 st = 3298 st = 3514 st = 3572 s









0 10 20 30 40 500

01

02

03

04

05

06

07

08

09

1

Time (t) [s]


Min

imum

dim

ensi

onle

ss fi

lm th

icne

ss (

Hm

in)

[ndash]




IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE







References














New York (1970)




IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE



02

04

06

08

1


Non

-dim

ensi

onal

film

thic

knes

s (h

h0)

t = 057 s t = 3082 st = 3298 st = 3514 st = 3572 s









0 10 20 30 40 500

01

02

03

04

05

06

07

08

09

1

Time (t) [s]


Min

imum

dim

ensi

onle

ss fi

lm th

icne

ss (

Hm

in)

[ndash]




IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE







References














New York (1970)




IP 95257666Tue 10 Jul 2012 204801

RESEARCHARTICLE







References














New York (1970)



characteristic times for pressure and electrostatic force driven thin film drainage

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