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Effects of capillary waves on the thickness of wettinglayers

S. Chatterjee, E.S.R. Gopal

To cite this version:S. Chatterjee, E.S.R. Gopal. Effects of capillary waves on the thickness of wetting layers. Journal dePhysique, 1988, 49 (4), pp.675-680. �10.1051/jphys:01988004904067500�. �jpa-00210742�

https://hal.archives-ouvertes.fr/jpa-00210742https://hal.archives-ouvertes.fr

675

Effects of capillary waves on the thickness of wetting layers

S. Chatterjee and E. S. R. Gopal (*)

Indian Institute of Astrophysics, Bangalore-560 034, India(*) Physics Department, Indian Institute of Science, Bangalore-560 012 India

(Reçu le 2 juillet 1987, révisé le 9 décembre 1987, accepte le 10 decembre 1987)

Résumé. 2014 On calcule la contribution des ondes capillaires à l’énergie libre d’un film mouillant une surface, eton montre que cette contribution dépend fortement de l’épaisseur du film liquide lorsque celle-ci est faible. Onmontre également que ces oscillations peuvent jouer un rôle important pour la stabilité thermodynamique d’unfilm mouillant si le liquide est un mélange binaire proche de son point critique, que les forces soient à courte ouà longue portée. En particulier, l’épaisseur du film tend vers zéro lorsque la température T s’approche deTc.

Abstract. 2014 The free energy contribution of capillary waves is calculated to show its significant dependence onthe thickness of the liquid layer, when the thickness is very small. It is shown that these oscillations can play animportant role in determining the thermodynamic stability of a wetting layer, close to the critical point of abinary liquid mixture in the case of both short range and long range forces. In particular, the thickness of thewetting layer goes to zero as the temperature T approaches Tc.

J. Phys. France 49 (1988) 675-680 AVRIL 1988,

Classification

Physics Abstracts68.10 - 64.70J

1. Introduction.

Wetting phenomena, close to the critical point of abinary liquid mixture, present an interesting situ-ation, where a heavier liquid can wet the walls of thevessel and can reside over a lighter one [1, 2]. Thisphenomenon has been observed recently in a numberof binary liquid mixtures and a typical situation isdepicted in figure 1. The thickness of the top layer ofthe liquid has to be determined from the minimis-ation of the total energy of the system, whichconsists of (1) work done against bouyancy and (2)the interfacial energy due to the mutual interactionof the surfaces AB and CD. In the present paper, weconsider the effect of the surface oscillations of the

interface CD in determining the equilibrium thick-ness of the layer.The importance of these surface oscillations in the

mechanical stability of the wetting layer has beenrecently studied [3]. It was shown experimentallyand theoretically that unless the radius R of thecontainer vessel is sufficiently small(R2 4 l’ /g I dp [ where Ap = PI - P2 is the differ-ence between the densities of the two liquids and

y is the surface tension) the Rayleigh-Taylor insta-bility makes the wetting layer unstable against theappearance of surface disturbances of small wavenumbers. The problem posed in this paper is asfollows : given a system, confined in a sufficientlynarrow vessel, such that the Rayleigh-Taylor insta-bility is absent, how do the surface oscillations comeinto play to determine the thickness of the wettinglayer ? In other words, we determine here the freeenergy contribution of the surface oscillations and

study whether the topmost film grows thicker orthinner when the surface oscillations are included.

By necessity, the calculations refer to an idealisedmodel and are approximate. However, they clearlyshow the important role played by the free energy ofthe capillary waves.The present paper is divided into four parts. Part 2

is devoted to a calculation of the free energycontribution from the capillary waves in terms of thethickness f of the wetting layer. In the subsequentsection, a solution for f is sought for the short rangeand long range forces. Some scaling laws forf are also pointed out, while the concluding sectionis devoted to a discussion of the results.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004904067500

http://www.edpsciences.orghttp://dx.doi.org/10.1051/jphys:01988004904067500

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Fig. 1. - The relative position of the wetting layer in thebinary liquids in which the liquid (1) is heavier than theliquid (2).

2. Calculation of the free energy.

Let us consider a vertical displacement q (x, y) atany point (x, y) on the surface CD of the wettinglayer. Writing

it can be shown that the system obeys a dispersionrelation [4, 5]

where

The peculiarity of the dispersion relation (2) is thatin the small k£ limit, w2 has a linear dependence onf. This dependence on f vanishes when k£ --+ oo.

Further because of the absence of fluid flow at thewalls of the container, k satisfies the eigen valueequation

where J,, are the Bessel functions.In the present case, since p 1 > p 2 we observe from

the equation (2) that the system is unstable unless

For a vessel of radius R, equation (3) gives theminimum eigen value to be km - 2/R and hence thesystem is mechanically stable if

This condition, related to the Rayleigh-Taylorinstability, has been recently demonstrated to be ofspecial significance in maintaining the stability ofwetting layers [3].

In the following, we consider the system to bestable against the Rayleigh-Taylor instability i.e.R’ -- 4 y /9 (P 1 - P 2) and calculate the thickness ofthe wetting layer when the capillary waves are takeninto account. As first approximation, one assumesthat the total free energy of the system is obtained byadding the free Fe of the capillary waves to the bulkof the free energy i.e.

where Fo is the Van der Waalsian or Cahn-Hillardtype free energy and F, is that due to the capillarywaves.

The equilibrium value of f is determined byminimising Ftot (f ) from equation (Sa) :

As is known, the free energy contribution due tocapillary waves [6, 7] is given by

where k corresponds to the eigen frequencies of thecapillary waves as given by equation (3). Such acalculation was first performed by Frenkel [7].Recently Tarazona et al. [8] have used the samemethod to study the displacements of the dividingsurface between two liquids. These authors, how-ever, do not study the system in the low f limit andtheir results are valid in the thick film range, where

the Q-dependence of the dispersion relation is ratherweak as seen from equation (2). In our calculationthe dependence in the low f range is considered.

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in which, for f --+ 0,

while for f - oo

The results of Tarazona et al. [8], Evans [9] andDavis [10] corresponds to the equation in which thef dependence is found to fall off exponentially as inequation (9). Their results are thus naturally veryinsensitive to the variations in f. In the presentcontext, however we are interested in the stability ofthe thin wetting layers, so that the correct expressionwould be the one given in equation (8). In this casethe expression on the right hand side of (8) has avery strong variation with f in the small f limit. Thisgives a significant contribution to the free energy, aswill be shown in the following calculations.For f -+ 0 one obtains from equations (7) and (8)

where the summation is to be performed over theeigen values of the system. In the limit of a cell oflarge diameter (but still consistent with the conditionof hydrodynamic stability), the above sum can beconverted into an integral over k. Thus we find, forunit area retaining only the term proportional toR2 and after some straight forward algebraic manipu-lations,

where

and

and the variable of the integration Z is defined as

Since we have chosen a case which precludes theRayleigh-Taylor instability the limits of integrationin equation (13) have an implicit dependence on theradius R of the vessel. However Zmin corresponds towmin which is close to zero. For the continuum

approach to be valid, k-1max = the interatomic dis-tance in the fluid, which leads to the upper limit ofintegration Zmax -+ 00. Alternatively, one sees thatas per the calculations of Buff, Lovett, Stillinges andothers [1], one has k- 1 =:z interfacial thicknessf- So Zmax - fl’2 k2max = f - 3/2 and hence Zmax -> 00is a good approximation. Using these limits forZ and writing

the evaluation of the integral is quite straightforwardand the different limiting expressions will be pre-sented below.

Close to the critical point, equation (12) becomes

where

Inserting some typical values yo - 20 erg/cm,f - 0.1 cm, AP 0 - 0.001 gmlcc, (p1 + p2) ~ 1 gm/cc,f3 = 0.33 and JL = 1.33 we have a = 10-14 t- 1/3.This shows a 1 unless t - 10- 40. The latter indeedcorresponds to an experimentally inaccessible situ-ation ; however, for the sake of completeness wepresent the expression for both a > 1 as alsoa 1.

For a

and

Sufficiently close to 7c the Rayleigh-Taylor mode ofinstability will destroy the overhanging layer. As willbe seen below, the capillary wave free energy termalso produces a similar effect from a thermodynamicpoint of view.For a 1 but finite,

and

Equation (18) above shows that for a > 1 the freeenergy of the capillary waves has no explicit depen-dence on the surface tension but is completelydetermined by the density difference while in the

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other limit the free energy is predominently deter-mined by the surface tension. It is to be rememberedhere that equations (18, 20) corresponds to the casewhere the thickness of a wetting layer is very small.For a thick layer, equation (9) is to be employed.The exponential term in (9) imposes a dependenceexp (- kf) on afclaf in this case, making thethickness dependence insignificant.We proceed to find the equilibrium value of

f, by solving equation (5b), and using the infor-mation available about afolaf.

3. Equilibrium thickness.

In solving for f the explicit form for F0 (f) is to beincorporated in equation (6). Two cases may arise,depending upon the bulk forces being of short rangeinteractions or long range interactions. The results inthese two cases are different and are thereforediscussed seperately.

3.1 CASE I : SHORT RANGE FORCES. - In this

situation, discussed by Cahn and Hilliard [11],

where y’ is a constant of the interaction andA is a measure of the range of the interaction. So the

equation (5b) yields

in the extreme small t limit, and

for higher values of t, where al and a2 are constants,which can be identified from equations (18) and(20). The left hand side of equations (22) and (23)are positive. The first terms on the right hand side ofthese equations are positive and second terms arenegative. Thus real solutions for f exist if values ofthe quantities on the left hand side are less than themaxima of the functions on the right hand sides ofthese equations.Thus equation (22) has a real solution if

where l1 is the real solution of

Equation (25) can be written as

No real solution of fi 1 can be obtained if the righthand side is greater than the maximum value of theleft hand side which is 27 exp (- 3 ). Thus there canbe no real solution if

whereas a real solution can exist if the inequality(24) is satisfied.

Similarly equation (23) has a real solution if

where f 2 is the real solution of

Equation (28) can also be written as

No real solution of f2 can be obtained if the righthand side is greater than the maximum of the lefthand side which is (5/2 )512 exp (- 5/2 ). Thus no realsolution exists if

though for higher t a real solution can exist if theinequality (27) is valid.These discussions show that capillary waves intro-

duce some stringent conditions for the ther-modynamic stability of the wetting layer when theforces are of short range type. The phase diagram inthis system depicting (f, t) relationship can bestudied by numerical solution of the equations (22)and (23).

3.2 CASE II : LONG RANGE FORCES. - It has beenobserved by several authors [12] that the long rangeforces give rise to a power law term instead of anexponential decay. Thus

where

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a 1, a 2 being the specific polarizabilities of the twophases 1 and 2. We consider a2 > a 1 so that W ispositive. Equation (5b) then reads :

in the extreme low t limit, while

for higher t.

This introduces two different regimes for f. Forsmall t,

For larger t,

which because of equation (31) is independent of t.Equation (36) corresponds to the case where the

stability is determined by the competition betweenthe Van der Waals and the capillary waves freeenergies which impose dependences on the freeenergy which are of power law type. In this case, thepotential energy necessary for raising the heavierliquid by a height L is considered to be small.Equation (37) is associated with the reverse case :free energy of the capillary waves oscillations aresmall and are hence neglected, giving a result aspredicted by the previous workers. Further in ourcalculations we have considered the quantity W to bepositive, i.e. the interaction to be repulsive. In thecase of attractive interaction the stability is notguaranteed unless a sufficiently strong short rangerepulsion is added.

Equation (32) and (34) concern inaccessible reg-ions of t. Nevertheless it is interesting that veryclose to T the scaling follows f - to-65 dovetailingwith the functional form of equation (36). Thedynamic aspects of the Rayleigh-Taylor instabilityand the thermodynamic form of the capillary wavesituation appear to be consistent.

JOURNAL DE PHYSIQUE. - T. 49, ? 4, AVRIL 1988

In equation (32) for very small values of t, we canneglect the first term, assuming L to be not toolarge ; this yields,

so that using equation (31) we have

For equation (33) we have the roots of the quadraticequation

4. Discussions.

The results of the present calculations can be sum-marised as follows. When the thickness of the

wetting layer is small, the capillary waves give rise toa free energy which increases with the thickness ofthe wetting layer as - - f - 112. For short range forceswe find that the capillary waves preclude the exist-ence of the wetting layer except in a restricted rangeof physical parameters. For a long range interactionwe predict a dependence of f on t as is given infigure 2, provided the long range interaction isrepulsive. For attractive long range forces, thewetting layer is found to be unstable unless asufficiently strong short range repulsive force is alsopresent. The important result that arises out of the

Fig. 2. - The dependence of the layer thickness f on thereduced temperature t.

44

680

theory is that the thickness of the wetting layer isexpected to go to zero as the system approachesTc. With increasing t, the layer thickness would growuntil it reaches saturation.There are a few experiments which indicate the

disappearance of the wetting layer as one approachesTc closely [3, 13, 15] ; but these experiments havenot been pursued very much since the more conven-tional theories predict that wetting should be ex-tended all the way to Tc [1, 2, 16,17]. In some sensethese theories make the behaviour close to Tc asbeing dependent on the surface interaction. In thelimit of small t and hence small f, we find thecapillary free energy to depend on a bulk propertylike the density difference between two liquids. Thepresent calculations emphasize that the capillarywaves which play an important role in maintaining adynamic stability of the wetting layer [3] have anequally important role to play in deciding thethermodynamic stability of these layers.The calculations are admittedly based on many

approximations. For instance the bulk and thecapillary wave free energies are merely addedtogether. An idealised slab configuration of thewetting layer is assumed. Precise numerical compu-tations have not been performed. Nevertheless themain features are likely to be preserved in any otherrealistic analysis because of the thickness depen-dence of the dispersion reiation of the capillarywaves. This would be true in other thin layerconfigurations also where the dynamical Rayleigh-Taylor instability may not be significant.

A number of experiments could be suggested tostudy the present model. Apart from the observationof the appearance or disappearance of the wettinglayer, one can measure the thickness variations byellipsometric techniques when the temperatures arechanged. Light scattering studies will give the disper-sion relations of the capillary waves, whose fre-quencies should follow equation (2) while the ampli-tude of every mode k should have 17 2( k» ’"KB Tlw 2(k) similar to the situation in the Debye-Waller functions in solids. The onset of the instabilitylike the Rayleigh-Taylor mode invokes an exper-imental growth of the amplitude with time. We liketo emphasize that the capillary waves which arealways present in the system because of the thermalmotions deserve further studies. As per the presentcalculations, they make the behaviour in the criticalregion to be determined also by some bulk propertieslike density difference between the two liquids,instead of being influenced markedly only by thesurface layers and the walls. Surface tension effectscome into play at relatively larger values of t.

Acknowledgements.

The authors would like to thank their colleagues Dr.V. C. Vani and Dr. S. Guha for discussions. Theyare also indebted to the D.S.T., D.A.E. and I.I. Sc.ISRO STC for financial help. They specially thankthe referees for their suggestions, which have im-proved the present work.

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