Effects of capillary waves on the thickness of wetting layers

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Effects of capillary waves on the thickness of wetting layers

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 de

Physique, 1988, 49 (4), pp.675-680. �10.1051/jphys:01988004904067500�. �jpa-00210742�

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.

On calcule la contribution des ondes capillaires ^à l’énergie libre d’un film mouillant une surface, ^et

on montre que ^cette contribution dépend fortement de l’épaisseur ^{du film} liquide lorsque ^{celle-ci est faible. On}

montre également que ^ces oscillations peuvent jouer ^{un rôle} important pour la stabilité thermodynamique ^d’un

film mouillant si le liquide ^{est un} mélange ^binaire proche ^{de son} point critique, que les forces soient à ^{courte ou} à longue portée. ^En particulier, l’épaisseur du film tend vers zéro lorsque ^la température ^T s’approche ^de Tc.

Abstract.

The free energy contribution of capillary ^waves is calculated to show its significant dependence ^on

the thickness of the liquid layer, ^{when the} thickness is very small. It is shown that these oscillations ^can play ^an important ^{role in} determining ^the thermodynamic stability ^{of a} wetting layer, ^{close to} the critical point ^{of a} binary liquid mixture in the case of both short range and long range forces. In particular, the thickness of the

wetting layer goes ^{to zero as} the temperature ^T approaches Tc.

Classification

Physics ^Abstracts

68.10

64.70J

1. Introduction.

Wetting phenomena, ^{close to} the critical point ^{of a} binary liquid ^mixture, present ^an interesting ^situ- ation, ^{where a heavier} liquid ^{can wet the walls of the}

vessel and can reside over a lighter ^one [1, 2]. ^This phenomenon ^{has been observed} recently ^{in a number}

of binary liquid mixtures and a typical situation is

depicted ⁱⁿ figure 1. The thickness of the top layer ^of

the liquid ^{has to be} determined from the minimis- ation of the total energy of the system, ^which consists of (1) ^{work done} against bouyancy ^and (2)

the interfacial energy due ^to the mutual interaction of the surfaces AB and CD. In the present paper, ^we consider 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 been} recently ^studied [3]. ^{It was shown} experimentally

and theoretically ^that unless the radius R of the container 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 ^the

appearance of surface disturbances of small wave

numbers. The problem posed in this paper is ^as follows : given ^a system, confined in a sufficiently

narrow vessel, such that the Rayleigh-Taylor ^insta- bility ^is absent, how do the surface oscillations come

into play ^to determine the thickness of the wetting layer ? ^{In other} words, ^we determine here the free energy contribution of the surface oscillations and

study whether the topmost film grows thicker or

thinner when the surface oscillations are included.

By necessity, the calculations refer to an idealised model and are approximate. ^However, they clearly

show the important ^role played by the free energy of the capillary ^waves.

The present paper is divided into four parts. ^{Part 2} is devoted to a calculation of the free energy contribution from the capillary ^{waves in terms of the} thickness f of the wetting layer. ^{In the} subsequent section, ^{a solution} for f is sought for the short range and long range forces. Some scaling ^{laws for} f are also pointed out, ^{while the} concluding ^section

is devoted to a discussion of the results.

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

676

Fig. ^1.

The relative position ^{of the} wetting layer ^{in the} binary liquids in which the liquid (1) is heavier than the

liquid (2).

2. Calculation of the free energy.

Let us consider a vertical displacement q (x, y) ^at

any point (x, y) ^on the surface CD of the wetting layer. Writing

it can be shown that the system obeys ^a dispersion

relation [4, 5]

where

The peculiarity ^{of the} dispersion ^relation (2) ^{is that}

in the small k£ limit, w2 has ^a linear dependence ^on f. This dependence ^{on f vanishes} when k£ --+ oo.

Further because of the absence of fluid flow at the walls of the container, k ^satisfies the eigen ^value equation

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 ^the

minimum eigen ^{value to be} km - 2/R and hence the system ^is mechanically ^{stable if}

This condition, ^{related to the} Rayleigh-Taylor instability, ^{has been} recently demonstrated to be of

special significance ⁱⁿ maintaining ^the stability ^of wetting layers [3].

In the following, ^we consider the system ^{to be} stable against ^the Rayleigh-Taylor instability ^i.e.

R’ -- 4 y /9 (P 1 - P 2) and calculate the thickness of the wetting layer ^{when the} capillary ^{waves are taken}

into account. As first approximation, one assumes

that the total free energy of the system is obtained by adding ^{the free} Fe ^{of the} capillary ^{waves to the bulk}

of the free energy i.e.

where Fo is the Van der Waalsian or Cahn-Hillard type free energy and F, is that due to the capillary

waves.

The equilibrium ^{value of f} is determined by minimising Ftot (f ) ^from equation (Sa) :

As is known, the free energy contribution due ^to

capillary ^waves [6, 7] ^is given by

where k corresponds ^{to the} eigen frequencies ^{of the} capillary ^{waves as} given by equation (3). ^{Such a}

calculation was first performed by ^Frenkel [7].

Recently ^{Tarazona et al.} [8] have used the same

method to study ^the displacements ^{of the} dividing

surface between two liquids. ^These authors, ^how-

ever, do not study ^the system in the low f limit and

their results are valid in the thick film range, where

the Q-dependence ^{of the} dispersion relation is rather

weak as seen from equation (2). ^{In our} calculation

the dependence in the low f range is considered.

in which, for f

0,

while for f - oo

The results of Tarazona et al. [8], ^Evans [9] ^and

Davis [10] corresponds ^{to the} equation in which the f dependence ^{is found to fall off} exponentially ^{as in} equation (9). Their results are thus naturally very insensitive to the variations in f. In the present context, ^{however we are} interested in the stability ^of

the thin wetting layers, ^{so that the correct} expression

would be the one given ⁱⁿ equation (8). ^{In this case}

the expression ^{on the} right hand side of (8) ^{has a}

very strong ^variation with f in the small f limit. This

gives ^a significant contribution to the free energy, ^as will be shown in the following calculations.

For f

0 one obtains from equations (7) ^and (8)

where the summation is to be performed ^{over the} eigen values of the system. In the limit of a cell of

large ^diameter (but still consistent with the condition of hydrodynamic stability), the above sum can be converted into an integral ^{over k. Thus we} find, ^for

unit area retaining only ^{the term} proportional ^to R2 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 ^the Rayleigh-Taylor instability the limits of integration

in equation (13) ^{have an} implicit dependence ^{on the}

radius R of the vessel. However Zmin corresponds ^to

wmin 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 of

integration Zmax -+ ^00. Alternatively, ^{one sees that}

as per the calculations of Buff, Lovett, Stillinges ^and

others [1], ^{one has} k- 1 =:z interfacial thickness

f- So Zmax - fl’2 k2max = ^{f - 3/2 and hence} Zmax -> ⁰⁰

is a good approximation. Using these limits for Z and writing

the evaluation of the integral ^is quite straightforward

and 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 indeed

corresponds ^{to an} experimentally inaccessible situ-

ation ; however, for the sake of completeness ^we present ^the expression for both a > 1 as also

a 1.

For a

and

Sufficiently ^{close to} 7c ^the Rayleigh-Taylor ^{mode of} instability ^will destroy ^the overhanging layer. ^{As will}

be seen below, ^the capillary ^wave free energy ^term also produces ^a similar effect from a thermodynamic point ^{of view.}

For a 1 but finite,

and

Equation (18) above shows that for a > 1 the free energy of the capillary ^{waves has no} explicit depen-

dence on the surface tension but is completely

determined by ^the density difference while in the

678

other limit the free energy is predominently ^deter-

mined by the surface tension. It is ^to be remembered here that equations (18, 20) corresponds ^{to the case}

where 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 dependence exp (- kf) ^on afclaf ^{in this case,} making ^the

thickness 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 be}

incorporated ⁱⁿ equation (6). ^{Two cases} may arise, depending upon the bulk forces being of short range interactions or long range interactions. The results in these two cases are different and are therefore discussed 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 and A 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 of these equations ^are positive and second terms are

negative. Thus real solutions for f exist if values of the quantities ^on the left hand side are less than the maxima of the functions on the right hand sides of these 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} right

hand side is greater than the maximum value of the left hand side which is 27 exp (- 3 ). Thus there can

be 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 right

hand side is greater than the maximum of the left hand side which is (5/2 )512 exp (- 5/2 ). ^{Thus no real}

solution exists if

though ^for higher t ^a real solution can exist if the

inequality (27) ^{is valid.}

These discussions show that capillary ^{waves intro-}

duce some stringent conditions for the ther-

modynamic stability ^{of the} wetting layer ^{when the}

forces are of short range type. ^The phase diagram ⁱⁿ

this system depicting (f, t) relationship ^{can be}

studied by numerical solution of the equations (22)

and (23).

3.2 CASE II : LONG RANGE FORCES.

It has been observed by several authors [12] ^{that the} long range forces give ^{rise to a} power law ^term instead of an

exponential decay. ^Thus

where

a 1, a 2 being ^the specific polarizabilities ^{of the two} phases 1 and 2. We consider a2 > a 1 so that W is

positive. Equation (5b) then reads :

in the extreme low t limit, ^while

for higher ^t.

This introduces two different regimes ^{for f. For} small 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 ^between

the Van der Waals and the capillary ^{waves free} energies ^which impose dependences ^{on the free}

energy which ^are of power law type. ^{In this} case, the

potential energy necessary for raising the heavier

liquid by ^a height ^{L is} considered to be small.

Equation (37) is associated with the reverse case :

free energy of the capillary ^waves oscillations are

small and are hence neglected, giving ^{a result as} predicted by ^the previous workers. Further in our

calculations we have considered the quantity ^{W to be} positive, i.e. the interaction to be repulsive. ^{In the}

case of attractive interaction the stability ^{is not} guaranteed ^{unless a} sufficiently strong short range

repulsion ^{is added.}

Equation (32) ^and (34) ^concern inaccessible reg- ions of t. Nevertheless it is interesting that very close to T the scaling ^{follows f -} to-65 dovetailing

with the functional form of equation (36). ^The dynamic aspects ^{of the} Rayleigh-Taylor instability

and the thermodynamic form of the capillary ^wave

situation appear ^to be consistent.

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

In equation (32) for very small values of t, ^{we can} neglect the first term, assuming ^{L to be not too} large ; ^this yields,

so that using equation (31) ^{we have}

For equation (33) ^{we have the roots of the} quadratic equation

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 to}

a free energy which increases with the thickness of the wetting layer ^{as - -} f - 112. For short range forces

we find that the capillary ^waves preclude ^{the exist-}

ence of the wetting layer except ^{in a} restricted range of physical parameters. ^{For a} long range interaction

we predict ^a dependence ^{of f on t as is} given ⁱⁿ figure 2, provided ^the long range interaction is

repulsive. For attractive long range forces, ^the wetting layer ^{is found to} be unstable unless a

sufficiently strong short range repulsive force is also present. ^The important result that arises out of the

Fig. ^2.

^The dependence ^{of the} layer ^{thickness f on the}

reduced temperature ^t.

44

680

theory is that the thickness of the wetting layer ^is expected ^to go ^{to zero as} the system approaches Tc. ^With increasing t, ^the layer thickness would grow until it reaches saturation.

There are a few experiments which indicate the

disappearance ^{of the} wetting layer ^{as one} approaches Tc closely [3, 13, 15] ; but these experiments ^have

not 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 sense}

these theories make the behaviour close to Tc ^as being dependent ^on the surface interaction. In the limit of small t and hence small f, ^{we find the} capillary free energy ^to depend ^{on a bulk} property like the density difference between two liquids. ^The

present calculations emphasize ^{that the} capillary

waves which play ^an important ^{role in} maintaining ^a dynamic stability ^{of the} wetting layer [3] ^{have an} equally important ^{role to} play ⁱⁿ deciding ^the thermodynamic stability ^{of these} layers.

The calculations are admittedly ^{based on} many

approximations. ^For instance the bulk and the

capillary ^{wave free} energies ^are merely ^added together. An idealised slab configuration ^{of the} wetting layer is assumed. Precise numerical compu- tations have not been performed. Nevertheless the main features are likely ^{to be} preserved in any other realistic analysis because of the thickness depen-

dence of the dispersion reiation ^{of the} capillary

waves. This would be true in other thin layer configurations ^{also where the} dynamical Rayleigh- Taylor instability may ^not be significant.

A number of experiments ^{could be} suggested ^to study ^the present ^model. Apart from the observation of the appearance ^or disappearance ^{of the} wetting layer, one can measure the thickness variations by ellipsometric techniques ^{when the} temperatures ^are changed. 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 instability

like the Rayleigh-Taylor mode invokes an exper- imental growth ^{of the} amplitude with time. We like to emphasize ^{that the} capillary ^{waves which are} always present ^{in the} system because of the thermal motions deserve further studies. As per the present calculations, they make the behaviour in the critical

region ^{to be} determined also by ^{some bulk} properties

like density difference between the two liquids,

instead of being influenced markedly only by ^the

surface layers and the walls. Surface tension effects

come 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. They

are also indebted to the D.S.T., D.A.E. and I.I. Sc.

ISRO STC for financial help. They specially ^thank

the referees for their suggestions, which have im-

proved ^the present ^work.

References

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