Size effect contributions to the chemical potential of a film

Surface Science 172 (1986) 311-318 North-Holland, Amsterdam

311

SIZE E F F E C T C O N T R I B U T I O N S T O T H E C H E M I C A L P O T E N T I A L O F A FILM N.K. M A H A L E and M.W. COLE Physics Department, The Pennsylvania State University, 104 Davey Laboratory, University Park, Pennsylvania 16802, USA Received 7 January 1986; accepted for publication 27 January 1986

The contributions of zero point and thermal excitations to the chemical potential of a liquid film are computed. The continuum approximation is made. Both bulk and surface waves are included. The effects are found to be small relative to the Van der Waals contribution in the case of a He film.

1. Introduction

It has long been of interest to understand the nature of physically adsorbed multilayer films [1-5]. A central problem is to predict the adsorption isotherm, relating coverage to vapor pressure P at a given temperature T. Entering the analysis are the nature of the substrate, its interaction with the adsorbate, and intrinsic properties of the adsorbate, such as density and interparticle interactions. Considerable effort has been addressed recently, in particular, to the problem of wetting [2-4]. The occurrence of wetting depends sensitively on differences in interaction strengths and length scales. Contributions to the free energy which are ordinarily negligible can tip the balance in favor of one phase instead of another [5]. For example, the substrate-atom interaction has the asymptotic form V-

-- C 3 z

3;

(1)

this dependence shows that subtle effects can become important at large distance z. These may be observed through the adsorption isotherm. For an ideal vapor the equivalence of the film chemical potential #f to that of the vapor yields t~ ~ I ~ f - ~

= fl-a ln( P / P ~ ) .

(2)

Here the subscript infinity represents the value at saturation, which is our reference, and /3-1 is Boltzmann's constant times T. 0039-6028/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

312

N.K. Mahale, M.W. Cole / Chernicalpotential of a film

Eq. (2) provides an opportunity to test model calculations of ~f as a function of film thickness d. A canonical example of the resulting prediction is the Frenkel-Halsey-Hill ( F H H ) relation [6-8], (3)

_ y / d 3 = f l - 1 ln( P / p o ) "

The coefficient y depends on differences in interaction strengths [6-9]. Although the assumptions underlying eq. (3) concerning the similarity of film and bulk material become more realistic at large d, experimental confirmation is lacking even then [9,10]. This paper reports a study of the influence on /~f of size effects associated with the film's excitations. The idea is that the thermodynamic properties' of the system depend on the excitation spectra. Specifically included are both bulk phonons and surface excitations. While this idea is quite general, our calculations will be applied specifically to the case of liquid 4He. In that case, there has been extensive previous study of these excitations for both the bulk surface and the film [11-13]. Our method involves the thermodynamic relation between ~f and the film free energy Ff at constant area A (4)

t~f = ( O F f / O N ) T , p , A + P / n ,

where n is the film density. The second term is negligible at low T and we omit it below. The calculation consists of evaluating the contributions of the various modes q; rq = ½~'l~q + f l - i ln(1 - e-~h'%),

(5)

where Wq is the excitation frequency. These are taken from ref. [11], as described in the next section, where/~f is evaluated. Section 3 summarizes our results.

2. Calculations 2.1. General relations

An inviscid, compressible film of thickness d supports two types of hydrodynamic modes - bulk and surface waves. The latter, which we call ripplons, have velocity potentials of the form [11] ~q

~

e ~ " cosh(lz),

o~2 = l( g + o q 2 / p ) t a n h ( l d

(6) ),

(7)

q2 _ ~ 2 / s 2 = l 2 > 0,

(8)

g(d)=

(9)

I(OV)

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N.K. Mahale, M.W. Cole /Chemicalpotential of a film

Here s is the speed of sound, P is the density, o is the surface tension, m is the atomic mass, and g(d) is an "effective" gravitational field arising from the substrate (which occupies the region z < 0). Limiting forms of the ripplon mode include third sound [14] and capillary-gravity waves [15]. We invoke a Debye-like restriction of q to the domain q < qm~x = 1 ,~-1 [16]. The " b u l k " solutions in this finite geometry are identified as phonons since their motion extends throughout the film; ~q --- e iq'" cos(kz),

(10)

2 = - k ( g + oq2/p)tan(kd), ~Oq

(11)

o~2/s 2 _ q2 = k 2 > O.

(12)

For a given q, k has an infinite number of solutions which satisfy [11] (2n+l)<2k,

d/rr<2(n+l),

n=0,1,2

.....

(13)

The constraint k < kmax = 1 .~-1 is imposed here [16]. The phonons described by these relations have been seen experimentally by Sabisky and Anderson and by Blackford [10,17,18]. The resolution, however, was not sufficient to detect deviation of the k values from the bulk limit k , = (2n + 1 ) , r / ( 2 d ) [11]. From eqs. (4) and (5), we obtain independent contributions from the two types of mode (denoted by r and p superscripts); we divide these into zero-point and thermal contributions. Letting N = Apd/m,

/£ = ~L0 q'- ~T,

[

(14)

]

hm 3 /'qmax (,.0(r) (~.0(qp)( kn ) dqq + ~'~ ,

/~0

4~rp~--~J°

ttT

m 3d,o 3 fqm,Xdq q { ln[1 _ exp(-Bh~o~qr))] 2~oB

k,,

kmax

+ ~ ln[1-exp(-Bh~o~r'(k~))]

(15)

)

.

(16)

kn

2.2. Analytic results We may anticipate the numerical results in a few regimes where analytic approximations are accurate. At high T, for example, /~T dominates/to; since then Z ln(1 - e -#h'~) ---* 3 In ~ Od 3d ' we find that/x T asymptotes to a linear dependence on T.

(17)

N.K. Mahale, M. I'14 Cole / Chemical potential of a film

314

In the opposite, low T, regime the ripplon contribution may be easily evaluated. From eqs. (7) and (8), we conclude that the relevant q and / are both "small" at low T. Suppose first that the film is sufficiently thin that we may take M << 1; from (1), (7), and (9) ~o ~ cq, (18) c = ( g d ) 1/2= ( 3 C 3 / m d 3 ) 1/2.

(19)

The regime of validity of this approximation is confined to cases where both of the following inequalities are satisfied: qT d << 1,

(20)

qTac << 1.

(21) We have used here a capillary length a c and thermal wave vector qT defined by [191 (22)

a c = ( o / o g ) 1/2, ~h60(qT)

=

1.

(23)

For 4He, on graphite, eq. (22) yields ac = 0.044d 2 A, if d is expressed in [20]. For the acoustic regime under discussion, eq. (23) becomes qT = 1//~hc. The relations (21) and (22) limit applicability to very low T; a 20 A film satisfies them only below 10-3K. Subject to these conditions, eqs. (16) and (19) yield [21] ~)

~

m 2 qrp/~ 3c3h 2

3c t(3) /'(3). Od

(24)

Here ~" is the Riemann zeta function and F is the gamma function. This T 3 dependence dominates the low T behavior of ~£T [22]. Consider next the problem of thick films, for which the acoustic regime of q is of negligible importance. This means that the opposite inequalities are applicable. Suppose that qT >> a~-1

and

d 1,

(25)

in which case eq. (7) approaches the capillary wave relation (26)

% = (o/p)'/2q3/2.

Then the thickness dependence of the dispersion relation is well approximated by 3~0(r)

-2~_-2(

/~)_-2mgf~'m., 3---rrod-fi x, Xmax ~

0 ll/2g,

dx ex -

[ ~ h ( o q 3 / p ) 1/2.

(27)

(28) 1 ' (29)

N.K. Mahale, M.W. Cole / Chemicalpotential of a film

315

Here x c -~ f l h c q T is a cutoff associated with deviation from eq. (25) at small q. We observe that eq. (28) is essentially proportional to T if Xmax >_ 1. Finally, we note that the zero temperature behavior of the chemical potential, eq. (15), is also dominated by the large q modes. Then the ripplon contribution should give a d-5 dependence, according to eq. (27). 2.3. N u m e r i c a l r e s u l t s

The calculations conform to the preceding analytic predictions in the appropriate limiting regimes. Fig. 1 shows the zero temperature contributions to the chemical potential of a 4He film of variable d on a graphite surface. These include for comparison the Van der Waals contribution to/~, given by eq. (1). The latter is seen to dominate at all thicknesses, but the phonon contribution becomes non-negligible at large d. Figs. 2 and 3 show the thermal contribution of the ripplons and phonons, respectively. The phonons play a smaller role here because of the gap in the spectrum at q = 0. Over the T range shown, however, even the ripplon contribution to # is less than a few percent of the Van der Waals term.

i0 o

I

*x

I

I

I

I

I

%

I 0- I

iX "-........ \\.

o10-2

iO- 3 -

\

\.

\

"X "o\.

io -4

0

I

I0

I

20

I

IN.

30

40 d

I

50

I

60

70

(~)

Fig. 1. Zero temperature contributions to the chemical potential of the p h o n o n s (solid curve) and ripplons (dash-dotted). Shown for comparison is the Van der Waals contribution C3d-3 (dashes) and the ratio of the total excitation contribution at T = 2 K to the Van der Waals term (plusses).

316

N.K. Mahale, M. W. Cole / Chemical potential of a film i0 -i

I

I

I0

I 20

I

I

I

I

K•-2

~ ,_

I0 "3 -

i(y 4

I0

0

I I 30 40 d (~)

%1 % I \ 50 60

70

Fig. 2. Thermal ripplon contribution to the chemical potential at various temperatures; to this L, added the T = 0 value in fig. 1 to obtain the net /~.

iO-~

I

I

[

I

I

I

40

50

6o

IO-" T=2K IK 10-4

0.SK

10-5 0

io

2o

30

"tO

d (~) Fig. 3. Thermal phonon contribution to the chemical potential at various temperatures.

N.K. Mahale, M.W. Cole / Chemical potential of a film

317

3. Discussion T h e largest effect of these e x c i t a t i o n s is the zero p o i n t p h o n o n c o n t r i b u t i o n . T h i s varies a p p r o x i m a t e l y as d - 3 o v e r the r a n g e of 2 0 - 7 0 A i n fig. 1 a n d is ~nus difficult to d i s t i n g u i s h f r o m a n --- 15% e n h a n c e d v a l u e of t h e V a n d e r W a a l s c o n s t a n t . S i n c e its m a g n i t u d e is s e n s i t i v e to the artificial large q cutoff, a realistic, q u a n t i t a t i v e p r e d i c t i o n is n o t possible. T h e c a l c u l a t i o n s p r e s e n t e d h e r e h a v e a s s u m e d T - i n d e p e n d e n t v a l u e s of the p a r a m e t e r s a n d n e g l e c t e d viscosity. M o r e r e l i a b l e e s t i m a t e s m a y give a larger c o n t r i b u t i o n , for e x a m p l e , of /~-), since o d e c r e a s e s w i t h T. T h e s m a l l m a g n i t u d e s c a l c u l a t e d here do n o t s e e m to j u s t i f y s u c h a c a r e f u l study.

Acknowledgments T h i s r e s e a r c h was s t i m u l a t e d i n p a r t b y h e l p f u l d i s c u s s i o n s w i t h G r e g D a s h , J a c k i e K r i m a n d M i c h e l Bienfait. S u p p o r t f r o m N S F G r a n t D M R - 8 4 1 9 2 6 1 is gratefully acknowledged.

References [1] W.A. Steele, The Interaction of Gases with Solid Surfaces (Pergamon, Oxford, 1974); D.F. Brewer, in: The Physics of Liquid and Solid Helium, Part 2, Eds. K.H. Bennemann and J.B. Ketterson (Wiley, New York, 1978). [2] R.J. Muirhead, J.G. Dash and J. Krim, Phys. Rev. B29 (1984) 5074: P.G. De Gennes, Rev. Mod. Phys. 57 (1985) 827. [3] R. Pandit, M. Schick and M. Wortis, Phys. Rev. B26 (1982) 5112. [4] M. Bienfait, to be published. [5] L.W. Bruch and X.-Z Ni, Disc. Faraday Soc. 80, in press. [6] J. Frenkel, Kinetic Theory of Liquids (Oxford Univ. Press, New York, 1949). [7] G.D. Halsey, Jr., J. Chem. Phys. 16 (1948) 931; T.L. Hill, J. Chem. Phys. 17 (1949) 590. [8] I.E. Dzyaloshinskii, E.M. Lifshitz and L.P. Pitaevskii, Advan. Phys. 10 (1961) 165. [9] W.A. Steele, J. Colloid Interfaces Sci. 75 (1980) 13; W.Y. Lee and L.J. Slutsky, J. Phys. Chem. 86 (1982) 842. [10] The strongest evidence supporting this relation is that of E.S. Sabisky and C.H. Anderson, Phys. Rev. A7 (1973) 790; the effect of retardation is seen to modify eq. (3) at very large d, in agreement with ref. [8]. [11] W.F. Saam and M.W. Cole, Phys. Rev. Bll (1975) 1086. [12] W.F. Saam, Phys. Rev. B12 (1975) 163; C. Ebner and W.F. Saam, Phys. Rev. B12 (1975) 923. [13] D.O. Edwards and W.F. Saam, in: Progress in Low Temperature Physics, Vol. 7A, Ed. D.F. Brewer (North-Holland, Amsterdam, 1978) ch. 4. [14] K.R. Atkins and I. Rudnick, in: Progress in Low Temperature Physics, Vol. 6, Ed. C.J. Gorter (North-Holland, Amsterdam, 1970). [15] L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Pergamon, London, 1958), ch. VII.

318

N.K. Mahale, M. IV. Cole / Chemical potential of a film

[16] The limiting values k m a x = 1 ,~-1 = qmax are estimated on the basis of the n u m b e r of allowed modes, but the c o n t i n u u m approximation fails for this large wave vector region anyway. See D.O. Edwards, J.R. Eckardt and F.M. Gasparini, Phys. Rev. A9 (1974) 2070, and J.R. Eckardt, D.O. Edwards, S.Y. Shen and F.M. Gasparini, Phys. Rev. B16 (1977) 1944. [17] E.S. Sabisky and C.H. Anderson, Phys. Rev. Letters 30 (1973) 1122. [18] B.L. Blackford, Phys. Rev. Letters 28 (1972) 414. [19] The capillary length in ref. [15] differs by the substitution for g in eq. (22) of the gravitational constant divided by 2. [20] We have used p = 0.145 g / c m 3, o = 0.37 d y n e s / c m , s = 240 m / s and C 3 = 2.9 × 10 37 erg cm 3, calculated by G. Vidali, M.W. Cole and C. Schwartz, Surface Sci. 87 (1979) L273, and measured by J.A. Roth, G.J. Jelatis and J.D. Maynard, Phys. Rev. Letters 44 (1980) 333. [21] We have here extended the upper limit of the q integral to infinity, as permitted at low T. [22] The p h o n o n contribution is proportional to T 4 at very low T, i.e., when only the smallest k n value is excited.