616
Langmuir 1989, 5 , 616-625
Physical Adsorption of Multilayer Films on Planar Substrates and inside Cylindrical Pores? E. Cheng* and M. W. Cole 104 Davey Laboratory, Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802
Received October 27,1988. In Final Form: February 1, 1989 We summarize and extend our previous study of multilayer film adsorption on planar surfaces. The density profiie of the adsorbed film is studied by a mean-field approximation. This leads to a self-consistent potential and a correction in thickness prediction to the result of Lifshitz and co-workers. The correction is determined by the van der Waals interaction and film compressibilityand is found to be important for He films. A simple model is presented also for the physisorption isotherm inside a cylindrical pore. Comparisons with experimental data are provided.
I. Introduction The nature of a physisorbed film is a subject of much current attention.'-ls In the case of a thick liquid film which wets the substrate, multilayer film adsorption becomes a particularly interesting system to study because of its apparent simplicity. One might expect here the substrate to present only a minor perturbation on the thermodynamic properties of the film. In the geometry of a planar surface, then, the film thickness d ought to be determined in a simple way by ita coexisting vapor pressure P and temperature T . Such a relation between T , P, and d on a planar substrate, as well as inside a cylindrical pore, is the subject of this paper. It has been shown that the difference in chemical potentials between the adsorbed film and bulk liquid is the key function in this relation:16 All = ll - Po (1) If the coexisting vapor is ideal at saturation, then the corresponding difference in the vapor phase satisfies" where Po is the saturated vapor pressure. The atoms in the film,on the other hand, experience an external potential V,(r)due to the substrate as well as the interaction among themselves. The substrate potential V&r) is dependent on the substrate geometry. For simplicity, we defer the discussion of physisorption in a cylindrical pore until section V and discuss here only the planar substrate, which is simply assumed to be a continuum occupying the region z < 0. In this simple case, the potential V , ( r ) is a function of z at large z:18 v,(Z)= - c 3 8 d Z - 3 (3) h C3a-s= dw a(iw)g,(io) (4) 47r 0
1
The quantity E , here is the dielectric function of the substrate, and a is the adatom polarizability. While evaluated at imaginary frequency iw, a and E , are real, positive, and monotonically decreasing functions of w , as we will see below. V,(z) given in eq 3 contains only the attractive part of the total substrate potential since the 'Presented at the symposium on "Adsorption on Solid Surfaces", 62nd Colloid and Surface Science Symposium, Pennsylvania State University, State College, PA, June 19-22, 1988; W. A. Steele, Chairman.
0743-7463/89/2405-0616$01.50/0
repulsive part is not relevant here. Furthermore, we note that this simple formula works in the nonretarded interaction region only. The retardation effect, due to the finite speed of light, will reduce it to a z4 dependence at large z.18 This effect will be discussed below. A very simple and qualitatively successful model, the so-called Frenkel-Halsey-Hill (FHH) model,lg was proposed about 40 years ago. Ignoring implicitly many body effects, it considers & to be the difference in the potential energies of adding an additional atom to the film and to its bulk liquid. AP =
Vs(d)- Va(d)
(6)
(7) Here Va(d)is the analogous potential of V,(d),with a bulk liquid substituted for the substrate. Also omitted in the model is the difference in entropy between the film and bulk. In general, however, retardation effects must be considered, and the simple proportionality Ap a d4 will not be valid. For later convenience, we define a general(1) Sullivan, D. E.; Telo da Gama, M. M. In Fluid Interfacial Phenomena; Croxton, C. A., Ed.;Wiley: New York, 1986. (2) Dietrich, S. In Phase Transitions and Critical Phenomena; Domb, C., Lebowitz, J., Eds.; Academic: New York, 1987; Vol. 12. (3) Brewer, D. F. In The Physics of Liquid and Solid Helium; Bennemann, K. H., Ketterson, J. B., Eds.; Wiley: New York, 1978; Part 11. (4) Lee, W. Y.; Slutaky, L. J. J. Phys. Chem. 1982,86, 842. (5) Bienfait, M. Surf.Sci. 1985, 162, 411. (6) Kayser, R. F. Kinam 1987, BA, 87. (7) Taborek, P.; Senator, L. Phys. Rev. Lett. 1986,57, 218. (8)Taborek, P. Phys. Reu., in press. Private communication. (9) Zimmerli, G.; Chan, M. H. W. Phys. Rev. B 1988,38,8760. (10) Blumel, S.; Findenegg, G. H. Phys. Reu. Lett. 1985,54, 447. (11) Lea, M. J.; Spencer, D. S.; Fozooni, P. Phys. Rev. B. 1987, 35, 6665. (12) Bartoech, C. E.; Gregory, S. Phys. Rev. Lett. 1985,54,2513. Drir, M.; Hess, G. B. Phys. Rev. B 1986,33,4758. (13) Migone, A. D.; Krim, J.; Dash, J. G.; Suzanne, J. Phys. Reu. B 1985,31,7643. Migone, A. D.;Dash, J. G.; Schick, M.; Vichea, 0.E. Phys. Rev. B 1986,34,6322. (14) Krim, J.; Watts, E. T. Phys. Rev. B, in press. (15) Smith, D. T.; Hallock, R. B. Phys. Rev. B 1986,34, 226. (16) Cheng, E.; Cole, M. W. Phys. Reu. B 1988,38, 987. (17) A small correction is needed if the vapor in equilibrium with the film is not ideal. In the case of 'He near 2 K, for example, the correction is about 5% in chemical potential; see ref 16. (18) Dzyaloshinskii, 1. E.; Lifshitz,E. M.; Pitaevskii, L. P. Adv. Phys. 1961, 10, 165. (19) Frenkel, J. Kinetic Theory of Liquids; Oxford: New York, 1949. Halsey, G. D., Jr. J.Chen. Phys. 1948,16,931. Hill, T.L. J. Chem. Phys. 1949, 17,590.
0 1989 American Chemical Society
Langmuir, Vol. 5, No. 3, 1989 617
Physical Adsorption of Multilayer Films
ization of this FHH relation: Ab
E
-~(d)d-~
(8)
where y ( d ) is a function to be derived. The FHH model then leads to (9) r ( d ) E YFHH = AC3 A more general approach has been presented by Dzyaloshinskii, Lifshitz, and Pitaevskii (DLP); it is believed to include quite general many body effects and retardation.le This theory is based on quantum field theory at finite T for the electromagnetic field in the presence of uniform media characterized by their bulk dielectric properties. It assumes that the density of the film is a constant, identical with its bulk value nL;a correction to this will be presented in section 111. Instead of the constant (YF"), the DLP model predicts r ( d ) as a function of d:20
11. Retardation and Many Body Effects a. Limiting Cases of ~ ( d )When . the film thickness d is very small, the retardation effect will become negligible and y ( d ) becomes a constant, $0). The characteristic thickness here is the distance light travels in a characteristic vibrational period (G-l), which depends in a complicated way (eq 15 below) on both adsorbate and substrate. Using an estimate hij 20 eV, CC found that retardation may be safely neglected only when d > 1, y *
Langmuir, Vol. 5, No. 3, 1989 619
Physical Adsorption of Multilayer Films
n(z)=nL =nv
0 Iz 5 d z>d
(35)
This neglects entirely any variation of the film density with z as well as the non-zero widths of the interfaces near z = 0 and z = d. The actual film density, of course, is a
smooth function of the distance z from the substrate surface. Due to the attractive substrate potential, moreover, the film layers near the substrate are denser than those near the liquid-vapor interface. This has a quantitative effect which we now consider. Since the adatoms in the film are in the field of the substrate potential V,(r) as well as the van der Waals (VDW) interaction u(r) among themselves, this is a nontrivial many particle problem. A mean field approximation should be adequate away from a phase transition. In this model, each adatom is considered to interact only with the average field of the surrounding atoms. In other words, the system acts as an “ideal” (hard-sphere) fluid in an effective potential field Veffdetermined self-consistently from the attractive interactions. The chemical potential of the fluid can then be expressed as a function of the film density profile n(r):l
h
5
3
U
II
- Veff(r)= d n ( r ) , n
D
Veff(r)= V,(r)
(.&+ d,
(A>
Figure 3. Graph of d I l zdefined by eq 32 versus the summation of d, and d, (defined by eq 34b and 34c, respectively). The solid line is the simple ansatz, eq 34. Data points are those of different adsorption systems, whose values are tabulated in Table I1 in ref 16. Substrates include graphite (filled circles), Si (plusses), A1 (crosses), and Au (open circles).
is proportional to d-’, the theoretical fully retarded behavior. An alternative form, used elsewhere,29corresponds to y* = (1
+ d*)-’
(33)
This dependence is seen in Figure 2 to be qualitatively, but not quantitatively, consistent with our results. Equations 29-32 provide the general behavior of y ( d ) and hence the physisorption chemical potential. They make it possible to make a general prediction from the knowledge of only two parameters, y(d) and d1/2. It has been shown that $0) has a small TJ expansion, eq 21, of which the leading term, AC3, represents the FHH theory. This expansion works very well in predicting the value of y(0) (see Table I in CC). A simple ansatz for dl12 has been proposed by CC as = da + 4
(344
da = c/[~a(~a(o))’/~I
(34b)
4 1 2
d, = c/w,
(34c)
i.e., a sum of propagation lengths in the two media, including the refractive index in the film. As seen in Figure 3, the estimate agrees with the numerical results, the average relative error being 4%. 111. Density Profile Effects
As stated above, one of the basic approximations in DLP theory is the use of a step-function density profile: (29) Smith, D.T.;Godshalk, K. M.; Hallock, R. B. Phys. Reu. B 1987, 36, 202. Putterman, S. J. Superfluid Hydrodynamics; North Holland: Amsterdam, 1974.
+ Jdr’
u(lr-r’l) n(r’)
(364 (36b)
where &,(n,n is the chemical potential of a reference (e.g., ”hard-sphere”) fluid in a state of uniform density n. u(r) is the attractive VDW pairwise interaction potential between adatoms, which has a simple form in the nonretarded region:
u(r) = C6r4
(37)
Equation 36 provides the expressions of the chemical potential as a functional of its density profile. Notice that Veffin eq 36b has been defined as a direct summation of the substrate and the adsorbate potentials, which completely ignores the many body effect discussed in the previous section. Using the “slab” model, eq 35 and the nonretarded substrate potential eq 3, simply leads to the FHH model, eq 6.’ On the other hand, the density profile can also be determined by these equations at any given chemical potential 1.1. In any case, these equations have to be solved numerically due to their complexity. In the thick-film region, however, the film density will be very close to its bulk value nL,and a perturbation method can be used to determine the density function n(z) n(z) = nL 6n(z) (39)
+
The unperturbed system is now just the bulk liquid at uniform density nL. For the perturbation potential Veff, the density response can be written as30 -kBT6n(r) = dr’ (no(r’)6(r-r’) + no(r)no(r’)ho(r,r’))Veff(r’) (40) Lid,
where no and ho are the density and total correlation function of the “unperturbed” system, respectively. The integration runs over the film region only, since we have made the approximation that nv = 0. In this case, the unperturbed state is the uniform bulk liquid, and we havem no(r) nLO(d-z) (41) 1
+ n L l d r ho(r) = kBTnLK
(42)
620 Langmuir, Vol. 5, No. 3, 1989
Cheng and Cole
where K is the isothermal compressibility of the bulk liquid. Since V e B is weak and slow varying in space, a linear and long-wavelength approximation can be applied. Then we simply have31 6 n ( z ) = -nL2KVeff(z)
(43)
This “additional” density will, of course, alter the adsorption potential at the top of the film. The correction in potential due to 6n is then (from eq 36b): 6V(d) =
dr’ u(lr-r’1)6n(z?
film
(44)
where u is the pairwise VDW potential between adsorbate atoms, eq 37. Inserting eq 43 and keeping the linear response limit again, we get 6 V ( d ) = - n L 2 ~ V e f f ( d ) l / , i d (-Cs/R6) R
(45)
The factor ‘ I 2is necessary when we extend the integration region from the film region to the whole space, neglecting the extra contribution from far distance since u is a power law decaying function. A lower limit cutoff in R is needed in the integration. The result of the integration is simply the constant a appearing in the VDW equation of characterizing the strength of VDW interaction. Then we have 6 V ( d ) = f/znL2KaVeff(d)
(46)
Calculations with alternative cutoffs give similar answers, as discussed in Appendix B. On the other hand, V,, itself is the total self-consistent physisorption potential. Since in lowest order perturbation theory the shifts in all thermodynamic energies coincide, we have Ag(d) = - ~ ( d ) d + - ~6 V ( d ) -7 (d) d-3 =-
t t
= 1 - t/2nL2Ka
(47) (48)
The “dielectric” constant t is defined in the usual way (customary in many body theory), so that screening is the medium’s self-consistent rearrangement effect. Notice here that t < 1 because the system is actually “anti-screening”; this is purely a property of the film, independent of the substrate. It should also be noted that this formula does not work near the critical point, since K m there and the vapor density was assumed negligible compared to that of the liquid. Near this point, singular effects due to the substrate’s compression will occur (as have been observed e~perimentally~~). The corresponding correction in film thickness d is found from the difference between eq 47 and 8 to be (to first order in 1 - e):
-
6d -_ d
l - E
3 - dT’/y
(49)
where dDLPis the film thickness predicted by DLP theory: (30) Rowlinson, J. S.;Widom, B. Molecular Theory of Capillarity; Clarendon: Oxford, 1982; p 77. (31) A more general form of 6n(z)can be found Steele, W. A. In The Solid-Gas Interface; Flood, E. A., Ed.; Marcel Dekker: New York, 1967. (32) Findenegg, G.H.; Loring, R. J . Chem. Phys. 1984,81,3270. (33) Pathria, R. K.Statistical Mechanics; Pregamon: Oxford, 1972. The coefficient a can be calculated from simple potential parameters (e.g., Lennard-Jones c and u); see: Israelachvili, J. N. Intermolecular and Surface Forces; Academic: New York, 1985.
Table I. Values of Compressibility K , Density n L, and Calculated by Eq 48 of He Films at Different Temperatures’ T,K nL, loz2~ r n - ~ K , lo-” cm2/dyn 1.50 2.18 1240 2.00 2.19 1330 2.50 2.18 1400 3.00 2.12 1490 3.50 2.04 1740 4.00 1.93 2150 4.20 1.88 2540
1
-e
l-€ 0.28 0.30 0.32 0.32 0.34 0.38 0.43
“ T h e K data are taken from ref 34 and those for nL from Table of Liquid and Solid Helium; Clarendon: Oxford, 1967.
A1 of Wilks, J. T h e Properties
eq 2, 8, and 10. The prime means differentiation with respect to d , which can be obtained simply from eq 29; all functions are evaluated at d = dDLp. The relative shift 6 d l d is only weakly dependent on the thickness dDLp: 6d 1-t -=(50) ~ D L P 3+ t { = (1
-
+ 1.64d*’.4 L
)
-
l
-
Notice that the denominator in eq 50 varies between 3 ( d 0 ) and 4 ( d m). The “dielectric” constant e and 6d are functions of temperature through variation of nL2K in eq 48. This leads to a temperature-dependent correction in predicted thickness. Since most of the experiments measuring film thickness of classical gases are carried out in rough vicinity of their triple points, we can estimate e for one gas there; the “law” of corresponding states says that this should be universal. The f i b of Ar, for example, has nL-’ = 47.63 A3, a = 3.76 X lo-% erg cm3,= and K = 200 X 10-l2 cm2/dynMand thus yields t = 0.83. This leads to a -5% correction in thickness according to eq 50. The helium film is a special case because of its significant quantum nature, which eliminates the triple point and leads to a relatively large compressibility and volume per particle. The temperature-dependent values of the compressibility and 1- e are tabulated in Table I. The value% a = 9.5 X lo-% erg cm3 is used in the calculation. It is easy to see that the corrections in thickness, from eq 49, are about 10% at T = 2.5 K and 15% at T = 4.2 K. This effect will be discussed in the next section when experimental results are compared. Here we only note that Migone et al.13 have observed in their experiment an increase in their empirical C3 values with temperature, which is consistent with our prediction since in the nonretarded regime of the experiment y / ( l - t) is proportional to C3. It should also be noted that an additional correction will enter experiments in which the total mass of the film is actually measured. This requires an integration of the density profile over the thickness of the film and is dependent on the experimental geometries. The effect of this additional contribution will make the correction even bigger. The calculation of this additional contribution is not included in this paper for simplicity. IV. Comparison to Experiments. Flat Surfaces Experiments measuring physisorbed film thickness have been carried out for 50 years.% Apart from an experiment of Sabisky and Anderson,36-37 the DLP theory has not been (34) Hallatt, A. C. H. In Argon, Helium and the Rare Gases;Cook, G. A., Ed.; Interscience: New York, 1961; Vol. 1. (35)Daunt, J. G.; Mendelssohn, K. Proc. R. SOC.A 1939, 170,439. (36) Sabisky, E.S.;Anderson, C. H. Phys. Reu. A 1973, 7,790.
Langmuir, Vol. 5, No. 3, 1989 621
Physical Adsorption of Multilayer Films
I 4.0
-
I
I
I
I
-
2.0
A
E
-
1.0
0
Y
I
0.5
0.25
-
t
1"
The ellipsometric method is one which can measure the thickness directly. A measurement of 4He film thickness has been done by Hemming using this t e ~ h n i q u e .A~ ~ rather big discrepancy was reported between his data and the DLP theory. Here, in Figure 4, we compare this data with the corrected DLP prediction (including the density profile effect). An excellent agreement is found in the case of quartz substrate. For the Au substrate, the agreement is also fairly good. The quartz crystal microbalance method has been used by several g r o ~ p s , ~Lea ~ J et ~ Jal.ll ~ have measured 4He film thickness and used an empirical AC, whose value decreases with the thickness in their (FHH) theoretical calculation. At best, this only qualitatively confirms the retardation effect predicted by the DLP theory. Using the same technique, Krim and Wattd4 measured the film thickness on metal substrates. Their data for classical gases (e.g., Xe, Kr, Ar, N2,and 0,)on the surface of Au(ll1) are quantitatively consistent with our calculations over the region d I50 A. The data of helium film thickness, however, are found to be consistently greater than the DLP prediction. In Figure 5, we show their measurements with the comparison of the theoretical prediction. We observe that the density profile correction has improved the situation, but quite large discrepancies still remain. Krim and Watts attribute this quite plausibly to the role of surface roughness. The vibrating fiber is a newly developed technique which has received much recent a t t e n t i ~ n . ' - ~ +It~ involves a nominally cylindrical fiber (about 10 pm in diameter) as a substrate. The fiber is driven to oscillate at a certain resonant frequency. The vibration frequency decreases when a film is adsorbed on the fiber due to the extra inertia (including the hydrodynamic mass of the displaced gas). This frequency drop can be measured to determine the mass, and hence the nominal thickness, of the film. The cylindrical geometry of the fiber introduces a further correction to p because of the surface energy of the film. A t saturation, instead of eq 52, we have6 r(d)d-, = mgh + u / n L r (53)
? 150
200
300
400
(a) Figure 4. Comparison between calculated results and the exd
perimental data (all points) from ref 38 of 'He film thickness d on quartz and gold substrates. Dashed curves are the DLP prediction calculated in ref 16 from eq 1,2, and 10. Solid curves are those with density profile corrections determined by eq 50. particularly successful in general in predicting film thickness. CC discussed several experimental difficulties in film thickness measurements and also reviewed some current work. Here, therefore, we mainly review some of the updated results and discuss the density profile corrections when applicable. The retardation effect, as in section 11, becomes important a t d 1 100 A. This coincides with the saturated film region, where bulk fluid coexists a vertical distance h below the film. A correction in p due to the gravitational energy difference is then needed in the relation
where u is the surface tension of the fim and r is the radius of the fiber. The last term in eq 53 is questionable, however, since it is a macroscopic curvature contribution, while the experimental surface could be locally flat. In CC, we concluded this effect should not be considered since it was believed that the fiber has a lateral extended scale of about 1p (>>d). Better pictures of the fiber surfacelghowever, r ( d ) d - , = mgh (52) indicate that the domain size is actually much smaller than that. If this scale be less than 500 A, the local curvature where m is the atomic mass and h is of order a few ceneffect will be important, but it is not the same as the timeters. In this thick-film region, experiments are difmacroscopic form in eq 53. The actual prediction is thus ficult because of the'requirements of high P and T resolution and uniformity, which are estimated as AP/P i= 1 0 ~ ~ambiguous. Ignoring this term completely, the results of saturated film thickness by Taborek were compared in CC at saturation or ATIT i= with theoretical predictions of DLP. For the 4He film, Basic techniques in measuring film thickness include however, the density profile correction, which is a 11% film mass (crystal microbalance, vibrating fiber, etc., see increase in thickness at T = 2.7 K (from eq 49),should be below), acoustic interferometry,%ellipsometry,38and third included in the theoretical prediction. Based on the DLP sound velocity meas~rernents.~~ The last is based on the theory, the corrected theoretical value is then dtheol= 340 fact that the third sound velocity in a thick He film is a A, which agrees the experimental value within 12%. This function of its thickness. As discussed in CC, it is very agreement, however, may be fortuitous due to the effects difficult to deduce the role of retardation from this meaof surface heterogeneity, established by Zimmerli and surement. Chan: who also measured the 4He film thickness. Some qualitative consistency with the predicted retardation (37) In their experiment, the density profile has been considered as effect was observed as discussed in their paper and CC. a phase-shift correction. This correction has been found experimentally to be about 5% of the thickness; see ref 36. Bruschi et aLa measured the Ar film thickness, where an (38) Hemming, D.Can J. Phys. 1971,49, 2621. More precisely, the
measured quantity is a kind of optical thickness. (39) van Spronsen, E.; Verbeek, H. J.; van Beelen, H.; de Bruyn Oubuter, R.; Taconis, K. W. Phsyica 1974, 73, 621.
(40) Bruschi, L.; Torzo, G.; Chan, M. H. W. Europhys. Lett. 1988, B6, 541.
622 Langmuir, Vol. 5, No. 3, 1989
40
i
Cheng and Cole
11 “1
Xe/Au( 1 1 1)
t
0
0
1
0.5
P/PO
coi
0
,”
He/Au(lll)
.
Un/Aii
0.5
1
P/PO
Figure 5. Comparison between calculated results and the experimentaldata (all points) from ref 14 of (a, left) Xe and (b, right) 4He film thickness d on Au substrate. Dashed curves are the DLP prediction calculated from eq 1, 2, and 10. Solid curve is that with density profile corrections determined by eq 50.
agreement with the FHH prediction is found for films up to 30 layers. They observed no effect of retardation, contrary to our expectation; eq 8 would predict y to fall by 40% and d to decrease by 10% at the upper limit of this experiment.
-
V. Physisorption inside a Cylindrical Pore While the discussion in previous sections concerns planar substrates, it is also very interesting to study multilayer physisorption in a restricted g e ~ m e t r y .In ~ this section, we propose a simple model of the adsorption energetics, from which we derive the adsorption isotherm. The cylindrical pore problem is much more complicated than that of a planar substrate. Besides the VDW potential, the curvature of the liquid-vapor interface gives an additional contribution to the chemical potential. We find it is convenient to calculate these two contributions separately. In doing so, we are led to consider an intermediate, nonexistent system, the “bubble” system, which is a cylindrical vapor cavity (with radius r = R - d and vapor pressure P ) inside the bulk liquid. The chemical potential of this “bubble” system is p(r). Then we have ACC = All(1) + AP(11)
(54)
where H(d,R) is the chemical potential of the actual adsorption system and po is that of bulk liquid; see Figure 6.
It is easy to see that, when d is small, the first term Ap, in eq 54 is the dominant term because it includes the adsorbatembstrate interaction, which is the largest energy here. When the film gets thick, however, the substrate will become less important, and Ap will be dominated by the curvature contribution ApLcI1), as we will see below. The retardation and many body effects we discussed in the planar substrate case also exist here. In section 11, we found that many body corrections to simple FHH theory (in the thin-film region) are very small for inert gases; see
(C>
Figure 6. To calculate Ap = pa - pC,we set this equal to pa - pb + pb - p c with (a) the physisorption system, (b) the ‘bubble” system, and (c) the bulk, with a flat interface.
eq 21 and Table I in CC. This is because the correction terms are of higher order in the small parameter 7 and a substantial cancellation occurs. It seems reasonable to assume here that this many body correction is also small in the cylindrical geometry. Thus, we omit this effect hereafter and consequently regard the VDW interaction in the film as a pairwise additive one. Under the assumption that many body effects due to film adatoms are negligible, a FHH-type calculation will be appropriate here. In calculating A ~ ( I )then, , we can remove the film slab (of thickness d ) from both systems in Figure 6a,b. This leads to a simple expression for Ap(1) as in eq 6: A/L(~)= Va*(d,R) - Va-*(d,R) (57)
Lafigmuir, Vol. 5, No. 3, 1989 623
Physical Adsorption of Multilayer Films where Va* is the adsorption potential of an adatom at distance d from the surface of a cylindrical pore substrate of radius R. Va* is the analogous potential with the substrate replaced by a bulk liquid. If retardation can be neglected, we have41-43
30
r-----
with
where I, and K , are modified Bessel functions. The function Va' can be obtained simply by replacing the substrate dielectric function e,(iw) with that of the adsorbate, c,(io). If we use this nonretarded potential and let R m, eq 57 will become the FHH flat surface formula, eq 7, as one would expect. The general formula of Va* (including retardation) has been given by Nabutovskii, Belosluskii, and K ~ r o t k i k h : ~ ~
-
"0
4m2q12q~2z22(€n1)*m(z)[*m(zl) - @ m k J I
1
(59)
Here the prime in the sum over m means to multiply the first term by one-half and z =
+ q2)'/2
+
z2 = p(x2 q2€*)1/2 z1 = pz q = (w/c)r p = pmxq12(€,- 1) q1 = p q 60' = P' + q12z12z22(@12 - *2l)(€n+12 - *zl) (x2
61° = P2 + 412~12ZZ2(@,2 - *21)('S@12 60'
5
10
15
20
25
30
Figure 7. Curve a and the plusses denote the absolute value of & calculated from eq 54 and &a defied by eq 55 and calculated from eq 57, respectively, for He/graphite. Curve b and the crwes denote the corresponding data for He/glass. Curve c shows the value of Ap(m defined by eq 56 and calculated from eq 62. All the calculations have been done at R = 50 A.
In the vicinity of the saturated pressure Po,we can expand the chemical potentials of both the vapor and liquid phases to first order:
- @21) - QZI)
= p 2 + q12z12z,2(@12 - W(en@12 *ik = Zi\km(Zk) @ik = Z i a m ( Z k )
Limiting cases of this general formula are discussed in the% paper, including the nonretarded limit, which yields eq 58.43 It is now easy to calculate Apa) with this general expression, using the same model dielectric functions we used for a planar substrate. The curvature contribution Apcccm, on the other hand, has nothing to do with the substrate. It will provide basically a geometric contribution to A p , which becomes relatively important in the thick-film region (i.e., d not small compared to R ) . Due to the existence of the curved surface, the equilibrium liquid pressure in the "bubble" system,PL, will be less than that of the vapor P by a small quantity determined by the liquid surface tension Q:
PL = P - c / r
(60)
(41)Schmeits, M.;Lucas, A. A. J. Chern. Phys. 1976,65,2901; Surf. Sci. 1977, 64,176. (42)Cole, M.W.;Schemite, M. Surf. Sci. 1978, 75, 529. (43)Nabutovskii, V. M.; Belosludov, V. R.; Korotkikh, A. M. Zh. Eksp. Teor. Fir. 1979, 77, 700 (Sou. Phys. JETP 1979,50(2), 352).
Using eq 60 with the coexistence relations, we get
(62) Notice that this formula works only in the thick-film region, where the vapor pressure is very close to Poand the expansions in eq 61 are adequate. Another approach to estimating Apucmby considering directly the VDW interaction potential difference is given in Appendix A. That less reliable result is comparable to eq 62. Figure 7 shows the behavior of AN(^) and Apcmfor these adsorption systems. We see that Ap(1) dominates at small d but rapidly decreases when the film becomes thicker. The curvature effect, Ap(=), becomes important at d = 20 A (with R = 50 A) for He film and dominates when d is even larger. This agrees with our previous discussion. We note that the region in which Ap(11)dominates is just the region in which the capillary condensation happens,44
624 Langmuir, Vol. 5, No. 3, 1989
Cheng and Cole
Table 11. Values of A, and A, for Several Films Calculated by Eq A5 and A6, Respectively” gas ‘He
HZ Ne Ar
CH, Xe
A1
A2
13 100 140 550 700 1250
9.2 97 110 460 630 900
Appendix A. Alternative Calculation of Apcm We can also estimate of eq 56 by considering interaction potential energy differences. Under the assumption that the VDW interaction in the liquid film is pairwise additive, we can see that the chemical potential A p m is due to the interaction between the adatom and the atoms in the shaded region in parts b and c of Figure 6. As the simplest approach, we use the nonretarded pairwise interaction, eq 5:
a The parameter values of CBand u are taken from ref 27 and 8, respectively. The unit is KA.
at which point this calculation becomes inapplicable. Retardation has been shown as an important effect in planar geometry for films -100 A To investigate this effect inside a cylindrical pore, we have calculated Va*(r) of the Hefgraphite system ( R = 100 A) by using eq 58 and 59, respectively. To our surprise, these two equations give essentially the same results for d up to 70 A, which means the retardation effect is not very important here. This result may be understood qualitatively by realizing that the cylindrical geometry provides a relatively larger contribution of nearby atoms, with nonretarded interactions, than the planar geometry. This can be verified directly by adding up these respective contributions, which is justified in the small q case. Finally, we note that when the film becomes thick, A p will be dominated by the curvature effect Ap(m instead of Thus, we conclude that the retardation effect included in eq 59 is not an important effect here. One can thus use the simpler eq 58 to calculate A P ( ~ ) .
where i; is the distance between the interacting atoms and the factor of ‘I2is to avoid double counting. The integration regions b and c are the shaded areas in parts b and c of Figure 6, respectively. In order to avoid the divergence at i; = 0, we introduce a minimum distance between the atoms in the liquid, Fm, which is determined by
Then we have
where y = max(P,2r sin e), y’ = min(P,2r sin e), and z’ = (Fm2
VI. Summary and Discussion This paper has summarized and extended the previous study (CC) of multilayer film adsorption. The widely used FHH model has been shown to be the leading term of a many body expansion of the DLP theory in the nonretarded limit. While the FHH model is a good approximation in the thin-film region, retardation has been shown to be important at d 2 100 A. Equations 29-32 provide the general retardation behavior with the quantities ~ ( 0 ) and dlI2 well determined by eq 21 and 34, respectively. The parameters for common adsorption systems have been tabulated in CC. In the present paper, we have also shown that the film density profile is an important effect in the theoretical prediction of He film thickness, while this correction is negligible for classical gases. A simple model of film adsorption inside a cylindrical pore is also presented, which, we believe, will be helpful in experimental studies of physisorption in porous media. Despite great experimental effort in this area, the status of these theories remains ambiguous. While some measurements show agreement with the theory, great discrepancies are also observed. We urge that experiments be done to address this unsolved problem, especially perhaps with capacitance or ellipsometric methods, where few results have been provided thus far on high-quality surfaces in the thick-film regime. Acknowledgment. We are grateful to Bill Steele, Jay Maynard, Moses Chan, Rich Kayser, Greg Zimmerli, Peter Taborek, Leon Slutsky, Jackie Krim, and Flavio Toigo for helpful discussions and preprints. This research was supported in part by NSF Grant DMR-8718771. (44) Evans, R. In Course of Lectures at Les Houches School on ‘Liquids at Interfaces”; May/June, 1988. (45) Saam, W. F.; Cole, M. W. Phys. Rev. B 1975, 11, 1086. (46) Awschalom, D. D.; Warnock, J.; Shafer, M. W. Phys. Rev. Lett. 1986, 57, 1607.
-p
y .
Since Pfr
