J. Phys. Chem. 1993,97, 937-943
937
Solvation Pressures for Simple Fluids in Micropores Perla B. Balbwna.,+David Berry, and Keith E. Gubbins' School of Chemical Engineering, Cornel1 University, Ithaca, New York 14853 Received: September 22, 1992
We report a comprehensive study of the effects of pore size, temperature, bulk fluid density, and intermolecular potentials on the solvation pressure for simple fluids in slit pores. The calculations are based on nonlocal density functional theory. The interactions involving the fluid and wall molecules are modeled using the Lennard-Jones potential, and the effects of varying the interaction parameters (esf/eff and uSf/um,where f and s denote fluid and solid molecules, respectively) are studied. Results are obtained for pores with both identical and nonidentical walls, and qualitative features of the results are compared with available experimental information.
1. Introduction We consider a pure fluid confiied between parallel solid surfaces separated by a distance Hand in equilibrium with a reservoir of bulk fluid at temperature T and chemical potential p. For a large separation, H,the force per unit area exerted by the fluid on the walls is the pressureof the bulk fluid+, but as Hdecreases and becomes of the order of the range of the intermolecular forces, the force per unit area, p ~differs , from the bulk fluid value. The difference,f = PH - p m , is usually called the solvation pressure, solvation force, or disjoining pressure. This solvation pressure plays a central role in adhesion, lubrication, pore swelling, and colloid stability. Positive values off correspond to mutual repulsion between the two walls; negativef corresponds to mutual attraction. The development of an experimentaltechnique to measure the solvation pressure by Israelachvili and co-workers was the starting point for a series of experiments using inert organic liquids,' liquid crystals.2 and electrolyte solutions3 between two crossed cylindrical surfaces of molecularly smooth mica. The effect of temperature and the presence of impuritieswas also in~estigated.~ Christenson et al. have made systematic studies with the Israelachvili apparatus for a series of n ~ n p o l a r ,polar: ~ and hydrogen-bonding liquids? binary mixtures of nonpolar liquids,8 and a thin film of nonpolar liquid containing small amounts of water between mica surface^;^ the effect of temperature was also studied.10 More recently, the experimentsby Parsegian et al.I1-l3 using surfactant solutions having improved the understanding of complex liquids. A comprehensive review on both the experimental and theoretical work has been given by 1~raelachvili.l~ Among the most important featurest4 shown by the solvation pressureis its oscillatory behavior as His varied. As Hincreases, the maxima in the oscillations off show an exponential decay of the form A exp(-H/K) where K is usually in the range 1.2-1.70 ( a is the diameter of the fluid molecule). The magnitude of the oscillations as determined experimentallyseems to be insensitive to changes in temperature but sensitive to the chemical nature of the surfaces and very sensitive to the presence of impurities. On the theoretical side, the works by Lane and S p ~ r l i n g l ~ - ' ~ and Snook and van Megen'8-20 were the first Monte Carlo simulations showing this phenomena. They studied the case of a Lennard-Jones (LJ) liquid and a wall-fluid LJ interaction. Later simulation studies have included Molecular Dynamics calculations for LJ2' and hard sphereZ2fluids and Monte Carlo simulations for LJ23 fluids in slit pores. Several authors have used density functionalthe0ry2~~25 or integral equation the or^^^.^' to study solvation pressure, and Hendersonz8has derived several exact conditions and suggested approximations to solve them. On leave from INTEC, Universidad Nacional del Litoral, Santa Fe, Argentina. +
0022-36S4/5S/2097-0937~04.oo/o
The previous simulation and theoretical studies of solvation pressure havebeen for a limited range of the independent variables involved, generally a single temperature and fluid-wall system. In this work we report a comprehensivestudy of solvation pressure for LJ fluids in slit pores over a wide range of temperature, pore size, and LJ interaction parameters (esf/eff and a,~/a~,where f and s indicate fluid molecule and solid molecule, respectively). We also study the effect of two different slit walls and make qualitative comparisons with experiment. Our calculations are based on a nonlocal density functional theory due to TarazonaZ9 that has successfully represented the behavior of inhomogeneous fluids under several conditi0ns.3~In a previous study3IJ2 we have reported similar results for the influence of these system variables on the class of adsorption behavior, wetting, and heats of adsorption. 2. Theory
1. Mean Field Density F U I I C ~ ~Theory. O M ~ The system is defined as two semiinfiniteparallel walls separated by a distance H and immersed in a very large reservoir of constant volume containing a bulk fluid at fixed chemical potential p and temperature T. At equilibrium, the fluid adsorbed on the walls reaches a chemical potential equal to the one in the bulk, p. The walls act as an external potential, creating an inhomogeneity in the densityof the fluid confined between them. For the particular geometrywe are considering, the densityvaries only as a function of the perpendicular distance z between the solid surfaces, i.e., p(r) - p ( z ) . As mentioned in the Introduction, we use a nonlocal density functional theory taken from Tarazona, and we refer to the original papers29933for details. The equilibrium density profiles for the confined fluid are obtained by numerical minimization of the grand potential free energy functional, Q ( p ( z ) ) . A perturbation approach is used to describe the free energy functional, the reference hard sphererepulsive part being a nonlocal functional of the density. To take into account the nonlocal nature of the functional, a smoothed density p(r) is introduced and it is postulated that the free energy density for the reference hard sphere fluid at some point r in the real system equals that for a homogeneous fluid at density p. In the Tarazona prescription p(r) is written as a weighted average of the local density, and the weighting function is then expanded in powers of the density; the coefficientsin theexpansion are obtained by matching the resulting hard spheredirect correlation function to the Percus-Y evick result for hard spheres. The attractive part of the free energy functional is calculated in the mean-field approximation, Le., molecular correlations due to attractive forces are neglected. The inputs needed for calculations are the intermolecular potentials (see below), the hard sphere diameter d, and an equation for the exccss Helmholtz free energy for the hard sphere fluid. For d we use
(d
1993 American Chemical Society
Balbuena et al.
938 The Journal of Physical Chemistry, Vol. 97, No. 4, 1993
a temperature-dependent diameter, using the expression of Lu et al.,34 and the CarnahanStarling expression3sis used for the excess Helmholtz energy for hard spheres. More completedetails of the theory have been given elsewhere.36 2. Solvation Pressure and the Grand Potential. As discussed by Lane and SpurlingI5the thermodynamics of a one-component fluid confined between two parallel walls of area A is described in terms of the grand free energy: QsU-TS-pN (1) If the system undergoes a reversible infinitesimal change, the change in the free energy is given by dn=-p,dV-SdT-Ndp+ 2ydA-(Aj) d H (2) where pa is the bulk fluid pressure, S is the entropy, N is the number of fluid molecules, and 7 is the solid-fluid interfacial tension. The first three terms are the usual ones for a bulk system, while the last two are due to the presence of the surfaces and are the free energy changes due to changes in the area and in the separationHbetween walls. Consideringthe same system without plates, eq 2 reduces to dQb = -pm dV- Sbd T - Nb d p (3) where the superscript b indicates values for the bulk fluid. Defining excess functions, xCx = X - xb,where X and xb are at the same p, T,and V, it follows from eqs 2 and 3 that dQeX= -FX d T - (AI')dp + 2 7 dA - (Aj)dH (4) where A r = N - IP and r is the excess adsorption per unit area. Taking the second differential form from this expression,it follows that2s
f = -2(ay/aH)T,g The integral form for the surface excess grand potential
(5)
flex= 2yA (6) Since the bulk fluid is recovered in the limit of infinite separation,
Ob = -p,AH
+
~(HA T ) = P,(H,P, T ) - P-(CC, T) (14) 3. IntermdeculuPotenUdWaUModel Acutandshifted LJ potential (Weeks, Chandler, and Andersen3*)is used to describe the fluid-fluid interactions; the cut-off distance r, was taken as 2 . 5 ~ ~The . interactions of the fluid molecules with the wall are given by the integrated form39of the LJ potential written in dimensionless form as
(%f/@ff)4
+
3A*(0.61A* z * ) ~ where A* = 2 ~ p s * ( c s ~ / e ~ ~ ) ( u s ~ / u ~ ~ps* ) z (isAthe * ) / reduced T.; solid density ~ ~ ( u fz* f )=~z/um, , A* = A / q , and T.is the reduced temperature kT/cfr. This model represents a surface with layers separated by a distance A. We have used the following parameters corresponding to a graphite surface: us = 0.340 nm, cu = 28 K, p s = 114 nm-3, and A* = 0.8793. For the slit geometry, the external potential becomes Vext(z,H) = 4sf(Z) + 4JH - Z) (16) The cross Lennard-Jones parameters (usfand c,f) are calculated according to the Lorentz-Berthelot rules, csf = (es/em)l/z; usf = (us uff)/2. For the special case of methane as the fluid (considered as an example in some of the later results) the fluidfluidparametersareMerf/k= 148.12Kandum=0.381 nm.Once the equilibrium density profile is determined, the number of molecules per unit area, or total adsorption rs*,is calculated according to
+
rS*= ( 1 / 2 ) r p * ( z * )
dz*
(17)
(7)
where rS*= rsuffZ,p*(z) = p(z)ud, Ns = H/uR, and z* = z / u ~ . Coexistence of more than one phase is determined when more than one minimum has equal values for the grand free energy.
(8)
3. Results and Discussion
Then, from eq 6, 2yA = s2 p,AH and from eqs 5 and 8 we obtain
conditions of p and T , the solvation pressure f is computed by subtracting the bulk pressure from p ~ ,
(9) Thus, since f = p~ - pa, we have for pH,
In the limit of H+ m, the fluid-solid interfacial tension tends to a constant value independent of H,and from eqs 5 and 9,f 0, and p~ pa. p~ is the normal component of the pressure tensor that results from confinement of the fluid. Derjaguin and Churaev37 have called the net force f the disjoining pressure. The free energy functional is given by the sum of the intrinsic Helmholtz free energy F(p(r)) and two other terms for the contributions of the chemical potential and the external potential,
+
+
Once the equilibrium density profile is determined for the given
The solvation pressure in the slit pore depends on five independent variables: the bulk fluid density (or equivalently the relative pressure PIP", where PO is vapor pressure), pore width H , temperature T, and the potential parameter ratios usf/uffand csf/cff. We define H to be the distance separating the centers of the first layer of solid atoms in each wall. All variables are reduced with the fluid-fluid LJ parameters: pb* = Pbbd, n+ = H/uff, '7 = kT/cff, and f = fur?/cff. The experimental studies5J0J4correspond to Ns values in the range 2-9 and T. values between 0.7 and 1.0. The results forf are presented as a function of either bulkdensity (section 1) or pore width (section 2). In section 2 we also consider the effect of varying temperature and potential parameters on the plots of solvation pressure vs pore width. After presenting these results for pores with identical walls we briefly consider the case of pores with nonidentical walls in section 3. 1. Effect of Bulk Fluid Density. Initially we studied the effect of varying the bulk fluid density on the solvation prmsurcs, at fixed H*and T..The LJ parameters were chosen to correspond to CH4 on graphite. Results for a pore of H* = 2.6 (H= 0.99 nm) at several temperatures are shown in Figure la, while the corresponding adsorption isotherms are presented in Figure 1b. At T.= 0.7.0.8, and 0.9 there is a discontinuity in the solvation pressure, a first-order phase transition (capillary condensation) that corresponds to the micropore filling. For this fluid-pore system the capillary critical temperature, Tal. is about 0.95. The
Simple Fluids in Micropores
fl
The Journal of Physical Chemistry, Vol. 97, No. 4, I993 939
a
a
f"-:i_
Io-' iom4 0.5 H*= 5.4
IO+
lom2
0.0
-0.5 10-0
16'
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loo
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IO0
P/ PO
0.8
0.6
c
0.4
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IO-^
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Figure 1. (a) Solvation pressures as a function of the fluid pressure P
(relative to the bulk saturation pressure P)for CHI on graphite in a micropore: H* = 2.6(H = 0.99 nm), at P = 0.7, 0.8, 0.9, and 1. (b) Adsorption isotherms.
transition is determined by computing the grand free energy a( p ) for all the points on the 'high-density" and the 'low-density" adsorption branches on Figure 1b and finding the point where both values coincide. The mean-field theory predicts a van der Waals loop in the two-phase region, so that we find points of metastable local minima in the free energy; these local minima have higher values of the free energy than those for the stable global minima. At 7" = 1, the transition becomes second order, and both the pore density and the solvation pressure change in a continuous way, as seen from Figure 1, parts a and b. At very low pressures, of the order of 10-7-10-5,corresponding to large negative values of the reduced excess chemical potential, the number of moles adsorbed per unit area, r,*,is very small, and the net force per unit area is also low, increasing in magnitude with the increasein pressure, remaining attractive until the phase transition takes place. At higher pressures, the fluid first condenses by capillary effects, and then for still higher pressures the repulsive forces (from the fluid-fluid and fluid-wall interactions) become dominant, and the solvation pressure becomes positive. At much higher pressures, the adsorptionisotherm tends to level off and the solvation pressure tends to a constant value of 8.74 (7" = 0.7), 8.6 (7" = 0.8). 8.4 (7" = 0.9), or 8.0 (7"
= 1).
P/PO
Figure 2. (a) Solvation pressures as a function of the fluid pressure (relative to the bulk saturation pressure) for CHI on graphite in a micropore: H* = 5.4(H= 2.06nm),at P =0.6,0.8,and 1.1. Theinset shows an expanded scale for low pressures. (b) Adsorption isotherms.
Similar results are shown for Hs = 5.4 (H= 2.06 nm) in Figure 2. The principal difference in this case is that at the lower temperatures (e+, 7" = 0.6) there is a 0-1 layering transition at low pressure ( P / P = 8 X 10-5 for 7" = 0.6) followed by capillary condensationat higher pressures (see Figure 2b). This layering transition has a relatively small effect on the solvation pressure, as seen in Figure 2a, since the fluid density is low in the center of the pore under these conditions. At the limit of P P,fr tends to 1.56 (7" = 0.7), 0.7 (7" = 0.8), 0.22 (7" = l), or 4 . 1 2 (T*= 1.1). The effect of changing the pore width at a constant reduced temperature 7" = 0.8 is shown in Figure 3a for CH4 on graphite, where the solvation pressure for Hs = 2.6 (0.99 nm) is compared to two other micropores, IP = 3.6 (1.37nm), and IP = 5.4 (2.06 nm) and to a mesopore of Hs = 7.6 (2.9 nm). As the pore size increasesfrom IP = 2.6, we pass from a fmt-order phase transition to cases of intermediate pore width with continuous filling and back to Cases of capillary condensation again (Figure 3b). For pore sizes between Hs = 2.6 and 3.6 there exists (at this particular F and e,!/cn) a critical pore size above which no 0-1 layering transition exists. At somewhat larger Hs another critical radius occurs, above which we find capillary condensation. Thus, at IP = 3.6 the adsorption isotherm (Figure 3b) is reversible, and the
-
Balbuena et al.
940 The Journal of Physical Chemistry, Vol. 97, No. 4, 1993
a
IO-^
I
iom2
IO-'
ioo
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,
,
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00
I 15
25
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Figure 4. (a) Solvation pressure for CH4 on graphite as a function of pore width H* for a bulk density pb* = 0.005 and 'P = 1.1. The solid line is the density functional theory calculation; the dashed curve is the analytic approximation g(H) (see text), with parameters K I= 80, K2 = K3 = 1.1 1. (b) Average density in the pore.
given by Figure 3. (a) Solvation pressures as a function of the fluid pressure (relative to the bulk saturation pressure) for CHI on graphite in slits of
H* = 2.6 (H= 0.99 nm), H* = 3.6 (H= 1.37 nm), H* = 5.4 (H= 2.06 nm), and H* = 7.6 (H= 2.9 nm) at 'P = 0.8. (b) Adsorption isotherms. density changes in a continuous way on micropore filling. This is reflected in thevariationof thesolvation pressure, whichchanges continuously from negative (attractive) values for low loadings to positive (repulsive) values at higher loadings, corresponding to a condensed phase. For H* = 3.6 the value off at the limit of P-. PO is 4.36 at Ts = 0.8. The largest micropore, H* = 5.4, again shows a first-order transition, this time corresponding to capillary condensation. In the mesopore, H* = 7.6, the first minimum observed in the solvation pressure corresponds to the completion of the first layer, which is filled in a continuous way, followed at higher pressures by a first-order transition, capillary condensation. However, for such larger pore widths the solvation pressures are smaller in magnitude than those for micropores. As P PO, f tends to 0.37 at this temperature for the mesopore. More detailed aspects of phase equilibria in such pores are discussed elsewhere.32 2. Effect of PoreWidth. In this section we present calculations forf vs H*, the separation between the pore walls, keeping other variables (pb*, TI,uSr/um,esr/cm) fixed. Experimental results are usually presented in this way. The calculatedf values oscillate as a function of H* for small slits; this behavior is similar to that found for other properties, such as adsorption and diffusion. For the 10-4-3 potential that we are using, the partial derivative of the external potential with respect to H,to be used in eq 13, is
-
(a,f/%)4
+
A*(0.61 A* H - z ) ~ where all the variables have been defined above. It is not simple to predict a priori the effect of changing a particular variable, e.g., u,r/ufr or esr/etr, by examining eqs 13,14, and 18, since both the density profile and the potential derivative are affected by such a change. Although this derivative, eq 18, is known analytically, the same does not hold for the density profile for this particular model. It is convenient to start with a physical picture of the phenomenon. The variation of the solvation pressure with wall separation H,is shown in Figure 4a, for a typical case (CH4 on graphite at Ts = 1.1); the corresponding mean density in the pore, pp* = 2I',*/H*, is presented in Figure 4b. The bulk fluid is a vapor reduced density pb* = 0.005. However, the effect of confinement is to cause the mean density within the pore to have values close to those of the bulk liquid state. It has been proposed by several author^*^-'^ that the maxima in the solvation pressure correspond to "ordered" states, where the molecules are efficiently arranged in layers, and the available space is utilized in the best way. These maxima, P I ,P2, and Ps on Figure 4a, correspond to points P I ,P2, and P3 on Figure 4b. They are near to, but to the left of, the maxima in the mean
Simple Fluids in Micropores density in the pore. For the special case of hard walls, Henderson28 has shown that the maxima and minima inf’ correspond to maxima and minima in the fluid density at contact with the wall. We believe the displacement of the extrema inf’ from those in the adsorption is a result of the attractive forces. When H- =, the solvation pressuref’ 0, and the average density in the pore tends to the bulk fluid value. Before continuing,we make several comments concerning the oscillatory behavior of the solvation pressures with H as found experimentally and in our calculations. First, in most of our calculations we have considered the behavior of a fluid at conditions corresponding to a bulk gas state, whereas many experimentsinvolve bulk liquids; however, the effect of confinement is to produce liquid-like densities within the walls, and our qualitative comparison with experimental results remains valid. As mentioned in the Introduction, the peak to peak amplitudes aremeasured as the difference in thevalue of the solvation pressure at the maximum and minimum points for an oscillation. Second, the amplitudes of oscillation off’ (e.g., fA* -fe* in Figure 4a) show an exponential decay of the form Af = A exp(-H/K), where H i s taken at the value of the middle point between a maximum and a minimum. The experimental value for the decay length has been reported to be of the order of 1.2-1.7 molecular diameters.5 Our calculations agree with this range. Third, it was established in the experiments4sSthat the oscillations do not appear to be sinusoidal, but for the first few oscillationsthe distance from a given maximum to the next minimum (e.g., PI to QI, P2 to Q2, etc. in Figure 4a) is less than half the periodicity expected for a sinusoidal curve. At larger values of H,however, the oscillations become less pronounced, with this separation approaching the sinusoidalvalue. Our calculationsare in agreement with these observations. To illustrate the nonsinusoidal nature of the oscillations, we have included in Figure 4a (dashed curve) an exponentially decaying cosine functional form, g(H), given by
The Journal of Physical Chemistry, Vol. 97, No. 4, 1993 941 9
+
3.81 4.25
3
ff 0
-3 -6
-9
2
3
4
6
5
H% ’ ,-
7
CH,
Figure 5. Effect of the ratio of molecular diameters, u,f/urf,on solvation pressures for different mokcules on graphite at T+ = 0.9, Pb* = 0.005, c,r/crr = 0.43478. The pore width is reduced with the uw for CHd, while
f a
rf(~CH4CH,)’/~CH4CH,~ Pb*
51
2
-7
g(H) = K,cos (2*(K*H)) exp(-(K,H))
(19) where K I , K2, and K3 are parameters that were adjusted to approximatelyfit g(H) to our results. A functional form similar to g(H) (with K1an arbitrary constant; K2 = ko, the maximum of the first peak in the fluid structure factor S(k0); and K3 = S(ko)/S”(ko),the ratio of the structure factor to its second derivative at the main peak), was derived by Tarazona and Vicente4’ based on an expansion of the free energy functional about the bulk value to describe the oscillations of the density profile p ( z ) for a fluid confined between parallel walls; it was shown that the solvation pressure will follow the same functional form. From Figure 4a it is clear that the calculated solvation pressure is markedly different from AH), except at large H, showing nonsinusoidal oscillations and a more rapid exponential decay. Another feature is the different slopes between the calculated pressures and g(H) both for the ascending and descending portions of each oscillation. ( a ) Effect of Potential Parameter us,/u,p From eq 15 it is seen that the value of the ratio u,f/urf will play a crucial role in the balance between the attractive and the repulsive interactions, since it determines the relative range of the fluid-solid and fluidfluid potentials. This range has been shown to be decisive on the order of the wetting transitions.I2J3 The solvation pressures calculated for a bulk density pb* = 0.005, e,f/eff = 0.43478, and a reduced temperature T* = 0.9 are shown in Figure 5 for three different values of u,r/urrfor the model adsorbate/graphite system. The range covered in our calculations implies a change in the fluid adsorbate molecules from uff = 0.425 nm and uSr/urr= 0.90, which approximately corresponds to CzH6 on graphite, to the value urf = 0.381 nm, u,f/uw = 0.9462, corresponding to CHI/ graphite, to a final value of urf = 0.354 nm, u,f/crrf = 0.98, corresponding to a small molecule like Ar on graphite. The physical pictures at a maximum and a minimum in f* are
- -0.946 0.900
-.-.-.
6
Pb(QCH4CH4)3’ I
kT/CCH4CH,.
I
I
I
5
6
7
iA v 1
2
3=
0.43470
I
I
I
3
4
Hf
bI
P”
014
ob
tlz
1.6
2.0
,
2.4
2”
Figure 6. (a) Effect of wall strength on solvation pressures, at Ts = 1.1, u,r/uff = 0.9, and bulk density Pb’ = 0.005. Values of the t,f/€ff ratio are indicated on the figure. All variables are reduced with the fluid-fluid parameters as described in the text. (d) Density profiles for a pore of H* = 5 and several wall strengths. For symmetry reasons, only half of the pore width is shown.
illustrated schematically in Figure 5 , for two values of the LJ parameter uff. The effect of molecular size is to cause a phase shift and also a change in amplitude of the oscillations. As the ratio of usf/ufrincreases, for constant err and esfr the number of these smaller fluid molecules per unit area is larger, and hence larger amplitudes are obtained in the solvation pressure oscillations. (b) Effect of Potential Parameters es,/tp The effect of trf/tft is shown in Figure 6a, for constant values of the other variables,
Balbuena et al.
942 The Journal of Physical Chemistry, Vol. 97, No. 4, 1993
r 0.65
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M p e 7. (a) Effect of temperature on solvation pressures for CHI on graphite (csf/cp= 0.43478, usf/up= 0.9462) at a bulk density pb* = 0.01 and temperatures T+ = 0.65,0.7,0.8.0.9, and 1.1. (b) Effect of bulk density on solvation pressures for CH4 on graphite (csf/cff = 0.43478, usr/urr= 0.9462) at temperature T+ = 0.7 and bulk densities pb* = 0.01, 0.005, and 0.OOO005. pb*= 0.005,T+ = 1.1and usf/um= 0.9. Asdiscussed by Sarman,26
in this case the oscillations remain in phase, but the amplitude of the oscillationsdecreases as c,r/cfris demeased. Density profiles showing thechangein adsorptionundertheseconditionsareshown on Figure 6b for a pore of Hs = 5 . From these results there is a clear relation between adsorption and the oscillatory behavior off vs Hs. We note also that the wetting temperature varies with the strength of the fluid-wall interaction energy. In previous studies,31,32 we have calculated the wetting temperatures as a function of the relative wall strength, e,f/tr, within the same model and for the same potentials used here. The wetting temperatures increase when the ratio csf/efrdecreases(at fixed u,f/urf), and we find that the wetting temperature T,* = 1.1 (the temperature for the results in Figure 6) corresponds to tsf/tfr= 0.33 for usr/ufr = 0.9. For values of c,f/cff < 0.33, the system will be below its wetting temperature, and we will have either partial wetting or nonwetting at the wall, even for small pores. Thus for e8r/tff = 0.1 in Figure 6a we have nonwetting and the solvation pressure is very close to zero, while for other values of t,f/tfr < 0.33 the solvation pressure is much reduced from the values for complete wetting (e.&, e,f/eff = 0.43478). The implicationsof wetting on vapor adsorption were recently addressed by Beaglehole and Christenson" based on a series of experiments that complement their direct force measurements. (c) EEfct of Temperature and Bulk Density. The effect of temperature is illustrated in Figure 7a, using parameters corresponding to methane in graphite pores at a bulk density pb* = 0.0 1. Lower temperatures lead to a more structured fluid in the pore, and this is reflected in the larger amplitude oscillations in
16
24
32
4.0
4.8
t" Figure 8. (a) Solvation pressures in a slit pore having nonidentical walls withpb* =0.01,P =0.9andcs,r/cfr=0.04. Theratioc,,f/crrisindicatad. (b) Density profiles for a pore of nonidentical walls of If* = 5; other conditions as in Figure 8a.
the solvation pressures. This clear distinction between the amplitude of the oscillations has not been seen (at least in the same order of magnitude) in experimental results. Christenson and Israelachvili'o have made careful measurements using a nonpolar silicon oil, pure octamethylcyclotetrasiloxaneOMCTS, at two different temperatures above and below the melting point. They concluded that the extent of layering remains within the experimental error when the temperature changes. The temperatures used in the experiments were 14 O C and 22 OC. We have made an estimation of the value of tu/& for OMCTS, using the corresponding states theory approximation,and conclude that these temperatures would correspond to reduced temperatures of about 0.78 and 0.80 respectively. In the case of cyclohe~ane,~ the reduced temperatures used in the experimentswere approximately 0.91 and 0.97. A possible explanation is that these temperature ranges are too small to detect appreciable changes in the structure of the fluids. Similar effects onf* are observed when the bulk fluid density is varied. Higher bulk densities give rise to more structured density profiles in the pore, and the solvation pressures usually increase in magnitude as shown in Figure 7b. 3. Nonidentical Walls. A theoretical study of the effect of having two nonidentical walls was carriedout by Parry and Evans4S for the particular case when the fluid wets one wall and dries the other. They concluded that the phase equilibria is very much influenced by the wetting conditions in this situation, and determined the regime where two phases can coexist in a slit geometry. We consider the case where each wall interacts with a potential of the form in eq 15 but with different e,,f parameters. The external potential that acts on the fluid particles inside the
The Journal of Physical Chemistry, Vol. 97, No. 4, 1993 943
Simple Fluids in Micropores
bulk density, and temperature on the solvation pressure. The qualitative trends are similar to those found experimentally,and we believe these results should provide a useful framework within which to discuss experimental data. In future work we plan to explore the effect of molecular variables and composition on the behavior of the solvation pressure for mixtures.
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Acknowledgment. We are grateful to the National Science Foundation for support of this work under Grant CTS-9122460.
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References and Notes
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Horn, R. G.; Israelachvili. J. N. Chem. Phys. Lett. 1980, 71, 192. Horn, R. G.;Israelachvili, J. N.; Perez, E. J . Phys. (Paris) 1981,42.
H*
b
Israelachvili, J. N.; Pashley, R. M. Narure 1983, 306, 249. Horn, R. G.; Israelachvili, J. N. J. Chem. Phys. 1981, 75, 1400. Christenson, H.K. J. Chem. Phys. 1983, 78,6906. Christenson, H.K.;Horn, R. G. Chem. Phys. Lett. 1983, 98, 45. Christenson, H. K. J. Chem. Soc., Faraday Trans. I 1984,80.1933. Christenson, H.K. Chcm. Phys. Lett. 1985, 118, 455. Christenson, H.K.; Blom, C. E. J . Chem. Phys. 1986.86, 419. Christenson, H. K.; Israelachvili, J. N. J. Chem. Phys. 1984, 80,
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4566. (11) Rand, R. P.; Parsegian, V. A. Biochim. Biophys. Acta 1989, 988, 351. (12) Parsegian, V. A,; Rand, R. P.; Fuller, N. L. J . Phys. Chem. 1991, 95, 477 I . (13) Parsegian, V. A.; Rand, R. P. Lungmuir, in press. (14) Israelachvili, J. Intermolecular andsurfaceforces, 2nd ed.;Academic Press: New York, 1991; Chapter 13. (15) Lane, J. E.; Spurling, T. H. Chem. Phys. Lett. 1979, 67, 107. (16) Lane, J. E.; Spurling. T. H. Ausr. J. Chem. 1980, 33, 231. (17) Lane, J. E.; Spurling, T. H . A u t . J. Chem. 1981, 34, 1529. (18) Snook, I. K.;van Megen, W. J. Chem. Phys. 1979, 70, 3099. (19) Snook, I. K.; van Megen, W. J. Chem. Phys. 1980, 72,2907. (20) Snook, I. K.; van Megen, W. J . Chem. Soc.. Faraday Trans. 2 1981. 77, 181. (21) Magda, J. J.;Tirrell,M.;Davis,H.T.J.Chem. Phys. 1985,83,1888. (22) Xiao, C.; Rowlinson, J. S.Mol. Phys. 1991, 73, 937. (23) Antonchenko, V. Y.; Ilyin, V. V.; Makovsky, N. N.; Pavlov, A. N.; Sokhan, V. P. Mol. Phys. 1984, 52, 345. (24) Freasier, B. C.; Nordholm, S. J . Chem. Phys. 1983, 79,4431. (25) Evans, R.; Marini Bettolo Marconi, U. J . Chem. Phys. 1987, 86, 7138. ... (26) Sarman, S.J. Chrm. Phys. 1990, 92, 4447. (27) Kjellander, R.; Sarman, S . Mol. Phys. 1991, 74,665. (28) Henderson, J. R. Mol. Phys. 1986, 59, 89. (29) Tarazona, P. Phys. Rev. A 1985,31,2672. See also ref 33 for some 8.1
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H*
Figure 9. Effect of temperature on solvation pressure in a slit pore with nonidentical walls, forpb* = 0.01, t,?r/tn = 0.4, a t T+ = 0.7 (dashed line) and 0.9 (solid line). (a) ts,f/ew = 0.2; (b) t,,r/ta = 0.3.
pore is given by
Typical results are represented in Figure 8a, where the resulting solvation pressures for a slit with er,f/en = 0.4 and four values of e,,f/enare compared. The ratio u,f/urris held constant and equal to 0.9, the bulk density is pb* = 0.01, and the temperature is 7"' = 0.9. The correspondingdensity profiles in Figure 8b show the evolution of these profiles as one of the walls is changed, for a fued separation between the walls, H+ = 5. The effect on the solvation pressures seems to be similar to the effect of changing the ratio t6f/en for identical walls (see Figure 6a), i.e., there is a change in magnitude in the amplitude of the oscillations. However, phase transitions can be induced as seen in Figure 8a for the e,,f/en = 0.1 for H+ = 5.4. For the higher e8,f/en values such phase transitions occur at H+ values higher than those shown in the figure. The effect of temperature on these systems is shown in Figure 9 parts a and b. Again, phase transitions occur at a particular H+ value; at 7"' = 0.7 this value is greater than those shown in the figure. 4. C o a c ~
We have used density functional theory to demonstrate the effects of pore size, wall strength, the size of the fluid molecule,
corrections to errors in the equations. (30) Evans, R. In Inhomogeneous Fluids; Henderson, D..Ed.; Dckker: New York, 1992; Chapter 5. (31) Balbuena, P. B.;Gubbins, K. E. Fluid Phase Equilib. 1992, 76,21. (32) Balbuena, P. B.;Gubbins, K. E. Lungmuir,submittcd for publication. (33) Tarazona, P.; Marini Bettolo Marconi, U.; Evans, R. Mol. Phys.
1987, 60, 573. (34) Lu, B. Q.;Evans, R.; Telo de Gama, M. M.Mol. Phys. 1985, 55, 1319. (35) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 35. (36) Tan, Z.; Gubbins, K. E. J. Phys. Chem. 1990, 94, 6061. (37) Derjaguin, B. V.; Churaev, N. V. J. Colloid Interface Sci. 1976,54, 157. (38) Weeks, J. D.;Chandler, D.; Andersen, H. C. J . Chem. Phys. 1971, 54, 5237. (39) Steele, W. A. Surf Sci. 1973, 36, 317. (40) Steele, W. The interaction of gases with solid surfaces; Pergamon Press: Oxford, 1974; p 56. (41) Tarazona, P.; Vicente, L. Mol. Phys. 1985, 56, 557. (42) Tarazona, P.; Evans, R. Mol. Phys. 1983, 48, 799. (43) Teletzke, G . F.; Scriven, L. E.; Davis, H. T. J . Chcm. Phys. 1983, 78, 1431. (44) Beaglehole, D.; Christenson, H. K. J . Phys. Chem. 1992, 96, 3395. (45) Parry, A. 0.;Evans, R. Phys. Rev. Lett. 1990, 64, 439.
