17004
J. Phys. Chem. 1996, 100, 17004-17010
Adsorption of Fluids in Disordered Porous Media from the Multidensity Integral Equation Theory. Associative Analogue of the Madden-Glandt Ornstein-Zernike Approximation Andrij Trokhymchuk† and Orest Pizio*,† Instituto de Quı´mica de la UNAM, Circuito Exterior, Coyoaca´ n 04510, Me´ xico D.F.
Myroslav Holovko‡ Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, LViV 11, Ukraine
Stefan Sokolowski Computer Laboratory, Faculty of Chemistry, MCS UniVersity, 200-31 Lublin, Poland ReceiVed: May 17, 1996; In Final Form: August 8, 1996X
The associative analogue of the integral equation theory developed by Madden and Glandt is applied to a model of methane adsorbed in a silica xerogel. The integral equations are solved in the associative PercusYevick approximation. The associative treatment of strong attraction between fluid particles and matrix species in this model yields a better description of the pair distribution functions at intermediate fluid densities and at low temperatures in comparison with the Percus-Yevick approximation. In contrast to the Percus-Yevick approximation, the associative Percus-Yevick theory gives solutions for some subcritical temperatures of the bulk fluid. The adlayer structure is discussed in terms of the fraction of bonded species and coordination numbers. The behavior of the distribution functions and internal energy coincides with the Monte Carlo simulation data of Kaminsky and Monson. It is shown that the model can be used for the description of different Lennard-Jones fluids in hard sphere matrices. Possible refinements of the theory by using the replica Ornstein-Zernike equations of Given and Stell are discussed briefly.
1. Introduction The study of fluid behavior in porous materials has received much interest during the last decade.1-13 Theoretical tools to investigate these systems include the Madden-Glandt Ornstein-Zernike (MGOZ) equations,1,2 which represent an approximation to the exact replica Ornstein-Zernike (ROZ) equations investigated by Given and Stell.5-8 The purpose of this communication is to initiate a study of adsorption of associating fluids (AFs) in disordered porous media. This report is part of an ongoing project to study AFs in porous materials by means of the integral equation method and computer simulations. Previous studies have focused on the nonassociating fluids;1-12 to our best knowledge the first attempt to consider the AFs in a hard sphere quenched matrix has been presented recently.13 However, the associative interactions in ref 13 have been investigated by using the simplest and not completely adequate tools, namely, the Percus-Yevick (PY) approximation. To include associative interactions (AIs) between the species in the fluid-matrix system is of much interest for several reasons. It is natural to consider AIs between the fluid species to investigate the chemical reactions in porous media, a problem of importance in chemistry (see, for example, ref 14). On the other hand, it is interesting to involve the AIs in order to generate the matrix subsystem with a special structure. We have in mind a matrix made of dimer species or polymer chains or a branched polymer structure. Finally, the associative approximations for the treatment of strong fluid-matrix correlations are also important to obtain an adequate description of adsorption of † Permanent address: Institute for Condensed Matter Physics, National Academy of Sciences of the Ukraine, Lviv 11, Ukraine. ‡ Visiting Scientist of the Instituto de Quimica de la UNAM. X Abstract published in AdVance ACS Abstracts, September 15, 1996.
S0022-3654(96)01443-8 CCC: $12.00
nonassociating and associating fluids in porous media. Associative interactions between fluid particles and a surface strongly influence fluid structure in the interface region,15 and it would be of interest to involve them also in the problem. Before proceeding with the description of the model and the procedure it is worth mentioning that one of the most successful approaches to study the association phenomena in fluids has been pioneered by Wertheim.16-19 The essential feature of this approach is its self-consistency in the treatment of density parameters that control participation of the fluid particles in bonds. This formalism excludes improper contributions arising in the usual cluster expansions of the total density and guarantees saturation of the contribution of the associative interaction into the properties of the system. Wertheim’s theory has been originally developed for the case of highly directional AIs; later it was reformulated for the treatment of AIs of spherical symmetry.20,21 The method includes thermodynamic perturbational theory and integral equations. We briefly discuss one of the problems that arises in the theory of associating fluids and that is related methodologically to the description of adsorption of fluids in porous materials. Very recently the associative version of the hypernetted chain approximation (AHNC) has been used to study highly asymmetric electrolyte (polyelectrolyte) solutions,22 which consist of large and highly charged macroions and small counterions. Due to a large asymmetry of diameters of these constituents as well as of their charges, the species are treated differently in the association theory; namely, the counterions are considered as singly bondable, whereas the macroions are permitted to form bonds with many counterions. The AHNC yields very good results and coincides well with the computer simulation data including the region of parameters of the model in which the usual HNC approximation fails to converge. Recently the © 1996 American Chemical Society
Adsorption of Fluids in Disordered Porous Media analytic solutions of the associative PY (APY) approximation for nonsymmetric hard sphere mixtures have been obtained also.23 One of the realistic models for adsorption of fluids in porous media has been developed by Kaminsky and Monson (KM),9 namely, for adsorption of methane in silica xerogel. The arrangement of matrix particles corresponding to the equilibrium hard sphere configuration is a reasonable initial approximation to the structure of this gel.10 This observation has been used in the model of Kaminsky and Monson. This model differs in two respects from the models of adsorption of hard spheres in hard sphere matrices studied previously.3,6 The attractive fluid-matrix and fluid-fluid interactions are included in the model. The effect of temperature on the structure of adsorbed fluid therefore is present. The fluid-matrix attraction is strong as compared with the fluidfluid interaction. The minimum of the potential energy, min min Ufm , is on the order Ufm ≈ 6f, where f is the minimum of the fluid-fluid interaction. The asymmetry of energies results in a strong accumulation of the fluid particles in the vicinity of a matrix particle. The model is characterized also by a quite large asymmetry of diameters of matrix particles (σm) and of fluid species (σf), σm/σf ≈ 7. The packing fraction of matrix particles at density Fm is chosen high, ηm ) πFmσm3/6 ) 0.386;11 therefore inclusion of the correlations through a matrix subsystem is important. All these features to a greater or lesser extent have to be shared by realistic models for adsorption of fluids in porous media. The fluid-matrix interaction in the model of Kaminsky and Monson is presented in a slightly more complicated form than the Lennard-Jones potential, but is given as an analytical function well suited for use in the integral equation theory. However, due to a large difference of diameters of the matrix and fluid particles and strong attraction, it is questionable to apply the usual integral equation techniques for these models. It has been well documented by Vega et al.11 for the model of Kaminsky and Monson. For low densities and at high temperatures, higher than the critical temperature of the bulk fluid, the PY approximation when applied to the MGOZ equations yields adequate predictions for the fluid-fluid and fluid-matrix correlation functions. However, at lower temperatures and higher densities, i.e. when the attractive forces play a dominant role in determining the microstructure of the system, the agreement between the PY and simulations deteriorates. Moreover, it becomes more and more difficult to obtain the solution of the MGOZ equations for lower temperatures, despite the absence of peculiarities in the results of the computer simulations, at least up to T* ) kT/f > 0.8. It is quite easy to find similarity between the model at hand, such as of Kaminsky and Monson (asymmetry of diameters and strong fluid-matrix attraction) and the problems arising in its description (discrepancies of the PY theory and simulations) on one hand, with some features of the models for polyelectrolyte solutions and problems in their study.22 One can expect that the application of associative methodology16-21 may improve the description of adsorption of fluids in disordered porous media either in the MGOZ or in the ROZ approach. Our intention in this communication is to propose the associative treatment of the fluid-matrix interaction. For exploratory purposes we shall start from the approach based on the associative analogue of the MGOZ equations and associative closures. A more sophisticated treatment involving the ROZ equations and the closures proposed in refs 5-8 will be presented elsewhere. The results obtained are discussed and
J. Phys. Chem., Vol. 100, No. 42, 1996 17005 compared with the data presented by Vega et al.11 from the usual PY closure and with the simulation data. 2. The Model and Theory Consider the model for adsorption of methane in a silica gel of Kaminsky and Monson.9 We consider it as a generic model for adsorption of fluids in a rigid matrix of hard spheres, which is characterized by a large asymmetry of diameters of the fluid and matrix particles and by a strong fluid-matrix attraction. The KM model proposed in ref 15 is given by the following potentials
Umm(r) ) Uhs(r,σm) )
{
∞, r < σm 0, r > σm
(1)
for the matrix particles and
[( ) ( ) ]
Uff(r) ) 4f
σf r
12
-
σf r
6
(2)
for the fluid particles, where σm ) 7.055σf. The fluid-matrix interaction is given as
Ufm(r) )
{
r < σm/2 ∞, φ(r), r > σm/2
(3)
where9,11
{ [ /192](r - σ
21 3 2 φ(r) ) πFsσm3gs σgs12 r6 + σm2r4 + σm4r2 + 3 20 16 σm6
2
2
m
}
/4)-9 - σgs6(r2 - σm2/4)-3 (4)
For convenience of the reader we list the set of parameters of the potentials: f/k ) 148.2 K, σf ) 0.3817 nm, σgs ) 0.33 nm, Fs ) 44 nm-3, gs/k ) 339 K.11 In our analysis the fluid-matrix potential is presented as a sum of two contributions, non as (r) Ufm(r) ) Ufm (r) + Ufm
(5)
where the first and the second term correspond to nonassociative and associative parts of the total potential, respectively. This division of the fluid-matrix potential is a crucial step in developing the procedure. It can be done, for example, according to the energetic criterion (see, for example, refs 21, 22 for detailed analysis of this issue). Each of the terms in eq 5 is defined as follows: non (r) Ufm
and
{
Ufm(r), 0 < r < r0 r0 < r < rc ) U0, Ufm(r), r > rc
{
0 < r < r0 0, as (r) ) Ufm(r) - U0, r0 < r < rc Ufm r > rc 0,
(6)
(7)
Here U0 is a parameter necessary to choose. The parameters r0 and rc are defined according to the condition Ufm(r0) ) Ufm(rc) ) U0. The choice of U0 is determined by the type of approximation one intends to apply in the theory. For the model considered, we do not use the associative treatment for the fluid-fluid attraction describing it in the framework of the usual Percus-Yevick approximation; that is, the attraction between fluid particles is assumed to be not very strong. Therefore it is
17006 J. Phys. Chem., Vol. 100, No. 42, 1996
Trokhymchuk et al.
natural to use the minimum of the fluid-fluild interaction as a natural criterion to choose the parameter U0, so we assume U0 ≈ f. The description of the system will become less adequate for the decreasing temperature. On the other hand, the results have to be weakly sensitive to the hoice of U0 as discussed in ref 22. At present we are not able to discuss the thermodynamic aspect of the proposed division of the fluid-matrix potential into two parts. This issue requires special investigations in view of the problems arising in deriving thermodynamic properties of fluids adsorbed in quenched matrices (see for example, refs 6-12 for detailed discussion of this and related issues). To proceed, we define the Mayer functions,
fmm(r) ) exp[-βUmm(r)] - 1 fff(r) ) exp[-βUff(r)] - 1
(8)
non (r) ) exp[-βUnonfm(r)] - 1 ffm
Ff )
( ) f Ff F0 Ff0 0
where Ff denotes the total density of fluid species, whereas Ff0 is the density of free particles. The self-consistency relation between Ff and Ff0 has a form similar to the polyelectrolyte problem discussed above (see, for example, refs 16, 17, and 22) and reads
Ff ) Ff0 + 4πFf0Fm∫drr2yfm 0 (r) Fas(r)
where Fas(r) is defined by eq 9 and yfm 0 (r) is the partial cavity distribution function.22 The partial correlation functions can be combined into the total correlation functions according to standard definitions:
0 fm hfm(r) ) hfm 0 (r) + xf h1 (r)
(9)
as in the usual Wertheim’s approach for the treatment of association.16 We shall use the concept of association for the fluid-matrix correlations in what follows. Assume formally that the fluid particles can be in “nonbonded” (free) and “bonded” states with respect to the matrix particles and construct the associative analogue of the MGOZ equation. It reads (see, for example, Vega et al.11 for the notations)
hmm(r12) ) cmm(r12) + Fm∫dr3cmm(r13) hmm(r32) (10) for the matrix-matrix correlation functions, fm fm m mm (r32) + hfm R (r12) ) cR (r12) + F ∫dr3cR (r13) h
f Fµν ∫dr3cRµff (r13) hfmν (r32) ∑ µν
(15)
hff(r) ) hff00(r) + 2xf0 hff01(r) + (xf0)2hff11(r)
and non as (r)]{exp[-βUfm (r)] - 1} Fas(r) ) exp[-βUfm
(14)
(11)
(16)
where xf0 ) Ff0/Ff. The associative analogue of the MGOZ equation must be supplemented by a closure relation. In this study we use the APY approximation. A particular reason to do that is to compare the results obtained from the usual PY approximation by Vega et al.11 and their simulation data. We are aware of the criticism by Stell and Given, who have shown that the MGOZ is consistent with the ROZ equation only with the PY or mean spherical approximations. We look forward to applying AHNC type closure besides the APY approximation in the rigorous ROZ theory. This approach in general requires, however, some approximations for the “tail” functions (bridge diagram contribution).5-8 At present this problem is difficult to address within the model of Kaminsky and Monson. The APY closure is
cmm(r) ) fmm(r) ymm(r) ff ff (r) ) fff(r) yRβ (r), cRβ
(17)
for the fluid-matrix correlations, and non fm fm cfm R (r) ) ffm (r) yR (r) + (1 - δR0)Fas(r) yR-1(r)
ff ff mf (r12) ) cRβ (r12) + Fm∫dr3cfm hRβ R (r13) hβ (r32) +
f Fµν ∫dr3cRµff (r13) hνβff (r32) ∑ µν
(12)
for the fluid-fluid correlations. We apply the two-density formalism for the fluid-matrix correlations; thus in eqs 11 and 12 each of the lower greek indices can take values 0 or 1, which correspond to nonbonded and bonded states of the particle, respectively. The fluid-fluid correlation functions are the matrices of the form
φff(r) )
(
φff00(r) φff01(r) φff10(r) φff11(r)
)
(13)
having the elements that are symmetric with respect to the permutation of indices, i.e. φff10(r) ) φff01(r). Moreover, there exists a symmetry relation for the matrix-fluid partial correlation functions, similar to that for the total correlation functions:5,6 mf fm fm φR (r) ≡ φmf 0R(r) ) φR0(r) ≡ φR (r). The matrix of fluid densities is
where y are the relevant cavity distribution functions. The pair correlation functions are related to the cavity distribution functions in the following way:
1 + hmm(r) ) [1 + fmm(r)]ymm(r)
(18)
ff ff (r) ) [1 + fff(r)]yRβ (r) δR0δβ0 + hRβ
(19)
non fm fm δR0 + hfm R (r) ) [1 + ffm (r)]yR (r) + (1 - δR0)Fas(r) yR-1(r) (20)
Equations 10-12, 15, and 17 form a complete set. We solve this system of equations numerically by implementing the iterative method very similar to that proposed by Labik et al.23 In the numerical work 4096 points have been used for each of the correlation functions with a grid size of ∆r ) 0.008σf. Convergence of the procedure was not a problem for T* ) 2.0 and for T* ) 1.2, which correspond to the supercritical and near critical temperature of the bulk fluid under consideration.9 However, for subcritical temperature, T* ) 1.0, in some cases
Adsorption of Fluids in Disordered Porous Media
Figure 1. Fraction of “dimerized” species, x1f , dependent on the depth min of the nonassociative term of the fluid-matrix interaction, U0/Ufm , for the KM model9 at T* ) 2.0 and F* ) 0.36. Division of the potential f as is shown in the inset. The associative term, Ufm (r), is denoted by a short-dashed line. The long-dashed line corresponds to U0. The solid line is the total potential Ufm(r).
we have realized the procedure with gradual inclusion of associative interaction in order to obtain the solution. 3. Results and Discussion Let us discuss now the results obtained. The division of the fluid-matrix potential into the associative and nonassociative contributions is arbitrary to some extent. However, according min | in order to to eqs 5-7 we have to choose |U0| < |Ufm improve the usual PY approximation. On the other hand the choice U0 ) 0 is not adequate for us. In the cae U0 ) 0, one would expect a possibility of the formation of more complicated complexes than those included in the two-density formalism. Our detailed investigation of the choice of U0 has shown that there exists a region of the values for U0 in which the theory is not very sensitive to the division of the potential. Similar trends have been observed for ionic systems.21,22 Evidently, the values of T*, F*f ) Ffσf3, and other parameters of the model influence the best choice of the values of U0. In Figure 1 we present the results of calculation of the fraction of singly bonded fluid particles xf1 ) 1 - xf0, dependent on the min at T* ) 2 and F* value U0/Ufm f ) 0.36. In the region 0.2 < min U0/Ufm < 0.4 the function xf1 changes quite weakly and the results for the pair distribution functions (pdfs) also are not very sensitive to the division of the potential. This region of U0/ Uminfm values also includes the value U0 ) f. The division of the potential for U0 ) f is given in Figure 1 too. This value of U0 is used to calculate the pdfs in what follows. We proceed to the discussion of the pdfs gff(r) ) hff(r) + 1 and gfm(r) ) hfm(r) + 1. We would like to noe that the usual PY approximation works remarkably well at low density and at T* ) 2.0. The shape of the pdfs from the PY coincides with the simulation data; however the values of the first maxima for gff(r) and gfm(r) are slightly underestimated. For T* ) 1.2, the PY approximation becomes less satisfactory. It follows from the simulation data that the first peaks of these functions increase. The PY approximation underestimates this effect. Besides, the PY theory does not reproduce well the shoulder arising on the curve of the gfm(r) on the decaying slope after the first maximum. We describe now the APY results. For low densities the APY results are slightly worse than the data from the PY approxima-
J. Phys. Chem., Vol. 100, No. 42, 1996 17007
Figure 2. Fluid-fluid and fluid-matrix pdfs for the KM model at F ) 0.36, T* ) 2.0. The pdfs from left to the right are gff(r) and gfm(r), respectively. The PY and GCMC results11 are denoted by a dashed line and points, respectively. The APY results are given by solid lines. (b) The coordination numbers, nfm(r) and nfm (r), obtained from the APY approximation, are shown by short-dashed lines from top to bottom.
tion. This effect has been expected. Usually the association leads to a lower value for the first maximum of the pdf in comparison with the association switched off. The shape of the pdfs from the APY theory is correct; however, at small interparticle distances the values of the pdfs are underestimated in comparison with the PY approach. It follows from a comparison of the PY and APY approaches that the choice of approximation to describe the model depends crucially on the fluid density. For the density F* f ) 0.36 and T* ) 2.0, the PY approach remains adequate only for the fluid-fluid correlations. It results, however, in the unphysical values for the gfm(r) for distances greater than the distance at which the first maximum is situated; it usually occurs in the PY theory for nonsymmetric mixtures. It follows from Figure 2 that the APY approach is successful and the APY curve for the function gfm(r) does not have a negative region and is closer to the simulation data. The value for the first maximum is higher than the GCMC value but lower than the PY result. In Figure 2 we also show the running coordination number, nfm(r), defined as follows:
nfm(r) ) 4πFf∫0 dRR2gfm(R) r
(21)
It describes the average number of fluid particles in the sphere of radius r around a given matrix particle. According to eqs (b) (r), can be 15, 16, and 20, the “bonded” contribution, nfm (b) extracted from the nfm(r). We define nfm (r) as follows: (b) (r) ) 4πFf0∫0 dRR2yfm nfm 0 (R) Fas(R) r
(22)
(b) (r) saturates for the distances r > rc It is worth nothing that nfm f and is equal to (Ff - F0)/Fm. This saturation yields a plateau region in the function nfm(r). From the results given in Figure 2, we can conclude that there exists an adsorption shell around the matrix particle determined by r ) rc ≈ 4.9σf. This shell contains approximately 120 fluid particles, 80 of which can be considered as “bonded”. It follows also from Figure 2 that the fluid-fluid distribution functions obtained from the APY and PY approximations and the GCMC simulations11 practically
17008 J. Phys. Chem., Vol. 100, No. 42, 1996
Trokhymchuk et al.
Figure 3. Fluid-matrix and fluid-fluid (inset) pdfs in the APY approximation for the KM model for T* ) 1.2 at F* f ) 0.156 (longdashed line), F* f ) 0.271 (short-dashed line), and F* f ) 0.301 (solid line).
coincide. In general, one can conclude that the APY theory is more successful than the PY approximation for a high-density fluid adsorbed in a hard sphere matrix. However, the factor of temperature is also important. In Figure 3 we present the pdfs for the temperature T* ) 1.2, for different fluid densities, namely, for F* f ) 0.156, 0.271, and 0.301. With increasing fluid density the height of the first maximum of gfm(r) increases; however, it increases more slowly in the APY approximation than in the PY. The value of the first minimum is higher in the APY than in the PY approximation for the function gfm(r); we do no observe the unphysical negative values of the gfm(r) from the APY closure up to the density F* f = 0.271. However, for higher fluid densities the APY also leads to the appearance of the region of negative values of gfm(r). Elimination of this defect can be reached by the application of more sophisticated closures than the two-density APY approximation in the AMGOZ equations. The first maximum of the gff(r) slightly increases with density (Figure 3), the shoulder on the decaying slope of the first maximum develops with increasing density, and for the highest density considered, F* f = 0.301, one more submaximum is observed. These trends demonstrate the difference in the packing effect of the fluid adsorbed in the matrix and “free” fluid at this density; obviously this behavior results from the influence of the rigid matrix. To conclude, we observe two adlayers of fluid particles at the surface of the matrix particle in the region of densities considered. It has been observed in the previous studies of the model of Kaminsky and Monson in the framework of the PY approximation that, for decreasing temperature T*, it is more and more difficult to obtain the solution of the MGOZ equations. In particular, for F*f > 0.1526,11 at T* ) 1, the PY solution has not been obtained. The long-range behavior of the function gff(r) from the PY approximation, at T* ) 1 and for F* f ) 0.1526,11 contradicts the GCMC simulation result; the latter does not show any phase transition for these values of temperature and density. In Figure 4 we consider even a slightly higher fluid density and show that the application of the APY closure essentially improves the results of the PY approximation for the pdfs. It is worth nothing that, in contrast to the case of a higher temperature, T* ) 2 (Figure 2), for T* ) 1 the pdf gff(r) becomes very sensitive to the consideration of a part of the fluid-matrix potential as the associative interaction. In all previous calculations we have used the universal choice of the value of U0, namely, U0 ) f (rc ) 4.9σf). However, the effects of association are overestimated under this choice of U0, and
Figure 4. The same as in Figure 2 but for F* f ) 0.155 and T* ) 1.0.
the function gff(r) does not agree well with computer simulations data. Unfortunately the GCMC dat afor gfm(r) are not available for this case. It can be seen from the pdf gff(r) given in Figure 4 that the choice rc ) 4.55 σf provides excellent agreement of the APY theory with the CCMC simulations. The only defect observable in this function from the APY approximation is that the first maximum is slightly underestimated. This value of rc corresponds approximately to the position of the first minimum of the function gfm(r). The function gfm(r) is positive for all distances at this division of the potential. The choice of associative contribution in the fluid-matrix interaction must be realized self-consistently. During the procedure of the solution of integral equations one must adjust the cutoff distance for associative potential, especially for low temperatures T*. The results of our calculations of the configurational part of the internal energy, of the fraction of bonded particles xf1 and coordination numbers, nfm(r), n(b)fm(r), obtained from the APY approximation are summarized in Table 1. There we give also the data obtained in the PY approximation and GCMC results from Table I of Vega et al.11 The internal energy per fluid particle, uf, can be obtained in the Madden-Glandt approach1-4 via the pdfs as follows:
uf ) uff + ufm )
Ff ∫drUff(r) gff(r) + Fm∫drUfm(r) gfm(r) 2 (23)
We begin with a discussion of the fraction of “bonded” species and coordination numbers. For increasing fluid density at fixed T* the fraction of “bonded” fluid particles with a matrix particle, xf1, decreases for T* ) 2.0 and for T* ) 1.2. This effect can be interpreted as follows. For low densities the fluid particles are mostly localized in the vicinity of the matrix particles due to a strong fluid-matrix attraction and weak influence of the fluid packing fraction effects. For increasing density the role of packing fraction becomes important; the adsorbed fluid becomes more structured. The adlayers are more pronounced at higher density. However, some of the particles in the first adlayer and all the particles in the subsequent adlayers are free; therefore the values for xf1 decrease. At fixed density the fraction of “bonded” fluid particles, xf1, increases for decreasing T*.
Adsorption of Fluids in Disordered Porous Media
J. Phys. Chem., Vol. 100, No. 42, 1996 17009
TABLE 1: Fraction of “Bonded” Particles xf1, the (b) Coordination Numbers nfm(r*), nfm (r*), the Fluid-Fluid uff and Fluid-Matrix ufm Contributions to the Internal Energy from the PY Approximation (Upper Row),11 APY Closure (Middle Row), and GCMC Simulation (Lower Row)11 T*
F*f
2.0 2.0
1.2
x1f
nfm(r*)
(b) nfm (r*)
0.03
0.7061
uff/kT
ufm/kT
22
10
0.6292
75
50
0.36
0.5097
122
84
0.0384
0.7061
23
17
0.1805
0.6292
77
65
-0.138 -0.137 -0.151 -0.728 -0.734 -0.736 -1.515 -1.530 -1.545 -0.425 -0.404 -0.516 -1.522 -1.495 -1.619
-3.174 -2.813 -3.350 -2.470 -2.402 -2.592 -2.178 -2.143 -2.156 -6.001 -5.444 -7.267 -4.092 -4.318 -5.267
0.1696
TABLE 2: Fraction of “Bonded” Particles xf1, the (b) Coordination Numbers nfm(r*), nfm (r*), the Fluid-Fluid uff and Fluid-Matrix ufm Contributions to the Internal Energy from the APY Approximation F*f 24
Ar f/k ) 119.8 K, σf ) 3.405 Å CO225 f/k ) 225.3 K, σf ) 3.8 Å Xe24 f/k ) 230.2 K, σf ) 4.05 Å MNP25 f/k ) 312.2 K, σf ) 5.8 Å
x1f
(b) nfm(r*) nfm (r*)
uff/kT
ufm/kT
0.15 0.6051 0.36 0.4986
70 125
43 82
-0.668 -2.227 -1.558 -1.981
0.15 0.5306 0.36 0.4633
68 123
38 77
-0.661 -1.852 -1.543 -1.684
0.15 0.5844 0.36 0.4919
68 121
42 81
-0.649 -2.086 -1.527 -1.866
0.15 0.8470 0.36 0.5355
72 116
60 83
-0.642 -4.065 -1.453 -3.141
In contrast to the xf1 that characterizes “bonded” states of fluid particles in the region of strong attraction, but by definition is normalized by the fluid density of the system, the coordination numbers presented in Table 1 characterize the first adlayer. (b) Obviously, the coordination numbers nfm(r*) and nfm (r*) (r* denotes the distance at which the first minimum of the fluidmatrix is located) increase for increasing density. The fraction (b) of “bonded” species in the first adlayer is determined by nfm (r*)/nfm(r*) and increases with increasing density. One should expect a decrease of the effective porosity of the system due to a strong accumulation of the fluid particles on the matrix species. From the comparison of the values for the configurational internal energy per particle for the model given by eqs 1-4, evaluated from the APY and PY approximation, and GCMC simulation data, we conclude that the trends of the behavior of the results are similar. In some cases the APY data agree slightly better with GCMC results, in particular at high densities. We are interested in the possibility of application of the model of Kaminsky and Monson for adsorption of other fluids in a hard sphere matrix. For this aim, besides th emodel for CH4, we use the APY approximation for Lennard-Jones fluids, such as Ar,24 CO2,25 Xe,24 and 2-methyl-2-nitrosopropane (MNP).25 The parameters for pure fluids (f, σf) employed in the following calculations are given in Table 2. The parameters gs, σgs, necessary as an input for the fluid-matrix potential given by eq 4, have been calculated according to the Lorentz-Berthelot combination rule (the parameters s, σs have been extracted from the fluid-matrix potential by using Lorentz-Berthelot combination rule also). For all the cases in question, the ratio of
Figure 5. Fluid-matrix and fluid-fluid (inset) pdfs from the APY approximation at F* f ) 0.15 and T* ) 2.0 for Ar, CO2, Xe, and MNP fluids, from left to right, in a hard sphere matrix, ηm ) 0.386 and σm ) 7.055σf.
diameters of the fluid and matrix species has been held fixed and equal to 7.055σf, similar to the model of Kaminsky and Monson. To complete the description of the model, let us note that the matrix packing fraction is held fixed, ηm ) 0.386; therefore the number of matrix particles differs in each case, but the fraction of excluded volume is constant. The results for the fluid-fluid and fluid-matrix pdfs at F*f ) 0.15 for different Lennard-Jones fluids are given in Figure 5. The positions of the first maxima of the pdfs gff(r) and gfm(r) coincide approximately with the positions of the minima of the corresponding interaction potentials. The height of the first maximum of the function gfm(r) in general (excluding the CO2) increases with increasing values of the Lennard-Jones parameters f and σf. The first maximum of the gfm(r) for MNP is much higher in comparison to other fluids in question. The gfm(r) for all cases has a second maximum; therefore, at high density all these fluids form two adlayers on a hard-sphere matrix particle. For larger distances, the fluid-matrix pdf is weakly structured at this density. The values of the first maximum for gff(r) decrease in the list of investigated systems from Ar to MNP. The fluidfluid correlation does not extend for large interparticle distances. We have used the pdfs to calculate the coordination numbers, the fraction of bonded species, and internal energy in a manner similar to that employed for the CH4 case. The data are given in Table 2. Worth nothing is that, despite the large diameter of MNP, this system is characterized by the highest xf1 (in contrast to what would be expected for the bulk hard spheres). The strongest correlation between MNP particles and matrix species and the weakest fluid-fluid correlation observed from the first maxima of the pdfs are reflected in the value of xf1. The CO2 model is characterized by the lowest value of xf1. These trends of behavior of the distribution functions are reflected also in the values of internal energy. Similarly to the CH4 model, the fluid-fluid contribution to the total internal energy is much lower for lower density; it becomes more important for higher density, as expected. The total internal energy is weakly sensitive to the changes of fluid density. It follows that the models of Kaminsky-Monson type can be used with success to describe adsorption of several fluids in porous media. 4. Conclusions The associative analogue of the Madden-Glandt OrnsteinZernike integral equations in the associative Percus-Yevick
17010 J. Phys. Chem., Vol. 100, No. 42, 1996 approximation is used to study the microstructure of fluids in a hard sphere matrix. The two-density formalism, which distinguishes between the fluid particles in the “bonded” and “nonbonded” states with the matrix particles, is applied for the model of Kaminsky and Monson for adsorption of methane in silica gel. For intermediate and high fluid densities, the adsorption results in the formation of the first and second adlayer on the surface of matrix particles. The first adlayer is composed of “bonded” with the matrix and nonbonded species. The fraction of “bonded” species, in the system as a whole, decreases with increasing fluid density. From the analysis of coordination numbers defined to characterize the first adlayer, it follows that the number of species “bonded” with matrix particles increases for increasing density. We presume that at high fluid density with decreasing temperature of the system, i.e. with increasing strength of attraction, the effect of saturation of the first adlayer by species bonded wtih matrix will be reached. The results are compared with the data from the usual Percus-Yevick approach and simulations.11 It has been shown that the APY approach is more successful than the PY for high densities and can be used for lower temperatures in the region where the PY theory faces severe difficulties. We have shown examples where the models of Kaminsky-Monson type are of wider applicability; different fluids adsorbed in porous media can be studied with the same success as methane. However, we would like to emphasize that the APY is also of limited applicability. More sophisticated closures are necessary to use, such as the associative hypernetted chain approximation (AHNC). To be free in the choice of closure relations, in particular, one would need to consider a more consistent treatment of the problem provided by the exact replica OZ equations. The closures necessary to apply in the ROZ scheme include, however, the contributions from the bridge functions. At present, it is difficult to find simple algorithms for the evaluation of bridge functions for the models of Kaminsky-Monson type. We plan to consider the application of the ROZ equation for the model at hand in a future publication. It would be of interest to consider by means of the theory presented a more sophisticated model for adsorption of fluids in porous media, including the effects of association in the bulk fluid (such as dimerization or polymerization) or association between fluid particles and matrix with formation of bonds. The study of models in which a fluid is adsorbed in
Trokhymchuk et al. porous media of special structure is important also. The theory of association in fluids must be involved in all these problems. Acknowledgment. M.H. is grateful to Instituto de Quimica de la UNAM for hospitality during his visit to Mexico. O.P. is grateful to M. C. Lozada Garcia for helpful discussions and continuous interest in this project. We are indebted to P. A. Monson for sending us the simulation data. Valuable discussions with D. Henderson, A. Patrykiejew, V. Vlachy are gratefully acknowledged. S.S. has been supported by KBN of Poland under Grant No. 3 T90A06210. References and Notes (1) Madden, W. G.; Glandt, E. D. J. Stat. Phys. 1988, 51, 537. (2) Madden, W. G. J. Chem. Phys. 1992, 96, 5422. (3) Fanti, L. A.; Glandt, E. D.; Madden, W. G. J. Chem. Phys. 1990, 93, 5945. (4) Fanti, L. A.; Glandt, E. D. J. Colloid Interface Sci. 1990, 135, 385, 396. (5) Given, J. A.; Stell, G. J. Chem. Phys. 1992, 97, 4573. (6) Lomba, E.; Given, J. A.; Stell, G.; Weis, J. J.; Levesque, D. Phys. ReV. E 1993, 48, 233. (7) Given, J. A.; Stell, G. Physica A 1994, 209, 495. (8) Given, J. A.; Stell, G. In Condensed Matter Theories; Blum, L., Malik, F. B., Eds.; Plenum: New York, 1993. (9) Kaminsky, R. D.; Monson, P. A. J. Chem. Phys. 1991, 95, 2936. (10) MacElroy, J. D.; Raghavan, K. J. Chem. Phys. 1990, 93, 2068. (11) Vega, C.; Kaminsky, R. D.; Monson, P. A. J. Chem. Phys. 1993, 99, 3003. (12) Rosinberg, M. L.; Tarjus, G.; Stell, G. J. Chem. Phys. 1994, 100, 5172. (13) Henderson, D.; Patrykiejew, A.; Pizio, O.; Sokolowski, S. Physica A, in press. (14) PerspectiVes in Catalysis; Thomas, J. M., Zamaraev, K. I., Eds.; Oxford-Blackwell: London, 1992. (15) Pizio, O.; Sokolowski, S. Phys. ReV. E 1996, 53, 820. (16) Wertheim, M. S. J. Stat. Phys. 1984, 35, 19, 35. (17) Wertheim, M. S. J. Stat. Phys. 1986, 42, 455, 477. (18) Wertheim, M. S. J. Chem. Phys. 1986, 85, 2929. (19) Wertheim, M. S. J. Chem. Phys. 1987, 87, 7323. (20) Kaluyzhnyi, Yu. V.; Stell, G. Mol. Phys. 1993, 78, 1247. (21) Kalyuzhnyi, Yu. V.; Holovko, M. F.; Haymet, A. D. J. J. Chem. Phys. 1991, 95, 9151. (22) Kalyuzhnyi, Yu. V.; Vlachy, V.; Holovko, M. F.; Stell, G. J. Chem. Phys. 1995, 102, 5770. (23) Labik, S.; Malijevski, A.; Vonka, P. Mol. Phys. 1985, 56, 709. (24) Vermesse, J.; Levesque, D. J. Chem. Phys. 1994, 101, 9063. (25) Kimura, Y.; Yoshimura, Y. J. Chem. Phys. 1992, 96, 3085.
JP961443L
