Langmuir 1994,10, 530-537
530
Modeling the Influence of Heterogeneous Adsorbent Microstructure upon Adsorption Equilibria for Binary Mixtures R. D.Kaminsky and P.A. Monson’ Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003 Received April 21, 1993. In Final Form: November 22, 199P
The influence of porous adsorbent microstructure on selective adsorption from binary mixtures is explored via Monte Carlo simulations of a molecular model representing simple mixtures adsorbed in silica gel. Two adsorbent structures are considered: a matrix of adsorbent spheres arranged in an equilibrium hard sphere configuration, and a matrix of adsorbent spheres arranged in a face-centered cubic configuration. At low pressures toward the Henry’s law limit, microstructure strongly influences both the total adsorbate density and composition. At higher pressures where substantial filling of the adsorbent occurs, microstructure influences the total density much more than the composition. The principal source of departures from ideal solution behavior for the systems studied is interpreted to be coupled effects of adsorbate size differences and of heterogeneity in the adsorbate-adsorbent potential field. 1. Introduction
Microporous adsorbents are commonly used in selective adsorption of components from gas mixtures. The selective adsorption of a given species is dependent on a complex combination of factors including confinement induced by the adsorbent structure, adsorbate-adsorbent intermolecular interactions, and adsorbate-adsorbate intermolecular interactions. Each of these factors in turn is significantlydependent on the adsorbent microstructure. In this paper we apply the framework of statistical thermodynamics to address the question of how adsorbent microstructure affects adsorption from mixtures. We are particularly interested in developing a fundamental qualitative understanding of adsorption in heterogeneous microstructures, i.e. structures without a simple pore geometry and with a complex distribution of energetic interactions and pore sizes. Common examples of such heterogeneous adsorbents include silica gels and activated carbons. Much of the previous theoretical work on the role of microstructure in adsorption has dealt with investigations of idealized pore structures. Through the application of mean-field theories, simulation, density functional theory, and integral equation theory, adsorption behavior in simple isolated pore geometries (e.g. slits, cylinders, and spheres) and zeolite structures has been studied.’-1° As a whole these studies of adsorption in various pore structures have greatly illuminated the nature of adsorption in confined regions, especially in regards to packing effects, inhomogeneous fluid structure, and wetting and layering phase transitions. Author to whom correspondence should be addressed.
* Abstractpublishedin Advance ACSAbstracts, January 15,1994.
(1)van Megan, W.;Snook, I. K. Mol. Phys. 1985, 54, 741. (2) Peterson, B.K.; Walton, J. P. R. B.; Gubbins, K. E. J.Chem. SOC., Faraday Trans. 2 1986,82,1789. (3)Peterson, B.K.;Gubbins, K. E. Mol. Phys. 1987,62, 215. (4)Evans, R.;Marini Bettolo Marconi, U.; Tarazona, P. J. Chem. SOC., Faraday Trans. 2 1986,82, 1763. (5)Ball, P. C.;Evans, R. Mol. Phys. 1988,63, 159. (6)Walton, J. P. R. B.; Quirke, N. Mol. Sim. 1989,2, 361. (7) Zhou, Y.;Stell, G . Mol. Phys. 1989, 66, 767. (8) MacElroy, J. M. D.; Suh, S.-H. Mol. Sim. 1989,2, 313. (9)Woods, G . B.;Panagiotopoulos, A. Z.; Rowlinson,J. S. Mol. Phys. 1988, 63, 49. (10)Woods, G. B.; Rowlinson, J. S. J. Chem. SOC.,Faraday Trans. 2 1989, 85, 765.
Although the study of ideal pore geometries is important in developing a fundamental understanding of adsorption in porous media, such systems are not particularly realistic models of the pore structures found in heterogeneous adsorbents, which have irregularly shaped and interconnected pores of varying dimensions. To probe these complex types of materials several researchers have developed and studied model adsorbents composed of randomly placed adsorbent particles. Analytic formulas for the Henry’s constant have been derived for a number of such structure^.^^-'^ For the study of adsorption at pressures above the Henry’s law regime, research has been largely focused on adsorption in purely hard repulsion systems. Some of the most extensive work in this area has been carried out by Fanti and Glandtl”ls who have investigated single component, hard sphere adsorption beyond the Henry’s law regime in both fibrous and spongelike materials. Such models although not particularly realistic do give basic insight into adsorption packing effects and high pressure adsorption. An interesting new approach to the theory of adsorption in heterogeneous media for arbitrary interaction potentials has been proposed by Madden and Glandt17-19 based on cluster expansions and integral equations describingthe molecular distribution functions in the system. This approach allows for rapid and in some cases quantitatively accurate study of complex heterogeneous adsorption. Recently Vega et aLZohave applied this approach to molecular models of the type considered in the present work. In this work we explore the role played by heterogeneous microstructure in binary mixture adsorption by application of Monte Carlo (MC) computer simulation to a variety of molecular models. The results of these simulations are interpreted with reference to the ideal adsorbed solution (11)Ogston, A. G. Trans. Faraday SOC.1958,54, 1754. (12)Giddinga, J. C.;Kucera, E.; Russel, C. P.; Myers, M. N. J. Phys. Chem. 1968, 72,4397. (13)Rikvold, P. A.;Stell,G. J. Colloid Interface Sci. 1986,108, 158. (14)Fanti, L.A.; Glandt, E. D. AIChE J. 1989,36, 1883. (15)Fanti, L.A.;Glandt,E. D. J. ColloidZnterfaceSci. 1990,136,385. (16)Fanti, L.A.;Glandt, E. D. J. ColloidInterface Sci. 1990,136,396. (17) Madden, W. G.; Glandt, E. D. J. Stat. Phys. 1988,51, 537. (18)Madden, W. G. J. Chem. Phys. 1992,96 (7),5422. (19)Given, J. A.;Stell,G. J. Chem. Phys. 1992,97 (6), 4573. (20)Vega, C.;Kaminsky, R. D.; Monson, P. A. J. Chem. Phys. 1993, 99 (4),3003.
0743-7463/94/2410-0530$04.50/00 1994 American Chemical Society
Langmuir, Vol. 10, No. 2, 1994 531
Influence of Porous Adsorbent Microstructure
(US) theory of Myers and Prausnitz.21 This paper represents a continuation of previous work22*23by the authors on the influence of adsorbent microstructure on Henry's constants, pure component isotherms, and adsorbate fluid structure inside porous adsorbents. We wish to address two general questions: (1)What is the role of heterogeneous porous microstructure in determining binary adsorption equilibrium? (2) How and when can a heterogeneous microstructure induce adsorption nonideality? The molecular models which are the focus of this work represent a balance between realism and utility. We have sought to develop models which are at least qualitatively realistic descriptions of the microstructure of silica gel. At the same time the models are sufficiently tractable to allow for extensive studies via Monte Carlo simulation. Moreover, the models are formulated to admit the possibility of developing approximate theories based on statistical mechanics. The remainder of the paper is organized as follows. Section 2 describes the molecular models employed in the study and some aspects of our methodology. Section 3 presents a comparison of our model with experimental data for single component systems to test the basic correctness of the model. Section 4 discusses our Monte Carlo simulation results for methane-argon mixtures adsorbed in silica gel. Section 5 extends the study to exploring the effects of molecular size differences. Lastly, section 6 reports our conclusions. 2. Molecular Model and Methodology The model adsorbent system considered in this work is that developed in our previous work22and is based on the structure of silica xerogel. The adsorbent consists of a fixed matrix of spherical adsorbent particles which are themselves uniform continuous composites of 12-6 Lennard-Jones (LJ) sites. We term these spherical adsorbent particles "composite spheres" (CS). The adsorbateadsorbent potential for such a composite sphere and a LJ adsorbate particle can be analytically determined22and is given by ucs(d) =
(( d6 + ?d4R2 + 3d2R4+ @)u"
-rcpR3 16 3
-
1
(d2-2R2)3 (1) where d is the adsorbate-adsorbent center-to-center distance, R is the CS radius, p is the LJ site density of the CS, c is the LJ potential strength for a site-adsorbate interaction, and 0 is the collision diameter for a siteadsorbate interaction. For d < R the potential is taken to be positive infinity, i.e. a CS particle has a hard core of radius R. The CS potential is very much in the same spirit as the commonly used 9-3 wall potential," which models an adsorbing flat surface as a composite of LJ sites. Indeed the 9-3wall potential represents the limiting behavior of the CS potential in the large R limit.22 MacElroy and Raghavan2s studied a very detailed molecular model of silica gel and suggested that for adsorption calculations the structure of silica gel may be taken to a first approximation as uniform spheres of silica arranged to correspond to an equilibrium configuration of
(d2- R2)'
(21) Myers, A. L.; Prausnitz, J. M. AIChE J. 1965, 11 (l), 121. (22) Kaminsky,R. D.;Monaon, P. A. J. Chem.Phys. 1991,95 (4), 2936. (23) Kaminsky, R. D.; Monson, P. A. Langmuir 1993,9,561. (24) Steele, W. A. The Interactions of Gases with Solid Surfaces; Pergamon: New York, 1974. (25) MacElroy, J. M. D.; Raghavan, K. J. Chen. Phys. 1990,93,2068.
Table 1. Potential Parameters and Henry's Constants (at !P = Id'/-& = 0.8) for Comwnents Used in This Work*
~
Me
0.3817 148.2 0.3405 119.8 0.4540 119.8 0.4540 119.8
0.3300 0.3100 0.3668 0.3668
339.0 19800 771 AI 305.0 283 50.0 B1 305.0 1.7 X 10" 41108 B2 180.5 283 133 u, and cg are 12-6 W adsorbate-adsorbate(gas-gas) interaction parameters. up and cp are CS adsorbate-adsorbent (gas-solid) interaction parameters. Each CS has a hard core diameter of 2.693 nm and U site density of p = 44/nms. Note: The Henry's constant valuea for the EHS system are calculated for the specific 32 CS configuration used in the simulations; not a thermodynamic liiit EHS system.
hard spheres. By reference to the work of MacElroy and Raghavan26 we determined22 suitable values for the parameters in eq 1 for methane interacting with a microporous (800m2/g) silica xerogel adsorbent. These values are listed in Table 1. We have previously shown in ref 22 that the CS model yields a satisfactory approximation to the Henry's law constants for methane in microporous silica xerogel. The adsorbateadsorbate interaction parameters for methane in Table 1 are LJ parameters taken from Hirschfelder et al.= The argon adsorbate-adsorbent parameters were determined via application of the Lorentz-Berthelot combining rules. It can be shown that the Lorentz-Berthelot rules imply for three components i, j, and k that
and (3)
Hence taking i to be argon, j to be silica, and k to be methane, the argon-silica interaction parameters were estimated from the methane-methane, methane-silica, and argon-argon interaction parameters. Thus, we have not optimized the adsorbate-adsorbent parameters for argon. We have considered adsorption in two adsorbent microstructures of composite spheres. Comparing the adsorption behavior observed in the two cases permits us to draw conclusionsabout the influence of the microstructure. The two microstructures employed are an equilibrium hard sphere (EHS) configuration and a face-centered cubic (FCC) configuration. The choice of these two structures was made since the first is a qualitatively realistic model of a heterogeneous adsorbent (e.g. silica gel) and the second allows for a comparison with a more ordered microstructure. Pure component adsorption in these structures was considered in ref 22. The hard core volume fraction of the composite-sphere adsorbent systems is taken to be 9 = 0.386, the volume fraction corresponding to our of a silica gel. The EHS structure was represented by a single 32-particle hard sphere configuration generated by a canonical (NVT) Monte Carlo simulation. The choice of configuration is the same one as that in ref 22 for the EHS structure. Figure 1presents a graphics image of the EHS structure of composite spheres employed in this paper. The binary adsorption was calculated via grand canonical ( p V T ) Monte Carlo simulation. The interactions between adsorbate particles were modeled using LennardJones 12-6potentials. Lorentz-Berthelot combining rules (26) Hirschfe!der,J.O.;Curtiss,C.F.;Bird,R.B. TheMoleculor Theory
of
Gases and Liquids; Wiley: New York, 1954.
Kaminsky and Monson
532 Langmuir, Vol. 10,No. 2, 1994
theory as the basis of comparison. The working definition Myers and Prausnitz21chose for an ideal adsorbed solution is Pyi = Pio(7r)xi
(constant )‘2
(5)
where P is the bulk pressure, yi is the bulk mole fraction of speciesi, xi is the excess adsorbed mole fraction of species i, and Pio(a)is the bulk pressure of the pure species i in equilibrium with a pure adsorbate i at a spreading pressure 7r. Equation 5 is analogous to Raoult’s law for a bulk solution. To solve for adsorption one needs only to know the spreading pressure of the pure species as a function of bulk pressure. Spreading pressure can be calculated via integration of the Gibbs adsorption isotherm21
Figure I. Computer graphics representation of our model heterogeneousmicroporous medium. The large spheres represent silica particles (arranged in an equilibrium hard sphere structure) and the small spheres represent adsorbed methane molecules.
were used for all adsorbate-adsorbate cross interactions. The adsorbate-adsorbate potentials were truncated at 2.50,~and the adsorbate-adsorbent potentials were truncated at 4.0ag, from the CS hard core surface, where agg is the adsorbate-adsorbate (gas-gas) LJ range parameter and ageis the adsorbate-adsorbent (gas-solid) CS range parameter. The chemical potential of each component must be explicitly set in the grand canonical simulation method. The activities were calculated by choosingthe desired bulk pressures and compositions and then applying van der Waals l-fluid (vdW-1)conformal solution t h e ~ r y . ~The ~,~s pure fluid equation of state used in the vdW-1 theory is that of a LJ fluid proposed by Nicolas29with a correction for the potential t r u n ~ a t i o n .The ~ ~ vdW-1theory is highly accurate for vapor densities, which is the case for all the bulk pressures considered in this work. Henry’s constants, KH were calculated independently from the MC simulations for the systems studied. Henry’s constants may be computed via direct integration of the one-particle configuration integral, 21, as (4)
where V is the system volume, @(r)is the adsorbateadsorbent potential at point r, k is the Boltzmann constant, and T is the absolute temperature. In the Henry’s law where limit the total adsorbate density is given by ZKH, z = Q exp(p/kT) is the activity, Q is the molecular partition function, and p is chemical potential. Note that for an ideal gas P = zkT. The integration in eq 4 can be done numerically via a Monte Carlo method.22 To gauge the influence of microstructure on producing nonideal behavior, we take ideal adsorbed solution (IAS) (27) Leland, T. W.; Rowlinson, J. S.; Sather, G. A. Trans. Faraday SOC.1968,64,1447. (28) Hansen,J. P.;McDonald,I. R. Theoryof Simple Liquids;Academic Press: New York, 1986. (29) Nicolas, J. J.; Gubbins, K. E.; Streett, W. B.; Tildesley, D. J. Mol. Phys. 1979,37, 1429. (30) Finn, J. E.; Monson, P. A. Phys. Reu. A 1989, 39 (12), 6402.
where A is the surface area and rio(P)is the pure component adsorption excess isotherm as a function of bulk pressure, P. Myers and Prausnitz show21that eq 5 can be rigorously derived by assuming no change in Gibbs free energy on mixing in the adsorbed phases at constant temperature and spreading pressure. For our purposes the adsorption excess required for evaluation of eq 6 was simply taken as (7)
where pi is the total density of species i in the porous system, pibulkdensity of species i, and is the CS hard core volume fraction. Equation 7 represents the definition of adsorption excess for porous materials analogous to the definition commonly used for flat walls.24 As is clear from the choice of definitions for an ideal adsorbed solution, IAS theory does not explicitly account for confinement effects, energetic heterogeneity effects, or unlike species adsorbate-adsorbate interaction effects, except in so far as such effects influence the pure component isotherm. As such, deviations of IAS predictions from experimental data suggest these effects are of some importance on the adsorption behavior. 3. Comparison with Experimental Data We have compared Monte Carlo simulation results from our model with experimental data by M a s u k a ~ a 3 ~for ~3~ methane adsorbed in silica gel. The silica gel used by Masukawa was Grade 15by Davison Chemical with specific area 803.5m2/g. This silica gel is similar to the Davison Chemical Grade 40 silica gel (834 m2/g) used for Gangwal et al.33on which MacElroy and R a g h a ~ a nbased ~ ~ their methane-silica gel model, upon which our own model is based. This being the case we feel justified in directly applying, with two minor modifications, our composite sphere/equilibrium hard sphere model of methane-silica gel to predict Masukawa’s experimental data. The two minor modifications we make are as follows. First, the adsorbate-adsorbent interaction strength, em, is chosen so that the Henry’s constant at a single temperature agrees exactly with Masukawa’s data, rather than with Gangwal‘s data as was originally done.22 Second, the LJ parameters for the methane-methane interactions are computed from critical point data rather than second virial ~~~~
~
(31) Masukawa, S. A Study on Two Phase Equilibria by Use of the Elution Gas Chromatographic Technique-The Methane-Ethane-Silica Gel System and the Methane-Normal Octane System; Rice University: Houston, TX, 1967. (32) Masukawa, S.; Kobayashi, R. J. Chem. Eng. Data 1968, 23 (2), 197. (33) Gangwal, S. K.; Hudgins, R. R.; Silveston, P. L. Can. J. Chem. Eng. 1979,57, 609.
Influence of Porous Adsorbent Microstructure
Langmuir, Vol. 10, No. 2, 1994 533
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coefficient data. By this method we find the methanemethane interaction parameters to be em/k = 154.8 K and ugg= 0.390 nm. To compare with Masukawa’s data it is necessary to specify the physical density of the porous silica gel (not including any large-scale pores due to the packing of bulk silica pellets or powder). The exact value of psilics is open to some question since the results from different measurement techniques are not always consistent. We chose the value of 1.129 g/cm3which Gangwal et al.33measured by mercury displacement for the silica gel studied in their work. Masukawareported the Henry’s constant for methane in the silica gel studied as 1.173 X 1od mol/g.psia at 298 K. Performing Monte Carlo calculations of Henry’s constantz2in our model we find that tdk must equal 349 K for KH to agree with Masukawa’s experimental value. We use this value of eB$k for the methane-silica interaction strength in our model to compare with Masukawa’s data. Figure 2 presents a comparison of grand canonicalMonte Carlo simulations of our model with Masukawa’s data for methane adsorbed in silica gel. From the figure we see that the simulation results are qualitatively similar to the experimental data, but the simulated isotherms seem to saturate at lower pressures than the experimental isotherms. Nevertheless, the spacing of the isotherms seems to agree fairly well with that in the experimental data. The agreement is not as good as in a previous c0mparison3~ of the model with another set of experimental data,36 but in that comparison compensating errors played more of a factor due primarily to uncertainty of the experimental Henry’s constant. There are several possible sources of the disagreement between the simulated and experimental isotherms. First, our simplified model system may not truly reflect the details of the real methane-silica system because of silica particle surface roughness or particle size polydispersity. (34)Knminsky, R. D.; Monson, P. A. Fundamentals of AdsorptionProceedings ofthe ZVth ZCFA,Suzuki,M., Ed.; Kodaneha: Tokyo,1993. (36)Haydel, J. J.; Kobayashi, R.Znd. Eng. Chem.Fundam. 1967,6, 546.
T (K) Figure3. Comparisonof Henry’sconstantbetween our CSmodel (squares)and Masukawa’s data (solidline)for methane adsorbed in silica gel. The methane-silica interaction strength in the CS model was set so that there is exact agreement at T = 298 K.
This as a source of error is given some support by comparing predicted and experimental Henry’s constants. Figure 3 presents such a comparison. From Figure 3 we see that although the agreement is moderately good, it is far from perfect. This is in contrast to the agreement in our previous comparisonzz with the data of Gangwal et al.33 and the results from MacElroy and Raghavan’s26more sophisticated molecular model. Since Henry’s constants depend solely on the adsorbent morphology and the adsorbateadsorbent interaction potential, the disagreement between the theoretical and experimental Henry’s constants is an indication of some model mismatch. A second possible source of disagreement between the simulated and experimental isotherms in Figure 2 is error in the value of pailia we chose. The lower the value of pasca, the greater the saturation density. The overestimation of adsorption by the Monte Carlo simulation in Figures 2 and 3, may also be due to overestimating the methane-methane interaction strength in the adsorbed phase by ignoring nonpairwise interactions. Moreover, the optimum LJ parameters for the adsorbed fluid may not be the same as those for the bulk. Although overall the agreement between the simulation and experimental data is only fair, the general quantitative and qualitative similarity lends credence to the idea that our model of silica gel is a reasonable one. 4. Binary Adsorption Equilibrium: Methane-Argon Mixtures The initial part of our investigation of the role of heterogeneity in determining selective adsorption consists of analyzing adsorption of model LJ methaneargon ( M e Ar) mixtures adsorbed into the two model microstructures described earlier. In presenting our results, all parameters are reduced by the ua and em Lennard-Jones values for methane, even if methane is not a component. We studied adsorption a t the temperature of !P = k!f”/EM+Me = 0.8. This choice of a subcritical temperature (with respect to the bulk critical temperature) was made with the assumption that a low temperature best highlights the adsorption physics and differences between systems. Three bulk pressures are considered P” = P ~ ~ M + M ~ ~ M + M = 0.OOO1,0.001,and 0.007. A t these pressures the bulk phase is essentially an ideal gas since the temperature is
Kaminsky and Monson
534 Langmuir, Vol. 10, No. 2, 1994
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,Y (b) Figure 4. Bulk methane-argon VLE behavior at T* = 0.8 (a) prewure-composition (P-x) behavior and (b)liquid compositionvapor composition (x-y) behavior. Solid lines, van der Waals 1-fluid (vdW-1)conformal solution theory predictions; dashed lines, Raoult's law predictions. well below the critical point (T,*r 1.23) and the pressure is below the saturation pressure. Figure 4 presents the vapor-liquid equilibrium (VLE) diagram for the a bulk LJ methane-argon mixture at T* = 0.8 with a potential ~ ~ ~highest . pressure studied (P'= truncation at 2 . 5 ~ The 0.007)for the methane-argon mixtures is just below the methane saturation pressure of P'(sat) = 0.00766 at T* = 0.8. The VLE was determined via the van der Waals onefluid (vdW-1) conformal solution t h e ~ r y . ~This ~ t theory ~~ has been shown to be accurate for bulk mixtures similar to those studied here.37 The comparison with Raoult's law suggests that the bulk solution is quite ideal. Figure 5 shows grand canonical Monte Carlo simulation results for the adsorbate composition vs the bulk composition for the methane-argon system. Also included in Figure 5 are the Henry's law predictions. Table 1lists the Henry's constants for the systems studied. Examining the adsorption behavior depicted in Figure 5 we note several features. First, all the composition curves are qualitatively similar showing a selectivity for methane. Second, Henry's law reasonably describes the composition behavior for the FCC case for the two lowest pressures, but not for the EHS case at any of the pressures. Third and somewhat surprisingly, the EHS and FCC cases show nearly quantitative agreement at the higher presaures,even though their microstructures and Henry's constants are considerably different. To get an alternate view of the adsorption behavior, Figure 6 presents adsorption in terms of selectivities, S = ( x M ~ / Y M ~ ) / ( x AIt ~ should / Y A ~ )be . noted that in the Henry's law limit selectivity is a constant given by, SO = K H ~ / K H From ~ Figure 6 we clearly see that selectivity for the EHS microstructure is quite dependent on both bulk composition and pressure whereas for the FCC microstructure the dependence is much less. In particular selectivity decreases as pressure or the argon mole fraction increases. Given the differences between the EHS and FCC microstructures, what then is the cause of the similarity
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1 o.2 P' (b) Figure 6. Monte Carlo simulation results for methane-argon selectivity,s = (xMdy&)/(xh/yh)at T* = 0.8 filled circles and solid lines, EHS structure; open circles and dashed lines, FCC structure. Part a presenta S as a function of bulk composition for P = O.OOO1 (top curves), 0.001 (middle curves), and 0.007 (bottom curves). Part b presents S as a function of pressure for bulk composition y~~ = 0.03 (top curves), 0.15 (middle curves), and 0.50 (bottomcurves). Circlesmark simulation results. Error bars are 1 % for EHS data and 4 % for FCC data. In the Henry's law limit selectivity is SO= 70.0 for the EHS structure and SO= 15.4 for the FCC structure.
of the composition behavior for the cases studied in Figure 5? The apparent insensivity of composition to structure for equal porwities impliesthat except for near the Henry's law regime, selectivity is primarily dependent on the gross (36)Shing, K. S.;Gubbina, K. E.Mol. Phys. 1983,49 (5), 1121. (37) Hariamiadis,V. 1.; Koutras, N.K.; Taseioe,D. P.; Panagiotopoulce, A. 2. Fluid Phase Equilib. 1991, 65, 1.
Influence of Porous Adsorbent Microstructure
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Langmuir, Vol. 10,No. 2, 1994 535
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,Y (b) Figure 7. Total adsorbed density versus composition for methane-argon system at T* = 0.8. (a) FCC, (b) EHS. Monte Carlo simulation results: squares, P = O.OOO1; triangles,P = 0.001; circles, P = 0.007. Ideal adsorbed solution predictions: solid lines.
features, not the details, of the adsorbent structure. This dependence may be understood by first considering the physics in the Henry’s law regime. As is discussed in ref 22, the regions which contribute most to the Henry’s law constant at low temperature constitute a quite small fraction of the total system volume. These are the regions having large negative potential energy. These regions of strong attractive potential are first to fill with adsorbate particles and possibly, as is the case for the EHS system, at extremely low pressure. After these regions fill, the higher pressure adsorption is dominated by the large-scale features of the adsorbent structure and potential field. In fact, at moderate to high coverage the selectivity behavior of the two systems becomes similar implying that possibly only the most grossfeatures of the systems, such as porosity and overall adsorbate-adsorbent interaction strength, are significant in determining the behavior. It must be remembered however that selectivity only depends on the ratio of the individual species adsorptions and hence total adsorption may significantly differ between adsorbent geometries even when the selectivities do not. The similarity of the composition curves for the EHS and FCC morphologies depicted in Figure 5 contrasts with the qualitatively different density vs composition behavior for the two morphologies which is shown in Figure 7. To judge the nature and the extent of nonidealities in the observed adsorption behavior we compared IAS theory predictions to our simulation results for the Me-Ar systems. Grand canonical Monte Carlo simulations were carried out to find the pure component isotherms of each species. The pure component isotherms for methane have been previously discussed in ref 22. As written, eq 5 assumes the bulk phase to be an ideal gas. Nonideality in the gas bulk phase may be readily included, as we do in our calculations, by substituting fugacity for pressure. For the methane-argon systems at temperature !P = 0.8 Figure 7 shows the comparison with IAS theory for the total adsorbed density and Figure 8 presents the comparison for the composition. Total adsorption, not adsorption excess, is reported. From Figures 7 and 8 we see
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,Y Figure 8. Comparison of adsorbed phase-bulk phase (2-y) compositionby IAS theory and Monte Carlo simulation for the methane-argon system at T* = 0.8 and the EHS morphology. Monte Carlo simulationresults: squares, P = O.OOO1; triangles, P = 0.001; circles, PC = 0.007. IAS predictions: solid lines.
that IAS theory produces good agreement with the simulation data. The IAS composition predictions in Figure 8 do show a slight systematic favoring of the methane relative to the simulation results. Calculations for the FCC structure exhibited similar agreement and trends. The small systematic deviations between IAS theory and simulation in the composition results are consistent with the deviations from Raoult’s law observed in the bulk VLE behavior (see Figure 4). Thus for this system IAS theory does an excellent job of capturing the main features of the adsorption behavior solely from knowledge of the pure component adsorption isotherms of methane and argon. 5. Binary Adsorption Equilibrium: Effect of Molecular Size Differences Since the molecular size difference in the methaneargon mixture is not very large, it is worthwhile to investigate what happens with a larger size difference. Hence, we investigated model binary mixtures of argon and a larger component adsorbed in the same silica adsorbent model used for the methane-argon study. The second component, which we will refer to as component B, has the same U parameter egg as argon but has a fourthirds larger cggthan argon. For the adsorbate-adsorbent interaction parameters we took ues to follow LorentzBerthelot combining rules and applied eq 3. For the adsorbate-adsorbent parameter tgswe considered two cases: (1)t,,(B-Si) = tW(Ar-Si) and (2) ees(B-Si) set so that for the EHS microstructure KH(B-Si) = &(Ar-Si). The second component in the former case will be referred to as species “Bl” and in the latter case as species “B2”. The parameters are given in Table 1. Of course it should be noted that the Si-Ar-B2 system is a somewhat unrealistic system since in general tesincreases monotonically with cgs. Nevertheless,this model system highlights the physics of adsorbate size contributions to adsorbed phase composition. Figures 9 and 10 show the adsorbed composition for both model microstructures for the Ar-B1 and Ar-B2 systems, respectively. All the systems considered in Figures 9 and 10 are at a bulk pressure of PC = 0.001 (with Preduced by the methane parameters) which is well below the saturation pressure for any of the components. Figure 11shows a VLE diagram for the Ar-B system. (Note that B1 and B2 have the same bulk behavior.) As before, the VLE diagram is calculated via vdW-1 theory. From Figure
Kaminsky and Monson
536 Langmuir, Vol. 10, No. 2, 1994 1 .o
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Figure 10. Argon-B2 adsorption at P = 0.8 and P = 0.001. Monte Carlo simulation results: squares, EHS triangles, FCC. Henry’s law predictions: short-dashed line, FCC. (The EHS case essentially corresponds with the x = y line and hence is not explicitly shown.) 9 we clearly see the Ar-B1 system produces the same qualitative behavior as seen for the M e A r system. Namely, the EHS and FCC cases show nearly identical selectivity. Note that the Henry’s law results for the EHS and FCC cases shown in Figure 9 are very different from the higher pressure results due to the extremelylarge values of KHfor species B1 (see Table 1). Turning our attention to the Ar-B2 system, we see quite different behavior from the previous cases. Specifically, the two adsorbent morphologies produce qualitatively different behaviors: the EHS case has an azeotrope (crossover in composition) whereas the FCC case shows significant favoring of B2 and some nonideality. What is the cause of the effects exhibited in Figure lo? In a multicomponent system where the components are adsorbed with similar affinity, any features of the mixture not present in the pure componentscan lead to azeotropes or other readily noticeable deviations from ideal solution behavior. With the ApB2 system we have specifically constructed a system where the two components adsorb similarly and any such effects will be highlighted. The Ar-B2 composition behavior in Figure 10 can be explained in terms of the coupling of the molecular size difference with the heterogeneity in the adsorbate-adsorbent po-
tential field and the stronger adsorbateadsorbate interactions for the B2 component. The argon component experiences a significantly stronger, but shorter range, interaction with the composite spheres than does the B2 component (see Table 1). Hence even though the Henry’s law limit adsorption for the two species is the same for the EHS case, at higher densities the argon is preferred to the B2 since because of its smaller size it can more effectively occupy the regions in the adsorbent structure where the adsorbateadsorbent interactions are stronger. At yet higher adsorbate densities where all these regions are occupied, addition of B2 is preferred over that of argon since B2 particles have stronger adsorbah-adsorbate attractive interactions. The B2 interaction range is significantly longer than that of argon even though their potential well depths are the same. The EHS microstructure case, which is at a moderate adsorbate density, exhibits a favoring of argon at low concentrations of argon since it can better fi the high energy regions of the microstructure. It shows a favoring of B2 at high concentrations of argon since all of the high energy regions are already filled by argon molecules and B2 has more favorable adsorbateadsorbate interactions. The FCC case shows similar curvature to the EHS case for the same reasons cited except that superimposedis a favoring of B2 since it has a larger Henry’s constant. We note that the effects seen in the k B 2 system play much leas of a role in the Me-Ar system. The large differences between the strengths of the methane-silica and argon-silica interactions dominate the adsorption behavior and obscure the more subtle effects. Applying IAS theory to the k B 2 system we observe that the agreementis qualitatively poor as shown in Figure 12. Figure 12a depicts the composition dependence for the EHS and FCC Ar-B2 systems. Although somewhat better than the Henry’s law predictions (see Figure lo), IAS theory fails to predict an azeotrope for the EHS w e or the proper curvature in the FCC case. This is because there are qualitative features of the adsorption behavior
Langmuir, Vol. 10, No. 2, 1994 537
Influence of Porous Adsorbent MiCtOStrUCtUre 1.0,
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sorption selectivity. This topic is of practical interest since many commonly used adsorbents do indeed have illd e f i e d porous microstructures. Statistical mechanical treatments of this problem have been traditionally hampered by the lack of tractable modela of complex porous media. We have previously developed a qualitatively correct adsorbent model based on silica gel which is quite suitable for Monte Carlo computer studies. By carrying out Monte Carlo simulations using this model, we have demonstrated that adsorbent microstructure plays a significant but sometimes subtle role in determining adsorption behavior. For the methane-argon system at low coverage, adsorption behavior is found to be quite sensitive to the details of the adsorbent microstructure. This is because the Henry's constant is highly dependent on the fine geometry of the pores and adsorbate-adsorbent potential field. Away from the Henry's law regime, however, adsorption selectivity appears to become very much a function of only the fluid properties (e.g. molecular size and overall adsorbateadsorbate interaction strength) and the gross features of the adsorbent (e.g. porosity and overall adsorbate-adsorbent interaction strength). We studied two porous microstructures: an equilibrium hard sphere (EHS) configuration of adsorbing spheres and a facecentered cubic (FCC) structure of adsorbingspheres. Both structures were constructed with identical porosity. Although the EHS and FCC are structurally quite different, they exhibit little difference in adsorption selectivity away from the Henry's law regime. However, the total amount adsorbed depends significantly on the microstructure, as is the case for the pure components. In addition to the methaneargon system, we studied two other model mixtures with larger molecular size differences. In such cases more complex behavior was observed, including the occurrence of azeotropes. In these systemsthe key effects are generatedby the coupling of molecular size differences, heterogeneity in the adsorbate-adaorbent potential energy, and attractive adsorbate-adsorbate interactions. A detailed comparison of the simulation results with the ideal adsorbed solution theory of Myers and Prausnitz was made. IAS theory does a good job of predicting the adsorption behavior of the methane-argon-silica system even though the adsorbent is a heterogeneous medium. We suspect the success of IAS theory is in large part due to the fact that if the pure componenta adsorb with significantly different affinities, as do methane and argon in silica gel, any nonideal behavior tends to be masked by the large differences in the strength of the adsorbateadsorbent interactions. The IAS theory is much less successful for the model systems with larger adsorbate size differences, which exhibit more complex selective adsorption behavior. Unfortunately, to our knowledge there are as yet no experimental data for the simple mixtures considered in this work with which we can compare our results. Extension of this work to nonsphericalmolecules is underway and should permit a wider comparison with experiment to be made. For systems with orientation-dependent intermolecular forces additional sources of adsorbed phase nonideality can be anticipated. We are also investigating applications of approximate theories'' for the adsorption equilibrium in these systems following from the work described in ref 20.
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(b) Figure 12. Ideal adeorbed solution (IAS) theory prediction6 of argon-B2 adsorption: (a) argon adeorbed phase-bulk phaee (xy) composition behavior, (b)totaladsorption-bulk compceition (PY) behavior. Monte Cwlo simulation reeulk squares, EHS structure; trianglea,FCC structufe. IAStheorypredictions: solid line, EHS structure;dashed line, FCC structure. All eyateme are at temperature !P = 0.8 and preseure P = 0.001.
of Ar-B2 arising from effects not readily accessible from only the pure component adsorption isotherms. In particular, as discuseed in the last section, we believe that the nonideality is due to the coupling of molecular size differences, heterogeneity in adsorbateadsorbent potential energy, and the relative strengths of the adsorbateadsorbate interactions. These effects are not explicitly included by IAStheory. Thh is more plainly seen in Figure 12b which depicts a comparison for the Ar-B2 system of total adsorbed density from ZAS theory and Monte Carlo simulation. IAStheory systematically undereathatea the total density. This underestimation is a direct reflection of a significant negative volume of mixing due to mixing spherical particless of greatly different sizes, which is of course ignored by IAS theory. Over& the nonidealities observed in this study can be interpreted in term of competition between different aspecta of the microscopic behavior. The important features of this behavior for the systems studied are (1) the interaction of the molecular size difference with the confinement effect of the adsorbent geometry, (2) the heterogeneity in the adsorbate-adsorbent potential field which directly reflects the adsorbent geometry, and (3) the relative strengths of the adsorbateadsorbate interactions. The first feature, confinement effects,promotes adsorption of the smaller component. The second feature, energetic heterogeneity, can lead to selective adsorption even for componenta with equal Henry's constant when coupled with the molecular size difference, which is in turn dependent on the f i t feature. The last feature, adsorbateadsorbate interactions, tends to promote the component with stronger interactions but is also coupled to the other two features. 6. Conclusions In this paper we have a d d r e d some aspecta of how adsorbent structural heterogeneity influences binary ad-
Acknowledgment. Thiswork was supportedby agrant from the National Science Foundation (Grant No. CTS9116297).
