Langmuir 1993,9, 561-567
Physical Adsorption in Heterogeneous Porous Materials: An Analytical Study of a One-Dimensional Model R. D.Kaminsky and P.A. Monson' Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003 Received July 23,1992. In Final Form: November 12,1992 The role of adsorbent heterogeneity in pore size in determining adsorption equilibria is explored using an analytically solvable molecular model. The model system is one-dimensionaland interactionsare via square-well potentials. Although the model is simple,it captures severalqualitativefeaturesof the behavior expected in real systems. By imposing various pore size distributions on the analytic model the effect of heterogeneity is investigated for both single component and binary mixture adsorption. The analytic nature of the model allows for an especially broad range of system parameters to be studied and reveals fundamentalinsightintothe phenomena involved. Heterogeneous pore size is found to play a very significant role in determining adsorption behavior in microporous systems. For single-component systems, heterogeneityimpacta the totaladsorption isothermboth through changesin Henry's constantand through changes in the adsorption capacity. These effects may be complementary or competitive depending on the pore size distribution. Heterogeneityimpacts binary selectivityby altering the ratio of the component Henry's constants and through excluded volume effects. This has a significant influence upon nonideal effecta in binary adsorption equilibria such as azeotropes. 1. Introduction
some extent polydisperse in size. This polydispersity, in addition to implyinga distribution of confinementeffects, further complicatesthe physics by producingadistribution of adsorbate-pore interaction potential energies, which are directly related to the pore size. Pore size polydispersity is referred to as structural heterogeneity and a pore potential strength nonuniformity is referred to as energetic heterogeneity. Partly as a consequence of the inherent complexity in modelingheterogeneousadsorbents,statistical mechanical investigations of microporous materials have primarily been focusedon studyingisolated pores. Singlepores allow principally for the investigation of confinement effects and the associated potential field effects. Several studies616 have examined adsorption in ideal confined regions such as slits, cylinders, and spheres or specific zeolite structures. These investigations have been accomplished via computer simulation, integral equations, and density functional theory. Often these studies examine adsorbent density profiles inside the pores, especially when wetting and layering phase transitions occur. These investigationsof porous materials have led to some important insights into various phenomena, including capillary condensation and layer transitions. Some work has also been donel7in extendingdensity functionaltheory to systems with a distribution of pore sizes and to pore
Industrial adsorption applications are often accomplished through the use of microporous materials. These materials have pore dimensions on the same order as the adsorbate molecular size. Since the length scalesinvolved in microporous physical adsorption are molecular, the phenomena are inherently best described by molecular thermodynamics. Indeed the common classical thermodynamic theories used in the prediction of physical adsorption (e.g. ideal adsorbed solution theory, vacancy theory, etc.) can produce quite incorrect results for microporous adsorption due to their essential ignorance of the adsorbent microstructure and the aasociated molecular scale phenomena.'"' This being the case, development of a statistical mechanical understanding and theory of the phenomena is quite desirable. This has been accomplished to some degree through the application of Langmuir (i.e. site) models and their generalizations.6Site models were of course primarily developed for modeling adsorption at a free surface. The concept of a site has less physicalsignificance in a pore than on an adsorbentsurface. This is the case since any adsorbed particles in a small pore are fairly nonlocalized and are strongly coupled with other adsorbed particles. Hence while site model methods can be fairly successful for fitting adsorption data, they tend to be less successful for predicting behavior.2 Sophisticated statistical mechanical modeling of microporous adsorbents is complicated by two key factors. First, confinement by its very nature leads to inhomogeneity in adsorbate density in the pore regardless of the magnitude of the adsorbate-adsorbent interactions. It is this issue which bas been the focus of the most theoretical studies. Second, in all real adsorbents the pores are to
(6) van Megan, W.; Snook, I. K. Mol. Phys. 1981,54, 741.
(7) Peterson, B. K.; Walton, J. P. R. B.; Gubbins,K. E. J. Chem. SOC., Faraday Trans. 2, 1986,82,1789. (8) Peterson, B. K.; Gubbins, K. E. Mol. Phys. 1987,62,215. (9) Evans,R.; Marini Bettolo Marconi, U.;Tarazona, P. J. Chem. SOC., Faraday Tram. 2, 1986,82, 1763. (10) Ball, P. C.; Evans, R. Mol. Phys. 1988, 63, 159. (11) Walton, J. P. R. B.; Quirke, N. Mol. Sim. 1989,2,361. (12) Zhou, Y.; Stell, G.Mol. Phys. 1989,66, 767. (13) MacElroy, J. M. D.; Suh,S.-H. Mol. Phys. 1989,2,313. (14) Woods, G.B.; Panagiotopolous,A. Z.; Rowlineon, J. S. Mol. Phys.
* Author to whom correspondence should be addressed.
(1) Valenzuela, D. P.; Myers, A. L.; Talu, 0.;Zwiebel, 1. AIChE J. 1988,34, 397. (2) Ruthven, D. M.F'rinciplesof Adsorption andddsorptionProcesses; Wiley: New York, 1984. ( 3 )Myers, A. L.; Prauenitz, J. M. AIChE J. 1961, 11, 121. (4) Suwanayuen, S.; Danner, R. P. AIChE J. 1980,26,68. (5) Steele, W. A. The Interactions of Gases with Solid Surfaces; Pergamon: New York, 1974.
0743-746319312409-0561$04.00/0
1988, 63, 49. (15) Woods, G. B.; Rowlinson,J. S. J. Chem. SOC.,Faraday D a m . 2 1989, 85, 765. (16) Tan, Z.; Gubbins,K. E.; van Swol, F.; Marini Bettolo Marconi, U.
Fundamentals of Adsorption; Meremann, A. B., Scholl, S. E., Eds.; Engineering Foundation: New York, 1991. (17) Ball, P. C., Evans, R. Langmuir 1989,5,714. (8
1993 American Chemical Society
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network effects in order to investigatethe molecular basis of hysteresis loops in adsorption isotherms. There has also been some progress with heterogeneous porous materials. These include studies of Henry's law limit1e20and finite con~entration.~l-~~ The finite concentration investigationsinvolved solely hard particles in adsorbents composed of hard particles. Madden and Glandt have derived an integral equation approach2' for calculating certain adsorption behaviors in randomly structured media using arbitrary interaction potentials. To date though it has only been applied to hard particle ~ y s t e m s Nevertheless, .~ the systemsand methods of study are sufficiently complex to severely hinder extensive investigationsnecessaryfor a broad overviewof the effects of adsorbent heterogeneity. Fully analytic studies of adsorption, especially those employing models including attractive interaction potentials, are exceptionallyscarcedue to the inherent difficulty in treating such systems. Nevertheless, the utility of analytic methods should be clear. First, analytic methods allow for rapid and broad phenomenologicalstudy. Second, they illuminate the mathematical functionality of the modeled phenomena revealing detailed fundamental insight. Finally, the role of all imposed assumptions is readily visible making very clear the applicability and limitations of the model. In this paper we propose to examine a wide range of adsorption aspects associated with heterogeneity through the investigation of a onedimensional (l-D) model of adsorption. The use of l-D models with nearest-neighbor interactions, including attractive interactions,admits analytic solutionof adsorption behavior. These models, though not at first glance particularly realistic, do in fact capture much of the essential physics of confined fluid thermodynamics and hence can produce results qualitatively similar to those in real systems. In particular one-dimensional models of confined systems have been shown to produce adsorbed fluids structures quite similar to those in ideal threedimensional slit and cylindrical pores.26 Thus, the study of l-D systems can be useful in drawing generalizations and gaining a fundamental understanding about real, complex systems. The study of one-dimensionalsystems has an extensive history starting a century ago with letters to Nature by R a ~ l e i g hand ~ ~Korteweg.2a Nevertheless, it is the work of T o n k P in 1936which is most cited as the original work on the hard rod equation of state. Analytic l-D models have been quite successfullyapplied to the study of fluid and solution thermodynamics by many researchers. The research involving l-D systems up to the mid-1960s has been given in a collection of reprints edited by Lieb and Mattkso Although in 1950 van Hove3l proved that any l-D system with a finite-ranged intermolecular pair potential is incapable of exhibiting phase transitions, investigations of such models were still actively pursued due to their clarity of insight. For example, in the 1950s (18) Ogaton, A. G. Tram. Faraday Soc. 1968,54,1754. (19) Giddinga, J. C.;Kucera, E.; Ruseel, C. P.; Myers, M. N. J. Phys. Chem. 1968, 72, 4397. (20) Rikvold, P.A.; Stall, G.J. Colloid Interface Sci. 1986,108, 158. (21) Fanti, L.A.; Glandt, E. D.AZChE J. 1989,35,1883. (22) Fanti, L.A.; Glandt,E. D. J. ColloidInterfaceSci.1990,135,385. (23) Fanti, L.A,; Glandt, E. D. J. Colloid Interface Sci. 1990,135,397. (24) Madden, W.C.; Clandt, E. D. J. Stat. Phys. 1988,51,537. (25) Fanti, L.A.; Glandt, E. D.; Madden, W. C. J. Chem. Phys. 1990, 93,5945. (26) Monson, P.A. Mol. Phys. 1990, 70,401. (27) Rayleigh, Lord Nature 1891,45,80. (28) Korteweg, D.T. Nature 1891,45,152. (29) Tonka, L. Phys. Reu. 1986,50,955. (30) Leib, E. H.; Mattie, D. C. Mathermatical Physics in One Dimemion; Academic Preee: New York, 1966. (31) van Hove, L.Physica 1950, 16, 137.
researchers such as P r i g ~ g i n eand ~ ~ L~nguet-Higgins~~ applied l-D models in the development of the molecular theory of solutions. During the 197Oethere was a renewed interest in l-D systems when they were recognized as a means of developing a fundamental understanding of inhomogeneous fluids. In particular by studying semiinfinite and finite l-D systems, detailed understanding can be gained of the physics of fluids near to walls or confiied in pores. In this connection PercusM-W has performed some of the most extensive research. The work of Percus was followed by other researchers, such as Robledo and Rowlineon37 and Vanderlick et al.,w who further explored the effect of confinement on hard rods. An important result of this work with l-D models has been development of improvements to 3-D fluid density functional theory.39 Recently Monsona has derived the analytic results for inhomogeneous square-well mixtures in l-D. This is a qualitatively realistic model of a fluid confined to a pore. The systemMonson studied is significant in that it includes both attractive and repulsive adsorbate-solid and adsorbate-adsorbate interactions. It is this model system which forms the basis of our study of adsorption. It includes several key features of real fluid-pore systems-confinement, adsorbate interparticle attractive forces, adsorbate particle excluded volume, adsorbateadsorbent attractive forces,and spatial inhomogeneity(i.e. no mean field assumptions). In this paper we examine heterogeneity in the context of pure and binary component adsorption in media composed of l-D pores. Using the analytic solution Monson derived for square-well adsorbate particles between square-wellpotential walls,we investigatethe effects of the physical heterogeneity (pore size polydispersity) and energetic heterogeneity (nonuniformpore potentials). In particular, we calculate adsorption behavior for a wide variety of porous systems. The pore size distributions in these systems range from monodisperse to broad polydisperse. Adsorption isotherms, x-y diagrams, and a variety of other results are presented to illustrate the various effects which occur. The remainder of this paper is arranged as follows. Section 2 reviews the analytic solution for adsorption in a single model pore. Section 3 describes the model of heterogeneity employed in our work. Section 4 presents results for pure component adsorption behavior. Section 5 extends the calculations to the question of selective adsorption from a binary mixture. Lastly, section 6 provides a summary of our results and conclusions. 2.
The Model
The model considered in this paper is that of a l-D fluid confined to a l-D pore (Le. a line segment). In particular, the l-D fluid is a single component or binary system consisting of particles interacting through square-well, nearest-neighbor potentials. Similarly, the walls at the ends of the pore interact with the fluid through squarewell, nearest-neighbor potentials. The two wall potentials (i.e. adsorbateadsorbent potentials) are symmetric. This (32) Prigogine, I. The Molecular Theory of Solutions; Intarscience Publiihere: New York, 1957. (33) Longuet-Higgine, H. C. Mol. Phys. 1958, I, 83. (34) Percue, J. K.J. Stat. Phys. 1976, 16, 605. (35) Percue, J. K. J. Stat. Phys. 1982, 28,67. (36) Percus, J. K.Liquid State of Matter; Montroll, E. W., Lebowitz, J. L., Eds.; North-Holland Ameterdam, 1982; p 31. (37) Robledo, A.; Rowlinson, J. S. Mol. Phys. 1986,58, 711. (38) Vanderlick, K.;Davis, H. T.; Percue, J. K. J. Chem. Phys. 1989, 91, 7136. (39) Kierlik, E.; Rosinberg, M. Phys. Rev. A 1990, 42, 3382.
Physical Adsorption in Heterogeneous Porous Materials
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Langmuir, Vol. 9, No. 2,1993 563
I
N-1
'.
l)e(N-l-n)@f(l - e8f)"[e28'v(1- nX)NB(l- nX)
n=O
+
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2e0~1 eBfw)(l- nA - &lNe(i - nX (1 - eB'32(1- nX - 2&,JNt9(l- nX - 2&)1 (4)
where B(x) is the Heaviside function. It is reasonably straightforward to extend this derivation to the more complex case of binary adsorption. These results are explicitly derived in ref 26. With the configuration integrals determined, any of the standard partition functions may be immediately computed. For adsorption the most appropriate partition function to work with is the grand partition function
'.-_________________---. Figure 1. Schematicof the 1-D model of adsorption in a pore. is perhaps the simplest model of adsorption in a pore which includes the effects of both confinement and attractive interactions. The adsorbate-pore system is characterized by severalparameters: L the pore length, u the adsorbate particle size, e the adsorbate-adsorbate potential well depth, e, the adsorbate-wall potential well depth, X the adsorbate-adsorbate potential range, and A, the adsorbate-wall potential range. Figure 1 schematicallydepicta this model. Distances are measured from the surface for walls and from the center of particles. The key to solving the statistical thermodynamics of these 1-D systems lies with the analytic derivation of the N-particle configuration integrals. Below we sketch the derivation of the configuration integrals for a single componentin a pore exhibitinga symmetricpotential field. Reference 26 presents a more detailed derivation. In general the N-particle configuration integrals, Z(N,V,T), are defined as
where V is the system volume, T is the absolute temperature, U is the total potential energy, and B = l l k T where k is the Boltzmann constant. The integral in eq 1is over the coordinates of all N particles in the volume V. Rscognizing that the pore potential acts only on the end particlee for a confiied 1-D nearest-neighbor interaction system,the configuration integral of N particles can be written in general ae
where L is the pore length, g(x) is the adsorbate-adsorbent (pore)potential field, and u(z)is the adsorbate-adsorbate potential. The restriction to nearestneighbor square-well interactions allows 'fop the analytic evaluation of eq 2. Specifically, the configuration integrals become convolutions which may be written as
where 1 L - Nu and yi = xi - (i - 1/2)u. Taking the Laplace transformation of eq 3 yields a product which is readily inverted after performing a binomial expansion. The final result for the configuration integral is
where p is the chemical potential and f is the activity (f = q exp(8p) where q is the molecular partition function). The adsorption is obtained from LII
Given eqs 4 and 6, evaluation of adsorption in the 1-D square-well pore model is readily accomplished. 3. Model Pore Size Distribution The general model of a microporous adsorbent we explore in this paper is that of an ensemble of 1-D pores each of which conformsto the model description in section 2 and overall with a specified continuous pore size distribution. We choose to define the adsorbent pore size distribution by use of a Pearson Type-I11distribution.40 This distribution is one of the simplest qualitatively reasonable forms. Furthermore, the distribution reduces to physically important limiting cases, as will be discussed below. The general form of the distribution function for our purposes is
where L is cavity length, r(x)is the gamma function, (L) is the mean of the distribution, and Y is a dimensionless measure of the distribution's variance. The product f ( L )dL is the fraction of a system's total length which is composed of cavities of lengths between L and L + dL. We note that if Y = 0, eq 7 reduces to a delta function, i.e. a monodisperse distribution of cavity sizes. Furthermore if Y = 1, eq 7 becomes identical to the well-known gamma distribution. The gamma distribution has special significance since it gives the distribution of cavity sizes in an equilibrium hard rod (EHR) system. This can be readily proved, as is outlined in the Appendix, baaed on work by Torquato et al." Figure 2 shows the distribution functions for severalvalues of Y using eq 7. The narrower distributions ( Y = 0.1, 0.25, and 0.5) portrayed in Figure (40) Fogiel, M., Ed. Handbook of Mathematical, Scientific, and Engineering Formula, Tablea,Functions,Graphs, Transforms; b e a r c h and Education Amxiation, 1984. (41) Torquato, S.; Lu, B.; Rubinatein, J. Phya. Reu. A 1990,41,2069.
Kaminsky and Momon
664 Langmuir, Vol. 9, No. 2, 1993 3.5
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2 are qualitatively similar to those found in certain real adsorbents such as silica gels, alumina gels, and carbon molecular sievese2 The broadest distribution, Y = 1.0, although not particularly realistic, can be considered a crude approximation to the pore distributions found in some activated carbon systems.2 The monodisperse case (v = 0) of course corresponds to uniform pore sizes such as those in zeolites.
Figure 3. Effect of pore size heterogeneity on pure component isotherms: solid line, v = 0; dotted line, v = 0.1; dashed line, v
4. Single-Component Adsorption
X = A, = u/2. (a) (Llo)= 2, d k T = 1; (b) (Llu)= 2, c,/kT = 4; (c) (L/u) = 6 , c,/kT 1; (d) (Llu)= 6, c,/kT = 4.
The starting point for our investigation of one-dimensional heterogeneous adsorption phenomena is a survey of adsorption behavior in single-component l-Dsystems. Using the analytic pore model and distribution function described in the previous sections, we have studied adsorption as a function of pressure and cavity (pore) size distribution. The degree of system heterogeneity directly follows from the imposed distribution of cavity sizes. We began by generating a series of isotherms for adsorption in a variety of l - D adsorbents to highlight the effects of the pore polydispersity and adsorbent potential field strength. Since in l - D there is no pore connectivity, adsorption isotherms for polydisperse pore size systems can be calculated by integrating the behavior for isolated pores over the pore size distribution. We have
where p1 is the adsorbate density in a single isolated pore (calculated via eqs 4 and 6). Note that p is the average number of adsorbent particles per unit length of cavity space. In practice the integration in eq 8 is necessarily done via numerical techniques. The calculated isotherms (cavity adsorbate density, pa, versus reduced pressure, PalkT) resulting from our adsorbent model are depicted in Figure 3. The figures reflect various combinations of adsorbent characteristics, specifically average pore size, (Lla) = 2 or 6, and adsorbatewall potential strength, cw/kT= 1or 4. The choice of the values 2 and 6 for (Lla) was made since in three-dimensional pores, two molecular diameters across and six molecular diameters across are reasonable values of small and large micropores, respectively. All the figures include isotherms for pore size distributions with Y = 0,0.1,0.25, and 1. Furthermore all the systems have tlkT = 1and X = A, = al2. From the isotherms in Figure 3 it is clear that polydispereity of pore size has a very significant effect for the small average pore size case ( ( L l a )= 2). On the other hand, for the large average pore size case ((Lla) = 6)the
0.1
00 0.0
0.2
0.4
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(d) = 0.25; dot-daehed line, v = 1.0. All systems have t/kT 5 1 and
effect of pore size polydispersity is minimal except for the broadest distribution (v = 1). Furthermore we note that the changes induced in the isotherms due to increasing the pore size polydispersity, Y, are not systematic. In the (Lla) = 2 cases, the Y = 0.1 and 0.25 isotherms show enhanced adsorption over the monodisperse cases (v = 0) whereas the isotherms corresponding to Y = 1 show only weakly enhanced or decreased adsorption with respect to the monodisperse cases. How do we reconcile this observed behavior? To do so we must consider the adsorption behavior in a single pore. The maximum adsorption in a cavity is not pa = 1,the maximum packing in the bulk phase. Rather, the maximum adsorption is less than unity since only an integer number of particles may be in a cavity. Furthermore if a cavity is exactly an integer multiple i of a, a maximum of only i - 1 particles can be in the cavity since the configuration integral associated with i particles is identically zero. Hence we see that the monodisperse isotherms in Figure 3 asymptotically approach 1/2 for the (Lla) = 2 case and 5/6 for the (Lla) = 6 case. We can readily calculate the maximum adsorbab density, p-, in a polydisperse system by noting that the maximum number of particles able to fit in apore is int(Lla),where the int(x) function gives the greatest integer less than or equal to x . Hence
where f(L) is the normalized pore size distribution. Applying this equation usingthe Pearson distribution (eq 71, we find that for Lla = 2, p-0 = 0.735,0.727,and 0.581 for Y = 0.1, 0.25, and 1.0, respectively, for the Lla = 6, pmua = 0.916,0.914, and 0.842 for v = O.f,0.26, and 1.0, respectively. The values for p- primarily explain the limiting behavior of the isotherms depicted in Figure 3 for high pressures.
Langmuir, Vol. 9, No. 2, 1993 566
Physical Adsorption in Heterogeneous Porous Materials
To completethe understandingof the isothermswe must include the fact that the adsorbatewall potential plays an important role in determining the degree of adsorption as a function of pressure. This is immediately clear by comparing the r,lkT = 1 and cw/kT = 4 monodisperse cases in Figure 3. The enhanced adsorption as r,lkT increases is of course directly related to the increasing Henry’s law constant. Henry’s constant, K H ,is defined as the value of the 1-particle configurationintegraldivided by the system volume. Hence via eq 1for a 1-D pore
For our square-well system the above integral becomes
1 0 ,
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1 0 ,
KH = 0 8-I
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