J. Phys. Chem. 1995,99, 14903-14906
14903
Sorption Capacities of Three-Dimensional Crystalline Microporous Materials with Defects Michael W. Deem**+and John M. Newsam BIOSYM Technologies Inc., 9685 Scranton Road, Sun Diego, Califomia 92121-3752 Received: June 9, 1 9 9 9
The effective sorption capacity of crystalline microporous materials with obstructed internal pore networks is calculated. We present Monte Carlo calculations of the sorption capacity of one-, two-, and three-dimensional networks with defects typical of zeolitic materials. Blockages in the windows and main pores lead to a reduction in sorption capacity. Interconnections between the main pores substantially moderate the reduction relative to that of isolated one-dimensional channels. A significant fraction of the micropore capacity, however, is available only below a percolation threshold in blocking probability. Explicit results for various crystallite sizes are presented.
Introduction
a
This paper discusses the pore volume that is available in crystalline microporous materials with random pore-blocking obstacles. Zeolites and related crystalline microporous solids are widely used in industrial sorptive, ion-exchange, and catalytic proc e ~ s e s . ' - ~The pore structures of these materials may be onedimensional (1-D), as in cancrinite, zeolite L, or AlPO4-5;twodimensional (2-D), as in decadodecasil 3R, TMA-E(AB), or heulandite; or three-dimensional (3-D), as in zeolites ZSM-5, chabazite, A, X, Y, and betae6q7 Although the specific pore architectures in real materials are quite diverse, in most systems they can be described in terms of either simple cubic or diamond pore connectivities. The cubic connectivity can be viewed as a square array of major channels along the c-axis with perpendicular interconnections along the a- and b-axes. This common pore topology is seen, for example, in zeolites A, ZSM5 , and beta. In the faujasite structure adopted by zeolites X and Y, the major cages occupy a diamond lattice with the tetrahedral nearest-neighborcages connected by windows. The degree of distinction between pores and windows necessarily depends on the size of the sorbent that is traversing the pore system. The structures of zeolite materials are rarely crystallographically perfect with, for example, point, line, and plane defects The effect of pore blockages on the occurring freq~ently.~ sorption capacity of a material with one-dimensional channels subject to planar defects has been analyzed analytically.8 Specific cases of blockages in various simple pore systems have been examined by Monte C a r l ~ ' ~ .and ' ' ~ percolation theory.'Ib We here consider point blockages in the major channels and windows of 2-D and 3-D pore systems. Such blockages can be caused by, for example, framework defects, coke, metal catalyst sintering, or siting of nonframework cations in aperture positions. In the present paper we consider 2-D square, 3-D cubic, and 3-D diamond pore connectivities (Figure 1). In the square and cubic case, the main channel is defined to be along the c-axis, with the interconnecting windows along the a-axis in the 2-D case and along both the a- and b-axes in the 3-D case. For the 3-D diamond connectivity, we use the h x x 2 lattice shown in Figure IC, with the main cages connected by smaller
* To whom correspondence should be addressed. Present address: Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138. Abstract published in Advance ACS Abstracts, September 1, 1995. @
Figure 1. Unit cells of the pore networks, with p denoting the probability of an accessible pore and q denoting the probability of an accessible window. The views are along the major crystallographic axis. (left) The 2-D square network of cylinders. (middle) The 3-D cubic network of cylinders. (right) The 3-D diamond network of spheres, with the integer n indicating that the z component of the center of the cage is given by z = nc/4.
windows. Without interconnecting windows, blockages in the main channels severely limit the pore volume accessible to sorbents at the surface of a crystallite.8 The extra connectivity afforded by the windows allows access into a blocked channel from adjacent unblocked channels and thereby increases the available volume. Above a certain blocking probability of the main channels, however, the accessible pore volume is limited to a shell near the surface of the crystallite. Below this limiting probability and for a high enough window density, the available pore volume percolates throughout the crystallite. In this case a finite fraction of the entire volume of the material is available even in the limit of a large crystal. Section 2 discusses the definition of the defects in the pore architectures and the simulation strategy used. Section 3 presents the results of the Monte Carlo calculations. Section 4 discusses the results. We conclude in section 5.
Simulation Strategy We define two distinct probabilities that characterize the statistics of the pore configurations. The main channels or cages are blocked with a probability 1 - p . The windows are blocked with a probability 1 - q. The probability of a free pore, p , is related in a real zeolite material to factors such as the fault density, level of coking, or metal cluster agglomeration. The probability of an open window or interconnecting channel segment, q, may depend also on the number and siting of nonframework cations. We assume sufficient separation between successive pores and windows so that there are no energetic interactions between them and consequently no correlations between the blocked sites. A square lattice configuration is used for the window connections (Figure 1). In the 2-D square case, there are n main channels, and each main
0022-3654/95/2099-14903$09.00/0 0 1995 American Chemical Society
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channel within the interior of the crystallite has windows to the two adjacent main channels. In the 3-D cubic case, there are n2 main channels, and each main channel within the interior of the crystallite now has windows to the four nearest-neighbor main channels. In the 3-D diamond case, there are n3/4 main cages, each connected to four nearest-neighbor main cages. The available pore volume is defined as the integrated volume of the main channels or pores that is accessible from outside the crystallite. This computation essentially assumes orthogonal crystallite shapes and that the edges of the crystallites terminate in windows where possible. In order to analyze the role of defects in these crystalline microporous materials, we generate typical configurations within an ensemble and calculate the sorption capacity for each configuration. The relevant experimental quantity is the average of the computed capacity over the entire ensemble. We use the fast and robust linear feedback shift register method to generate the pseudorandom numbers required to create pore configurations. A square lattice is employed computationally to define the pore configuration. It is possible to specify on such a lattice both the main channel or cage and window occupancies. In the 2-D square case, we use a (2n - 1) x n lattice; in the 3-D cubic case, we use a (2n - 1) x (2n - 1) x n lattice; and in the 3-D diamond case, we use a n x n x n lattice. In the square and cubic cases, the crystallites are composed of n unit cells in each direction. In the diamond case, the crystallites are composed of n/2 unit cells along the a-axis and b-axis and n/4 unit cells along the c-axis. Once a lattice of occupancies is created by introducing random pore or cage and window blockages, the efficient Kopelman algorithm is used to identify and quantify the connected cluster^.'^ While the windows are not part of the main pore structure, they do define the connections between the main pore unit cells. The algorithm was modified to account for the effect of the windows. The algorithm was also modified to perform garbage collection on the cluster index array to reduce the memory required. The number of main channels or cages accessible from the external surface of the crystallite is generated from the results of this algorithm. ReSUltS
Figure 2 presents the Monte Carlo results for the 2-D square pore architecture. All results are averages over 100 independent runs. Figure 2a indicates the available capacity when the probability of free pores, p , is high. The number of main pores or channel segments accessible, N , is shown as a function of window probability, q, for each value of p . Moreover, results are shown for various model crystallite sizes. For heulandite the corresponding crystallite sizes are some 150 8, (n = 20), 600 8, (n = 80), and 0.2 p m (n = 320). Figure 2b displays the corresponding plots of the available capacity when the probability of free pores is low. The standard deviation of these results is rather small on the displayed scale. Figures 3 and 4 present the available capacity in the 3-D cubic and 3-D diamond pore architectures, respectively, using the same format as in Figure 2. In these 3-D cases, the results for n = 20 are averaged over 50 runs, the results for n = 40 are averaged over 20 runs, the results for n = 80 are averaged over 10 runs, and the results for n = 160 are averaged over two runs. The values of n chosen in these figures correspond to crystallite sizes for zeolite A of roughly some 250 8, to 0.2 pm.
Deem and Newsam
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0.0 0.2 0.4 0.6 0.8 1.0 9 Figure 2. Loading capacity of a 2-D square n x n lattice. (a, top) Capacity f o r p = 0.9,0.8,0.7, and 0.6. Curves for n = 20 (+), 40 (A), 80 (O), 160 (A), and 320 (0)are shown within each family of p . The symbols are shown only for p = 0.9. The capacity increases with increasing p . (b, bottom) Capacity as in (a) for p = 0.1, 0.2, 0.3, 0.4, and 0.5. The symbols are shown only for p = 0.5.
Discussion The results in Figures 2-4 apply directly to zeolites such as TMA-E(AB), chabazite, A, or Y in which the interconnecting windows are simple apertures, without significant volume. For some other zeolitic materials, such as ZSM-5 and beta, the interconnects are actually channel segments with substantial pore volume. In these cases, the accessible volume computation needs to include the volume of these channels. Figures 2-4 present only the volume associated with the accessible main channels or cages. The volume associated with the windows can be estimated, however, in a mean-field approximation that ignores correlations between the accessible main cages or channel segments. We find that the number of windows is given by N, = Nzq(1 - 8/2). Here N is the number of main cages or channel segments accessible, z is the number of nearestneighbor channels or cages, and 8 is the fraction of main cages or channel segments that are occupied. Thus z = 2, 8 = N/n2 for the 2-D square case; z = 4, 8 = N/n3 for the 3-D cubic case; and z = 4,8 = 4N/n3 for the 3-D diamond case. The most striking feature of the present results is the increased pore volume accessible as a result of the windows, Le., N increases with 4. With no windows, each main channel is independent, and the capacity of the one-dimensional channel as a function of blocking probability is known analytically.* The square and cubic Monte Carlo results agree with this known result. For high blocking probabilities of the main channel or
Sorption Capacities of Crystalline Microporous Materials
J. Phys. Chem., Vol. 99, No. 40, 1995 14905
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Figure 3. Loading capacity of a 3-D cubic n x n x n lattice. (a, top) Capacity for p = 0.9, 0.8, 0.7, 0.6, 0.5, and 0.4. Curves for n = 20 (+), 40 (A),80 (0).and 160 (A) are shown within each family of p . The symbols are shown only for p = 0.9. The capacity increases with increasing p . (b bottom) Capacity as in (a) for p = 0.1, 0.2, and 0.3. The symbols are shown only for p = 0.3.
pores (Figures 2b, 3b, y d 4b), the windows increase the available volume in a quantitative way only. In these cases, the available pore volume is proportional to the exposed surface area of the crystal in the large n limit. This behavior is the reason for the scaling of the accessible volume in these figures. For lower blocking probabilities of the main channel or cages, the windows can have a dramatic, qualitative effect on the accessible pore volume (Figures 2a, 3a, and 4a). In these cases, for a high enough window probability, the accessible volume can percolate throughout the entire crystal. This percolation transition is clearly evident in Figures 2a, 3a, and 4a as the point at which the capacity curves rise from negligible levels. In the limit of large n,the fraction of the pore volume available is zero below qC@)and finite above qc(p). This behavior is the reason for the scaling in these figures. Below qc(p), in fact, the available pore volume is proportional to the exposed surface area. Exactly at qc(p),the available pore volume is fractal with a fractal dimension d - 1 < df < d. The percolation threshold q&) can be reliably identified from a finite-scaling analysis. We define q(p,n) as the point where N(q,p,n) reaches one-half N( l,p,n). Extrapolating the value of q(p,n) so determined as a function of l/n to n leads to the asymptotic value of &). Figure 5 shows this asymptotic critical window probability, qc(p),for the relevant main-channelblocking probabilities for each of the architectures considered. Two points on the square and cubic curves are known. In the limit of no windows (q = 0), percolation occurs only for p =
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Figure 4. Loading capacity of a 3-D diamond n12 x n12 x n14 lattice. (a, top) Capacity for p = 0.9,0.8,0.7,0.6, and 0.5. Curves for n = 20 (+), 40 (A),80 (0).and 160 (A) are shown within each family of p . The symbols are shown only forp = 0.9. The capacity increases with increasing p . (b, bottom) Capacity as in (a) for p = 0.1, 0.2,0.3, and 0.4. The symbols are shown only for p = 0.4.
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o’2 0.0 ’ 0.0 0.2 0.4 0.6 0.8 1.0 P Figure 5. Critical probability of free windows, qc, above which percolation occurs as a function of the probability of a free pore, p . The function q@,20) is shown to illustrate the effect for small crystallites. Results for the 2-D square (O), 3-D cubic (M), and 3-D diamond lattices (0) are shown.
1. Thus, qc(l) = 0. Alternatively, in the limit of perfect windows (q = l), an isotropic square lattice is recovered, and percolation occurs at the known value for “site percolation”: p = 0.592 75 in 2-D andp = 0.31 17 in 3-D.I4 Thus, qc(0.592 75) = 1 in 2-D and qc(0.3117) = 1 in 3-D. The site percolation argument still holds for the 3-D diamond case, where p =
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14906 J. Phys. Chem., Vol. 99, No. 40, 1995
0.4299, so qc(0.4299) = 1. The case p = 1 is different, as windows are required to connect the main cages in this topology. This is a “bond percolation” problem, for which q = 0.3886, and so qc(l) = 0.3886. Our numerical results interpolate between these known end points in each of the architectures considered. The existence of such percolation transitions in zeolites was documented experimentally as early as 1956 by Breck et d . ’ O This archetypical system was zeolite A in which the Ca2+/Na+ ratio was varied. The Na+ ions block the windows in this cubic architecture. Substitutional Ca2+ions beyond the first displace the Na+ ions from these window positions and thereby open the windows. Using the value of pc = 0.2492 for bond per~olation,’~ one can calculate that when 29.13% of the Na+ ions have been replaced, a percolation transition should occur. Indeed, a transition is observed at 29% exchange.I0 A sudden transition also occurs in the intracrystalline diffusivity. This transition occurs at the same value of the Ca2+/Na+ ratio as does the sorption transition and has also been observed in numerous experiments.I0 There are several experimental conditions that will lead to a rounding out of the sharp percolation transition. The Monte Carlo results in Figures 2-4 indicate that finite crystallite size is one such factor. This factor can be quite significant for the rather small crystallite sizes used industrially. Observed blockage probabilities of the 1-D channels of PtKL zeolites have, in fact, been used to predict an optimal size for the catalyst ~rystal1ites.l~Another factor is mobility of the “obstacles”. If, for example, nonframework cations that block the windows were mobile due to, say, high temperatures, the sorbents would pass with some freedom between main cages or channel segments. Such behavior would lead to a rounding off of the bond percolation transition.
Conclusions Interconnecting windows or channel segments increase the number of main channels or cages accessible in crystalline microporous materials. For high densities of blockages in the main channels, the available pore volume is proportional to the exposed surface in the limit of a large crystal, no matter what the window probability. The windows increase the accessible volume only quantitatively. When there are fewer defect in
the main pores, the available pore volume can be proportional either to the exposed surface area or to the entire crystal volume in the limit of a large crystal. The percolation transition between these two limits is governed by the defect density in the windows, with low densities leading to an accessible volume that is proportional to the crystal volume. This sharp transition is rounded off in finite-sized crystallites. The present Monte Carlo approach can be applied to any crystalline, microporous material to calculate quantitative sorption capacities, and extensions to cases of correlated disorder are also possible. The results presented here, however, cover the most common pore topologies.
References and Notes (1) Breck, D. W. Zeolite Molecular Sieves: Structure, Chemistry, and Use; Wiley: London, 1974 (reprinted R. E. Krieger: Malabat, 1984). (2) Barrer, R. M. Zeolites and Clay Minerals as Sorbents and Molecular Sieves; Academic Press: London, 1978. (3) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; Wiley: New York, 1984. (4) Zeolite Chemistry and Catalysis: Rabo, J. A., Ed.; ACS Monograph 171; American Chemical Society: Washington, DC, 1976. ( 5 ) Zeolites and Related Microporous Materials: State of the Art 1994; Weithamp, J., Karge, H. G., Heifer, H., Holderich, W., Eds.; Elsevier: Amsterdam, 1994. (6) Meir, W. M.; Olson, D. H. Atlas ofzeolite Structure Types, 3rd ed.; Butterworth-Heinemann: London, 1993. (7) Newsam, J. M.; Treacy, M. M. J. Zeolites 1993, 13, 183. (8) Newsam, J. M.; Deem, M. W. J. Phys. Chem. 1995, 99, 8379. (9) Newsam, J. M.; Treacy, M. M. J.; Vaughm, D. E. W.; Strohmaier, K. G.; Melchior, M. T. In Synthesis of Microporous Materials Volume I. Molecular Sieves; Occelli, M. L., Robson, H. E., Eds.; Van Nostrand Reinhold: New York, 1992; pp 454-472. (10) Kirger, J.; Ruthven; D. M. Difision in Zeolites and orher Microporous Solids; Wiley: New York, 1992; Section 12.7. (1 1) (a) Frisch, H. L.; Hammersley, J. M.; Welsh, D. J. Phys. Rev. 1962, 126,949. (b) Reyes, S. C.; Scriven, L. E. Ind. Eng. Chem. Res. 1991,30, 71. (12) Sedgewick, R. Algorithms, 2nd ed.; Addison-Wesley: New York, 1986; Chapter 3. (13) Hoshen, J.; Kopelman, R. Phys. Rev. E 1976, 14, 3438. (14) Creswick, R. J.; Farach, H. A,; Poole, C. P., Jr. Zntroduction to Renormalization Group Methods in Physics; Wiley: New York, 1992; Section 9.8. (15) McVicker, G. B.; Kao, J. L.; Ziemiak, J. J.; Gates, W. E.; Robbins, J. L.; Treacy, M. M. J.; Rice, S. B.; Vanderspurt, T. H.; Cross, V. R.; Ghosh, A. K. J . Catal. 1993, 139, 48.
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