7665
J. Phys. Chem. 1993, 97, 7665-7671
Computer Simulation Studies of Ordered Structures Formed by Rare Gases Sorbed in Zeolite Rho Anthony Loriso, Mary J. Bojan, Alexei Vernov, and W. A. Steele’ Department of Chemistry, 152 Davey Lab, Pennsylvania State University, University Park, Pennsylvania 16802 Received: January 13, 1993; In Final Form: April 19, I993
Computer simulations of the energies and structures of krypton and argon in zeolite rho at high sorbate loadings are reported here. The results, taken together with previous simulations of xenon, show that all these gas atoms sorbed in the zeolite cages of diameter 11 %L exhibit ordered structures at high loadings. The ordering does not appear to be greatly affected by increases in temperature and thus is most likely due to packing considerations. Several types of order were observed, depending upon the size of the sorbate atom and the cage loading. Basically, all were of cubic symmetry, reflecting the symmetry of the cages. Omitting the central sorbate atom that was usually present, the structures can be described as simple cube (8 Xe atoms), distorted face-centeredcubic (14 Ar or Kr atoms), and a more complex but still cubic structure (20 Ar atoms). It is noted that cages of this size are found in many zeolite crystals.
-
6 ring
1. Introduction
Although zeolites have been the subject of experimental and theoretical studies for many decades, the use of computer simulation as a technique to produce thermodynamic datal4 (isotherms and heats) and dynamical constants34 (self-diffusion) for molecules physisorbed in zeolitic cavity systems is a relatively recent development. Both nonionic1.3ss and ionic29436 forms of zeolite crystals have been investigated, with the effect of strong electrostatic fields upon the properties of polar sorbates being a topic of particular interest. A second important area for study arises from the molecular sieve properties of these materials, which, as is well-known, depend upon the relative sizes of the sorbate molecules and dimensions of the cage openings; a secondary question here is the cage size itself and its relation to the maximum number of molecules that can be sorbed in the zeolite crystal. A recent study of the thermodynamic properties of xenon in nonionic zeolite rho has yielded simulationdata in good agreement with experiment.’ Zeolite rhois composedoftwointerpenetrating but unconnected cubic lattices of cages; the cages in each sublattice are connected by pairs of nearly circular rings of eight oxides. This cagewindow structure is illustrated in Figure 1. These windows are in the faces of the cubic environment of a cage, but the cages themselves can be considered as roughly spherical with a diameter of 1 1 A. In the previous study, a Xe-solid potential energy adjusted to fit the low coverage experimental data was found to give an interaction energy with a very deep minimum at the center of a window channel, amounting to 6.5 kcal/mol; in contrast, the interactions for a Xe in the center and on the interior wall of the cage were calculated to be slightly more than 1.4and 4.0kcal/mol, respectively. Consequently,xenon is sorbed first in the channels and is quite well localized there, at least for temperatures less than 300 K. Furthermore, it was found that the cages themselves are large enough to hold nine xenon atoms, in a model where the zeolite structure is assumed to be rigid. (This gives a maximum loading of 24 atoms per unit cell = 18 in two cages plus 6 in the window channels, in agreement with experiment.) It was also observed that, at the maximum loading, the xenons tended to form a well-ordered structure, executing primarily vibrational motions about their equilibrium positions and that this structure was quite insensitive to increases in temperature, with no sign of *melting”or significant disordering up to the highest temperatures considered. In the present work, we report a detailed study of the ordered structures that occur at high sorbate loadings for krypton and
-
K
m 8-ring
Figure 1. Diagram showing the cagewindow structureof nonionic zeolite rho. Oxides are located at the midpoints of the light lines and silicons, at their intersections. This structure is replicated by translation of the cube to form a lattice. The second nonconnectinglattice has cage centers at the corners of the cubes that define the lattice. The heavy linesindicate a cube diagonal and a face bisector.
argon in nonionic zeolite rho. The structures were first observed in the computer-generated plots of the sorbate atom positions as a function of time (i.e., the trajectories followed). Sincethe atomic positions during the course of the simulations were saved, they could be utilized for quantitative calculations of the paircorrelation functions for the sorbate atoms within a cage. Atoms in the windows were excluded from the calculation, so that one could evaluate the approximate number of neighbors by integrating under the peaks observed in this function. It will be shown that the simulations yield ordered structures for all the rare gases considered. The detailed nature and symmetry of the structures observed depend upon the size of the sorbate atom, and it is argued that these structures are produced by packing requirements for the sorbate atoms in a rigid case rather than their attractive interactions. In section 2, the details of the simulation study and the choice of gas-solid potential parameters are given; section 3 contains the results, and a brief discussion is given in section 4. 2. Simulations
In modeling the gas-solid interaction, a pure Si02 lattice was assumed, with oxide positions taken from the experimental diffraction measurements on zeolite rho.* Only the oxides were taken into account in a pairwise summation of gas-site Lennard Jones 12-6 energies. The gas-gas energies were also assumed to be given by a 12-6 function with parameters taken from the l i t e r a t ~ r e .The ~ new Ar and Kr results will be compared with the previous Xe-zeolite simulations. Thus, the Xe-0 potential
0022-3654/93/2097-7665$04.00/0 0 1993 American Chemical Society
Loriso et al.
7666 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993
a
-190
Fil
I
-350 3.75
0.00
7.50
R(A)
b I
-300I
5.00
0.00
Figure 2. Sorbate-zeolite interaction energies for the three rare gases as a function of atomic position within the cage-window structure. The gas atoms are moved outward from the cage center along a line intersecting a window (a) and a 6-ring in the cage wall (b); these lines are illustrated in Figure 1. The vertical line in (a) marks the distance of the 6-ring oxygens from the center of the cage (Rap).
TABLE I: Parameters of the Interaction Potentials
(4
rm
(A)
Ar-Ar Kr-Kr Xe-Xe Ar-O
4.10 3.03
4.0 4.6 3.4
Kr-0 Xe-0
3.15 3.30
3.5 3.1
3.40 3.60
3.8
elk (K) 122 170 221 138 170
151
is unchanged from that of the previous paper, since the Xezeolite interaction obtained there yielded Henry's law constants and zero-coverage heats of adsorption in agreement with experiment.' In the Ar and Kr cases, no data were available; the constants used were taken from papers by Kiselev et a1.10 Well depth and size parametes for all potentials are listed in Table I. Plots of the gas-solid energy for the three systems as a function of the gas atom position within the zeolite structure are shown in Figure 2. Here the potentials are calculated as a gas atom moves along the lines indicated in Figure 1. Deep minima are observed for atoms in the centers of the window channels (part
a); shallower minima are observed for atoms on the cage walls (Le., above the center of a 6-ring, in part b); at the cage centers, the interactions, while still negative, are rather weak. Finally, note that the cage radius Lgc = 5.5 A is essentially equal to the distance of the 6-ring oxygens from the cage center. The simulations were isokinetic molecular dynamics based on a predictor-corrector algorithm." The data-gathering parts of the simulation runs had durations slightly in excess of 60 ps for Ar and Kr (and about 34 ps for Xe). A time-step length of 3.2 (Ar),3.0 (Kr), and 1.7fs (Xe) was taken and equilibrationperiods of at least 20 ps were used, with the initial configuration taken to be either the final configuration for the previous run with the same number of sorbate atoms or from a configuration in which the sorbate atoms were placed approximately in their minimumenergy configuration in the zeolite cell. All runs were made with atoms distributed equally between the two (independent) zeolite sublattices; this gave two independent calculations of the behavior at high loadings, which was of some help in observing infrequent events such as window F? cage diffusionaljumps. In the Ar and Kr cases, a low temperature of 70 K was chosen along with a high temperature of 195 K. The Xe data were obtained at 300 K. In order to gain a quantitative picture of the atomic configurations within the zeolite cages, two distribution functions were evaluated: n,(R), the number of atoms in a spherical shell of volume 47r R2AR at a distance R from the cage center, and g(r), the correlation function for pairs of atoms on the cage walls that are separated by a distance r. In both cases, atoms in the window channels are excluded from the calculation. Formally, if r is the sorbate-sorbate separation distance,
where N(r f Arf2) is the number of atoms on the cage walls in a spherical shell extending from r - Arf2 to r + Arf2 when an atom is at the sphere center, Vis the volume of the shell (equal to 47r/3(3r2 + Ar2/4)Ar), and PO is the surface density in the cage equal to Nsurf/4~R2cagc, where Lgc is the cage radius and Nsurfisthe total number of atoms sorbed on the cage wall. It will be shown that g(r) is a strongly peaked function for the cage loadings used in this work. Consequently, one can evaluate number of nearest neighbors, etc., by integrating under successive peaks (after weighting by 47rr2Ar). Of course, the locations of the peaks correspond to the nearest-neighborseparation distances, etc. Similarly, evaluation of n,(R) and integration under the curve enables one to know whether there is an atom (or atoms) in the interior of the cage, as well as a value for the average number of atoms on the cage walls. Finally, plots of the trajectories for the sorbed atoms will be presented in which the atoms in the window channels have been excluded. (Of course, the use of periodic boundary conditions means that all cages in a given sublattice are identical.) 3. Results
Although there is in principle no limit to the number of atoms that could be inserted in a simulation for this system of cages, a practical limit is reached when the sorbed atoms become sufficiently tightly packed that their mutual interactions begin to be repulsive rather than attractive. For example, in the previous study of xenon in rho, it was found that the average (negative) xenon-xenon energy begins to decrease sharply as the loading approaches nine atoms per cage. Plots of this average energy as a function of the number of atoms in the cage are given in Figure 3 for all three systems considered here. This figure is given in units of reduced energy and reduced number that might be expected to give a common curve for all three sorbates. We will
Rare Gases Sorbed in Zeolite Rho
The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 7667
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4 1201
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Figure 3. Average values of the sorbate-sorbate interaction energy as a function of the density of atoms in the cage. Energies are reduced by the gas- as well depth tu; density N,,,/V,, (where Vase is taken to be 700 is reduced by multiplying by the size parameter us3.
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44 Figure 4. The cage-wall pair correlation function g(r) for 9 Xe atoms per cage at 300 K. The specific definition of n,(R) and g(r) is given in the text. The pair correlation function is evaluated only for those atoms in the central cage. show that the differences shown are in the most part due to packing differencesthat give rise to different numbers of nearest neighbors in the high-density ordered structures. Note that theincreasingly negative gas-gas interactions in the cages tend to go through maxima at roughly the same loading which is in the region of N,, qro3/VaScequal to 1.5-2.0. At a temperature of 300 K and a loading of nine atoms per cage, the Xe density plot in ref 7 shows a central atom surrounded by eight atoms on the cage wall as indicated by the single narrow peak at R = 4.0 A. Figure 2b shows that this position is a little further from the cell center than the distance (3.8 A) which corresponds to the minimum sorbate wall energy. This is consistent with the energy plot of Figure 3 that shows an increase in the X e X e energy (relative to the minimum) for this packing. The trajectory plot for this system shown in ref 7 indicates an ordered structure even at 300 K. Visually, the structure is that of a simple body-centered-cubic array. It should also be noted that the observed relative perfection of the structure of the atoms in one cell but not the other is a consequenceof the fact that two intercage diffusive displacements have occurred in the relatively poorly ordered case but none for the well-ordered structure. The simulated pair correlation for
0 3
4
5
44
6
7
a
Figure 5. Local particlenumber &R) and thecage-wall pair correlation function g(r) are shown for 13 Ar atoms per cage at 70 and at 195 K. The vertical line in (b) marks the location of r, for Ar. this system is shown in Figure 4; the number of neighbors and the relative positions of the peaks are listed in Table I1 and correspond well to those expected for a body-centered-cubic arrangement. When one excludes the central atom from the calculation, a simple cube results with three nearest neighbors at distance x , three second-nearest neighbors at d 2 x , and one neighbor across the cube diagonal at d 3 x . The numbers in the table have been rounded to thenearest integer and are approximate in any case, since the upper and lower limits of the integration over a peak are somewhat arbitrary, especially at high temperature. Consequently, the total number of calculated neighbors is not necessarily equal to the expected value of 7, for xenon. The maximum capacity of a cage for argon and krypton atoms is of course larger than that for xenon. The simulations for Ar were carried out at loadings of 13, 15, 20, and 22 atoms/cage. Results for n,(R) and g(r) at 13 atoms/cage are shown in Figure 5 for two temperatures. Although some local order is present, a fully ordered structure is not observed. When the loading is increased to 15 atoms/cage, the pair correlations for 70 K that are shown in Figure 6a begin to show structure for other than the nearest-neighbor shell. This structure is much more pronounced when 15 of the larger Kr atoms/cage are simulated as shown in Figure 6b. These curves, together with the trajectory plot of
Loris0 et al.
7668 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993
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Figure 8. Local density of atoms is shown for Ar at 195 K for three different loadings.
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Figure 6. Cage-wall pair correlation g(r) is shown for a loading of 15 atoms per cage at two temperatures for (a) Ar and (b) Kr.
Figure 9. Same as Figure 8 but for Kr at 70 K and a loading of 21 atoms/cage.
TABLE II: Number of Neighbors per Atom for Rare Cases in Zeolite Rho Excluding the Central Atom (Also Shown Are the Semration Distances of the Atoms) ~~
I
~
system Xe, 300 K 9 atoms/cage
Kr, 70 K 13 atoms/cage 15 atoms/cage Ar,70 K 15 atoms/cage Kr, 70 K 20 atoms/cage
21 atoms/cage I
I
Figure 7. Trajectory plots for 15 Kr/cage at 70 K. The central cage shows a highly ordered structure which is a distorted face-centered-cubic structure plus a central atom.
Figure 7, show clearly that Kr (and Ar as well, although the ordering is not as pronounced) forms a distorted face-centered cube with a central atom at a loading of 15 atoms/cage. The
~~
~~~~
~~
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shell 1
shell 2
3 at 4.7 A
2 at 7.3 A
6 at 3.8 A 6 at 3.7 A
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6 at 3.7 A
2 at 4.8 A
9 at 3.5 A 9 at 3.4 A
6 at 5.9 A 6 at 5.9 A
distortion is an outward displacement of the atoms in the face to givea structurein which the 14 outer atoms are almost spherically distributed. As a consequence, this structure does not have the three nearest neighbors at separation distance x plus three secondnearest neighbors at 4 2 x expected for fcc packing but six nearest neighbors (three distorted face plus three nearest corners) which correspond to the initial peak in g(r). Numbers of atoms under this peak for this and more distant shells at 70 K are shown in Table I1 for Ar and Kr at 13 and 15 atoms. These data indicate that the packing for 13 atoms are the same as for 15 at 70 K, but
Rare Gases Sorbed in Zeolite Rho
The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 7669
* a
7.5 n
7*5
-15 atoms/cage - - - - 20 atams/cage
-T=’IOK - - - T=185 K
.
~
~
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4.5
4)
6.5
(-a)
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Figure 11. Same as Figure 10 but for Kr at 70 K for three loadings: (-) 15 atoms/cage; 20 atoms per cage; (- -) 21 atoms per cage.
b -T=7OK
4.5
8.5
8.5
r(A) Figure 10. Cage-wall pair correlation function g(r) for Kr at both 70 and 195 K for loadings of 20 atoms/cage (a) and 21 atoms/cage (b). with missing atoms. At 195 K, there is insufficient structure in g(r) to allow one to split the curve into peaks associated with neighbor shells more distant than the nearest. Even though reasonably well-ordered structures are found at 15 atoms/cage for Ar and Kr, it is possible to go to considerably higher loadings in the simulations. Specifically, systems with 20 and 21 atoms/cage were also studied, with the idea that 21 might correspond to dodecahedral packing plus a central atom and 20, possibly, the same structure without the central atom. However, a glance at the plots of %(R)for argon at 195 Kin Figure 8 shows that a central atom is present at 20 atoms/cage, but in the case of 21 atoms/cage, one sees two atoms orbiting tightly about the center. Also of interest is the appearance of a shoulder in the peak for the wall atoms, indicativeof a deviation from the spherical cage shape that has been implicitly assumed so far. At 2 1atoms/ cage of krypton at 70 K, this leads to the pronounced splitting of the wall peak in %(I?), as shown in Figure 9. Also of interest is the fact that only one central atom is present in this case. A careful analysisof the results obtained for the highest loadings of krypton in these cases reveals that the effect of an increase in temperature from 70 to 195 K is minimal (Figure lo), the wall pair correlation functions for 20 and 21 atoms per unit cell at 70 K are very nearly the same (Figure 1l), and integration of these curves yields (rounded-off) values of nine nearest neighbors and
-
six second neighbors at both temperatures. In fact, the atomic arrangements for these high loadings are not dodecahedral as expected for a spherical cage but have a defect cubic structure determined primarily by the oxide ring structures of the zeolite. For example, there are three nearest neighbors at distance x and six second neighbors at 1 . 6 for ~ an atom in a dodecahedron (and six third neighbors). However, there is no sign of splitting of the first peak of the simulated g(r) into six and three neighbors. Furthermore, a detailed inspection of the trajectory plots for 21 atoms/cage of Kr at 70 K does not show this structure but rather one with basically cubic symmetry. Three views of the trajectories along directions parallel to each of the cube edges of the zeolite cell views are shown in Figure 12. The first feature to be noted is that the “wall” atoms are found at various distances from the cell center, which is of course reflected in the %(R) plot of Figure 9. We emphasize that the trajectories of the atoms sitting at the centers of the window channels are not shown in Figure 12-also, the straight lines that lie parallel to the edges of the squares and between the squares and the octagons indicate a side view of the 8-rings of oxides that make up the openings to the windows. In fact, there are atoms lined up with the centers of these 8-rings but still lying inside the cage walls that are formed in part by these rings-in Figure 9, these are the atoms that are most distant from the cell center. The next group of atoms lie adjacent to the centers of the 6-rings (which, as can be seen in Figure 1, are intersected by the diagonals of the cube surrounding each cage), In the projections of Figure 12, the krypton atoms associated with the 6-rings lie close to the centers of the octagon edges that are at 45O to the edges of the outer square. There are two such sites per edge, but not all sites are occupied-we estimate that onepercageisvacant for aloadingof 21 atoms/cage. Thenumber of atoms accounted for at this point is 6 (near the 8-ring centers) 8 - 1 = 7 (near the 6-ring centers) + one (central), which leaves five atoms unaccounted for. These are near the 4-ring centers and appear in the trajectory plots as the innermost atoms (excluding thecentral atom, of course). Since thereare 12 4-rings per cage (see Figure l), it is evident that roughly half of these are occupied. This might lead one to believe that even more Kr atoms per cage would be feasible, but the rapid change in Kr-Kr energy shown in Figure 3 indicates that significantly higher loadings might not be energetically stable. (Presumably, the “4-ring site” occupancy is correlated to minimize the Kr-Kr energy.) Note that this argument would lead to an %(R) for the wall atoms that is split into three peaks. The peaks for the simulations shown in Figure 9 are tentatively assigned as 4-ring, 6-ring, and (split) 8-ring, in order of increasing Kr atom distance
+
Loriso et al.
7670 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993
a
a
1
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-
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b
I
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Figure 12. Trajectory plots for Kr at 70 K with 21 atoms per cage. The
trajectories are viewed along the three directions parallel to the cube axes. from the cage center. The reason for the splitting of the 8-ring peak is unclear but is not inconsistent with the locations implied by the trajectories shown in Figure 12. 4. Discussion
These simulations show rather clearly that rare gases sorbed in zeolite rho form ordered structures at high loadings. The ordering is not particularly sensitive to temperature, and, for example, there is no sign of “melting” or of an “order-disorder” transition. Since the simulations are actually run at constant density, no thermal expansion (or volume change of melting) is allowed. Although one might guess that melting is prevented for this reason, simulations carried out at the varying cage loadings studied here do not show intracage disorder. (Of course, chaotic,
Figure 13. Same as Figure 12, but for 21 argon atoms per cage at 195 K.
liquidlike motions of xenon on the cage walls have been observed at 300 K in simulations at a considerably lower loading of 5 atoms/cage; see ref 7.) It is concludedthat the ordering observed in the systems reported here is a consequence of tight packing in a rigid zeolite cage-the role of the attractive part of the gas-gas interactions is to help stabilize these structures in the sense of lowering the vapor pressure relative to the hypothetical noninteractive sorbate. The observation of the structures simulated here of course requires that this vapor pressure be in a convenient range and that the zeolite crystal itself remain stable at the loading and temperature of interest. Whether these requirements are attainable is unknown at present, except for xenon where loadings of 9 atoms/cage have been observed at moderate pressures.12 The general conclusion reached in the analysis of the observed structures is that the predominant symmetry is cubic rather than spherical. The body-centered structure for the 9 xenon atoms/
The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 7671
Rare Gases Sorbed in Zeolite Rho cage occurs with a specific orientation which places an atom just inward from the centers of the &rings in the cage walls. The quasi-fccstructures for 15 krypton atoms/=@ are oriented with sixatomsin thehingopenings. Theremainingeight are located inward from the &rings (plus one atom at the cage center). The structures of krypton and argon (see Figure 13) at 21 atoms/ cage are more complex, but still basically cubic. In these cages, it appears that the 15 atoms/cage are essentially in the distorted fcC structure as at the lower loadings and the final five or six atoms are to be found near the centers* The energies of these simulated systems that are shown in Figure 3 indicate that this loading is very close to the maximum value for a stable system in thecase of krypton-for thesmaller argon atoms, slightly larger Of the loading possibly be in the laboratory. Because of the specific packings and ‘‘crystal’’ orientations of these sorbatezeolite systems, it seems likely that their diffraction signatures might be measurable. Such data have not yet been reported. Note that the cages in zeolite rho are hardly unique. A cage size of 11 A is characteristic of the cubooctahedron units that are the basic units in many zeolita-the structures of the various crystals differs primarily in the way in which these units are connected. Thus, zeolite A, X,sodalite, and numerous others possessvery similar cages and would be expected to exhibit ordered structures for small molecules sorbed at high loadings in these materials.
-
Acknow’edgment* This work was DMR-9022681.
by NSF Grant
References and Notes (1) June, R. J.; Bell, A. T.; Theodorou, D. N. J. Phys. Chem. 1990,94, 1508. Kaminsky, R. D.;Monson, P.A. J.Chem.Phys. 1991,95,2936.Pickett, S. D.; Nowak, A. K.; Cheetham, A. K.; Thomas, J. M. Mol. Simul. 19%9,2, 353, Snurr, R. Q.; June, R. L.; Bell, A, T.; Theodorou, D. N. ~ 0 1Simul. . 1991,8,73. Woods, G. B.; Panagiotopoulos, A. Z.; Rowlinson, J. S . Mol. Phys. 1988,63,49.Woods,G. B.; Rowlinson, J. S.J. Chem. Soc., Faraday Trans. 2 1989,85,765.
(2) Schrimpf,G.; Schlenkrich,M.; Brickman, J.; Bopp,P. J. Phys. Chem. 1992,96,7404.KaraVias, F.; Myers, A. L. Mol. S i t d 1% 8,23951. Van Tassel, P. R; Davis, H. T.; McCormick, A. V. Mol. Phys. 1991,73, 1107. Yashonath, S.;Thomas, J. M.; Nowak, A. K.; Cheetham, A. K. Nature 1988, 331,601. Soto, J. L.; Myers, A. L. Mol. Phys. 1981,42,971.Soto, J. L.; Fisher, P. W.; Glessner, A. J.; Myers, A. L. J. Chem. Sm., Faraday Trans. 1 1981,77, 157. Ramus, D. M.; Hall, C. K. AIChE J. 1991,37,769. (3) Santikary, P.; Yashonath, S . J. Chem. Soc., Faraday Trow. 1992, 88,1063. June, R. L.; Bell, A. T.; Theodorou, D. N. J . Phys. Chem. 1992, 96,1051.MacElroy, J. M. D.; Raghavan, K. J. Chem. Soc., Faraday Trans. 1991,87,1971. Yashonath, S. Chem. Phys. Lett. 1991,177,54. June, R. L.; Bell, A. T.; Theodorou, D. N. J . Phys. Chem. 1990,94,8232.MacEIroy, J. M. D.; Raghavan, K. J. Chem. Phys. 1990,93,2068.Den Ouden, C. J. J.; Smit, B.; Wielers, A. F. H.; Jackson, R. A.; Nowak, A. K. Mol. Simul. 1989,4,121. Nowak, A. K.; Cheetham, A. K.; Pickett, S . D. Mol. Simul. 1981,1, 67. (4) Leherte, L.; Andre, J.-M.; Vercauteren, D. P.; Derouane, E. G. J . Mol. Catal. 1989,54,426;J. Chem. Soc., Faraday Trans. 1991,87,1959. Yashonath, S.;Demontis, P.; Klein, M. L. J. Phys. Chem. 1991,95,5881; Chem. Phys. Lett. 1988,153,551;1.Phys. Chem. 1989,93,5016.Nowak, A. K.; Cheetham, A. K. Proceedings of the 7th International Zeolite Conference; Murakami, M., Iijima, A., Ward, J., Kodansha, Eds.; Elsevier: Amsterdam, 1986;p 475. ( 5 ) Demontis, P.;Fois, E. S.; Suffritti,G. B.; Quartieri, S . J . Phys. Chem. 1990, 94, 4329. Friturche, S.; Haberlandt, R.; Urger, J.; Heifer, H.; Heinzinger, K. Chem. PhP. Lett. 1992,198,283.June, R. L.; Bell, A. T.; Theodorou, D. N. J . Phys. Chem. 1991,95,8866.Pickett, S. D.; Nowak, A. K.; Thomas, J. M,; A. K. Zeolites 1989,9,123. (6) Leherte, L.; Lie, G. C.; Swamy, K. N.; Clementi, E.; Derouane, E. G.; Andre, J.-M. Chem. Phys. Lett. 1988,145,237. (7) Vernov, A. V.; Steele, W. A.; Abrams, L. J . Phys. Chem., preceding paper in this issue. (8) Meier, W.M.;Kokotailo,G.T.Z. Kristallogr.1965,121,211. Meier, W. M. Molecular Sieves; Societyof Chemical Industry: London, 1968;p 10. Parise, J. B.; Gier, T. E.; Corbin, D. R.; Cox, D. E. J. Phys. Chem. 1984,88, 1635. (9) Hirschfelder, J. 0.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954. (10) Kiselev, A. V.; Lopatkin, A. A.; Shulga, A. A. Zeolites 1985,5,261. Kiselev, A. V.; Du, P. Q. J. Chem. Soc., Faraday Trans. 2 1981,77, 1. (11) Evans, D. J.; Morriss, G. P. Chem. Phys. 1983,77,63.Hoover, W. G.Phys.Rm.A1985,31,1695.G a , C . W.NumrriccrllnitialValucProblemc in Ordinary Differential Equations; Prentice-Hall: Englewood cliffs, NJ, l9l1. (12) Tsaio, C.; Kauffman, J. S.;Corbin, D. R.; Abrams, L.; Carroll, Jr., E. E.; Dybowski, C. J. Phys. Chem. 1991,95,5586.
