J. Phys. Chem. 1996, 100, 967-973
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Diffusion and Percolation on Zeolite Sorption Lattices D. Keffer, Alon V. McCormick, and H. Ted Davis* Department of Chemical Engineering and Materials Science, UniVersity of Minnesota, 421 Washington AVenue SE, Minneapolis, Minnesota 55455 ReceiVed: July 21, 1995; In Final Form: October 16, 1995X
Percolation is investigated using Monte Carlo lattice dynamics on two unusual lattices of sorption sites suggested by recent zeolite simulations. The lattices are modifications of the simple cubic and the tetrahedral lattice. We investigate the percolative behavior caused by these modifications and compare this behavior with that of the parent lattice from published simulation results for simple cubic lattices and the approximate (but analytical) effective medium approximation (EMA) theory. The modifications of the parent lattices include (1) the addition of an octahedral sublattice at each node within a simple cubic superlattice and (2) partial blocking of three adjacent bonds by a single bond-blocker on a tetrahedral lattice. For the former, we examine both random and correlated placement of superlattice bond-blockers. We also examine random placement of sub-lattice bond-blockers. We also investigate the effect of activation energies on percolation. We ascertain to what degree regular lattice percolation behavior holds for the modified lattices, finding that modifications to regular lattices change the general transport behavior in easily predicted ways. This result suggests that regular lattice percolation theory can be relevant and applicable to a wide variety of zeolitic transport problems even when it is not obvious that the zeolite presents a regular lattice of sorption sites.
I. Introduction The cage-and-window structure of many zeolite nanopores (e.g., of zeolite A and zeolite Y) lends itself to the node-andbond lattice modeling approach of percolation theory. Typically the zeolite cages are represented as structureless nodes, and the zeolite windows are represented by bonds, which connect the nodes. Large, strongly adsorbed species or cations are represented as bond-blockers, which serve to reduce the diffusivity of small, more mobile adsorbates. In the percolation theory description of a diffusion process through such a lattice network having randomly distributed blockages of bonds, p denotes the fraction of bonds that are blocked and pc denotes the percolation threshold, which is the smallest such fraction that reduces the conductance of the lattice to zero. Percolation on three-dimensional simple cubic lattices has been investigated by theory and computer simulation.1,2 An approximate but analytical treatment of percolation, the effective medium approximation (EMA) theory, accurately predicts the simulated behavior for p < pc. Near the percolation threshold, though, simulated transport behavior deviates significantly from EMA predictions. EMA predicts that diffusivity linearly approaches zero, whereas three-dimensional simulations find that diffusivity near zero is proportional to (p - pc)1.6.1 We are interested in whether EMA predictions and previous simulations of the simple lattices match the results of the more complex lattices that are suggested by molecular simulations in zeolites. The sorption lattices differ from the related regular lattices (which we will refer to as the parent lattices) by (1) decoration with a sublattice and (2) exhibiting multiple blocking behavior. Analytical solutions to Bethe trees decorated with sublattices show that, in those cases, the presence of the sublattice does change the percolation threshold.3 The complex lattices we will describe below are treated by Monte Carlo lattice dynamics (MCLD) simulations. In MCLD, adsorbates are confined to a lattice of sites and motion occurs as a series of activated hops along the lattice.4-18 MCLD has X
Abstract published in AdVance ACS Abstracts, December 15, 1995.
0022-3654/96/20100-0967$12.00/0
been previously applied to adsorbates on flat surfaces6,11-13 and in zeolites.14-18 Most of the studies in zeolites have treated a cage as a featureless node that is either empty or occupied by a single adsorbate. Recent zeolite sorption simulations, though, have suggested that modifications to simple cubic and tetrahedral lattices would more closely model the relationship between adsorbate diffusivity and blocker content in zeolite. In light of these studies, we have examined the lattices described below. A. Cubic Superlattice/Octahedral Sublattice Hybrid. The pore network of zeolite A is composed of cages connected in a simple cubic arrangement by windows (Figure 1a). The presence of localized sites within a given cage, though, creates a sublattice within each node of the superlattice. An example we will consider here is suggested by Monte Carlo computer simulations that define the localized adsorption sites in the cages of cation-poor zeolite A.19 Six adsorption sites are arranged in an octahedron within the pore. The locations of the sites do not change with loading up to an occupation of six adsorbates per cage (Figure 2). This sort of intracage structure can be modeled by including a sublattice within each node of the cubic superlattice. Percolation on this network can take several forms, and we investigate each. Each of these modifications may cause deviations in the transport behavior from the predictions of EMA theory for the parent simple cubic lattice. (1) Superlattice Bond-Blockers. In zeolite A, some of the cations occupy positions in the plane of the windows linking adjacent cages; these cations can be thought of as superlattice bond-blockers, naturally leading to a percolation threshold with an increase in cation content. The presence of a cation effectively blocks the window in which it resides (Figure 3b). We examine a range of cation blocker concentrations spanning from that of a cation-poor R cage (all six windows open) to a cation-rich R cage (all six windows blocked).19 For simplicity, we neglect the indirect effects of the additional cations on the location and depth of the adsorption sites within the cage (i.e., the structure of the sublattice). We also study the effects of incorporating a free energy of activation along sublattice bonds, a free energy of activation along superlattice bonds, and adsorbate-adsorbate attraction or repulsion energy. © 1996 American Chemical Society
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Figure 1. Shematics of zeolite cages: (a, top) zeolite-A; (b, bottom) zeolite-Y.
Figure 3. Models of blocking on the simple cubic superlattice/ octahedral sublattice hybrid: (a) simple cubic lattice bond blocking; (b) superlattice bond blocking (with and without free energies of activation); (c) correlated superlattice bond blocking; (d) sublattice bond blocking.
Figure 2. Lattice model in zeolite-A.
(2) Correlated Superlattice Bond-Blockers. Cations may not be distributed uniformly throughout the zeolite. There may be distinct regions of cation-poor and cation-rich cages, which we represent as correlated placement of bond-blockers (Figure 3c). We examine both positive correlations (grouping of ions) and negative correlations (dispersal of ions) in our study. (3) Sublattice Bond-Blockers. We have encountered in our adsorption studies situations where two coadsorbed species occupy different sites within the cage. The presence of the less mobile species can effectively block some sublattice bonds linking the adsorption sites of the smaller, more mobile species (Figure 3d). In this case, we are interested in bond-blocking on the sublattice. B. Partial Blocking of Adjacent Bonds by a Single Blocker on the Tetrahedral Lattice. In some zeolites, it is possible for a blocker to reside between several windows22 or in channel intersections.23,24 The blocker then partially hinders diffusion along multiple adjacent bonds. For example, in the cages of zeolite Y (Figure 1b), where the cages are connected tetrahedrally, it is suspected that benzene resides between three of the four windows,21 partially blocking these three windows (Figure 4b). Moreover, it has been shown that the benzene has a very low diffusivity and can be considered a stationary blocker.22 This partial blockage of multiple windows is a
Figure 4. Bond blocking by benzene in zeolite-Y; (a) single bond blocking; (b) multiple bond blocking.
suspected cause of severely diminished diffusivity of methane in zeolite Y in the presence of benzene.22 The incorporation of multiple blocking on the tetrahedral lattice can be solved numerically at infinitely dilute loading, so we do not use MCLD in this case. Despite the shortcoming of simple EMA that it does not account for the possibility of a single blocker affecting more than one bond, we have previously used EMA to model methane diffusion with benzene blocking in zeolite Y with good agreement.22 By incorporating multiple blocking into EMA, we hope to show that a model with a more reasonable physical similarity to the actual system performs as well, if not better, than simple EMA.
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II. Effective Medium Approximation A. EMA-Bond-Blocking on a Simple Cubic Lattice. We use EMA theory to obtain a simple, analytical relationship between the mean adsorbate diffusivity, Dm, and the fraction of windows blocked, p, at an infinitely dilute adsorbate loading. For the superlattice/sublattice studies, we use a bimodal distribution of bond conductances:
g(D) ) pδ(D - Db) + (1 - p)δ(D - Do)
(1)
where p is the fraction of blocked windows; 1 - p, the fraction of open windows; δ, the Dirac delta function; Db, the diffusivity of a blocked window; and Do, the diffusivity through an unblocked or open window. For the sublattice study on the cubically connected (z ) 6) lattice, the simple two-valued distribution function given in eq 1 closely represents the situation of a window in a zeolite A (either the window is occupied by a cation (or other blocker) or it is not). We define f ) Db/Do, the ratio of diffusivities through blocked bonds to that through unblocked bonds. If we consider that the placement of a cation in a window completely blocks the window, then we set Db ) 0 (thus f ) 0) and set Do equal to the average diffusivity of a system with no windows blocked, i.e., when p ) 0. The lowest order effective medium approximation for the mean diffusivity, Dm, can be obtained via physical arguments1,25,26 or via lattice Green functions.27 The result is
D -D
∫0∞((z/2) -m 1)D
m
+D
g(D) dD ) 0
(2)
where z is the average coordination number. Inserting eq 1 into eq 2 and integrating, one obtains
{ [
]}
Dm 1 4f ) A + A2 + Do 2 ((z/2) - 1)
1/2
(3)
Figure 5. Probabilities {pi} for a window having i blockers as a function of p, number of blockers per number of windows.
TABLE 1: Blocked Window Probabilities in a Single Cage as a Function of Cage Occupancy of Blockers n
p′0,n
p′1,n
p′2,n
p′3,n
0 1 2 3 4
1 1/ 4 0 0 0
0 3/ 4 1/ 2 0 0
0 0 1/ 2 3/ 4 0
0 0 0 1/ 4 1
has i ) 0, 1, 2, or 3 blockers, {pi}, are defined as follows. In zeolite Y, there are four windows and there are four distinct sites where a blocker sits. For a given blocker occupancy in a cage, n ) 0-4, we give the probabilities of finding a window with i blockers for a single cage, p′i,n, in Table 1. The singlecage probabilities, {p′i,n}, are needed to calculate the overall ensemble probabilities, {pi}. In order to do this, we must take into account that for a system of C distinguishable cages and B indistinguishable blockers, there will be a statistical distribution of blockers per cage, so the {pi} are given by 4
pi ) ∑FCB(n)p′i,n
where
A ) 1 - p + fp -
(f + p - fp) ((z/2) - 1)
(4)
For f ) 0, eq 3 becomes
Dm 3p )1Do 2
(7)
n)0
(5)
which states simply that the average diffusivity, in a simple cubic system with a fraction of windows blocked equal to p, is linearly related to the diffusivity of the unblocked system and declines to zero when 2/3 of the windows are blocked. In simulations of simple cubic lattices it has been shown that this linear relationship between Dm and p holds true for most p less than pc, but as p reaches 2/3, the relationship changes to (p pc),1.6 where pc is about 3/4.1 B. EMA-Multiple-Partial Bond-Blocking on a Tetrahedral Lattice. For the multiple-partial blocking study, we use zeolite Y, tetrahedrally connected (z ) 4), as our model. In this case, the pores can be blocked to different extents. The distribution function of window diffusivities is composed of four elements, including fully open windows and windows partially blocked by one, two, or three blockers:
g(D) ) p0δ(D - Do) + p1δ(D - D1) + p2δ(D - D2) + p3δ(D - D3) (6) The four-element vector of probabilities that a given window
FCB(n) (given by Chmelka et al.28) is the hypergeometric distribution for B blockers and C cages with a maximum occupancy of K ) 4 blockers per cage
FCB(n) )
(Bn)(KCK --nB)/(KC K)
(8)
The number of configurations, and thus the set {pi}, depends on both B and C. As C goes to infinity, though, the {pi} depend only on B/C, the average blocker occupancy. For this system, the fraction of all blocked bonds in the effective medium approximation, p, is equal to B/(4C) as there are four windows per cage. Thus, to fully specify the distribution, {pi}, only p needs be set, just as is the case for the unmodified EMA. We calculated {pi} from eq 7 for C )100, B ) 0-400, and K ) 4. The calculated {pi} are given in Figure 5 as functions of p. Equation 6 has three values of diffusivity through windows blocked to various extents, D1, D2, and D3. We examine two methods to choose {Di}. The first method, an additive method, considers equal blocking by the first, second, and third blockers to obstruct a window. Given f then f3 ) D3/Do ) f, then f2 ) D2/Do ) (1 - f)/3 + f and f1 ) D1/Do ) 2(1 - f)/3 + f. The second method allows that the first, second, and third blockers of a given window may contribute unequally to the total blocking. In this case we have three independent parameters f1, f2, and f3, which are constrained to Σfi e 1. The physical basis for this method is that the first blockers are the most important, diminishing the diffusivity by the greatest
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amount. Additional blockers reduce the diffusivity further but to a lesser extent. Substituting these values of the diffusivities into eq 6 and eq 2 yields a fourth-order polynomial for Dm. We used the Newton-Raphson method to solve for the roots and zero-order continuation to track the roots as a function of p. This polynomial has three negative roots, which are neglected on physical grounds, and one positive root, Dm. Note that having posed this problem in the above manner, we place up to 4C blockers to block 2C bonds, seeming to doubly block them. EMA accounts for this double counting by averaging the blockage across a bond, e.g., the diffusivity of a bond which is blocked by two blockers in one cage and one blocker in the other is (D2 + D1)/2.
an intracage hop or GAW for an intercage hop), Nc(n) is the number of occupied nearest neighbor sites, and EAA is the adsorbate-adsorbate interaction energy. With this time increment and with the Einstein relation for self-diffusivities, the adsorbate diffusion coefficient can be calculated
D ) lim tf∞
〈∆r2〉 6t
) lim Nhopf∞
〈∆r2〉 6Nhop∆τ
〈∆r2〉(∑∑Pnm)Γ ) lim Nhopf∞
n
m
6Nhopz (12)
A. Theory. For the (simple cubic superlattice/octahedral sublattice) hybrid lattices, no EMA solution is available yet. Additionally, to account for nonzero adsorbate loading, we must perform MCLD. MCLD simulations allow self-diffusion coefficients to be calculated from Monte Carlo simulations. The MCLD simulations used here are in canonical ensembles, where the volume, number of adsorbates, and temperature are specified. In an MCLD simulation, the positions of adsorbate sites are fixed. Adsorbates move from site to site via a series of activated hops. The hops need not all be equivalent; in zeolite A, there are two types of hops (inter- and intracage), each with different free energies of activation. To employ MCLD, one creates a set of possible hops which includes the hop of the ith adsorbate from the nth site to a neighboring mth site, excluding all hops into occupied sites or along blocked bonds. With this method, all hops are successful so greater computational efficiency is achieved.12-14 One hop from all possible hops is selected at random using a probability weighted by the free energy of activation for that hop. The increment of time associated with one MC hop is ordinarily
where Nhop is the number of hops in the simulation. B. Implementation. For each data point, a minimum of 20 simulations were run with random initial placement of the adsorbates and blockers. Each run consisted of at least 5000 initial moves, which were discarded and then 20 000-100 000 moves (depending on the number of adsorbates) for data collection. Although previous studies reported data taken from a system of 2 × 2 × 2 cages with periodic boundary conditions, we found anomalous results in the percolation threshold from a 2 × 2 × 2 system. Our results from systems of 3 × 3 × 3, 4 × 4 × 4, and 5 × 5 × 5 cages agreed within statistical deviation, so we use at least 3 × 3 × 3 cages. The loadings, θ, are given as Nads/(6C) where Nads is the total number of adsorbates, C is the number of cages, and there are six sites per cage. The fraction of windows blocked by cations, psuper, is reported with respect to the total number of windows, B/(3C), where B is the total number of blockers and there are three windows per cage. The fraction of site-site bonds which are blocked, psub, is reported with respect to the total number of intracage bonds, B/(12C), where B is the total number of blockers and there are twelve intracage bonds per cage. In the case of correlated placement of the cations, the first cation is placed randomly in a window. Each additional cation has a choice of any of the remaining empty windows and is placed in window m with probability
∆τ ) 1/NadsΓ
Pm ) exp[-(EccNw(m))/kT]
III. Monte Carlo Lattice Dynamics on the Modified Simple Cubic Lattice
(9)
where Γ is the vibrational frequency of an adsorbate in a site and Nads is the number of adsorbate molecules. However, since we attempt only successful hops, we must scale the time by the ratio of the number of possible hops (including hops to occupied sites or along blocked bonds with zero probability of success) to the number of probable hops (which excludes hops with zero probability of success):
∆τ )
1
(
possible hops
) ( )
NadsΓ probable hops
)
1
Nadsz
NadsΓ
Pnm ∑n ∑ m
)
z
(∑∑Pnm)Γ n
(10)
for an unoccupied window m and Pm ) 0 for an occupied window, where Ecc is the interaction energy between neighboring cations (negative for attractive interactions) and Nw(m) is the number of occupied neighboring windows next to window m. Ecc is zero for random placement and infinite for separate regions of cation-rich and cation-poor cages. A simulation at infinite dilution (θ ) 0) with no blockers (psub ) psuper ) 0), no barriers to motion (GAC ) GAW ) 0), and no adsorbate-adsorbate or cation-cation correlations (EAA ) Ecc ) 0) is used as the reference state. The diffusivities reported here are scaled to the reference case, and the vibrational frequency, Γ, cancels. This is useful because the vibrational frequency may not be known. IV. Results and Discussion
m
where Pnm is the unnormalized probability of a hop from site n to site m. The unnormalized probability for a hop from site n to site m, when m is unoccupied and the bond linking sites n and m are unblocked, is
Pnm ) exp{-[Gnm + Nc(n)EAA]/kT}
(13)
(11)
where Gnm is the free energy of activation for the hop (GAC for
A. Octahedral Sublattice/Cubic Superlattice Hybrid. (1) Superlattice Bond Blocking. Figure 6 shows the self-diffusivity as a function of the fraction of blocked windows for the base case (all activation and interaction energies are zero) at an average loading, θ, of one adsorbate per cage (1/6). The percolative behavior of the modified lattice is in good agreement with the simple cubic analysis from previous studies.1 The presence of the sublattice does not alter the percolative behavior of the system at this low loading. Figure 7 shows the self-diffusivity as a function of the fraction
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Figure 6. Percolation via superlattice bond blocking on the super/ sublattice as a function of the system size at θ ) 1/6. The simple cubic data points are from ref 1. The EMA is for a simple cubic lattice. Kirkpatrick’s data points and EMA theory have been linearly scaled to our data point at psuper ) 0.
Figure 9. Percolation via superlattice bond blocking with nonzero adsorbate-adsorbate interaction energies (EAA ) 0.46kT). Positive EAA is repulsive. The EMA is for a simple cubic lattice.
Figure 7. Percolation via superlattice bond blocking as a function of system density at θ ) 1/6 and θ ) 1/2. The EMA is for a simple cubic lattice.
Figure 10. Percolation via superlattice bond blocking with correlated placement of blockers at θ ) 1/6. The simple cubic data have positive correlations and are taken from ref 1. The EMA is for a simple cubic lattice without correlations.
Figure 8. Percolation via superlattice bond blocking with nonzero intercage (GAW ) 1.0kT) and intracage (GAC ) 1.0kT) free energies of activation at θ) 1/6. The EMA is for a simple cubic lattice.
The self-diffusivity was also calculated for adsorbateadsorbate interaction energies of EAA ) +0.46, 0, and -0.46kT (Figure 9). While adsorbate-adsorbate interactions increase or decrease the diffusivity at p ) 0, they do not affect the functional form of D with p and hence do not affect agreement with EMA of the simple cubic parent lattice. (2) Correlated Placement of Superlattice Bond-Blockers. Figure 10 plots the self-diffusivity when Ecc ) +4.0kT, 0kT, and -4.0kT at an average loading of θ ) 1/6. A negative correlation energy corresponds to clustering of the cations, and a positive energy corresponds to dispersal of the cations. Also shown are the results of EMA for a simple cubic lattice with random placement and for Kirkpatrick’s correlated bond percolation model on a simple cubic lattice. This comparison is qualitative only; Kirkpatrick’s correlation is negative (clustering of open bonds), but his method of correlation is different from ours.1 Clustering of blockers postpones the percolation threshold to higher values of p (pc ) 0.75 for simple cubic without correlation and pc > 0.85 with negative correlation) because, outside the blocker-rich region, there must be blocker-poor regions. These allow at least one large cluster of cages, through which the adsorbates can diffuse over the time scale of the simulation. The behavior of the modified and simple lattice is qualitatively similar, and we conclude that the sublattice has little effect on percolative behavior with blocker correlation when only superlattice bonds are blocked. Surprisingly, the dispersal of cations also delays the percolation threshold beyond that of random cation placement. This is because, by dispersing the cations, we block equal numbers of windows in all cages. This leaves all cages with two open
of open windows for the base case at two loadings: one and three adsorbates per cage. As one expects, the diffusivity decreases with loading but the value of the percolation threshold is unchanged. EMA of the simple cubic lattice captures the percolative behavior well for both loadings. Despite the fact that EMA is rigorously valid only for infinite dilution, crowding on the sublattice has no effect on percolation. Figure 8 shows the self-diffusivity as a function of the fraction of open windows for the case when the free energy of activation for intracage motion (on the sublattice), GAC, is 1.0kT and when the free energy of activation for intercage motion (on the superlattice), GAW, is 1.0kT, at θ ) 1/6. Both activation free energies lower diffusion. The intercage activation free energy, GAW, has a stronger effect than GAC, as shown previously.13
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Figure 11. Percolation via sublattice bond blocking at θ ) 1/6. For comparison, the EMA is for a simple cubic lattice.
bonds at p ) 2/3, which allows entrance from one adjacent cage and exit to another. (3) Sublattice Bond Blocking. In many situations, it is unknown whether cations or coadsorbates are placed in windows or rather are placed inside the cage, where they can block only sublattice bonds. In Figure 11, percolation by sublattice bond blocking is demonstrated. In the octahedral sublattice there are twelve sublattice bonds (Figure 2) which can be blocked. There is a slight but surprising negative curvature to the plot of diffusion with respect to blockage which may be due to the finite size of our simulated system, but remained present up to systems of 6 × 6 × 6 cages. The percolation threshold predicted by EMA is greater for sublattice blocking because, ignoring the (always open) superlattice bonds, the connectivity of this lattice is 8 (cf. 6 for the simple cubic superlattice). It is interesting to note how important it is to know the placement of the blocker. The importance arises from the fact that there are twelve sublattice bonds per cage but only three superlattice bonds per cage. If one erroneously gauged this percolation in terms of superlattice bonds, they would find an apparent percolation threshold 4 times the actual value. B. Partial Blocking of Adjacent Bonds by a Single Blocker on a Tetrahedral Lattice. Since we have established that for the hybrid lattice adsorbate loading does not change percolation behavior from that of the parent lattice (instead it affects only the magnitude of the diffusivity), in the following examination of the tetrahedral lattice we look only at infinitely dilute loading. Furthermore, since we have also shown above that the presence of a sublattice of adsorption sites does not change the percolation behavior for superlattice bond blocking, we do not include a sublattice within the nodes of the tetrahedral superlattice but consider each cage as a node. Parts a and b of Figure 12 compare the diffusivity on a tetrahedral lattice with multiple bond-blockers (both equal and unequal blocking) with that on a tetrahedral matrix with ordinary single blocking. In Figure 12a, we demonstrate complete blocking (f ) 0.0) for the single and multiple blockers. The two multiple blocking curves represent equal blocking (f1 ) 2/ , f ) 1/ , and f ) 0) and unequal blocking (f ) 0.3, f ) 3 2 3 3 1 2 0.1, and f3 ) 0.0). Whereas the percolation threshold for a network with z ) 4 from EMA for single blocking is pc ) 0.5, the percolation threshold for both multiple blocking cases is 0.795. The percolation threshold is delayed in multiple blocking simply because it takes three blockers to fully block a bond. This plot also demonstrates that, depending on how the multiple blocking behaves (i.e., how one chooses the {fi}), one can obtain diffusivities at small p that fall more quickly or more slowly than for ordinary single blocking. In Figure 12b, we examine incomplete (95%) blocking of a bond with three blockers. For the single blocking case, f )
Figure 12. Percolation on the tetrahedral lattice with additive and nonadditive multiple blocking and single blocking for (a, top) f or f3 ) 0.0 and (b, bottom) f or f3 ) 0.05.
Figure 13. Least squares fit of equal multiple blocking (f1 ) 2/3, f2 ) 1 /3, and f3 ) 0.0) and unequal multiple blocking (f1 ) 0.6, f2 ) 0.1, and f3 ) 0.061) and single blocking (f ) 0.061). The data points are for methane diffusivity in the presence of coadsorbed benzene in zeolite-Y from Nivarthi et al.
0.05; for the equal multiple blocking case, f1 ) 2 × 0.95/3 +0.05 (0.683), f2 ) 0.95/3 + 0.05 (0.367), and f3 ) 0.05; for the unequal multiple blocking case, f1 ) 0.3, f2 ) 0.1, and f3 ) 0.05. Because f (or f3) is not 0, the bonds are never fully blocked and there is no percolation threshold. In Figure 13, we show that this model can explain experimental data. We make a least squares fit of the single blocking EMA and the multiple blocking EMA with the experimental data points of methane coadsorbed with benzene in zeolite Y from Nivarthi et al. The least squares fit yields f ) 0.061 for the single blocking model, f1 ) 2/3, f2 ) 1/ , and f ) 0 for the equal multiple blocking model, and f ) 3 3 1 0.6, f2 ) 0.1, and f3 ) 0.061 for the unequal multiple blocking. While we suspect that there is multiple blocking in this system, the single blocking model performs just as well as the unequal multiple blocking model. Clearly, though, the obstructive effects of several blockers are not equal.
Zeolite Sorption Lattices V. Conclusions Comparing percolation on a simple cubic lattice with that of an octahedral sublattice/cubic superlattice hybrid (a model of adsorption sites in zeolite A), we find that the inclusion of a sublattice of adsorption sites substantially preserves the percolative behavior of the simple cubic lattice. Factors such as adsorbate density, inter- and intracage activation energies, and adsorbate-adsorbate interaction energies change the magnitude of the diffusivity but do not alter its percolative behavior. Either clustering or dispersing cations in the model zeolite A network extend the percolation threshold compared to random placement in the same manner as correlated blocking on the simple cubic lattice. Blockage of the octahedral sublattice bonds shows a slight negative curvature with respect to p, in constrast with the linear behavior of the superlattice bond blocking. If, while blocking sublattice bonds, one gauges percolation based on the number of superlattice bonds, we find an unreasonable percolation threshold 4 times greater than the actual value. It is important, therefore, to understand where the blocking occurs in the zeolite. The application of three EMA models to methane coadsorbed with benzene in zeolite Y indicates that the a multiple blocking model captures the diffusive behavior of the methane with increasing blocker (benzene) concentration as well as but not better than a single blocking model. Although physical arguments favor a multiple blocking model, the EMA theory for this model gives no improvement over the single blocking EMA theory when compared with experiment. We have shown, however, that the effects of multiple blockers are not equally distributed among the three blockers. The first two blockers decrease the diffusivity through that window by the largest amounts, followed by the third blocker, indicating that small amounts of coadsorbed blocker can strongly influence the diffusivity of the more mobile species. Acknowledgment. We wish to acknowledge the Minnesota Supercomputer Institute and National Science Foundation (Grant CTS-9058387) for partial support of this work. D. Keffer was partially supported by a Chevron Fellowship. Nomenclature B C Dc Di Dm Do EAA Ecc g(D) GAC GAW f FCB(n) k K Nads
number of blockers number of cages diffusivity through a blocked window diffusivity through a window multiply blocked by i blockers mean diffusivity diffusivity through an unblocked window adsorbate-adsorbate interaction energy cation-cation correlation energy distribution of bond conductances free energy of activation for an intracage hop free energy of activation for an intercage hop ratio of Dc/Do hypergeometric distribution for B blockers and C cages Boltzmann's factor maximum number of blockers per cage number of adsorbates
J. Phys. Chem., Vol. 100, No. 3, 1996 973 Nhop Nc(n) Nw(n) p pc pi p′i,n Pnm Q 〈∆r2〉 t T 〈X〉 z
number of successful hops number of occupied nearest neighbor sites to site n number of occupied nearest neighbor windows to window n fraction of windows blocked percolation threshold fraction of windows multiply blocked by i blockers in overall network fraction of windows multiply blocked by i blockers in a single cage with n blockers probability of hop from site n to site m probability normalization factor mean square displacement time temperature ensemble average of X coordination number of the lattice
Greek Letters ∆τ time associated with a hop Γ site vibrational frequency θ fraction of sites occupied
References and Notes (1) Kirkpatrick, S. ReV. Mod. Phys. 1973, 45, 574. (2) Hieba, A. E.-A. Porous Media: Fluid Distributions and Transport with Applications to Petroleum Recovery. Ph.D. Thesis, University of Minnesota, 1985. (3) Fisher, M. E.; Essam, J. W. J. Math. Phys. 1961, 2, 609. (4) Bowker, M.; King, D. A. Surf. Sci. 1978, 71, 583. (5) Murch, G. E. Philos. Mag. A 1980, 41, 157. (6) Reed, D. A.; Ehrlich, G. Surf. Sci. 1981, 105, 603. (7) Reed, D. A.; Ehrlich, G. Surf. Sci. 1982, 120, 179. (8) Kutner, R.; Binder, K; Kehr, K. W. Phys. ReV. B 1982, 26, 2967. (9) Kutner, R.; Binder, K.; Kehr, K. W. Phys. ReV. B 1983, 28, 1846. (10) Sadiq, A.; Binder, K. Surf. Sci. 1983, 128, 350. (11) Mak, C. H.; Anderson, H. C.; George, S. M. J. Chem. Phys. 1988, 88, 4052. (12) Kang, H. C.; Weinberg, W. H. J. Chem. Phys. 1989, 90, 2824. (13) Bowler, A. M.; Hood, E. S. J. Chem. Phys. 1991, 94, 5162. (14) Van Tassel, P. R.; Somers, S. A.; Davis, H. T.; McCormick, A. V. Chem. Eng. Sci. 1994, 49, 2979. (15) Ruthven, D. M. Can. J. Chem. 1974, 52, 3523. (16) Theodorou, D.; Wei, J. J. Catal. 1989, 83, 205. (17) Aust, E.; Dahlke, K.; Emig, G. J. Catal. 1989, 115, 86. (18) Nelson, P. H.; Kaiser, A. B.; Bibby, D. M. J. Catal. 1991, 127, 101. (19) Van Tassel, P. R.; Davis, H. T.; McCormick, A. V. Mol. Phys. 1991, 73, 1107. (20) Ohgushi, T.; Yusa, A.; Takaishi, T. J. Chem. Soc., Faraday Trans. 1 1977, 73, 613. (21) Fitch, A. N.; Jobic, H.; Renouprez, A. J. Phys. Chem. 1986, 90, 1311. (22) Nivarthi, S. S.; Davis, H. T.; McCormick, A. V. Chem. Eng. Sci., in press. (23) Caro, J.; Bulow, M.; Karger, J.; Pfeifer, H. J. Catal. 1988, 114, 186. (24) Forste, C.; Germanus, A.; Karger, J.; Pfeifer, H.; Caro, J.; Pilz, W.; Zikanova, A. J. Chem. Soc., Faraday Trans. 1 1987, 83, 2301. (25) Davis, H. T.; Valencourt, L. R.; Johnson, C. E. J. Am. Chem. Soc. 1975, 58, 446. (26) Kirkpatrick, S. Phys. ReV. Lett. 1971, 27, 1722. (27) Sahimi, M.; Hughes, B. D.; Scriven, L. E.; Davis, H. T. J. Chem. Phys. 1983, 78, 6849. (28) Chmelka, B. F.; Raftry, D.; McCormick, A. V.; de Menorval, L. C.; Levin, R. D.; Pines, A. Phys. ReV. Lett. 1991, 66, 580.
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