J. Phys. Chem. 1996, 100, 833-836
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A Model for Random Si/Al Distribution in a Zeolite Framework Restricted by Loewenstein’s Rule Alexander J. Vega DuPont Central Research and DeVelopment, P.O. Box 80356, Wilmington, Delaware 19880-0356 ReceiVed: August 8, 1995X
Analytical expressions are derived for a description of random Loewenstein Si/Al ordering in the faujasite zeolite framework. These formulas served as a random distribution reference in the analysis by Melchior et al. of 29Si NMR spectra of faujasites for the determination of Si/Al ordering in framework building blocks. The equations are based on a model where correlations between the Al occupancies of T-sites belonging to different four-rings are ignored. This simple approach gives reliable results for 29Si peak intensities when the Si/Al ratio is larger than 1.5.
Introduction In a recent publication,1 Melchior et al. presented a thorough analysis of the intensity distributions of the line shape components in magic-angle-spinning 29Si NMR spectra of zeolites having the faujasite structure. By assuming that the spectra are superpositions of components arising from as many as 22 distinct local environments for a silicon atom, and by employing an elaborate mathematical description of a classification scheme of short-range Si/Al siting patterns in the zeolite framework, the analysis led to compelling conclusions regarding specific deviations of framework-Al distributions from a random distribution model. The results have strong implications concerning the building units involved in faujasite crystallization. To determine the nonrandom nature of the Al distributions, the numerical values of several mathematical functions of the peak intensities were compared with corresponding theoretical values derived for a random distribution. However, the random model used in those evaluations has not yet been discussed in the literature. Instead, Melchior et al. referred to it as an unpublished private communication by the present author. A description of this model is now given. The concept of randomness is not trivial in this particular case, because we are considering a situation where the Al atoms are randomly distributed over the tetrahedral sites (T-sites) of the framework under the restriction of Loewenstein’s rule, which states that two adjacent sites cannot both be occupied by Al atoms.1,2 The difficulty is that Loewenstein’s rule imposes a certain degree of local order on the site distribution. This ordering becomes more prominent as the fractional Al occupation of the T-sites increases and eventually results in a perfectly alternating -Al-O-Si-O-Al-O-Si- pattern when the Al occupation is 50% (Si/Al ratio is 1). Hence, a “random Loewenstein distribution” is random to a limited extent only. Since Melchior’s work was directed toward the identification of local Al ordering patterns induced by driving forces which act in addition to the adjacent site avoidance, a precise characterization of the state of optimum randomization under the sole restriction of Loewenstein’s rule was needed as a reference. The Model In the framework topology of the faujasites (zeolites X and Y), every T-site is the meeting point of three edge-sharing fourX
Abstract published in AdVance ACS Abstracts, December 1, 1995.
0022-3654/96/20100-0833$12.00/0
rings, as is shown in Figure 1a. The chemical shift of a 29Si placed at the center of this pattern is, to a first approximation, determined by the number of Al atoms occupying its four nearest-neighbor sites.3 This results in an NMR spectrum composed of five main peaks with intensities In (n ) 0, ..., 4) which are proportional to the probabilities of finding the Si atoms surrounded by n Al atoms. A previous publication4 described a Monte Carlo calculation in which Al atoms were randomly distributed over the T-sites under the restriction of Loewenstein’s rule. This provided values for In as functions of the fractional Al concentration x ) [Al]/([Si] + [Al]). Those calculations were performed on a structural unit containing 1296 T-sites (at least 11 T-O-T linkages across in all directions) with periodic boundary conditions. In the computational procedure, the required number of Al atoms was first randomly distributed over the T-sites without regard to nearest-neighbor avoidance. They were subsequently allowed to exchange places with Si atoms at randomly chosen sites of lower Al coordination, until full compliance of Loewenstein’s rule was achieved. For further details, see ref 4. In order to arrive at a truthful description of random Al siting in zeolites, the Monte Carlo calculation must work with a structural unit which is larger than the correlation length of the Al site occupancies. This correlation is imposed by the Loewenstein condition. Its characteristic length (measured, e.g., in numbers of T-O-T linkages) increases with increasing x until it approaches ∞ when x ) 1/2. It was found that an 11linkage-wide unit is amply sufficient to describe Loewenstein randomness when x e 0.44 In the following analytical approach algebraic expressions are derived for In taking into account Al siting correlations within each of the individual four-rings, while ignoring the correlations that exist among different four-rings and within larger topological features. Comparison with Monte Carlo results (see below) will show that this bold assumption turns out to be quite reliable for x e 0.4 (Si/Al g 1.5). An indication that the correlations resulting from structural elements larger than four-rings have no appreciable effect on the Al distributions is obtained from comparisons among Monte Carlo results of the framework topologies of the zeolites faujasite, ZK-4 (isomorphous with zeolite A), chabazite,5 and zeolite RHO.6 These four zeolites have the same short-range connectivity pattern as shown in Figure 1a, but they differ in the interconnectivities of these unit on a longer length scale via additional four-rings, six-rings, eight-rings, etc. Table 1 shows that the peak intensities for a random Loewenstein distribution with x ) 1/3 (Si/Al ratio ) 2), as calculated with © 1996 American Chemical Society
834 J. Phys. Chem., Vol. 100, No. 2, 1996
Vega abilities u1, ..., u4 are shown in Figure 1c. For the time being, the occupation of the fourth corner is left unspecified (shaded circle). The probabilities are assumed to be independent of the manner in which the four-rings is incorporated in the zeolite framework. For instance, the asymmetric environment of the upper left four-ring in Figure 1a is ignored. This allows the assignment of identical probabilities u2 to the two cases with one Al shown in Figure 1c. The probabilities obey three relations dictated by normalization, the average occupation of the center site, and the average occupation of the other two sites:
Figure 1. Short-range connectivities in a zeolite framework. Open circles designate Si, filled circles Al, shaded circles unspecified T-sites, and lines oxygen bridges. (a) The local connectivity pattern of faujasite, ZK-4, chabazite, and RHO. (b) The probabilities of the three ways in which a T-site pair can be occupied in compliance with Loewenstein’s rule. (c) Same for the five occupation patterns of a triad in a fourring. (d) Scheme showing that the probability to find two Al’s in a four-ring is u3P (see text). (e) Scheme showing how the probability of a three-neighbor occupancy can be derived from an already established two-neighbor occupancy, assuming correlation within fourrings only.
the Monte Carlo procedure, are virtually identical in these four topologies. This indicates that the differences in connectivities beyond the immediate vicinity of the three four-rings are immaterial for the short-range Al ordering at this level of Al occupancy. One could attempt to simplify the model even further and consider structural units consisting of only a central site with four, mutually uncorrelated, neighbors. This would reduce the smallest correlated unit to just one pair of T-sites. The 29Si NMR peak intensities for random Loewenstein distribution in this model can be evaluated by first considering the three allowed ways in which two adjacent sites can be occupied. These and the designations of their probabilities are shown in Figure 1b, where open and filled circles represent Si and Al, respectively. From the requirement that the average occupation of one of the sites is x, we find p2 ) x, which combined with the normalization condition, p1 + 2p2 ) 1, gives p1 ) 1 - 2x. Consequently, when a site is known to be Si, then the probability of finding Al in the other site is given by r ) p2/(p1 + p2) ) x/(1 - x). It has been noted that r is the inverse of the Si/Al ratio.7 The assumption of uncorrelated occupancies of the four neighbors then leads to the binomial distribution
I0 ) (1 - r)4; I1 ) 4r(1 - r)3; I2 ) 6r2(1 - r)2; I3 ) 4r3(1 - r); I4 ) r4 (1) Unfortunately, this simple model is unrealistic,1 because the occupancies at opposite corners of four-rings are strongly coupled. The extent of the errors introduced by this simplification can be seen by a numerical comparison of the x ) 1/3 case (r ) 1/2) with the Monte Carlo results mentioned above (see Table 1 and data presented in the Results section). Our statistical treatment of the model based on internally correlated four-rings begins with the five Loewenstein-allowed ways in which a triad within a four-ring can be occupied by Si and Al atoms. The designations of their four distinct prob-
u1 + 2u2 + u3 + u4 ) 1
(2)
u3 ) x
(3)
u2 + u4 ) x
(4)
These equations are rigorously valid and are independent of the ordering scheme. (If the symmetry between the two u2’s is broken, the equations still hold provided u2 stands for their average.) To fully determine the four probabilities, we need a fourth relationship. If we identify u4 (the probability of having two Al’s at opposite corners of a four-ring) with a new, tentatively independent, variable y, the solution is
u1 ) 1 - 3x + y; u2 ) x - y; u3 ) x; u4 ) y
(5)
From Figure 1c we can further derive the probability P of finding a site in a four-ring occupied by Al, given that both adjacent sites are Si’s. Continuing to assume 4-fold symmetry, this is easily determined from u1 and u3:
P ) u3/(u1 + u3)
(6)
Figure 1d illustrates how we can use the property P to evaluate y. The probability for the occurrence of the Si-Al-Si triad shown in the left diagram of Figure 1d is per definition u3. Randomness then predicts that the probability to find an additional Al in the fourth site is given by the product of u3 and P. The site occupancy of the resulting, fully determined, four-ring is identical to that corresponding to u4, because the fourth unspecified corner in the pattern illustrating u4 in Figure 1c must be Si in accordance with Loewenstein’s rule. This yields u4 ) u3P as the fourth relationship. It leads to a quadratic equation in y with the solution
y ) 1/2[x1 - 4x + 8x2 - 1 + 2x]
(7)
and by substitution in eq 5 to fully determined dependencies of u1, ..., u4 on x. (An alternative approach, using u2 ) u1P, gives the same results.) We can now develop the probabilities for the various nearestneighbor coordination schemes of a Si atom occupying the central position in the pattern of Figure 1a. Figure 1e shows an example of how this is done. The placement of three Si’s at the indicated sites in the pattern on the left, while leaving the other sites unspecified, has a probability u1. This is also the sum of the probabilities of the two patterns on the right. Since the site occupancies of the four-rings are assumed to be uncorrelated, the ratio of the latter two is u1:u2 (compare Figure 1c). From this we derive the indicated expressions for the probabilities of the two occupation schemes. Applying this procedure to all nine ways in which the same four sites can be occupied, we obtain the seven distinct probabilities V1, ..., V7 indicated in Figure 2. The next step is to specify the occupation of the fourth neighboring site, i.e., the one below the central position. This is worked out in Figure 3, where all 17
Si/Al Distribution in a Zeolite Framework
J. Phys. Chem., Vol. 100, No. 2, 1996 835
Figure 2. Nine ways in which the four indicated specified sites in the three edge-sharing four-rings can be occupied in compliance with Loewenstein’s rule. The indicated probabilities are calculated with the model of mutually uncorrelated four-rings.
possibilities are shown with their distinct probabilities w1, ..., w11. Finally, the 29Si NMR peak intensities, evaluated for a central Si in Figure 3, are given by
I0 ) w1/(1 - x)
Figure 3. Seventeen ways in which the five indicated specified sites in the three edge-sharing four-rings can be occupied in compliance with Loewenstein’s rule. The indicated probabilities are calculated with the model of mutually uncorrelated four-rings.
I1 ) (2w2 + 2w3)/(1 - x) I2 ) (2w5 + 2w6 + w7 + w8)/(1 - x) I3 ) (2w9 + 2w10)/(1 - x) I4 ) w11/(1 - x)
(8)
Results and Discussion We have considered two analytical models and one Monte Carlo model with increasing levels of topological constraint on random Loewenstein ordering. They were based on mutually uncorrelated pairs, mutually uncorrelated four-rings, and internally correlated clusters of 1296 T-sites, respectively. To appreciate the ordering effect due to topological constraints within a four-ring, we compare the two analytical models. In particular, we consider a four-ring with one of its corners assumed to be occupied by Si, and we evaluate the probability to find one Si and one Al at the two neighboring sites within that ring. This probability is given by 2r(1 - r) for the pair model and by 2u2/(1 - x) for the four-ring model. Numerical values of these expressions (see Figure 4a) show that the intrafour-ring correlation substantially suppresses the probability of finding a Si with just one Al neighbor within a four-ring. Conversely, the probabilities of finding a Si with two Si neighbors or with two Al neighbors (not shown) are both enhanced by the intra-ring correlation. On the other hand, additional ordering due to coupling between adjacent four-rings appears to have a negligible effect on the predicted 29Si peak intensities. This is seen in Table 1, where numerical values of equation 8 are compared with the
Figure 4. Numerical results of random Loewenstein distribution models. (a) Probability to find one Si and one Al in two sites adjacent to a central site presumed to be Si, assuming that correlations are limited to independent pairs or that the three sites belong to a four-ring and inter-four-ring correlations can be ignored. (b) Intensities of the five 29Si NMR peaks (I ) of a faujasite framework. Symbols: Monte Carlo n calculation on a structural unit containing 1296 T-sites. Lines: Mutually uncorrelated four-rings, eq 8.
Monte Carlo results for x ) 1/3 (Si/Al ) 2), and in Figure 4b, where the two sets of values are plotted for 0.05 e x e 0.4 (1.5 e Si/Al e 19). The agreement between the results derived
836 J. Phys. Chem., Vol. 100, No. 2, 1996
Vega
TABLE 1: 29Si NMR Peak Intensities (%) Calculated for a Random Loewenstein Al Distribution in Zeolites with a Local Topology of Three Edge-Sharing Four-Ringsa and Si/Al ) 2b model faujasitec
Monte Carlo, Monte Carlo, ZK-4c Monte Carlo, chabazitec Monte Carlo, RHOc Monte Carlo, averaged analytical, pairse analytical, four-ringsf
I4
I3
I2
I1
I0
12.0 11.9 11.9 11.6 11.9 ( 0.2 6.25 11.8
23.8 24.1 24.1 24.4 24.0 ( 0.2 25 23.6
28.5 28.1 28.4 28.8 28.5 ( 0.3 37.5 29.2
23.5 24.0 23.3 22.9 23.4 ( 0.5 25 23.6
12.2 11.9 12.3 12.3 12.2 ( 0.2 6.25 11.8
a Figure 1a. b x ) 1/ . c Calculated with the procedure outlined in ref 4 on structural units containing about 1300 T-sites. d Average and standard 3 deviation of the four values listed above. e Equation 1, neglecting interpair correlations. f Equation 8, neglecting inter-four-ring correlations.
from the single-four-ring model and those derived from the cluster containing 1296 T-sites is remarkable. It explains why the four zeolite connectivity schemes listed in Table 1 give nearly identical 29Si intensity distributions. The expressions in Figures 2 and 3 and in eqs 5-8 served to generate the random Loewenstein reference data used by Melchior et al.1 It should be emphasized that the five 29Si peak intensities I0, ..., I4, which measure the probabilities of nearest-neighbor site occupancies, do not provide a full picture. The analysis of Melchior et al. has demonstrated that intermediate-range ordering patterns exist in actual faujasites and that these are reflected in the fine structures of peaks.1 The present paper merely shows that correlations within topological units which are larger than four-rings do not affected random Loewenstein Si/Al ordering on the shortest length scale. However, it is clear that connectivities between adjacent four-rings impose intermediate-range ordering on the random Loewenstein distribution. Examples of this were shown by Melchior et al., who applied the expressions derived above to calculate probabilities of specific Al/Si configurations, such as Al-Al pair densities in four-rings and in six-rings.1 Intermediate-range order can also be quanti-
fied in terms of a distance-dependent correlation function of the site occupancies.4 Finally, we note that the present analytical model does not work for a hypothetical square planar lattice where every T-site is the meeting point of four edge-sharing four-rings.4 In that case, the interconnection between the fourrings is too strong to ignore its effect on nearest-neighbor distributions. Acknowledgment. I thank Dr. Michael T. Melchior for stimulating conversations and for providing a preprint of ref 1. References and Notes (1) Melchior, M. T.; Vaughan, D. E. W.; Pictroski, C. F. J. Phys. Chem. 1995, 99, 6128. (2) Loewenstein, W. Am. Mineral 1954, 39, 92. (3) Engelhardt, G.; Michel, D. High Resolution Solid-State NMR of Zeolites and Related Systems; Wiley: London, 1978. (4) Vega, A. J. ACS Symp. Ser. 1983, No. 218, 217. (5) Breck, D. W. Zeolite Molecular SieVes; Wiley: London, 1974. (6) Robson, H. E.; Shoemaker, D. P.; Ogilvie, R. A.; Manor, P. C. AdV. Chem. Ser. 1973, No. 121, 106. (7) Klinowski, J.; Ramdas, S.; Thomas, J. M.; Fyfe, C. A.; Hartman, J. S. J. Chem. Soc., Faraday Trans. 2 1982, 78, 1025.
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