J. Phys. Chem. 1995,99, 8379-8381
8379
Effect of Faulting on Sorption Capacities of Microporous Solids J. M. Newsam* and M. W. Deem? Biosym Technologies Inc., 9685 Scranton Road, San Diego, Califomia 92121 -2777 Received: February 22, 1995@
The effective sorption capacity of a crystalline microporous material with one-dimensional channels statistically blocked by planar faulting is calculated. This capacity depends on the channel length. Even low faulting levels cause a dramatic reduction in the accessible micropore volume for channel lengths typical of those in zeolitic materials such as natural gmelinites.
1. Introduction Planar faults are common in zeolites and related crystalline microporous solids.IJ Advances in diffraction and electron microscopy experimentation, coupled with improved methods for interpreting such analytical data on the basis of particular faulting models, have resulted in an increasing number of successful characterizations. Recent topical examples include zeolite p,3materials in the FAU-EMF and ABC-6 framework fa mi lie^,^ ferrierites,6 SSZ-26 and SSZ-33,' babelite,* and, most recently, the titanosilicate ETS-10.9 Although we can now simulate powder or single-crystal diffraction patterns from such systems straightfonvardly,I0 as far as we are aware there has not yet been as detailed a description of the manner in which planar faulting is manifested in other experimental characteristics. We provide here an analysis of the reduction in sorption capacity of one-dimensional channels in a crystalline microporous solid with pore-blocking planar defects.
2. Development Planar faults can influence the sorptive characteristics in any one of several ways: (i) they can have little influence on the overall accessibility or capacity, but alter the pore architecture, accessibility, or diffusional constraints; (ii) they can reduce the limiting dimensions of pore windows while leaving the total pore volume unaffected; (iii) they can block channels. It is evident that, if we describe the last quantitatively, the second follows naturally; smaller molecules will sample the full volume, while larger molecules will suffer the same constraints as deduced for the complete pore-blocking example. The analysis is presented for the one-dimensional channel case. The same analysis applies also to 2D and 3D channel systems in which planar faulting blocks channel access along one unique direction. 2.1. Single Direction Access. We consider first the case of a one-dimensional pore system in which access is possible from only one entry direction. This scenario is pertinent to configurations such as zeolite membranes, sensors based on electrodesupported zeolites, binderless zeolite-on-zeolite aggregates, or, on longer length scales, for electrochemicallyproduced ceramic membrane supports. The single-end case (Figure 1) can be pursued analytically. We define 1 - p to be the per sheet probability of fault occurrence and assume no correlation between the faults in different layers. Then the probability of the accessible channel length being zero, p(O), is the probability of a fault in the f i s t Present address: Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138. Abstract published in Advance ACS Abstracts, April 15, 1995. @
(c) Figure 1. The gating influence of planar faults on the effective sorption capacities of a material with a one-dimensional pore system; (a) supported or single-direction access, (b) dual-directional access with one fault, (c) dual-directional access with two faults.
cell, (1 - p ) . The probability of the length being i , for i > 0, p(i), is the probability of a fault in the ith unit cell multiplied by the probability that each of the preceding cells has no fault, p(i) = pi(l - p ) . We are particularly interested in the accessible region of a channel of length n unit cells. The expectation channel length, (L), for a total of n cells is given by n-l
The second term is the probability that all of the cells will be
0022-365419512099-8379$09.00/0 0 1995 American Chemical Society
Newsam and Deem
8380 J. Phys. Chem., Vol. 99, No. 20, 1995 fault-free. This sum can be rewritten as
h
3 *e
(L), = pl-p" 1-P
(3)
-
2a Q
which tends to p/(l - p ) as n -. Disorder, either in the overall size or aspect ratio of the crystallite in question, can be sampled by this average formulation. As in the case of diffraction simulations, statistical disorder about this mean has little effect on the statistical pore capacity results, other than at very short channel lengths. Explicitly,
u
0
9
0 0.00
0.20
0.40
0.60
0.80
1.00
Fault Probability
In the limit of n
-
-, this
Figure 2. Accessible sorption capacity, ((L)Jn)plotted against poreblocking fault probability, (1 - p ) , for a system with dual-access channels of length 20, 100, 200, and 500 unit cells. The curves uniformly decrease with increasing n. The lines are analytical results from eq 8. The squares are Monte Carlo simulation results.
fluctuation tends to
-
2.2. Dual Direction Access. In a more typical configuration, the one-dimensional pores are accessible from both ends of a crystallite (Figure la). When faults are rare, they have little influence on the sorption capacity because the bidirectional pore access allows essentially all of the pore volume to realized (Figure lb). At higher faulting probabilities (Figure IC), the likelihood of each crystallite or pore having more than one poreblocking defect becomes significant. The interior region between the two faults becomes inaccessible, and the sorption capacity is substantially reduced. The higher the prevalence of faulting, the lower the effective adsorption capacity of the material. The bidirectional access case is pursued analytically as follows. The probability of an entire channel of n cells being fault free, p(n), is p". The probability, p(n-1), of the accessible number of cells in this total being (n - 1) is the probability of a single fault, n( 1 - PIP-';the factor n reflects the n possible locations for this single fault. The probability, p(n-2), of the accessible number of cells being (n - 2) is the probability of two faults side by side, (n - 1)(1 - p)2P-2.In general the probability of the accessible number of cells being i is p ( i ) = (i
+ 1)(1 -p12pi;
iPn
-2
n-2
+
+ (n - l > n ( l - p)pn-' + npn
i=O
+ (n - 2n2 - 2p - 4np + 4n2p + 4p2 + 3np2 - 2n2p2)pn+ (2n2p - n2 - 4np - 4p2 +
( ( L - (L)J*),,= [2p
2 2
2n
4np2 - n p ) p 141 - P)* (10) In the limit of large n, this fluctuation tends to
2.3. Monte Carlo Simulations. We perform Monte Carlo simulations to illustrate these results. For the simulations, we define a succession of single pores of predefined length, introduce a fault at each position along the channel according to the specified statistical faulting probability, and then inspect the resulting model for the total number of unit cells traversed before encountering a fault. For the single-directionaccess case, the sampling is made from only a single entry direction; in the bidirectional case the sampling is made from both ends. 3. Results and Conclusion
(6)
To obtain the expectation length, (&, we multiply p ( i ) by i for each i from 0 to n - 2 and add explicitly the p(n) and p(n-1) terms
(L), = Ci(i 1)(1 - p>$'
Note that the average accessible length for a two-sided tube is twice that of a single-ended tube in the limit n -. Again, the fluctuations in this length can be evaluated with the result
(7)
Evaluating this sum, the expectation accessible length for n cells becomes
This average has the limiting form
(9)
Evaluations of the accessible volume according to expression 8 for channels of length n = 20, 100, 200, and 500 unit cells, corresponding for zeolite gmelinite to c-axis dimensions of some 0.02,O.1, 0.2, and 0.5 pm, are presented in Figure 2. The results of rapid Monte Carlo evaluations for these same parameters, and using 50 repetitions in the Monte Carlo sampling at each increment in faulting probability, are overlaid on the analytical results in Figure 2. For each parameter set the agreement between analytical and Monte Carlo results clearly confirms the appropriateness of eq 8. The statistical spread of the Monte Carlo results is an inherent property of the model, as suggested by eqs 4 and 10, and would be reduced with more extended samplings. The present results can aid in the interpretation of sorption data both from materials whose framework structural principles are reasonably known and from samples that are potentially multiphasic or show variable sorptive characteristics. Offretites' and gmelinitesi2 are classic examples. For gmelinites, although natural crystallites vary from submicron up to some
Effect of Faulting on Sorption Capacity cm in dimension^,'^ sorption experiments are typically performed on polycrystalline materials composed of smaller crystallites, likely of not much less than 0.5 ym crystallite dimension along the c-direction. The observed factor of some 10 in the reduction of n-alkane capacity over that expected for the unblocked ~ t r u c t u r e ~ ~is' *thus . ~ ~ indicative of a faulting probability of around 0.04 (Figure 2). Better quantitation beyond this rough estimate would require sorption uptake measurements on samples for which quality SEM, HRTEM, or diffraction data are available. The analytical results apply also to other phenomena in which the magnitude of a macroscopic observable depends in some fashion on the extent of contiguous length in a one-dimensional system that can be sampled from one or both ends. Carbon nan~tubules,'~ ion ~ ' channels, ~ conducting polymers, data communication lines, and vascular transport are illustrative of such systems.
Acknowledgment. The Biosym Catalysis and Sorption Project is supported by a consortium of industrial, academic, and govemment institutions; we thank the membership for their guidance, input, and many stimulating discussions. References and Notes (1) Barrer, R. M. Zeolites and Clay Minerals as Sorbents andhlolecular Sieves; Academic Press: London, 1978.
J. Phys. Chem., Vol. 99, No. 20, 1995 8381 (2) Newsam, J. M.; Treacy, M. M. J.; Vaughan, 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. (3) Newsam, J. M.; Treacy, M. M. J.; Koetsier, W. T.; deGruyter, C. B. Proc. R. SOC.(London) 1988, A420, 375-405. (4) Treacy, M. M. J.; Vaughan, D. E. W.: Strohmaier, K. G.; Newsam, J. M. To be submitted. (5) Lillerud, K. P.; Akporiaye, D. M.; Szostak, R. M. J . Chem. SOC., Faraday Trans., in press. (6) Rice, S. B.; Treacy, S. B.; Newsam, J. M. Zeolites, in press. (7) Lobo, R. F.: Pan, M.; Chan, I.; Li, H. X.; Medrud, R.; Zones, S. I.; Crozier, P. A.; Davis, M. E. Science 1993, 262, 1543. (8) Szostak, R. M.; Lillerud, K. P. Submitted. (9) Anderson, M. W.; Terasaki, 0.;Ohsuna, T.; Philippou, A,; Mackay, S. P.; Ferriera, A.; Rocha, A.; Lidin, S. Nature 1994, 367, 347-351. (10) Treacy, M. M. J.; Newsam, J. M.; Deem, M. W. Proc. R . SOC. (London) 1991, 433, 499-520. (1 1) Chen, N. Y.; Schlenker, J. L.; Garwood, W. E.; Kokotailo, G. T. J. Catal. 1984, 86, 24-31. (12) Barrer, R. M. Trans. Faraday SOC.1944,40, 555-564. (13) Tschemich, R. W. Zeolites of the World; Geoscience Press: Phoenix, AZ, 1992. (14) Breck, D. W. Zeolite Molecular Sieves: Structure Chemistry and Use; Wiley (reprinted by R. E. Krieger: Malabar, FL, 1984): New York, 1973. (15) Iijima, S.; Ichihashi, T. Nature 1993, 363, 603-605. (16) Bethune, D. S.; Kiang, C.; de Vries, M. S.; Gorman, G.; Savoy, R.; Vazquez, J.; Beyers, R. Nature 1993, 363, 605-607. JP9505082
