J. Phys. Chem. 1991,95, 5558-5560
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Random Walk through TweChannel Networks: A Simple Means To Correlate the Coeffkktnts of Anlsotropic DMusion in ZSM-5 Type Zeolites Jiirg Karger Sektion Physik der Universitdt Leipzig, 7010 LinnPstrasse 5, Leipzig, Germany (Received: October 2, 1990)
A three-dimensional random walk through a network of two intersecting channel systems is shown to be characterized by a diffusion tensor with interdependent principal elements. By application of this model to ZSM-5 type zeolites, the predicted relations are found to be in agreement with the results of both MD calculations and initial, PFG NMR diffusion experiments with oriented zeolite crystallites.
Introduction Diffusion anisotropy may occur as an immediate consequence of an anisotropic arrangement of the diffusants as is well-known from anisotropic solids' and liquid crystals.* In such cases the prediction of the anisotropy parameters necessitates a detailed knowledge of the interaction energy between the diffusants. A completely different origin of diffusion anisotropy may occur with molecules adsorbed on microporous crystals (zeolites).*5 In such "molecular sieves" diffusion anisotropy is brought about by the interaction between the diffusants and the zeolite framework, rather than by the mutual interaction of the diffusants. In most cam, zeolites represcnt an arrangement of interconnected cavities of molecular dimension, and diffusion of the molecules adsorbed therein becomes anisotropic as soon as the escape rate out of these cavities is different for different directions. Diffusional anisotropy is then determined by the interaction energy between the diffusants and the zeolite framework along the different diffusion paths. In the last few years a certain class of microporous crystals, the ZSM-5 type zeolites (and their aluminum-free analogues, silicalites6)have attained especial con~enr.4~ In these adsorbents diffusion anisotropy may be considered to be brought about by the steric confinement exerted on the diffusants by a system of interconnected channels, leading to the effect that the diffusivities in the three principal directions of the diffusion tensor are no more uncorrelated. In the present communication a correlation between the principal tensor elements will be derived by simple random walk arguments and compared with experimental data on diffusion anisotropy in microporous crystals. Random Walk Model The framework of ZSM-5type zeolites forms a system of intersecting channels composed of slightly elliptical (diameters -5.1 and 5.7 A), straight channels along y , cross-linked by nearly circular (-5.4 A), zig-zag channels along x . Figure 1 shows the axes of these channels within three adjacent unit cells. Obviously, molecular propagation along z, i.e. in the direction perpendicular to the directions of the two existing channel systems, may only proceed in alternating periods of migration through the segments of both channel systems. This fact is illustrated by Figure 2, showing a view along the x axis of the channel system: Succeeding steps between the intersections in the z direction do only lead to a resulting horizontal diffusive motion if between two steps the diffusant has carried out one (or 3, 5 , 7, ...) step(s) in the y direction. A random walk implies the assumption that the "memory" of the diffusants is sufficientlyshort so that a molecule being situated ( I ) Miller, P. H.;Banh, F. R. Phys. Rm. 1941, 59, 943. (2) KrUger, G. J. Phys. Rep. 1982, 82, 229. (3) Breck, D.W. Zeolite Molecular Sieuds; Wiley: New York, 1974. (4) Ruthven, D. M. Principles of Adsorption ond Adsorption Processes; Wilev: New York. 1984. (5) Mtier, W. M.;Olson, D. H. Atlas of Zeolite Structure Types; Butterworth: Stoneham, U.K., 1988. (6) Flanigen, E. M.; Bennett, J. M.; Grose, R. W.; Cohen, J. P.; Paton, R. L.; Kirchner, R. M.;Smith, J. V. Nature 1978, 271, 512.
0022-3654/91/2095-5558$02.50/0
within one channel intersection (network vertex) will proceed to one of the four adjacent channel segments with a probability independent of the history, i.e. independent of the channel segment from which it has come from. This assumption corresponds to the basic supposition of Knudsen diffusion4 that molecules striking a solid surface are reemitted in a random way and is justified by the random character of momentum transfer between the diffusants and the solid as well as, in the present case, between the individual diffusants. The probability that the succeeding intersection on the diffusion path is connected with the preceding one by a straight channel shall be p l . Correspondingly, the probability that a diffusion step is along a zig-zag channel segment is p2 = 1 - p l . The interdependence of the diffusivities in different directions may be derived from the mean-square displacements in the three principal directions during a certain time interval t . For this purpose we introduce the number of steps between adjacent vertices along straight channel segments (nl) and along zig-zag channel segments (n2),which are related to each other by 4/n2 =PdP2 (1) Thus,from elementary random walk arguments the mean-square displacements during t along x and y are
(A?) = n * ( ~ / 2 ) ~ (P) = nl(b/2)* (2) where we have taken into account that b/2 coincides with the step length between adjacent vertices along the straight channels and that a/2 is the x component of the displacements during steps along the zig-zag channels (cf. Figure 1). In contrast to the displacements in the x and y directions, succeeding steps in the z direction are not uncorrelated, and on calculating the meansquare displacements after n2 steps along segments of the zig-zag channels we must also consider the cross terms. In analo y to the treatment of correlation effects in solid-state diffusion'* one thus has
i
(22)= ((21 + 22 + =
+
n2(z,Z) 2(n2
..I
+ 2"y)
- ~ ) ( Z , Z , + ~ )+ 2(nz - 2)(z,z,+,) + ... + 2 (Z/Z,+,'
(3)
where z, denotes the z component of the ith step along a zig-zag channel segment. For the evaluation of the cross terms we introduce the probabilities p+ and p- (= 1 p+) that succeeding steps along segments of the zig-zag channels are carried out in the same or in opposite directions, respectively. The probability of any particular sequence of k steps in the z direction is given by a product of the type P+~-'p-l. For I = even, the directions of the first and kth steps will be identical, for 1 = odd they will be opposite to each other. One may write, therefore
-
(Z,Zl+k)
= (P+ - P.-Y(C/2)*
(4)
(7) Manning, J. R. Phys. Reu. 1959, 116, 819. (8) Manning, J. R. D~fusionKinettcsfor Atoms in Crysrals; van Nostrand: Princeton, 1968.
0 1991 American Chemical Society
The Journal of Physical Chemistry, Vol. 95, No. 14, I991 5559
Random Walk through Two-Channel Networks
TABLE I: Priacipnl Ekwab of the DiffigiOa Tcllror of Ad"te Molecuks In ZSM-5 Type Zcdita AS by MD Calculations md Vdw of DzFOIIOWIO~ from Dx rad D, O. the Barb of the Random Walk Model (Eq 12) adsorbate xenonio methaneI2 methane" a=
a.mA
TIK 298 298 400
D, 1.3 3.6 27
D. 4 14.9 31.6
D./lV m 2 k 0.28 1.2 6.3
D.
(a12) 0.44 1.3 6.5
c
Figure 1. Outline of channel network of Z S M J type zeolites over three adjacent unit cells. The straight and zig-zag channels are oriented along the y and x directions, respectively.
This result corresponds to the fact that the correlation between succeeding steps in the z direction decreases (corresponding to correlation factors approaching one) with increasing jump prob ability along the straight channels. The diffusivities in the principal directions are related to the mean-square displacements by Einstein equations ( S )= 2Dxt (P) = 2 0 , ~ (22)= 2D,t (11) Thus, combining eqs 2 and 9, one obtains (C/2Y -=Dz
0,
(12)
This equation implies the simple result that the mean diffusion time in the z direction is the sum of the diffusion times in x and y directions over the same multiples of the elementary unit. The discussion of this relation is facilitated by considering diffusivities normalized with respect to the step lengths in the principal directions:
b
Yt
+-( bD/Y2 I 2
D,(") = D , / ( U / ~ ) ~Dy(n) = D y / ( b / 2 ) 2 DP)= D , / ( C / ~ ) ~
'
leading to l / D p ) = l/D,(") + l / D / )
Z Figure 2. Projection of channel network of ZSM-5type zeolites onto the y-z plane spanned over an area of 3 X 3 unit cells.
where c / 2 (cf. Figure 1 ) is the z component for steps along the zig-zag channels. Inserting eq 4 into eq 3 yields for sufficiently large step numbers n2
In the limiting cases of no correlation @+ = p- = 1 / 2 ) and of totally correlated migration within a single zig-zag channel (p+ = 0), eq 5 degenerates to the trivial cases
(in analogy to eqs 2) and (22)= 0, respectively. The probabilities p+ and p- depend on the number of steps along the straight channel segments between succeeding steps along zig-zag channel segments. If there is an even number of steps in the direction of the straight channels, the following step along the zig-zag channel will be carried out with a z component opposite to that of the preceding step in z direction. Summing the prob abilities of all these jump sequences yields P-(= 1
(13)
- P+) = p2 c (PI2)" n-0
I
=2-Pz
(7)
from which one obtains P+-P-=
-P2 2-Pz
and finally, inserting these results into eq 5 yields
(22)= ~
Z ( C / ~ ) ~ P ~
(9)
Equation 14 implies the following, easily comprehensible conclusions: (i) The (normalized) diffusivity in the z direction is smaller than in either of the other principal directions. (ii) For a given mean diffusivity l/z(D,(") Dy(a))in the x-y plane, 0,'") is a maximum for D,(") = Dy(n),i.e. for equal (normalized) diffusivities in the two-channel systems. (iii) For different mobilities in the two-channel systems, Dp)approaches the (normalized) diffusivity of the channel system with the lower mobility.
+
Comparison with Experiment In comparison to studies of neat liquids, computer experiments with adsorbate-adsorbent systems are substantially complicated by the fact that molecular dynamics (MD) is determined by the interaction of the diffusants with both each other and the adsorbent f r a m e ~ o r k .To ~ avoid unacceptably long computation times, in initial MD studies of diffusion in zeolites1*l2 the adsorbent framework has therefore generally been assumed to be rigid. Table I summarizes the results of recent MD calculations for the self-diffusion of adsorbed molecules in ZSM-5. Applying cq 12, we use the MD data for D, and Dy to calculate the value of D, to be expected on the basis of the random walk model. For methane, these data are found to be nearly identical; for xenon there is agreement within a factor of two. One may conclude, therefore, that despite the assumption of a rigid framework, MD calculations imply the correct randomization of molecular migration. In ref 13 this effect has been shown to be brought about by the mutual interaction of the diffusants. In agreement with (9) Allan, M. P.; Tildaley, T. S. Computer SImulatlon of tlqulds; Clarendon: Oxford, 1987. (10) Pickett, S.D.; Nowak, A. K.; Thomas, J. M.;Peterson, B. K.; Swift, J. F. P.; Cheetham, A. K.;den Ouden, C. J. J.; Smit, B.; Port, M.F. M.J. Phys. Chem. 1990,94, 1233. (1 1) June, R. L.; Bell, A. T.; Theodorou, D. N. J . Phys. Chem. 1990,94,
..---.
87'13
(12) Demontis, P.; Fois, E. S.; Suffritti, G. B.; Quartieri, S. J. Phys. Chem.
Combining q s 6 and 9 yields the correlation factor7**
f = (m/(m"nc =PI
(14)
199'0, 94,4329.
(10)
(13) Fritzsche, S.; Hakrlandt, R.; Urger, J.; Pfeifer, H.; Wolfsberg, M. Chem. Phys. Lett. 1990, 171, 109.
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J. Phys. Chem. 1991,95, 5560-5567
F m OlMCthv I.ZSM-5 at 298 K h Determid by M D C . l d r t k r ud PIG NMR Experilcab lad C o m q d " with E.thrte o f t k R"Walk Model
TABLE U
method MDI2
PFG NMRl6 (powder samples) PFG NMR" (oriented crystals) random walk model
anisotropy factor, 1/2(Dx+ DJDZ 7.7 55
determination of the diffusivity in the direction of the longest of ~ J ~in) the the crystals axes (corresponding to the z d i r e c t i ~ n ~and plane perpendicular to it became possible. Table I1 provides a + Dy)/Dzob comparison between the anisotropy factor 1/2 (0% tained in these PFG NMR experiments with the results of MD calculations (Table I). Combining eqs 2 , 9 , and 11 for the random walk model yields an anisotropy factor
4.c
>4.4
this conclusion model calculations for nonrigid frameworks show that the diffusivities are essentially independent of the framework rigidity.I2 A direct measurement of the principal elements of the diffusion tensor is complicated by the small size of the zeolite crystallites. First attempts to measure orientation-dependent diffusivities in zeolite ZSM-5 have been made by means of pulsed field gradient (PFG) NMR s p e ~ t r o s c o p y . ~In~ *measurements ~~ with methane adsorbed on ZSM-5 in powder samples, no deviation from the pattern of isotropic diffusion could be observed. In this way, a factor of the order of 5 was estimated as an upper limit for the ratio between the diffusivities in different directions.I6 Recently, for the same system PFG NMR measurements have been carried out with oriented ZSM-5 crystals," so that a separate (14) Urger, J.; Pfeifer, H.;Heink, W. Adu. Mag. Reson. 1988, 12, 1. (1 5) KHrger, J.; Pfeifer, H. Zeolites 1987, 7,90. (16) Zibrowius, B.; Caro, J.; KHrger, J. 2.Phys. Chem. (Leipzig) 1988, 269, 1101.
With a = b (cf. Figure l), the anisotropy factor becomes a minimum for pI = p2 = 0.5, attaining a value of 4.4. This result is in complete agreement with the MD calculations, which have led to values above this lower limit. In the PFG NMR measurements with oriented crystallites the anisotropy factors are found to be of the order of this lower limit. A tendency toward anisotropy factors slightly below the values of the random walk model might be related to lattice imperfections in the real zeolite crystallites and/or to a less perfect adjustment of the zeolite crystallites than was assumed in the PFG NMR experiments. Acknowledgment. Stimulating discussions with Prof. Harry Pfeifer are gratefully acknowledged. (17) Hong, U.; Kirger, J.; Kramer, R.; Pfeifer, H.; Milller, U.;Unger, K. K.;Lllck, H.-B.; Ito, T. Zeolites, in press. (18) Price, G. D.; Pluth, J.; Bennett, J. M.;Patton, R. L. J . Am. Chem.
Soc. 1982, 104, 597 1.
(19) Hayhunt, D. T.; Aiello, R.; Nagy, J. B.; Crea, F.; Giordano, G.; Nastro, A.; Lee,J. C. ACS Symp. Ser. 1988. No. 368, 277.
Dependence of the Electrooxldath Rates of Carbon Monoxide at Gold on the Surface Crystabgraphk Orlentation: A ComMned Kinetlc-Surface Infrared Spectroscopk SWY Si-Chung Chug, Antoinette Hamelin,+ and Michael J. Weaver* Department of Chemistry, Purdue University, West Lafayette, Indiana 47907 (Received: October 16, 1990) Rate parameters evaluated by linear sweep voltammetry are reported for the electrooxidation of carbon monoxide on six oriented monocrystalline gold surfaces in acidic aqueous perchlorate electrolytes. At a given electrode potential, the rate amstants depend markedly upon the crystallographic orientation in the sequence Au (1 11) < Au(533) Au(100) S Au(221) < Au(210) < Au( 110). up to 100-fold differencesin rate being observed. Marked inhibition of the notably facile electrode kinetics are observed if 0.1 M H m 4is substituted for 0.1 M HCIO, electrolyte. Parallel voltammetric measurements coupled with real-time surface infrared spectroscopy at four surfaces, Au( 1 1l), Au( loo), Au( 1IO), and Au(210), enable the role of CO reactant adsorption in the surface-dependent catalysis to be assessed. A low-coverage, yet reactive, form of adsorbed CO was detected on Au(ll0) and (210) from a C-O vibrational ( v a ) band at 2100-21 15 cm-' which disappears at the onset of the voltammetric wave. The sequence of CO surface concentr@ons as discerned from infrared spcstmcopy, Au( 111). Au(100) < Au(ll0) < Au(210), differs from the above reactivity sequence, signaling the presence of additional factors in the electrocatalysis. Relationships are explored between the surface-dependentrates and the potentials of zero charge or the density of "broken bonds" in the surface lattice (Le., the average surface coordination number). These correlations suggest that the rate-determining electrooxidation step between coadsorbed CO and H20(or OH) species is favored either at step sites or on rows of low-coordination metal atoms, such as on Au( 110). This is speculated to be due to the ability of such sites to engender CO adsorption and nearby H20 (or OH) coadsorption.
-
An emerging topic in surface electrochemistry involves examining the sensitivity of electrochemical reaction kinetics to the surface crystallographic orientation.' The practicality of such experiments has increased substantially in recent years with the advent of reliable procedures for preparing ordered singlecrystal e l e c t r ~ d e s . ~A * ~central motivation behind these studies is to understand the manner and extent to which electrocatalytic 'Permanent address: Labontoire d'Electroohimie Interfaciale du C.N.R.S., I , Place A. Briand, 92195 Meudon, France. OO22-3654/91/2095-5560$02.50/0
processes are influenced by the surface bonding and stereochemical factors. For this purpose, it is highly desirable to supplement macroscopic kinetic data with microscopic information on the adsorbed intermediates as can be furnished by surface spectra(1) For a recent miew, we: Adzic, R. In Modern Aspects of Electmchrmfstry; White, R.E., Bock&, J. O'M.,Conway, B. E.,EQ.;Plenum Pre3.s
New York, 1990, Vol. 21, Chapter 5. (2) Hamelin, A. In Modem Aspects of Electrwhemistry; Conway, B. E., White, R. E., Bockris, J. O'M.,Eds.;Plenum Pruur: New York, 1986; Vol. 16, Chapter 1. (3) Clavilier, J. ACS Symp. Ser. 1988, 378, 202.
Q 1991 American Chemical Society
