J. Phys. Chem. 1992,96,6027-6030 (8) Ludeiia, E. V. J . Chem. Phys. 1978,69, 1770. (9) Ludeiia, E. V.; Gregori, M. J. Chem. Phys. 1979, 71, 2235. (10) Gorecki, J.; Byers Brown, W. J. Phys. B At. Mol. Opt. Phys. 1988, 21, 403. (1 1) Ley-Koo, E.; Cruz, S.A. J. Chem. Phys. 1981, 74, 4603. (12) Le Sar, R.; Herschbach, D. R. J . Phys. Chem. 1981,85, 2798. (13) Gorecki, J.; Byers Brown, W. J . Chem. Phys. 1988,89, 2138. (14) Baldini, G. Phys. Reu. 1964, 136, A248. (15) Whittaker, E. T.; Watson, G. N. A Course of Modern Analysis, 4th ed.;Cambridge University Press: London, 1962; Chapter (XVI). (16) Condon, E. U.; Shortley, G. H. The Theory of Atomic Spectra; Cambridge University Press: London, 1963; Chapter V. (17) Slater, L. J. Confluent Hypergeometric Functions; Cambridge University Press: London, 1960.
6021
(18) Abramowitz, M., Stegun, I. A. Eds. Handbook of Mathematical Functions, Dover Publications Inc.: New York, 1972; Chapters 13 and 14. (19) Bransden, B. H.; Joachain, C. J. Physics of Atoms and Molecules; Longman Group UK Ltd.: London, 1983. (a) p 147; (b) p 55 and Appendix 5; (c) p 245. (20) Fernlndez, F. M.; Castro, E. A. In?. J . Quantum Chem. 1982, 21, 741. (21) Fernlndez, F. M.; Castro, E. A. J . Math. Phys. 1982, 23, 1103. (22) Fernlndez, F. M.; Castro, E. A. Hypemirial Theorems; Springer Verlag: Berlin, 1987. (23) Mizushima, M. Quantum Mechanics of Atomic Spectra and Atomic Structure; Benjamin: New York, 1970. (a) Chapter 5; (b) Chapter 9. (24) Hirschfelder, J. 0.;Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954; p 914.
Effectlve Dimensionality of Layered Diffusion Spaces Roberto A. Garza-Upez and John J. Kozak* Department of Chemistry, Franklin College of Arts and Sciences, University of Georgia, Athens, Georgia 30602 (Received: December 23, 1991)
The stochastic master equation dpi/dt = -&Cupj for the evolution of the probability pi, with transitions between nearest-neighbor sites governed by the transition matrix Gii' is solved for finite k X k X k cubic lattices, subject to the temporal boundary condition pi(0) =,,6 m being an interior slte of the lattice, and two sorts of spatial boundary conditions (periodic and strictly confining). From the regime where the entropy function S(t) = - [ z i p i ( ? ) In pi(?)] grows linearly with In t , we extract the spectral dimension d, of the lattice. It is found that the Euclidean dimension de of the defining k X k X k lattice effectively converges to de 3 when k = 11. Taken together with the results reported earlier on k X k square-planar lattices and insights drawn from calculations of the mean walk length ( n ) before trapping of a random walker on finite de = 3 , 4 cubic lattices subject to periodic boundary conditions, the study allows the identification of the minimal size of diffusion space such that, in studying diffusion-mediatedprocesses in microheterogeneous media, the dynamics may be sensibly described by a Fickian equation with a Laplacian defined for integer dimension.
-
I. Introduction In classical, continuum theories of diffusion-reaction processes based on a Fickian parabolic partial differential equation of the form
ac(T,t)
= DV2C(7,t)+ f [C(T,t)] at specification of the Laplacian operator is required.' Although this specification is immediate for spaces of integral dimension, the problem becomes rather more complicated for spaces of intermediate or fractal As examples of problems in chemical kinetics where the relevance of theoretical predictions using an approach based on eq 1 is open to question, one can cite the avalanche of work reported over the past decade on diffusion-reaction processes in microheterogeneous mediaH (zeolites, clays, and organized molecular assemblies such as micelles and vesicles) where the (local) dimension of the reaction space is (often) not clearly defined. It is to examine one aspect of this general problem that the work reported in this paper is directed, To introduce the problem considered here, it has been shown in earlier w ~ r k ~that . ' ~the entropy function S(t) = - [ z i p i In pi] constructed from the solution of the stochastic master equation (withthe initial boundary condition that the probability is localized on a single site) increases linearly with In t (t time) (see the discussion in section 11). The slope of the linear regime is dJ2, where d, is the spectral dimension of the underlying lattice. If attention is focused on Euclidean lattices, one is assured that the dimension de of the Euclidean space, the fractal (Hausdorff) dimension d,, and the spectral dimension d, are equal.11J2Consider, then, a squareplanar lattice (coordination number or valency v = 4); in calculations reported in ref 9, we showed numerically that d, = 2.00, already for lattices 21 X 21. Then, in ref 10, in our lattimtatistical study of diffusion-controlled processes in smectite clays, we showed by studying a series of layered square-planar lattices of increasing spatial extent that the Eu0022-3654/92/2096-6027$03.00/0
-
clidean dimension of the expanded (two-layer) system remained effectively de 2. Now, intuitively, one expects that if square-planar lattice layers continue to be stacked one above the other (generating, eventually, a lattice of cubic symmetry, i.e., one for which each site has a valency v = 6), the dimensionality of the space should converge, eventually, to d, = d f = de = 3. The chief aim of the present work is to determine the (minimal) size of k X k X k cubic lattices such that an effective Euclidean dimension de 3 is realized. Our calculations are carried out assuming first periodic boundary conditions and then strictly c o n f i i g boundary conditions, the latter class of spatial boundary conditions defined operationally such that if the randomly diffwing particle is on a boundary site of the defining k X k X k lattice, its next displacement can be only back to the same site, to an adjacent boundary site, or to an internal site of the lattice. Once this "minimal" cubic lattice has been determined, the expectation is that one will then have a means of gauging the validity of theoretical studies of diffusion-reaction processes occurring in compartmentalized systems of finite spatial extent when analyzed using an approach based on eq 1, with the Laplacian defined for integer dimension.
-
11. Formulation In recent work, we have solved numerically the stochastic master equation
dpi(t)/dt = - [ ? G , j ~ j ( t ) I
(2)
for various Euclidean and fractal lattices subject to periodic (or confining) spatial boundary conditions and to the temporal boundary condition pi(t=O) = 6im (3) where m is an interior site of the lattice?JO Here, the matrix Cij describes the transition rate of the probability p i ( t ) to the site i from a nearest-neighbor site j . Thus, if Mij is the probability of 0 1992 American Chemical Society
Garza-LSpez and Kozak
6028 The Journal of Physical Chemistry, Vol. 96, No. 14, 1992 TABLE I: Effective Dimensionality of time range pts slope 2 0.8940599 0.05-0.10 0.05-0.15 3 0.94148729 0.05-0.20 4 0.96514414 0.05-0.25 5 0.97325305 0.05-0.30 6 0.97089867 7 0.96143504 0.05-0.35 8 0.947 16693 0.05-0.40 0.05-0.45 9 0.92971302 0.05-0.50 10 0.91022936 11 0.889 547 71 0.05-0.55 12 0.868 267 39 0.05-0.60 13 0.84681927 0.05-0.65 14 0.825 51069 0.05-0.70
the Lattice 3 corr coeff 1.ooo 000 0.999 192 57 0.999 165 97 0.999 378 60 0.999 532 25 0.999 456 00 0.999 054 03 0.998 277 98 0.997 111 17 0.995 558 16 0.993 637 59 0.991 377 05 0.988 809 36
X
3 X 3" eff dimen
1.946 506 1
"In N(calc) = 3.295836. TABLE II: Effective Dimensionality of the Lattice 5 X 5 X 5" time range pts slope corr coeff eff dimen 0.05-0.10 2 0.996 15522 1.000000 0.05-0.15 3 1.0836373 0.99793023 0.05-0.20 4 1.146 9090 0.996 947 20 0.05-0.25 5 1.1932218 0.99657891 0.05-0.30 6 1.2270532 0.996 566 39 0.99675495 0.05-0.35 7 1.251 3961 8 1.268 3598 0.997 046 68 0.05-0.40 0.997 376 94 0.05-0.45 9 1.2794899 0.05-0.50 10 1.2859512 0.997701 86 11 1.2886427 0.997991 03 2.5772854 0.05-0.55 12 1.2882702 0.998223 17 0.05-0.60 13 1.285 3965 0.998 383 38 0.05-0.65 0.998461 33 0.05-0.70 14 1.2804760
" In N(calc)
with Mii = 0 and M . . = l / v j for i # j . From the solutions generated, it was founa after an initial interval of time (- 1) that the function
grows linearly with In t ; for finite lattices, departures from this linear behavior occurred (eventually) on longer time scales and S(t) was found to approach asymptotically the limiting value In N, where N is the number of sites in the fundamental lattice unit. Let R, be the number of distinct sites visited in an n-step random walk and let Pk be the probability of visiting site k in an n-step walk (where Pk = ik/nif site k is visited ik times); in earlier work,I3-l6 it had been established numerically that the spectral dimension d, d, In R, -=n - w 2 In n could be computed by calculating the quantity -[Ck(PkIn P k ) ] / l n n, Le., that (7)
This seminal observation led us to suggest (and ~ e r i f y ~ the ,'~) following relation for t
s
(8) in the linear regime noted above. As was pointed out in ref 9, the significant advantage gained in using the relation (8) to determine d, is that earlier determinations of ds based on eq 7 required very lengthy Monte Carlo simulations, whereas the results obtained based on solution of the stochastic master equation for the same problem could be derived in a very time-efficient way. 1
1.000 000 0.997 768 5 1 0.996 578 64 0.995 978 12 0.995 726 72 0.995 687 00 0.995 776 17 0.995 942 66 0.996 153 51 0.996387 17 0.996 629 34 0.996 870 35 0.997 103 54 0.997 324 26 0.997 529 16 0.99771576 0.997 882 22 0.998 027 07 0.998 149 17 0.998 247 55 0.998 321 43 0.998 370 15 0.998 393 15 0.998 389 95
2.803 1932
TABLE I V Effective Dimensionality of the Lattice 9 X 9 X 9"
going from site j to a nearest-neighbor site i in a single displacement and uj is the coordination number (or valency) of the lattice site j , then G.. IJ = (6.. V - M IJ, , ) Jv . (4)
ds/2 = S(t)/ln t
of the Lattice 7 X 7 X 7a corr coeff eff dimen
"In N(calc) = 5.836 122.
= 4.828 3 1 1
k
TABLE 111: Effective Diwnsklity time range pts slope 2 0.05-0.10 1.OOO 7286 0.05-0.15 3 1.092 3015 0.05-0.20 4 1.1609415 0.05-0.25 5 1.213 7976 0.05-0.30 1.2552216 6 7 0.05-0.35 1.288 0673 0.05-0.40 1.3143036 8 9 0.05-0.45 1.335 3382 10 1.352 2049 0.05-0.50 1.365 6803 0.05-0.55 11 0.05-0.60 1.376 3584 12 1.384 7008 13 0.05-0.65 1.391 0716 0.05-0.7 0 14 0.05-0.75 15 1.3957616 0.05-0.80 16 1.3990061 1.4009979 0.05-0.85 17 1.401 8966 0.05-0.90 18 1.401 8356 0.05-0.95 19 1.4009276 20 0.05-1 .OO 1.399 2693 21 0.05-1.05 1.3969439 22 0.05-1.10 1.3940235 23 0.05-1.15 1.390 5711 24 0.05-1.20 0.05-1.25 25 1.386 6428
time range 0.05-0.10 0.05-0.15 0.05-0.20 0.05-0.25 0.05-0.30 0.05-0.3 5 0.05-0.40 0.05-0.45 0.05-0.50 0.05-0.55 0.05-0.60 0.05-0.65 0.05-0.70 0.05-0.75 0.05-0.80 0.05-0.85 0.05-0.90 0.05-0.95 0.05-1.00 0.05-1.05 0.05-1.10 0.05-1.15 0.05-1.20 0.05-1.25 0.05-1.30 0.05-1.35 0.05-1.40 0.05-1.45 0.05-1.50
pts 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 21 28 29 30
slope 1.0008598 1.092 6527 1.161 6727 1.215 1056 1.257 3326 1.291 2306 1.3187849 1.341 4123 1.360 1509 1.375 7763 1.388 8777 1.399 9079 1.4092199 1.4170908 1.423 7408 1.429 3462 1.4340490 1.437 9650 1.441 1889 1.443 7996 1.445 8623 1.4474320 1.448 5554 1.449 2725 1.4496176 1.449 6204 1.449 3072 1.448 7009 1.447 8222
corr coeff 1.ooo 000 0.997 759 25 0.996 552 20 0.995 926 06 0.995 640 90 0.995 560 14 0.995 602 3 1 0.99571751 0.995 874 53 0.996 053 71 0.996 242 64 0.996 433 53 0.996 621 52 0.996 803 7 1 0.996 978 36 0.997 144 58 0.997 301 94 0.997 450 29 0.997 589 69 0.997 720 27 0.997 842 19 0.997 955 63 0.998 060 75 0.998 157 68 0.998 246 52 0.998 327 34 0.998 400 18 0.998 465 03 0.998 521 88
eff dimen
2.899 2408
"In N(ca1c) = 6.563 695.
For example, Pitsianis et al.14 performed a Monte Carlo simulation on a Sierpinski gasket using 1OOOOOrandom walkers on a gasket of 29 526 sites and obtained a value of the spectral dimension of 1.354 (as compared to the exact value of 1.365), whereas in our stochastic approach? we obtained a value of 1.367 by solving the stochastic master equation on a gasket of 366 sites. 111. Results
Reported Tables I-V are the results obtained for the effective Euclidean dimension deof the lattice spaces 3 X 3 X 3, 5 X 5 X 5 7 x 7 x 7 , 9 X 9 X 9,and 11 X 11 X 11. As hasbeen brought out in our earlier discussion, the value of d, (and hence de for Euclidean lattices) is determined from the slope of the cume, S vs In t . By inspection of the results generated for S vs In t for
Effective Dimensionality of Layered Diffusion Spaces
The Journal of Physical Chemistry, Vol. 96, No. 14, 1992 6029
TABLE V: Effective Dimensionality of the Lattice 11 X 11 X 11“
time range 0.05-0.10 0.05-0.15 0.05-0.20 0.05-0.25 0.05-0.30 0.05-0.35 0.05-0.40 0.05-0.45 0.05-0.50 0.05-0.55 0.05-0.60 0.05-0.65 0.05-0.70 0.05-0.75 0.05-0.80 0.05-0.85 0.05-0.90 0.05-0.95 0.05-1 .OO 0.05-1.05 0.05- 1.10 0.05-1.15 0.05-1.20 0.05-1.25 0.05-1.30 0.05-1.35 0.05-1.40 0.05-1.45 0.05-1.50 0.05-1.55 0.05-1.60 0.05-1.65 0.05-1.70 0.05-1.75 0.05-1.80 0.05-1.85 0.05-1.90 0.05-1.95 0.05-2.00
pts 2 3 4 5 6 7 8 9 10 11 12 13 14 15
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
slope 1.0008598 1.0926608 1.1617006 1.2151691 1.2574555 1.2914442 1.319 1280 1.341 9316 1.3609007 1.3768180 1.3902790 1.401 7435 1.411 5668 1.4200318 1.4273618 1.4337358 1.4392977 1.444 1643 1.4484308 1.4521755 1.4554631 1.4583470 1.4608720 1.4630756 1.4649896 1.4666407 1.4680517 1.4692423 1.4702292 1.4710272 1.471 6488 1.472 1052 1.4721052 1.4725607 1.4725761 1.4724595 1.4722174 1.4718554 1.4713786
corr coeff 1.OOOOOO 0.007758 89 0.996 550 85 0.99592295 0.99563495 0.995550 1 1 0.995586 86 0.99569520 0.979315 58 0.99691358 0.996 191 71 0.996370 69 0.996545 96 0.99671475 0.996875 59 0.997027 82 0.997171 27 0.997306 10 0.99743264 0.997551 32 0.99766263 0.997767 05 0.997865 05 0.99795709 0.998043 58 0.99812490 0.998 201 40 0.998273 38 0.998341 1 1 0.998404 83 0.998464 73 0.998521 00 0.998 521 00 0.998623 17 0.998669 30 0.998712 24 0.99875205 0.998788 77 0.99882243
eff dimen
N (n) (n)MW 27 30.46 40.94 125 157.32 189.55 343 455.27 520.12 729 997.4 1105.4 1331 1856.1 2018.3
8“ 34.42 20.49 14.24 10.83 8.74
d,(calc) d,(integer) %* 1.9465061 3 35.12 2.5772854 3 14.09 2.803 1932 3 6.56 2.8992408 3 3.36 2.945 152 3 1.83
“ 8 = [((n) - (n)Mw)/(n)]lOO,with (n)MWdrawn from ref 18. ‘ 8 = [(d,(intger) - d,(calc))/d,(integer)]100. TABLE VII: Effective Dimensionality of the Lattice 3 X 3 X 3 X 3” time range pts slope corr coeff eff dimen
0.05-0.10 0.05-0.15 0.05-0.20 0.05-0.25 0.05-0.30 0.05-0.35
2 3
4 5 6 7
1.1920805 1.2553162 1.2868584 1.2976703 1.2945313 1.2819141
1.000000 0.99919259 0.999165 99 0.99937861 0.999532 26 0.99945604
2.5953406
“In N(ca1c) = 4.394449. TABLE VIII: Effective Dimensionality of the Lattice 5 X 5 X 5 X 5“ time range pts slope corr coeff eff dimen
2.945 152
0.05-0.10 0.05-0.15 0.05-0.20 0.05-0.25 0.05-0.30 0.05-0.35 0.05-0.40 0.05-0.45 0.05-0.50 0.05-0.55 0.05-0.60
2 3 4 5 6 7 8 9 10 11
12
1.3282089 1.4448512 1.5242135 1.5909631 1.6360717 1.6685287 1.691 1471 1.7059871 1.7146022 1.7181908 1.7176942
1 .om000 0.997930 25 0.996947 22 0.996578 95 0.996566 42 0.99675498 0.997271 98 0.997376 96 0.997701 88 0.997991 04 0.998 223 18
3.436 3816
“In N(ca1c) = 6.437752.
“In N(ca1c) = 7.193686.
each of the five k X k X k lattices considered in this study, it is clear that the linear regime in the plot of S vs In t becomes more and more extended with an increase in N = k X k X k. For example, the linear regime for the 3 X 3 X 3 lattice persists over the time interval 0.05465, whereas for the largest cubic lattice considered here, the 11 X 11 X 11 lattice, the linear regime persists over the time range 0.05-2.00. From this observation, one anticipates that the determination of d, should become more and more accurate as the “size” N of the diffusion space increases. Also recorded in these tables are the limiting values of In N for each case, calculated in the regime of long time; the correspondence of these latter data with the theoretically predicted value of In N (where N = 27, 125, 343, 729,and 1331, respectively) is a critical check on the accuracy of the calculations. A further check on the accuracy of the calculations is provided by comparing the results obtained using the two spatial boundary conditions cited earlier, periodic and strictly confining (seesection I); if the motion of the diffusing particle is initiated from the centrosymmetric site of the defining lattice (as it is in this study), the results obtained for de and In N should be exactly the same-and they are. It is evident from the results reported in Table I that, with respect to the diffusion problem studied here, the 3 X 3 X 3 lattice is still ‘behaving” like a Euclidean space of dimension de 2. However, already for the 5 X 5 X 5 lattice, there is a distinct ‘jump” in the value of de, and with the 7 X 7 X 7 lattice, the 9 X 9 X 9 lattice, and, finally, the 1 1 X 1 1 X 1 1 lattice, it is seen that one is approaching asymptotically a spatial regime for which the dimensionality is de 3 (see Tables 11-V). The asymptotic behavior noted in the preceding paragraphs is interesting in light of the behavior of the average walk length ( n ) before trapping of a random walker migrating on a k X k X k cubic lattice with a centrosymmetrictrap and subject to periodic
-
-
TABLE VI: Values of ( n ) and d , Calculated for k X k X k Cubic Lattices (See Text)
boundary conditions. Recorded in Table VI are values of ( n ) calculated previously using the theory of finite Markov processes” and those calculated using the Montroll-Weiss (asymptotic) expression’* for infinite, periodic cubic lattices ( u = 6), namely, (n)
N
1.516386N
(9)
As is seen from these data, the percent differences between the numerically calculated values of ( n ) and values calculated using eq 9 mirror the percent differences between the values of de calculated using the stochastic approach described in section I1 and the expected integer Euclidean dimension de = 3. Taken together, these calculations of ( n ) and de suggest that the anticipated threedimensionalityof diffusion-reaction space for small, and finite, k X k X k cubic lattices is realized only gradually with an increase in N. Finally, given the broad applicability of (d - €)-expansion methods in renormalization group theory,I9it is of interest to ask whether the behavior uncovered in dimensions d = 2,3 also persists in dimension d = 4, namely, whether there occurs an asymptotic convergence to the integer dimension d = 4 with an increase in spatial extent of k X k X k X k lattices subject to periodic boundary conditions. As is seen from data presented in Tables VII-IX, the convergence to d = 4 is once again a strong function of N k X k X k X k. From the further data reported in Table X, we also find that this convergence is mirrored by the convergence to the exact (numerically calculated) ( n ) vs asymptotic ( n ) , quantities calculated using methods laid down in ref 20; convergence of the random walk variable is somewhat more rapid.
IV. Discussion In our earlier work?JOwe found that the dimensionality d, of a sequence of square-planar lattices (v = 4) of increasing spatial extent (7 X 7, 9 X 9, 1 1 X 11, 21 X 21, and 31 X 31) subject to periodic boundary conditions systematically approached the integer value de = 2. Moreover, upon considering a sequence of
6030 The Journal of Physical Chemistry, Vol. 96, No. 14, 1992 TABLE I X Effective Mmeadonrlity of the Lattice 7 X 7 X 7 X 7“ time range pts slope corr coeff eff dimcns 0.05-0. 10 2 1.334 3086 1.om000 .456 4040 0.997 768 58 0.05-0.15 3 .547 9232 0.996 578 72 4 0.05-0.20 5 -618 3977 0.995 978 19 0.05-0.25 6 .673 6293 0.995 726 79 0.05-0.30 7 .7174233 0.995 687 06 0.05-0.35 8 .752 4052 0.995 776 21 0.05-0.40 9 .7804513 0.995 942 70 0.05-0.45 .EO2 9402 0.996 15354 10 0.05-0.50 ,8209076 0.996 387 19 11 0.05-0.55 ,835 1453 0.996 629 35 12 0.05-0.60 3 4 6 2688 0.996 870 35 13 0.05-0.65 3547635 0.997 103 54 14 0.05-0.70 ,861 0169 0.997 324 26 15 0.05-0.75 ,865 343 1 0.997 529 15 0.05-0.80 16 0.05-0.85 3 6 7 9988 0.99771576 17 3.738 3936 1.869 1968 0.997 882 22 0.05-0.90 18 1.869 1151 0.998 027 07 0.05-0.95 19 1.867 9045 0.998 149 16 0.05-1.00 20
“In N(ca1cd) = 7.783641, TABLE X Values of ( n ) and de Calculated for k X k X k X k Cubic Lattices (See Text) d,(integer) %b d,(calc) (n)PKW %’ N (n) 93.60 4 35.1 81 99.62 6.4 2.595341 4 14.1 768.67 2.6 3.436382 625 749.38 2401 2923.45 2952.92 1.0 3.738394 4 6.5 ” % = [((n) - (n)pKw)/(n)]100, with (n)PKWdrawn from ref 20. b % = [(d,(integer) - d,(ca~c))/d,(integer)]~~~.
rwelayer, square-planarlattices (the “stacked” 7 X 7,9 X 9, and 11 X 11 lattices), we foundlo that the dimensionality remained, effectively, d, 2. In the review of that earlier work, a referee pointed out that, in the time regime before the asymptotic limit is reached, the dimension for “stacked” two-layer systems might well remain two-dimensional, provided the third dimension had an effective length 1 < (Dt)I/*,where D is the diffusion coefficient in eq 1. To explore the implications of this insight relative to the more general n-layer problem studied in this paper, suppose that the relevant (chemical/physical) time scale for a dynamical process in a given microheterogeneous medium is of the order of 1 ps. Then, for typical values of the diffusion coefficient of adsorbates migrating in microheterogeneous media [see, for example, the experimental study of Caro et al.21on the diffusion of methane, ethane, and propane in ZSM-51, 10“ cm2/s > D > cm2/s, the values of 1 corresponding to this range of diffusion coefficients
-
Garza-Lbpez and Kozak would be 10 A > I > 1 A. That is, for a 1-ps timescale and a diffusion coefficient D of lWi0 cm2/s, a “stacked”n-layer system will remain, effectively, two-dimensional provided the vertical lattice spacings were about 1 A; if, however, the diffusivity of the adsorbate species increases (sothat D 10-8 cm2/s), the vertical lattice spacings required to ensure that the process remains effectively two-dimensional would be 10 A. Since these values of 1 are characteristic of the spatial dimensions of the microchannels in many structured media, specifically smectite clays and zeolites, it is clear that caution must be exercised in interpreting dynamical processes in such systems using an approach based on a classical Fickian diffusion equation (1). In particular, use of the Fickian equation (1) demands specification of the Laplacian, and unless one is assured that estimates of 1 of the sort described above justify the specification of a Laplacian in a given integral dimension, an analysis based on a stochastic master equation (2) rather than a continuum diffusion equation (1) should be used in studying theoretically diffusion-controlled reactive processes in microheterogeneous media.
-
References and Notes (1) Ark, R. The Mathematical Theory of Difjusion and Reaction in Permeable Catalysts; Clarendon Press: Oxford, 1975; Vols. I and 11. (2) Mandelbrot, B. B. The Fractal Geometry of Nature; Freeman: San Francisco. 1983. (3) OShaughnessy, B. A.; Procaccia, I. Phys. Rev. 1985, A32, 3073. (4) Orbach, R. Science 1986, 231, 814. (5) Avnir, D. The Fractal Approach to Heterogeneous Chemistry: Surfaces, Colloids, Polymers; John Wiley and Sons: New York, 1989. (6) Fendler, J. H. Membrane Mimetic Chemistry; John Wiley and Sons: New York, 1982. (7) Thomas, J. K. The Chemistry of Excitation at Interfaces; American Chemical Society: Washington, DC, 1984. (8) Kalyanasundaram, K. Photochemistry in Microheterogeneous Systems; Academic Press: New York, 1987. (9) Rudra. J. K.; Kozak. J. J. Bull. Am. Phys. Soc. 1990,35,826. Rudra, J. K.; Kozak, J. J. Phys. Lett. 1990, A151, 429. (10) Garza-Lbpcz, R. A,; Kozak, J. J. J. Phys. Chem. 1991, 95, 3278. (11) Alexander, S.;Orbach, R. J. Phys. Lett. 1982,43, L625. (12) Rammal, R.; Toulase, G. J. Phys. Lett. 1983, 44, L13. (13) A!gyrakis, P. Phys. Rev. Lett. 1987, 59, 1729. (14) Pitsianis, G.; Bleris, L.; Argyrakis, P. Phys. Rev. E 1989,39. 7097. (15) Argyrakis, P.; Coniglio, A.; Paladin, G. Phys. Rev. Lett. 1988, 61, 2 156. (16) Argyrakis, P. Phys. Rev. Lett. 1988, 61, 2157. (17) Walsh. C. A.; Kozak, J. J. Phys. Rev. E 1982, 26,4166. Politowicz, P. A,; Kozak, J. J. Mol. Phys. 1987, 62, 939. (18) Montroll, E. W. Proc. Symp. Appl. Math. Am. Math. Soc. 1964,16, 193. Montroll, E. W.; Weiss, G. H. J. Math. Phys. 1965,6, 165. Montroll, E. W. J. Math. Phys. 1969, 10. 753. (19) Wilson, K. G.; Fisher, M. E. Phys. Rev. Lett. 1972,28,540. Fisher, M. E. Rev. Mod. Phys. 1974,46, 597. (20) Politowicz, P. A,; Kozak, J. J.; Weiss, G. H. Chem. Phys. Lett. 1985, 120, 388. Kozak, J. J. Phys. Rev. A 1991, 44, 3519. (21) Caro, J.; Blilow, M.;Schirmer, W.; Kiirger, J.; Heink, W.; Pfeifer, H.; Zdanov, S.P. J. Chem. SOC.,Faraday Trans. 1 1985, 81, 2541.
