3278
J. Phys. Chem. 1991, 95, 3278-3281
Calculation of the Effective Dimensionality of Layered Diffuslon Spaces. Application to Diffusion-Controlled Processes In Smectlte Clays Roberto A. Garza-L6pez and John J. Kozak* Department of Chemistry, Franklin College of Arts and Sciences, University of Georgia, Athens, Georgia 30602 (Received: July 2, 1990)
In theories of surface-mediated diffusion-reaction processes, it is usually assumed that the diffusing atom/molecule is strictly confined to a surface of Euclidean dimension de = 2. We study the consequences of relaxing this constraint by representing the surface by a network of (lattice) sites and providing to the diffusing species access to a companion layer of nearest-neighbor sites above the basal plane. The dynamics of a particle diffusing in this expanded space is then studied by solving numerically the stochastic master equation for the problem and the solutions generated used to construct the entropy S(t). From the slope of the curve S(t) versus In t , the spectral dimension d, of the flow can be determined. For Euclidean spaces, d, = de, and by studying a series of layered square-planar lattices of increasing spatial extent, we find that the Euclidean dimension of the expanded (two-layer) diffusion space remains effectively de 2. The implications of this result with respect to recent theoretical studies on the efficiency of diffusion-controlled reactive processes in intercalated (smectite) clay minerals are discussed.
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I. Introduction
In designing models of diffusion-reaction processes involving the migration of a coreactant on a surface in/on which are embedded one or more stationary reaction centers, a standard approximation has been to assume that, in its random (or biased) displacements, the coreactant is strictly confined to the surface. In recent work on the influence of surface imperfections (and multipolar correlations) on the efficiency of diffusion-controlled reactive processes on molecular organizates and colloidal catalysts,' and on the influence of swelling on reaction efficiency in intercalated clay materials,* an effort has been made to relax this assumption. In the former class of problems, one expects, owing to the discrete atomic or molecular geometry of the surface and/or fluctuations in the diffusing particle's motion (due to solvent buffeting, thermal perturbations, or the like), that the coreactant will likely ''skip" across the surface, i.e. undergo (small) displacements normal to, as well as across, the surface. In the second class of problems, increasing the interlamellar spacing between the layers of a smectite clay has the consequence of relaxing the motional constraints on a molecule diffusing through such a structured medium. A natural way of representing the discrete geometry of a structured medium is to construct a lattice model. For example, in ref 2, out-of-plane excursions of a coreactant moving across a surface were dealt with by envisioning one or two layers of (lattice) sites accessible to the diffusing areactant above the basal plane. In such a representation, the dimensionality of the reaction space will no longer be strictly d = 2; whether a fully developed d = 3 dimensionality is realized if the motional constraints on a diffusing coreactant are relaxed (by providing access to sites away from the basal plane) is the question addressed in this communication. This issue is of some theoretical importance because standard continuum theories of diffusion-reaction processes based on a Fickian parabolic partial differential equation of the form aC(i,t)/at = DV2C(i,r)+ fIc(i,t)] require the specification of the Laplacian, and this operator is specified for geometrical spaces of integral dimensionality. Hence, if a numerically precise determination of the effective Euclidean dimension de of the augmented diffusion space described above were significantly different from an integer value, this would limit the usefulness of insights drawn from theoretical studies based on a Fickian representation of the problem. 11. Formulation
Very recently, we have solved numerically the stochastic master equation To whom correspondence should be addressed.
0022-3654/91/2095-3278$02.50/0
for several (large) Euclidean and fractal, planar lattices subject to periodic (or confining) spatial boundary conditions and to the temporal boundary condition p,(t=O) = 6i, (2) m being an interior site of the lattice. In eq 1, the matrix Gij describes the transition rate of the probability pi(t) to the site I from a neighboring site j. In particular, if Mijis the probability of going from site j to a nearest-neighbor site i in a single jump and vj is the coordination or valency of the lattice site j, then G.. = (6,. - M.,) (3) IJ u u "j with Mii = 0 and Mij = I/vj for i # j. From the solutions generated, it was found3 that the function S ( t ) = -Cpi(t) In pi(t) (4) i
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after an initial interval of time ( t I), grows linearly with t; owing to the finite size of the lattices considered, departures from this linear behavior occurred on longer time scales, and S(t) was found to approach asymptotically the value In N , N being the number of sites in the fundamental lattice unit. Let R, be the number of distinct sites visited during an n-step random walk and Pk be the probability of visiting site k in an n-step walk (with Pk = ik/nif site k is visited i k times); earlier ~ t u d i e s ~ . ~ had established numerically that the spectral dimension d,, viz. dJ2 = (In R,)/ln n n m (5)
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could be evaluated by calculating the quantity n; Le., it was shown that ds/2 = -c(Pk In P k ) / h n k
In Pk)/h (6)
This led us to propose (and verify)) the following relation: d , / 2 = S(t)/ln t for t k 1 (7) in the linear regime of growth. The advantage gained in using the representation (7) to determine d, is that previous determinations of d, based on eq 6 required full-scale (and very lengthy) Monte Carlo simulations, whereas the implicit-function methods developed in our earlier work6 for solution of the stochastic master ( I ) Mandeville, J. B.; Hurtubise, D. E.; Flint, R.; Kozak, J. J. J . Phys. Chem. 1989, 93, 1816. (2) (a) Politowicz, P. A.; Kozak, J. J. J . Phys. Chem. 1988, 92, 6078. (b) Politowicz, P. A.; Leung, L. B. S.; Kozak, J. J. J. Phys. Chem. 1989, 93, 923. (3) (a) Rudra, J. K.; Kozak, J. J. Bull. Am. Phys. Soc. 1990,35,826. (b) Rudra, J. K.; Kozak, J. J. Phys. Letr. A 1990, f51, 429. (4) (a) Argyrakis, P. Phys. Reu. Lerr. 1987, 59, 1129. (b) Pitsianis, G.; Bleris, L.; Argyrakis, P. Phys. Rev. B 1989, 39, 7097. ( 5 ) Argyrakis, P.; Coniglio, A.; Paladin, G. Phys. Reo. Lerr. 1988.61.2156.
0 1991 American Chemical Society
Effective Dimensionality of Layered Diffusion Spaces TABLE I: Asymptotic Value8 of the Entropy S ( t ) lattice layer N S(t+-) 3 x 3 5 x 5 7 x 7 9 x 9 1 1 x 11 21 x 21 31 X 31 1 x 7 9 x 9 11 x 1 1
1 1 1 1 1 1 1 2 2 2
9 25 49 81 121 441 961 98 162 242
2.197225 3.218 879 3.891 820 4.394449 4.195191 6.089 045 6.867 914 4.584 961 5.081 596 5.488 938
The Journal of Physical Chemistry, Vol. 95, No. 8,1991 3219
8 In N 2.191225 3.218 819 3.891 820 4.394 449 4.495791 6.089 045 6.867 914 4.581 961 5.087 596 5.488 938
equation (1) allow a whole range of problems to be explored in a very time-efficient way. Suppose now we design a lattice model to explore theoretically the question posed earlier. In particular, consider a basal layer of sites accessible to the diffusing particle. For definiteness, we consider lattices of square-planar symmetry (coordination or valence Y = 4), subject to periodic boundary conditions. We then construct an overlayer of sites to allow for the possibility that the particle may undergo random excursions normal to the surface, as well as lateral excursions in the overlayer. For each finite lattice system considered in our study, the lateral boundaries of the composite system (i.e. basal plane plus overlayer) are subjected to periodic boundary conditions; from a given site in the basal plane, “up/down” excursions are restricted to the (single) nearest-neighbor overlayer site (i.e., periodic boundary conditions are not imposed in the “third” direction). Overall, then, to account for the possible “skipping” of a coreactant across the surface, the two-layer model is designed such that the coordination v of each (and every) site in the augmented reaction space is v = 5 . If attention is confined to Euclidean lattices, we are assured that the dimension de of the embedding Euclidean space, the fractal (Hausdorff) dimension dr, and the spectral dimension d, are equal.’ Intuitively, one expects that if 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 of v = 6), the dimensionality of the reaction space should converge to d, = de = df = 3. However, the “full” Euclidean dimension de = 3 may not be realized for only one overlayer, especially for systems of finite spatial extent. Thus, the specific question addressed in this communication using the theoretical tool (7) is, What is the effective dimensionality of a diffusion-reaction space modeled as a two-layer lattice system? 111. Results A (typical) profile of S versus In t is displayed in Figure 1 for
the case of a 31 X 31 square-planar lattice. It is seen that there is a linear regime (beyond t N I), which persists for an interval of time, subsequent to which higher order nonlinear behavior sets in. In fact, the asymptotic value of S reached in the limit of long time is an important check on the accuracy of the calculations reported here, inasmuch as it has been proved theoretically that (in this limit) the entropy S is given by In N (where, again, N is the total number of lattice sites). Reported in Table I are the calculated values of S in the regime of long time and corresponding values of In N for the lattice systems studied in this paper. As is plain from these data, the agreement between values of (asymptotic) S and In N is (numerically) exact. Detailed numerical study of the linear regime in the plot of S versus In t for a series of lattices of increasing spatial extent shows that the larger the lattice, the longer the time interval over which the linear behavior persists. A way of documenting this trend is to record the slope and the correlation coefficients of a linear least-squares fit of the data in this region. The smallest one-layer (6) (a) Boulu, L. G.; Kozak, J. J. Mol. fhys. 1987,62, 1449. (b) Boulu, L. G.; Kozak, J. J. Mol. fhys. 1988, 65, 193. (7) (a) Alexander, S.;Orbach, R. J . fhys., Lerr. 1982, 43, L625. (b) Rammal, R.; Toulase, G. J . fhys., Leu. 1983, 41, L13. (8) Politowicz, P. A.; Garza-Ldpcz, R. A.; Hurtubise, D. E.; Kozak, J. J. J . fhys. Chem. 1989, 93, 3728.
/
8J
1:oo 2:oo do0 LN T Figure 1. Plot of entropy (S) versus In r for a 31 X 31 square-planar lattice. 92.00
-5.00
d.00
lattice considered in this study is the 7 X 7 square-planar lattice, for which the linear regime persists over the relatively narrow range 1.O It I2.6. Displayed in Table I1 are the slopes and correlation coefficients for a sequence of (one-layer) planar lattices over this same range, 1 .O 5 t I2.6. As is evident from an examination of these data, the linearity of the profile of S versus In t becomes more and more pronounced, the larger the lattice considered. Actually, for the largest single-layer lattice studied here (the 3 1 X 3 1 square planar lattice), the linear regime persists over the range 1.0 It 520.0 (see Figure 1). It should be noted that although the asymptotic value of S is correctly given by the limiting value (In N) for the 11 X 11, 21 X 21, and 31 X 31 lattices [recall the (different) values in Table I], the values of the slope of S versus In t in the linear regime remain effectively the same; in fact, as is seen from the data in Table 11, the slope values of S versus In t for the 21 X 21 and 31 X 31 lattice are exactly the same (to eight significant figures). Since twice the calculated slope gives d, = de (see eq 7), it is clear from the data reported in Table I1 that an effective Euclidean dimension of de = 2 is realized for lattices of spatial extent n X n, where n 1 11. We turn now to the central question addressed in this paper: What is the effective Euclidean dimension of the augmented, two-layer lattice system described earlier? The data displayed in Table I11 give the results for the slopes and correlation coefficients of a sequence of two-layer (square-planar) lattices, each site of which has coordination v = 5 . It is reasonable to conclude from these data, especially the results for an 11 X 11 square-planar lattice with a companion 11 X 11 overlayer (a total of N = 2(121) = 242 sites), that the Euclidean dimension of the layered diffusion space considered remains effectively de 2. IV. Diffusion-Controlled Processes in Expanded Smectite Clays
In previous work: we studied theoretically the extent to which the efficiency of reaction between a fixed target molecule and a diffusing coreactant is affected when the interlamellar space separating the atomic layers defining a smectite-type clay is expanded (or “swollen”). We considered partially ordered layer lattices of hexagonalZBand triangular2bsymmetry, and for the case where the target molecule is positioned in the basal lattice at the
3280 The Journal of Physical Chemistry, Vol. 95, No. 8, 1991
Garza-Lbpez and Kozak
TABLE 11: Slop of the Linear Regime [ S ( t )versus In t ] for One-Layer Systems' At 1 .O-2.6
mb
1 .0-2.4
8
1 .o-2.2
7
1 .o-2.0
6
1 .o- 1.8
5
1.0-1.6
4
I .o-1.4
3
1.0-1.2
2
7 x 7 0.447 785 (0.979 862) 0.476 082 (0.984 328) 0.507 647 (0.988 556) 0.542 050 (0.992 167) 0.579 079 (0.995 017) 0.61 8 940 (0.997 122) 0.664 22 1 (0.998 880) 0.713 110 ( I .ooo 000)
9
9 x 9 0.744 49 1 (0.995 307) 0.769 195 (0.996 545) 0.794 571 (0.997 57 I ) 0.821 131 (0.998 473) 0.847 1 15 (0.999 095) 0.873 219 (0.999 542) 0.898 363 (0.999 789) 0.927 043 (1 .ooo 000)
11 x 11 0.906 475 (0.999 056) 0.920 830 (0.999 347) 0.935 052 (0.999 588) 0.948 087 (0.999 740) 0.960882 (0.999 857) 0.973 13 1 (0.999 939) 0.983 933 (0.999 983) 0.992 869 (1 .ooo 000)
21 x 21 1.007 620 (0.999 997) 1.008 382 (0.999 997) 1.009 I88 (0.999 998) 1.009 873 (0.999 998) 1.010 909 (0.999 998) 1.012 718 (1.00OOoo) 1.013 416 ( I .000 000) 1.014811 ( 1.ooo 000)
31 X 31 1.007 620 (0.999997) 1.008 382 (0.999 997) 1.009 188 (0.999 998) 1.009 873 (0.999 998) 1.010 909 (0.999 998) 1.012 718 ( I ,000000) 1.013 416 ( 1.OOo000) 1.014 81 1 ( 1.OOo 000)
'Listed under each lattice are the slope and associated correlation coefficient (in parentheses) for a linear least-squares fit of the data generated for S versus In 1. b m is the number of points considered in the specified range of At: the grid used is uniformly 0.2. TABLE III: Slop of the Linear Regime [ S ( t )versus In t ] for Two-Layer Systems' At mb 7 X 7c 9 x 9c 11 x 1 1 c 1 .0-2.6
9
1.0-2.4
8
1.0-2.2
7
1.0-2.0
6
1.0-1.8
5
1 .O-1 .6
4
1.0-1 .4
3
1.0-1.2
2
0.445 778 (0.978 614) 0.485 163 (0.983 219) 0.518 088 (0.987 598) 0.554 I89 (0.991 355) 0.593 240 (0.994 289) 0.637 697 (0.996914) 0.685 223 (0.998 654) 0.740538 ( 1 .ooo 000)
0.752 670 (0.994 794) 0.778677 (0.996 118) 0.805 783 (0.997 259) 0.833497 (0.998 159) 0.861 920 (0.998 854) 0.890 889 (0.999 344) 0.922 248 (0.999 747) 0.954471
0.91 5 005 (0.998 873) 0.930686 (0.999 209) 0.945 738 (0.999452) 0.960578 (0.999635) 0.975 589 (0.999 784) 0.990075 (0.999 880) 1.004934 (0.999 952) 1.020296 ( I .000OOO)
( I .000000)
'See Table 11, footnote a. "see Table 11, footnote 6. 'Spatial extent of basal plane: total number of lattice sites is 2 (n X n). centrosymmetric site, exact numerical solution of the underlying stochastic problem showed that in the limit of large planar arrays ( N 600 sites), the reaction efficiency (as calibrated by the mean walk length of the diffusing coreactant) decreased by a factor of 1.7 for a ("stacked") two-layer structure, for both symmetries. That is, if is the average walk length of a diffusing coreactant before it undergoes an irreversible reaction with a target molecule anchored at the centrosymmetric site of a d = 2 dimensional planar lattice and ( n)ll is the average walk length when one adds a second, companion layer of sites above the basal plane (keeping the site of the target molecule fixed), then for lattice units of hexagonala and triangular2bsymmetry, the ratio (n)ll/(n)l 1.7 was found for large lattices. To check further the apparent independence of the ratio (n)ll/(n)l of the valency of the defining lattice system, we report here calculations of (n)ll (and also verify earlier calculations9 of (n),)for the series of square-planar lattices (v = 4) considered in this study. As is evident from the data reported in Table IV, the limiting ratio, ( n ) l l / ( n ) l 1.7, for N large is also realized for these layered lattice systems. A similar invariance was foundZbfor "stacked" three-layer systems of triangular and hexagonal symmetry, where for large lattices the ratio (n)lll/(n)l 2.5 was found; here, (n)lll is the average walk length before the diffusing coreactant undergoes an irreversible reaction with a target molecule positioned at the centrosymmetric site in the basal Dlane. Taking together the results of (n)ll/(n)l and (n)lll/(n)l,'weconjectur& in ;ef 2b that there appeared to be a kind of "superposition principle" at play
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-
-
-
(9) Walsh, C. A.; Kozak, J. J. Phys. Reu. B 1982, 26, 4166.
in these calculations, one in which stochastic events in higher than two-dimensional spaces (specifically for layered diffusion spaces) are still representable in terms of spaces of dimension d = 2. Given the further reuslts reported in Table IV, it would appear that this conjecture is quite independent of the valency u of the defining lattice system, a t least for large lattices. The results reported in section 111 provide a quantitative rationale for the conjecture noted above inasmuch as we found that for a two-layer, square planar lattice system, the Euclidean dimension of the "stacked" two-layer system is effectively d, 2. Given this result, one would expect that the results for ( t ~ ) ~ ~ / ( n ) ~ should be essentially unity if one were to calculate this ratio for the case where both u and N were held fixed. This expectation can be analyzed by considering the data reported in Table I of ref 2a, where we listed values of the average walk length as a function of system geometry for one-, two-, and three-layer "stacked" hexagonal lattices. Focusing on the case of two hexagonal layers (i.e., a basal plane and one overlayer of sites), we note that for such a configuration, all sites are of valency v = 4; Le., each site in the basal plane or in the overlayer has a lateral coordination of v = 3 as well as a further degree of freedom ( u = 1) for excursions normal to the plane; thus, a particle diffusing in the basal plane has access to the set of sites defining the overlayer (and vice versa). The consequent valency u = 4 is therefore the same as for a one-layer square-planar lattice. Hence, one can use the results for (n)IIreported in ref 2a and the results for (n)l calculated by using the asymptotic theory of Montrolllo and den Hollander and Kasteleyn" to compute values of (n) corresponding to a fixed setting of u and N . The results of this calculation for the series of two-layer lattices considered in ref 2a are displayed in Table V. Given that the Montroll-den Hollander-Kasteleyn theory reproduces values of (n) to within 5% for large lattices (see the discussion in ref 12), it is reasonable to conclude that the results reported in Table V verify the effective de = 2 dimensionality of a two-layered (hexagonal) diffusion space. In other words, the near constancy of the ratio (n)lI/(n)l, shown here and in ref 2 to be independent of lattice valency for large lattices, is a consequence of the fact that, despite the presence of the overlayer, the flow of the diffusing coreactant remains predominantly two-dimensional.
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V. Conclusions In this paper we have examined the dimensional consequences of expanding this diffusion space available to a migrating atom or molecule. Specifically, for surface diffusion processes, we have found that adding an overlayer of sites accessible to a species is essentially equivalent to expanding the lateral extent of the surface (10) Montroll, E. W . J . Marh. Phys. (N.Y.) 1969, IO, 753. (1 1) den Hollander, W.Th. F.; Kasteleyn, P.W . Physica A (Amsterdam) 1982, 112A, 523. (12) Politowicz, P. A,; Kozak, J. J. bngmuir 1988, 4, 305.
The Journal of Physical Chemistry, Vol. 95, No. 8,1991 3281
Effective Dimensionality of Layered Diffusion Spaces
TABLE IV: (n)I and ( I I ) I I Value for Y = 4 SqurrcPhnar Lattice8 N (unit) N (one-layer) Wr" (4lb 3 5 7 9
x x x x
9 25 49 81 121 44 1
3 5 7 9
11 x 11
21 x 21
9.000 000 3 1.666 667 71.615385 130.604 452 209.937 058 942.192 176
8.000000 24.000000 48.000000 80.000 000 120.000 000 440.000 000
N (two-layer) 18 50 98 162 242 882
(nhl 20.1 I 7 641 63.3 11966 136.129 999 240.489 894 311.981 585 1601.018 886
(nh/(n)l 2.235 294 1.999 51 5 1.900 849 1.841 361 1.800952 1.698 167
e ( n ) , is the average walk length before trapping for walks initiated at the nearest-neighbor site relative to the trap. Montrolltoproved that ( n ) l = N - I exacrly, so the numerical values reported for ( n ) l are an important check on the accuracy of our calculations. *Values of ( n ) l here are in agreement with those reported in ref 9 except for the 21 X 21 case; we believe the value listed here (calculated on the UGA Cyber-205) is more accurate, since the ( n ) , value reported in ref 12 was 439.8, whereas here we calculate 440.000000.
TABLE V: Comparison of ( n ) Values for One-Layer and Two-Layer Y = 4 Lattices
N
( n )Gnc-l.ycP
(n)twcl.ycr b
wc
8 32
8.000 43.148 72.507 159.95 217.70 364.42 453.19 663.79 785.47 1063.34 1219.40
7.701 42.779 73.375 163.70 224.30 378.09 47 1.86 694.36 823.53 1 1 18.63 1284.91
-2.87 -0.86 1.20 2.34 3.03 3.62 3.96 4.40 4.62 4.94 5.10
so
98 128 200 242 338 392 512 578
( n ) values calculated from the asymptotic expression ( n ) = [ N / ( n I ) ] [ A , N In N A2N A3 A , / M for Y = 4, where A I = 0.318309886, A2 = 0.195062532, A3 = -0.116964779 (values taken from Montrol19), and A4 = +0.484065704 (value taken from den Hollander and KasteleynIO). b ( n ) values reported in ref 2b. c % =
-
a
+
[(n)twc~aycr
+
+
- ( n ) ~ " ~ ~ a y e r l / ( n ) ~ ~ c ~Xa y100. cr
domain by the number of added sites. This conclusion was reached after calculating the effective Euclidean dimension de of a basal lattice augmented by an overlayer consisting of a companion set of sites characterized by the same coordination or valency v. In particular, we mobilized a recently developed theoretical relationship involving the evolution of the entropy S(t) on a diffusion space free of reaction centers; the slope of the profile, S ( t ) versus In t , gives the effective (Euclidean) dimension of the lattices considered here. S(t) itself was calculated by solving numerically the stochastic master equation for the problem to determine the probability p ( t ) whence S(t) could be determined from the defining expression, eq 7. Once the effective dimensionality of the two-layer square-planar lattice system had been determined and found to be de 2, an earlier result involving "stacked" layer lattices was made understandabte. In ref 2, we found that, upon introduction of a reaction center or target molecule at the centrosymmetric site of the basal plane and calculation of the average number (n)of steps taken by a diffusing coreactant before an irreversible reaction takes place a t the target site, the ratio (n)ll/(n)I 1.7 was realized for large lattices. Again, ( t ~ ) is, ~the mean walk length calculated by assuming the diffusing coreactant has access to all sites in the basal plane and in the overlayer, with free migration between any pair of nearest-neighbor sites in the two layers; (n)I is the average walk length calculated by assuming that the diffusing coreactant is restricted to the basal plane. The ratio (n)II/(n)Iwas found to be 1.7 for two-layer lattice systems for which the lateral
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-
coordination was v = 3 (hexagonal lattices) or v = 6 (triangular lattice), with new calculations reported here (Table IV) confirming the universality of this limiting ratio for lattices of lateral coordination v = 4 (square-planar lattices). For each of the two-layer lattice systems, the valency v of each site in the diffusion space is, by construction, 1 greater than the lateral valency, this owing to our relaxing assumption that the diffusing coreactant be strictly confined to the basal plane. Thus, each site of the hexagonal system studied in ref 2 has a valency of v = 3 (for lateral excursions) and an additional degree of freedom (v = 1) for excursions to the companion layer, giving an overall valency of v = 4 for nearest-neighbor transitions. Consequently, results for ( n)IIcalculated for the two-layer, hexagonal system could be contrasted with values of ( n)l calculated for a single-layer, square-planar lattice system; in particular, these two lattice systems are characterized by a common valency v = 4, and by taking advantage of the Montroll-den Hollander-Kasteleyn asymptotic expression for (n)Ifor square-planar lattices, we can calculate values of ( n)IIand ( n)l for a common setting of v and N. Values of (n)IIand (n)I were found to be in good agreement (error 6 . 1 % for the largest lattices considered here). We turn now to the significance of these results vis-&vis the problem of determining the influence on the reaction efficiency resulting from expanding the interlamellar space separating the atomic layers of the host lattice of a smectite clay. "Swelling," by virtue of providing a larger domain within which the diffusing particle can migrate, ought sensibly to extend the time scale required for reaction. This conclusion can now be made precise: for a basal domain defined by a lattice of N sites, adding a single overlayer of sites accessible to the diffusing coreactant (N 2 N ) extends the time scale of reaction by a factor 1.7, a result apparently independent of the valency of the defining lattice system (see ref 2 and Table IV). Finally, an immediate theoretical consequence of the results reported in this paper is that continuum theories of surface diffusion-reaction processes, based on a Fickian equation for which the Laplacian is specified for de = 2 dimensions, ought to provide a reasonable, first-order description of the temporal and spatial changes in the concentration C(F,t)of the coreactant for geometries, local degrees of freedom, and initial conditions similar to those defined by the lattice model considered here. A reviewer has also pointed out that, in the time regime before the asymptotic limit is reached, the diffusion might well remain two-dimensional, provided the third dimension has an effective length I < (Dr)lI2, where D is the diffusion coefficient in the Fickian equation. It will be of interest to explore this and other factors that influence surface diffusion (e.g. the role of interparticle correlations) by using the theoretical approach described in this paper.
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