J . Phys. Chem. 1987, 91, 5555-5557
5555
Steady-State Chemical Kinetics on Fractals: Geminate and Nongeminate Generation of Reactants L. W. Anackert and R. Kopelman* Department of Chemistry, The University of Michigan, Ann Arbor, Michigan 48109 (Received: July 20, 1987)
Supercomputer simulations of the elementary A + B 0, diffusion-limited reaction were performed under steady-source conditions on a Sierpinski lattice. For geminate generation the steady-state reaction order has the classical value ( X = 2). However, a dramatic segregation appears for the nongeminate reactant generation, resulting in X = 2.14 and a drastically reduced rate constant. This is relevant to chemical reactions on heterogeneous catalysts, to the annealing of radiation damage, and to electron-hole recombination in inhomogeneous media. +
geminate AB pairs on a nonuniform surface followed by immediate dissociation into A and B. Reactions between A and B proceed on the surface with no activation barrier while reactions between two A or two B particles are forbidden. This reaction was simulated with random walkers of type A and type B on a planar Sierpinski gasket% (f = 0.68) under steady-source conditions. We report the kinetic rate laws observed when A and B walkers are started at randomly selected nearest-neighbor sites (geminate landing) and compare the results to those obtained when A and B start on independently selected random sites (nongeminate landing).
Introduction
A surprising effect of reactant segregation under steady-state conditions has been demonstrated recently.’ This is of much interest to technological and biological processes. It should not be confused with the previously documented segregationz-8 found for transient (“batch” or “big bang”) reactions at the limit of very long time (Le., minute reactant concentrations). The steady-state case, in contrast to the transient one, does include effective stirring via continuous particle generation. However, there is no convective m i ~ i n g . Indeed, ~ convective stirring is unlikely for solid-state, surface, or capillary media. In addition to chemical reactions, our formalism applies to radiation induced defects8J0and to charge Simulations recombination, soliton-antisoliton, exciton-exciton, and even matter-antimatter annihilation.’ 1-z3 Biological m o r p h o g e n e s i ~ ~ ~ ~ ~ ~In both the geminate AB and the nongeminate A + B simuand population biologyz6(and even sociological processes2’) may lations, exactly equal numbers of A and B random walkers are be related topics. added at a constant rate, R . Time is incremented after every Standard chemical kinetic relationships no longer apply for the diffusion-limited transient reactions’-23 (1) Anacker, L. W.; Kopelman, R. Phys. Rev. Lett. 1987,58, 289. A +A products (2) Ovchinnikov, A. A.; Zeldovich, Ya. B. Chem. Phys. 1978,28, 215.
-
or A+B-0 in low-dimensional media and on fractal s u r f a ~ e s . ~The ~ - ~re-~ action kinetics in these systems are well described by -dp/dt = kpX
t
-
(1)
m
where p = pA(t) = pB(t) is the density and
X = 1 +f’
(A
+ A reaction)
(A
+ B reaction)
or
X=
1
+ 2f’
-
with spectral d i m e n ~ i o n ~ds ~ ,=~ ’2f and 0 If I1.3z For the A, the system (nongeminate) steady-source reaction A + A is well described33 by the following rate law
- dp/dt = k p X - R
(2)
where R is the constant rate of walker addition, and X = 1 +f’. After the A + A reaction reaches a steady state, this same power law relation holds32-34
R
= kpSX
-
(3)
where pS is the steady-state density achieved under steady-source 0 reaction we report here some conditions. For the A + B unexpected results for the value of X , furthermore, there is a dramatic segregation of reactants a t the steady state for the nongeminate A + B reactant addition. One chemical reaction of interest involves the adsorption of ‘John von Neumann Center, Princeton, N J 08505.
0022- 365418 7 1209 1- 5 5 5 5 $01.50/0
(3) Toussaint, D.; Wilczek, F. J . Chem. Phys. 1983,78,2642. (4) Blumen, A.; Zumofen, G.; Klafter, J. J . Phys. Colloq. 1985,46,C7 (1985). ( 5 ) Blumen, A.; Klafter, J.; Zumofen, G. In Optical Spectroscopy of Glasses; Zschokke, I., Ed.; Reidel: Dordrecht, Holland, 1986, p 199. (6) Kang, K.; Redner, S. Phys. Rev.Lett. 1984,52, 955. (7) Meakin, P.; Stanley, H. E. J. Phys. A 1984,17,L173. (8) Kuzovkov, V.; Kotomin, E. Czech. J. Phys. 1985,835,541. (9) Argyrakis, P.; Kopelman, R. J . Phys. Chem. 1987,91,2699. (10) Waite, T. R. J . Chem. Phys. 1957,107,463. (11) de Gennes, P. G. J . Chem. Phys. 1982,76, 3316. (12) Evesque, P. J . Phys. (Paris) 1983,44,1227. (13) Evesque, P.; Duran, J. J . Chem. Phys. 1984,80, 3016. (14) Klymko, P. W.; Kopelman, R. J . Lumin. 1981,24/25,457. (15) Klymko, P. W.; Kopelman, R. J . Phys. Chem. 1982,86,3686. (16) Klymko, P. W.; Kopelman, R. J. Phys. Chem. 1983,87,4565. (17) Anacker, L. W.; Klymko, P. W.; Kopelman, R. J. Lumin. 1984, 31 132,648. (18) Kopelman, R.; Klymko, P. W.; Newhouse, J. S.; Anacker, L. W. Phys. Rev.B 1984,29, 3747. (19) Zumofen, G.; Blumen, A.; Klafter, J. J. Chem. Phys. 1985,82,3198. (20) Kopelman, R.; Hoshen, J.; Newhouse, J. S.; Argyrakis, P. J . Stat. Phys. 1983,30,335. (21) Torney, D. C.; McConnel, H. M. J . Phys. Chem. 1983,87, 1441. (22) Torney, D. C. J . Chem. Phys. 1983,79,3606. (23) Elyutin, P. V. J . Phys. C 1984,17,1867. (24) Turing, A. M. Philos. Trans. R. Soc. London, Ser. B 1952,237,37. (25) Field, R. J. Am. Sei. 1985,73, 142. (26) Lawton, J. H. Nature 1987,326,328. Kareira, P. Nature 1987,328, 388. (27) Maddox, J. Nature 1987,326,327. (28) Mandelbrot, B. B. The Fractal Geometry of Nature; W. H. Freeman: San Francisco, 1983. (29) Pfeifer, P.; Avnir, D. J. Chem. Phys. 1983,79,3558. (30) Alexander, S.; Orbach, R. J. Phys. (Paris) Lett. 1982,43, L-625. (31) Rammal, R.; Toulouse, G. J . Phys. (Paris), Lett. 1983,44, L-13. (32) Anacker, L. W.; Kopelman, R. J . Chem. Phys. 1984,81, 6402. (33) Anacker, L. W.; Parson, R. P.; Kopelman, R. J . Phys. Chem. 1985, 89, 4758. (34) Anacker, L. W.; Kopelman, R.; Newhouse, J. S. J . Star. Phys. 1984, 36. 591.
0 198 7 American Chemical Society
Letters
5556 The Journal of Physical Chemistry, Vol. 91, No. 22, 1987 0
TABLE I: Steady-State Densities‘
QlU
D
RM
geminate AB 0.007 11
nongeminate A 0.0246 0.0340
+B
0.012 33
0.0470 0.021 40 ‘p
=
p A = pB
3 “
..
m
U
are averaged over the last 500000 time increments.
0
“8,
TABLE II: Rate Constants and Reaction Orders geminate AB nongeminate A + B
k
X
1.95 i 0.01 0.145 f 0,001
1.99 i 0.01 2.14 i 0.01
walker on the gasket tries to move (with a probability of 0.25) to one of its nearest-neighbor sites. Only one walker is allowed to occupy each site; if an A (or B) walker tries to move to a site occupied by another A (or B) walker, then the move is forbidden. A reaction occurs when an A (or B) walker moves to a site occupied by a B (or A) walker; the reacting A and B walkers are removed from the system when the attacked B (or A) walker takes its turn to move. N o walkers are allowed to occupy the site where the reaction occurred, the “annihilation” site, for an average of a half-time step. Time is discretized by 1 / N where N is the number of walkers on the gasket. All random numbers referred to in these simulations use the uniform pseudorandom number generator RANF on the CDC Cyber 205. These computer simulations were performed on the Cyber 205 at the John von Neumann National Supercomputer Center. Simulations of the nongeminate A + B starting conditions (where A and B are not forced to land on nearest-neighbor sites) are performed by choosing, at random, independent starting positions for all A and B walkers; each walker lands with equal probability on any unoccupied site. Simulations of the geminate AB starting conditions (where A and B are forced to land on nearest-neighbor sites) use up to three sets of random numbers. The first random number selects with equal probability any unoccupied site on the gasket. The second random number selects one of the four nearest-neighbor sites for the other member of the arriving pair. This site can be either an annihilation site, an occupied site, or an unoccupied site. If the site is an annihilation site, then the process for selecting the starting positions of the landing pair is repeated, choosing randomly from the unoccupied sites on the gasket. If the site is occupied by an A (or B) walker, then an annihilation occurs with the B (or A) member of the arriving pair. The net effect is that the site first selected for the pair is now occupied by an A (or B) walker. If the nearestneighbor site is an unoccupied site, then a third pseudorandom number is used to assign (with a probability of 0.5) the relative AB or BA occupancy of the nearest-neighbor pair of sites. Simulations were followed for one million time increments on a planar Sierpinski gasket with M sites (vertices). Cyclic boundary conditions were imposed. Table I reports the steady-source rates of walker addition to an M site gasket in units of R M where an R M value of unity means that one A and one B walker were added to the gasket before every time increment. The simulations were performed on 8th-, 7th-, and 6th-order gaskets with respective values of M = 9843, 3282, and 1095. Simulations on the 8th-order gasket consisted of 25 realizations (independent runs) for the geminate AB constraint with values of RM = 1, 3, and 9. The simulations for the nongeminate A + B starting condition consisted of 10 realizations with R M = 1, 2, and 4. At ALL times the algorithm implemented here required that the number of A walkers was identically equal to the number of B walkers. The average steady-state densities, p, from simulations on the 8th-order planar Sierpinski gasket, are reported in Table I for both geminate AB (nearest-neighbor) starting conditions and for nongeminate A + B (totally random) starting conditions. Steady-state densities of the geminate AB simulations were independent of the size of the gasket; for a given value of R the same
Figure 1. A snapshot at the steady state (after the millionth time step) for one of the realizations of the geminate system (A, dot; B, square). RM = 9 (see Table I).
Figure 2. A snapshot at the steady state (after the millionth time step) of the walker distribution for one of the RM = 4 realizations of the A + B nongeminate system; this was the most dramatic segregation of A, dot, and B, square, walkers, but all of the realizations demonstrated
dramatic segregation. steady-state densities were obtained on the 6th-, 7th-, and 8th-order gaskets. For nongeminate simulations minor finite size effects were observed. The “effective” rate constant k and the “effective” reaction order X in eq 3 are calculated simultaneously by using a nonlinear regression with R as the independent variable and the average density as the only dependent variable; a fixed point ( R = 0 and p = 0) is included. Rate constants and X values are reported in Table 11. The error bounds in these regressions are within 1% on both k and X . Results and Discussion The results of the simulations are discussed in terms of three major categories: reactant segregation, reaction order, and rate constants. The distribution of A and B under steady-state conditions for the geminate AB landings are random; a typical illustration is given in Figure l . In contrast, a dramatically segregated distribution results from nongeminate A B landings as illustrated in Figure 2. This dramatic segregation has now
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The Journal of Physical Chemistry, Vol. 91, No. 22, 1987 5557
Letters
0
been observed for both reflective1 and cyclic boundary conditions. The reaction orders for both geminate AB and nongeminate A B are close to the classical value of two. The geminate case yields X = 2 within the 1% error bound (Table 11). However, the nongeminate case shows a small, though significant, deviation from two, X = 2.14 f 0.01. The most striking quantitative distinction between the two cases is exhibited by the rate constants (Table 11). The rate constant for the geminate case, k = 1.95 f 0.01, is 13 times greater than the rate constant for the nongeminate case, k = 0.145 f 0.01. This is consistent with the higher steady-state densities observed for the nongeminate case, e.g., Figure 3. Based on Table I we can see that nearly equal steady-state densities are obtained when walkers are added to the geminate system at a rate 9 times faster than those added to the nongeminate system. Similarly, for the nongeminate case, there appears to be a direct correlation between dramatic segregation and high steady-state densities (for a given rate of walker addition R); a higher steady-state density is required for the nongeminate A B system to maintain the same rate of reaction as occurs for the geminate AB system. Why is dramatic segregation observed for the nongeminate A B starting conditions and not for the geminate AB starting conditions? Perhaps this dramatic segregation is simply the result of converting initial local density fluctuations into large-scale density fluctuations ( " m o r p h ~ g e n e s i s " ~ ~With ~ ~ ~the ) . geminate AB starting conditions there are no initial local density fluctuations, in contrast to the initial density fluctuations which naturally occur in the nongeminate A B system where A and B start at independent sites selected at random. We emphasize that, for both geminate AB and nongeminate A B systems, these simulations assign the same microscopic diffusion rate (one nearest-neighbor jump per time increment) and the same reaction cross-section (one lattice site). Nevertheless, the macroscopic rate constants differ by more than an order of magnitude. This is in conflict with the classical formalism of diffusion-limited reactions where the rate constant depends only on the diffusion rate and the reaction c r o s s - ~ e c t i o n . ~ ~ - ~ ~ Our observation that the steady-state densities do not markedly depend on the lattice size (for given values of R)is consistent with the observation that the choice of boundary conditions (cyclic versus reflective) has little effect on the results. Both observations
0
8-
+
+
+
+
+
(35) Smoluchowski, M. V. Z . Phys. Chem. 1917, 92, 192. (36) Noyes, R. M. Prog. React. Kinet. 1961, I , 129. (37) Wilemski, G.; Fixman, M. J. Chem. Phys. 1973, 58, 4009.
0
!
I
01
0 -
-r--7-
0
7 -
250000 500000 750000 1000000
time Figure 3. M ( p A ( t ) .+ p B ( t ) ) versus t for geminate AB pairs (bottom curve) and nongeminate A + B (top curves). RM = 1 and M = 9843. Note that the rise times are too short to be discerned on this time scale.
indicate that finite size effects play a small role in these systems (this is not always the case38). We note that for the steady-state A A reaction on the Sierpinski gasket X = 2.45 which is consistent with the relation X = 1 +yl(see eq 1 and 3). Our results for the A B reaction do not show an analogous behavior (such as X = 1 2y1). This also seems to be the case for other low-dimensional topologies.39 On the other hand, for a geminate steady-state A A reaction we expect X N 2, as reported here for the A B geminate case.4o
+
+
+ + +
Summary Steady-state kinetics on fractals show new, unexpected patterns. Subtle differences, such as geminate versus nongeminate generation of reactants, have drastic effects. In the absence of analytical formalisms for diffusion-limited low-dimensional and heterogeneous media, computer simulations are important in providing both conceptual insights and quantitative results. Acknowledgment. This work was supported by NSF Grant No. DMR 8303919 and NSF special supercomputing allocation at JVNC. (38) Clement, E.; Kopelman, R.; unpublished work. (39) Harmon, L.; Li, L.; Parus S.; Kopelman R.; unpublished work. (40) Li, L.; Kopelman, R. J. Lumin., in press.
