PHYSICAL REVIE% B VOLUME 28, NUMBER 10 15 NOVEMBER 1983 Trapping and reaction rates on fractals: A random-walk study A. Blumen' and J. Klafter Corporate Research Science Laboratories, Exxon Research and Engineering Company, P. O. Box 45, Linden, New Jersey 07036 G. Zumofen Laboratorium fur Physikalische Chemic, Eidgenossische Technische Hochschule-Zentrum, CH-8092 Zurich, Switzerland (Received 8 August 1983) In this Rapid Communication we study random walks on Sierpinski gaskets, which are fractal structures of simple geometry. We determine the probability of the walker to be captured by traps randomly distri- buted on the gaskets. The general reaction process is modeled through a kinetic approach. The decay laws which we find are smooth extensions to dimensionalities between one and two. I. INTRODUCTION Many aspects of solid-state physics and chemistry are re- lated to random walks on periodic structures. ' ' In the last few years processes of energy transfer and of carrier recom- bination in disordered materials have attracted considerable attention. One way to introduce the disorder aspect has been the continuous-time random-walk approach, as dis- cussed in previous works. ' ' Another way of dealing with disorder is by realizing its self-similar nature, which then in- troduces the fractal concept. " A simple class of fractal structures are the Sierpinski gaskets. The gaskets are fairly regular: Their basic unit is the d-dimensional simplex from which the gasket is created by repeated dilatations. " In this Rapid Communication we deal with two- and three- dimensional gaskets (d =2 and 3) on which we perform random walks. We concentrate here on the decay laws and show, by comparison with former results, " that trapping on these fractals interpolates nicely between the decay laws found for one- and two-dimensional regular lattices. II. THEORY In this section we focus on the decay functions due to trapping and on the corresponding decay rates. For the evaluation of the decay laws we follow the procedure of Ref. 12: We take the traps to be randomly distributed on the gasket, occupying its sites with probability p. The micro- scopic transfer rates from a site to its neighboring sites are assumed to be equal, and the walker gets trapped at the first trap encounter. For a particular realization of the random walk on the trap-free gasket, let R„denote the number of distinct sites visited in n steps. Note, as is usual in disordered systems, the difference from the regular lattice: Here the stochastic variable R„depends both on the starting point on the gasket and on the sequence of directions of the steps; for a regular lattice the starting point is irrelevant. For the same realiza- tion of the walk let F„denote the probability that trapping has not occurred up to the nth step in the ensemble of lat- tices doped with traps. Thus F„ is also a stochastic variable, so that F„=(I-p) " assuming the origin of the walk not to be a trap, and, in standard fashion, having Ra=1. The measurable survival probability is 4„, the average of F„over all realizations of the random walk' d „=e "(e ") = — e "d „, (3) with where the Kj„are the cumulants of the distribution of R„. As an example, the first two curnulants are K&, „= (R„) — = S„ and (6) where S„and o-„' are the mean and the variance of R„. The knowledge of all cumulants allows the exact deter- mination of the decay function C„via Eqs. (3) and (4). In general, however, one has to restrict oneself to the first cu- mulants, since the distribution of R„ is not known in great detail: N The expression for à =1, C & „=exp( — XS„) (8) has been advanced in many areas' '; in the random-walk field it corresponds to the first-passage-time (FPT) approxi- mation ", in the fractal field it was recently used by de Gennes. '6 For N = 2 one obtains from Eq. (7) the (2) As mentioned, the average in Eq. (2) also includes the average over starting points, and may be viewed as a double average; we encountered a similar situation for the continuous-time random-walk model. ' Introducing )&. =— ln(l — p), Eq. (2) allows a straightforward cumulant expansion 6112 l983 The American Physical Society