Kinetics of the ‘ scavenger ’ reaction

We study the kinetics of a diffusion-controlled reaction in which perfect traps (scavengers) diffuse and consume randomly distributed particles. We find that the number of particles remaining after time t decays as exp(-actd”), for spatial dimensions d 1 2 , and as exp(-act) for d 3 2 , where U is a constant, and c is the trap concentration. These predictions are supported by Monte Carlo data in d = I and d = 2 . We also discuss the differences between the scavenger reaction and the reaction of particles diffusing and being consumed by randomly distributed static traps. Finally, we treat the situation where both the particles and the traps diffuse. Very recently, there has been considerable resurgence in studying a variety of diff usioncontrolled reactions. One important example is the ‘trapping’ reaction where particles diffuse in the presence of static, randomly distributed perfect traps. Although this problem has been studied for some time, it has only recently been recognised that the particle density p( t ) follows an anomalous long-time decay proportional to p ( t ) exp(-ac2/d+2td/d+2) where d is the spatial dimension, c is the trap concentration, and a is a constant (Balagurov and Vaks 1974, Donsker and Varadhan 1975, Tanaka 1978, Grassberger and Procaccia 1982, Movoghar et a1 1982, Zumofen and Blumen 1982, Redner and Kang 1983, Havlin et al 1984). This unusual decay law stems from the presence of large (but very rare) trap-free regions. In such a region, the particle lifetime is anomalously long, and the contribution of these long lifetimes leads to the decay of equation (1). If such large fluctuations in the spatial distribution of the traps did not occur, the decay law would be exponential; this can be thought of as the mean-field limit of the decay law. While there is now a reasonable understanding of the trapping reaction, it appears that the general case where both the particles and the traps diffuse has not been considered in detail. Such a situtation may be useful to describe physical processes such as fluorescence quenching and catalysis (see e.g. Calef and Deutch (1983) for a review and extensive references). In this letter, we derive an approximate bound for the decay law of this general situation. In the extreme case of diffusing traps and stationary particles, which we term the ‘scavenger’ reaction, we argue that the decay law will be proportional to exp(-actd’2) for d < 2 and proportional to exp(-act) for d 3 2, where a is a constant. These predictions are supported by computer simulations in d = 1 and 2, where we also compare the decay laws of the scavenger and trapping 0305-4470/84/08045 1 +05$02.25 @ 1984 The Institute of Physics L45 1 L452 Letter to the Editor reactions. Finally we consider the general situation of both particle types diffusing and we argue that the long-time decay follows that of the scavenger problem. We begin by discussing the decay law of the scavenger and trapping reactions for short times. For the former reaction, the number of particles that are trapped after time t is proportional to the density of traps c and to the number of particles on the path of the moving trap. Hence P(0)A t ) CP(O)(S(t)) (2) where ( S ( t ) ) is the mean number of sites visited by the scavenger after time t. For small times, we can rewrite this as (3) On the other hand, for the trapping reaction, the exact expression for the average survival probability is ((1 c ) ~ ( ' ) ) (see e.g. Zumofen and Blumen 1982, Redner and Kang 1983). For short times, this average may be approximated by (1 c) '~") ) , and by writing 1 c = exp(-c), one sees that the decay laws of the two reactions coincide. This should hold until, in the scavenger reaction, the traps diffuse a distance of the order of their initial separation. This crossover time is given by c -* '~D; ' , where DT is the diffusion coefficient of the traps. For longer times, a particle can now be consumed by a trap which was initially very distant from the particle, and the decay in the scavenger reaction should be faster than that in the trapping reaction. Notice that the crossover time can also be obtained by equating the asymptotic decay laws of (1) and (3), and solving for the time. Now consider the long-time decay of the scavenger reaction. We first discuss the one-dimensional case and then generalise to higher dimensions. As in the trapping reaction, we focus on the rare events where a particle lies in a long trap-free region (Grassberger and Procaccia 1982). A particle is located at the origin, while the traps on one side of the particle are located at I , , 1 2 , I,, . . . (figure 1). The probability for the A t ) P(O)(l c ( S ( t ) ) > P(0 ) exp(-c(S(t))).