Segregation in annihilation reactions without diffusion: Analysis of correlations

Abstract

We study binary one-dimensional exchange-mediated A +/?-• 0 reactions in the absence of diffusion. Using the Kirkwood superposition approximation to decouple the infinite hierarchy of many-particle densities a closed set of integrodifferential equations is derived. A comparison to simulation calculations shows that the approximation reproduces the correct long-time asymptotic decay of reactant concentration as well as the correct spatial correlations of the segregation pattern.PACS numbers: 82.20.-w, 05.40.+J Recent years have seen a drastic increase in interest inthe A +B-• 0 reaction scheme, following the realization that segregation of like particles occurs, 1 " 4 which leads to long-time behavior different from the classical kinetic scheme. A recent review article summarizes the findings. 4 In contrast to the A+A-+ A, A+A-^>0 9 and A+B-+B diffusion-limited reactions, for which in the last years several exact ID solutions have become available, 5 " 9 the analytic work for A +B-> 0 reactions is still based (even in ID) on approximate (although increasingly reliable) analytical methods. 4,10Most of the analyses have centered on diffusionlimited reactions in which the reactants move (either diffusionally or as a random walk) before reacting on contact. By contrast, far less attention has been paid to reactions in which the particles annihilate via longerranged interactions, such as exchange, in the absence of motion. Since motion and reaction often cause adverse effects on segregation, 1,2,10 we consider here only reactions with immobile reactants. Such reactions, especially when taking place in spaces of low dimensions, provide a stringent test of approximate analytical approaches, since possible deviations of the theory from the computer experiments are not obscured by diffusion effects or by the (often milder) behavior in higher dimensions.Interestingly, we find expressions which use the Kirkwood superposition approximation 11 for the three-center correlation functions to perform exceedingly well even for short-range exchange reactions in ID, and to provide both the correct long-time asymptotic behavior of the reaction, as well as the correct ^42?-particle segregation pattern. This fact is quite remarkable, since then the method may represent a powerful analytical approach to the A +B-• 0 problem, even when diffusion is involved. By contrast, neglect of the segregation aspect leads to (even qualitatively) wrong asymptotic forms.In the following we consider randomly distributed A and B particles, which react via exchange, w>(r) -w(r) = woexp( -r/ro), where w is the probability rate that A and B situated at distance r react and ro is a constant, which determines the interaction range. The exponential

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