Reaction Kinetics of Clustered Impurities

TL;DR

This paper investigates how the initial spatial distribution of impurities affects their long-term survival in diffusion-controlled reactions, revealing a critical codimension that determines whether impurities decay or survive indefinitely.

Contribution

It introduces a theoretical framework linking impurity survival probabilities to codimension and system dimension, highlighting a transition at codimension two.

Findings

Survival probability depends on initial impurity codimension and system dimension.

A transition occurs at codimension two, separating decay and survival regimes.

Decay behavior follows algebraic or logarithmic laws depending on parameters.

Abstract

We study the density of clustered immobile reactants in the diffusion-controlled single species annihilation. An initial state in which these impurities occupy a subspace of codimension d' leads to a substantial enhancement of their survival probability. The Smoluchowski rate theory suggests that the codimensionality plays a crucial role in determining the long time behavior. The system undergoes a transition at d'=2. For d'<2, a finite fraction of the impurities survive: ni(t) ~ ni(infinity)+const x log(t)/t^{1/2} for d=2 and ni(t) ~ ni(infinity)+const/t^{1/2} for d>2. Above this critical codimension, d'>=2, the subspace decays indefinitely. At the critical codimension, inverse logarithmic decay occurs, ni(t) ~ log(t)^{-a(d,d')}. Above the critical codimension, the decay is algebraic ni(t) ~ t^{-a(d,d')}. In general, the exponents governing the long time behavior depend on the dimension as well as the codimension.