Diffusion-controlled annihilation $A + B \to 0$ with initially separated reactants: The death of an $A$ particle island in the $B$ particle sea

TL;DR

This paper studies the diffusion-controlled annihilation process where an island of particles A is surrounded by a sea of particles B, revealing universal scaling behavior and the distribution of particle death during expansion and contraction stages.

Contribution

It introduces a universal scaling regime for the annihilation dynamics with initially separated reactants and derives the reaction zone scaling for different dimensions.

Findings

Approximately 80% of particles die during island expansion.

Remaining 20% of particles die during island contraction.

Scaling laws for the reaction zone are obtained for different dimensional regimes.

Abstract

We consider the diffusion-controlled annihilation dynamics $A+B\to 0$ with equal species diffusivities in the system where an island of particles $A$ is surrounded by the uniform sea of particles $B$. We show that once the initial number of particles in the island is large enough, then at any system's dimensionality $d$ the death of the majority of particles occurs in the {\it universal scaling regime} within which $\approx 4/5$ of the particles die at the island expansion stage and the remaining $\approx 1/5$ at the stage of its subsequent contraction. In the quasistatic approximation the scaling of the reaction zone has been obtained for the cases of mean-field ($d \geq d_{c}$) and fluctuation ($d < d_{c}$) dynamics of the front.