The asymptotic behaviour of the initially separated A + B(static) -> 0 reaction-diffusion systems

TL;DR

This paper analyzes the long-time behavior of initially separated A+B->0 reaction-diffusion systems, deriving formulas for reaction zone dynamics and studying effects of diffusion constants through mean-field approximation and numerical solutions.

Contribution

It provides general analytical expressions for reaction zone properties and explores the impact of diffusion constants on reaction kinetics in separated species systems.

Findings

Derived formulas for reaction zone center and total reaction rate.

Identified how the reaction zone width depends on system parameters.

Numerical validation of mean-field analytical results.

Abstract

We examine the long-time behaviour of A+B \to 0 reaction-diffusion systems with initially separated species A and B. All of our analysis is carried out for arbitrary (positive) values of the diffusion constant D_A of particles A and initial concentrations a_0 and b_0 of A's and B's. We derive general formulae for the location of the reaction zone centre, the total reaction rate, and the concentration profile of species A outside the reaction zone. The general properties of the reaction zone are studied with a help of the scaling ansatz. Using the mean-field approximation we find the functional forms of `tails' of the reaction rate R and the dependence of the width of the reaction zone on the external parameters of the system. We also study the change in the kinetics of the system with D_B > 0 in the limit D_B \to 0. Our results are supported by numerical solutions of the mean-field reaction-diffusion equation.