The long time behavior of initially separated A+B->0 reaction-diffusion systems with arbitrary diffusion constants

TL;DR

This paper analyzes the long-term behavior of segregated A+B->0 reaction-diffusion systems with arbitrary diffusion constants, deriving general formulas for concentration profiles, reaction zone location, and reaction rates, and studying the reaction layer properties.

Contribution

It provides a comprehensive analysis and explicit formulas for the asymptotic behavior of reaction-diffusion systems with arbitrary diffusion constants and initial conditions.

Findings

Derived general formulas for concentration profiles and reaction zone location.

Established conditions for stationary reaction front.

Showed scaling functions are independent of diffusion constants and initial concentrations.

Abstract

We examine the long time behaviour of A+B->0 reaction diffusion systems with initially segregated species A and B. All of our analysis is carried out for arbitrary (positive) values of the diffusion constants $D_A$, $D_B$, and initial concentrations $a_0$ and $b_0$ of A's and B's. We divide the domain of the partial differential equations describing the problem into several regions in which they can be reduced to simpler, solvable equations, and we merge the solutions. Thus we derive general formulae for the concentration profiles outside the reaction zone, the location of the reaction zone center, and the total reaction rate. An asymptotic condition for the reaction front to be stationary is also derived. The properties of the reaction layer are studied in the mean-field approximation, and we show that not only the scaling exponents, but also the scaling functions are independent of $D_A$, $D_B$, $a_0$ and $b_0$.