Diffusion-controlled coalescence, fragmentation and collapse of $d$-dimensional $A$-particle islands in the $B$-particle sea

TL;DR

This paper systematically analyzes the diffusion-controlled evolution, coalescence, fragmentation, and collapse of two spatially separated $A$-particle islands in a $B$-particle sea, revealing dimension-dependent critical parameters and universal behaviors.

Contribution

It introduces a dimensionless parameter governing island interactions and identifies critical values for coalescence and collapse, providing a unified framework for different dimensions.

Findings

Existence of a $d$-dependent critical value $\\Lambda_{\star}$ for coalescence.

Universal trajectory for island centers merging at a critical parameter.

In 1D, coalescence and collapse occur simultaneously at the same threshold.

Abstract

We present a systematic analysis of diffusion-controlled evolution and collapse of two identical spatially separated $d$-dimensional $A$-particle islands in the $B$-particle sea at propagation of the sharp reaction front $A+B\to 0$ at equal species diffusivities. We show that at a sufficiently large initial distance between the centers of islands $2\ell$ compared to their characteristic initial size and a relatively large initial ratio of concentrations island/sea the evolution dynamics of the island-sea-island system is determined unambiguously by the dimensionless parameter $\Lambda={\cal N}_{0}/{\cal N}_{\Omega}$, where ${\cal N}_{0}$ is the initial particle number in the island and ${\cal N}_{\Omega}$ is the initial number of sea particles in the volume ${\Omega}=(2\ell)^{d}$. It is established that a) there is a $d$-dependent critical value $\Lambda_{\star}$ above which island coalescence occurs; b) regardless of $d$ the centers of each of the islands move towards each other along a {\it universal} trajectory merging in a united center at the $d$-dependent critical value $\Lambda_{s}\geq\Lambda_{\star}$; c) in one-dimensional systems $\Lambda_{\star}=\Lambda_{s}$, therefore at $\Lambda<\Lambda_{\star}$ each of the islands dies individually, whereas at $\Lambda>\Lambda_{\star}$ coalescence is completed by collapse of a single-centered island in the system center; d) in two- and three-dimensional systems in the range $\Lambda_{\star}< \Lambda < \Lambda_{s}$ coalescence is accompanied by subsequent fragmentation of a two-centered island and is completed by individual collapse of each of the islands. We discuss a detailed picture of coalescence, fragmentation and collapse of islands focusing on evolution of their shape and on behavior of the relative width of the reaction front at the final collapse stage and in the vicinity of starting coalescence and fragmentation points.