Diffusion of particles in an expanding sphere with an absorbing boundary

TL;DR

This paper analyzes particle diffusion within an expanding sphere with an absorbing boundary, providing exact and asymptotic solutions for particle density and survival rates in different growth regimes.

Contribution

It offers new exact solutions for particle diffusion in a growing sphere and derives asymptotic solutions for different growth rate scenarios.

Findings

Exact solution for d=3 with parabolic growth

Asymptotic solutions for slow and fast sphere growth

Quantitative analysis of particle survival over time

Abstract

We study the problem of particles undergoing Brownian motion in an expanding sphere whose surface is an absorbing boundary for the particles. The problem is akin to that of the diffusion of impurities in a grain of polycrystalline material undergoing grain growth. We solve the time dependent diffusion equation for particles in a d-dimensional expanding sphere to obtain the particle density function (function of space and time). The survival rate or the total number of particles per unit volume as a function of time is evaluated. We have obtained particular solutions exactly for the case where d=3 and a parabolic growth of the sphere. Asymptotic solutions for the particle density when the sphere growth rate is small relative to particle diffusivity and vice versa are derived.