Time dependent diffusion in a disordered medium with partially absorbing walls: A perturbative approach

TL;DR

This paper analytically investigates the time-dependent diffusion coefficient in a dilute suspension of partially absorbing spheres, providing a perturbative expansion that captures short and long time behaviors with first-order accuracy.

Contribution

It introduces a perturbative approach using the exact single particle operator to derive an analytical expression for the diffusion coefficient in a partially absorbing medium.

Findings

Derived a closed-form expression for D(t) accurate to first order in volume fraction

Validated short and long time limits against known results

Found that long-time diffusion coefficient scales as 1/t with specific constants

Abstract

We present an analytical study of the time dependent diffusion coefficient in a dilute suspension of spheres with partially absorbing boundary condition. Following Kirkpatrick (J. Chem. Phys. 76, 4255) we obtain a perturbative expansion for the time dependent particle density using volume fraction $f$ of spheres as an expansion parameter. The exact single particle $t$-operator for partially absorbing boundary condition is used to obtain a closed form time-dependent diffusion coefficient $D(t)$ accurate to first order in the volume fraction $f$. Short and long time limits of $D(t)$ are checked against the known short-time results for partially or fully absorbing boundary conditions and long-time results for reflecting boundary conditions. For fully absorbing boundary condition the long time diffusion coefficient is found to be $D(t)=5 a^2/(12 f D_{0} t) +O((D_0t/a^2)^{-2})$, to the first order of perturbation theory. Here $f$ is small but non-zero, $D_0$ the diffusion coefficient in the absence of spheres, and $a$ the radius of the spheres. The validity of this perturbative result is discussed.