Diffusion on a Rearranging Lattice

TL;DR

This study uses computer simulations to analyze how the rearrangement rate of a heterogeneous lattice affects diffusion, revealing significant changes in diffusion coefficients depending on site properties and rearrangement times.

Contribution

It introduces a novel simulation approach for diffusion on a rearranging lattice with site-based diffusion control, differing from bond percolation models.

Findings

Diffusion coefficient can vary up to threefold with rapid lattice rearrangement.

Varying the fraction of low conducting sites significantly impacts diffusion.

Rearrangement effects diminish for timescales greater than 250 times the random walk time.

Abstract

In this paper we present a computer simulation of a random walk (RW) for diffusion on a rearranging lattice. The lattice consists of two types of sites -- one good conducting (type 1) and the other poor conducting (type 2), distributed at random. The two types of sites are assigned different waiting times ($\tau_{1}$ for type 1 and $\tau_{2}$ for type 2) . We assume that at intervals of time $\tau_{r}$, the site distribution changes. The effect of this rearrangement on the diffusion coefficient is studied with varying $\tau_{r}$. We study this effect for different ratios of dwell times of the two types of sites (R) and also for different fractions (X) of the less conducting sites. An empirical relation for D($\tau_{1}$ ,$\tau_{2}$,$\tau_{r}$,X) is suggested. We have employed the well model and considered diffusion controlled by sites, rather than bonds. So our approach is different from the dynamic bond percolation model, which studies these aspects. Our results show that the diffusion coefficient D may change by a factor of upto 3 (approximately) for rapid rearrangement, and there is a considerable effect of varying X and R on the range of variation of D, where X is the fraction of low conducting sites, and R is the ratio of the dwell times for the two types of sites. Further for $\tau_{r}$ > 250$\tau$ ($\tau$ is the time unit for the random walk) the effect of rearrangement becomes negligible. The results may be useful for studying diffusion and conduction of ion conducting polymers.