Modifying continuous-time random walks to model finite-size particle diffusion in granular porous media

Abstract

The continuous-time random walk (CTRW) model is useful for alleviating the computational burden of simulating diffusion in actual media. In principle, isotropic CTRW only requires knowledge of the step-size, $P_l$, and waiting-time, $P_t$, distributions of the random walk in the medium and it then generates presumably equivalent walks in free space, which are much faster. Here we test the usefulness of CTRW to modelling diffusion of finite-size particles in porous medium generated by loose granular packs. This is done by first simulating the diffusion process in a model porous medium of mean coordination number, which corresponds to marginal rigidity (the loosest possible structure), computing the resulting distributions $P_l$ and $P_t$ as functions of the particle size, and then using these as input for a free space CTRW. The CTRW walks are then compared to the ones simulated in the actual media. In particular, we study the normal-to-anomalous transition of the diffusion as a function of increasing particle size. We find that, given the same $P_l$ and $P_t$ for the simulation and the CTRW, the latter predicts incorrectly the size at which the transition occurs. We show that the discrepancy is related to the dependence of the effective connectivity of the porous media on the diffusing particle size, which is not captured simply by these distributions. We propose a correcting modification to the CTRW model -- adding anisotropy -- and show that it yields good agreement with the simulated diffusion process. We also present a method to obtain $P_l$ and $P_t$ directly from the porous sample, without having to simulate an actual diffusion process. This extends the use of CTRW, with all its advantages, to modelling diffusion processes of finite-size particles in such confined geometries.