Effective diffusivity in lattices of impermeable superballs

Magnus Röding

doi:10.1103/physreve.98.052908

Abstract

We perform simulations of obstructed diffusion in periodic lattice configurations of solid, nonoverlapping, impermeable superballs to study effective diffusivity as a function of shape and solid volume fraction for simple cubic (sc), body-centered cubic (bcc), and face-centered cubic (fcc) lattices. In the simulations, point particles diffuse in the intermediate space between the solid superballs. For sc lattices, we find that for a constant solid volume fraction, the optimal effective diffusivity is obtained for a particular superball between a sphere and a cube, whereas for bcc and fcc lattices, no such optimum exists. Further, we investigate the prediction of the effective diffusivity in these systems using an approximate analytical expression based on two- and three-point correlation functions and the microstructural parameter ${\ensuremath{\zeta}}_{2}$.

Highlights

Granular materials constitute a broad class of two-phase media with discrete, solid particles i.e. granules surrounded by a continuous void phase
We perform stochastic simulations of diffusion, where point particles diffuse in the intermediate space between the solid superballs, to estimate effective diffusivity as a function of shape and solid volume fraction for simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC) lattices, and study the performance of an analytical approximation based on twoand three-point correlation functions and the microstructural parameter ζ2
Point particles diffuse in the intermediate space between the solid superballs

Summary

INTRODUCTION

Granular materials constitute a broad class of two-phase media with discrete, solid particles i.e. granules surrounded by a continuous void phase. Optimal two-point bounds i.e. involving correlation functions for n ≤ 2 were derived by Hashin and Shtrikman for effective magnetic permeability in isotropic two-phase media [35], a case which is mathematically analogous to effective diffusivity. These two-point bounds were improved upon by the three-point bounds introduced by Beran [36], which were in turn simplified by Torquato [37] and Milton [38] to involve the solid volume fraction and a three-point parameter known as ζ2. We perform stochastic simulations of diffusion, where point particles diffuse in the intermediate space between the solid superballs, to estimate effective diffusivity as a function of shape and solid volume fraction for simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC) lattices, and study the performance of an analytical approximation based on twoand three-point correlation functions and the microstructural parameter ζ2

DIFFUSION SIMULATION

CHARACTERIZATION AND PREDICTION

RESULTS AND DISCUSSION

CONCLUSION AND FURTHER WORK