Percolation through Voids around Randomly Oriented Platonic Solids

TL;DR

This study analyzes the percolation transition in porous materials made of polyhedral grains, providing exact thresholds for different Platonic solids and revealing a universal subdiffusive scaling exponent at the transition.

Contribution

It offers an exact calculation of percolation thresholds for various Platonic solids, including both aligned and random orientations, and identifies a universal dynamical scaling exponent.

Findings

Percolation thresholds vary with grain shape and orientation.

Cube-shaped grains show distinct thresholds between aligned and random orientations.

Universal subdiffusive scaling exponent z = 0.19(1) at the percolation threshold.

Abstract

Porous materials made up of impermeable polyhedral grains constrain fluid flow to voids around the impenetrable constituent barrier particles. A percolation transition marks the boundary between assemblies of grains which contain system spanning void networks, admitting bulk transport, and configurations which may not be traversed on macroscopic scales. With dynamical infiltration of void spaces using virtual tracer particles, we give an exact treatment of grain geometries, and we calculate critical densities for polyhedral inclusions for the five platonic solids (i.e. tetrahedra, cubes, octahededra, dodecahedra, and icosahedra). In each case, we calculate percolation threshold concentrations $\rho_{c}$ for aligned and randomly oriented grains, finding distinct $\rho_{c}$ values for the former versus the latter only for cube-shaped grains. We calculate the dynamical scaling exponent at the percolation threshold, finding subdiffusive value $z = 0.19(1)$ common to all grain shapes considered.