Percolation through Voids around Randomly Oriented Polyhedra and Axially Symmetric Grains

Porous materials made up of impermeable grains constrain fluid flow to voids around the impenetrable inclusions. A percolation transition marks the boundary between densities of grains permitting bulk transport and concentrations blocking traversal on macroscopic scales. With dynamical infiltration of void spaces using virtual tracer particles, we treat inclusion geometries exactly. We calculate the critical number density per volume ρ_c for a variety of axially symmetric shapes and faceted solids with the former including cylinders, ellipsoids, cones, and tablet shaped grains from highly oblate (platelike) to highly prolate (needlelike) extremes. For the latter, results suggest a common asymptotic value identical to the counterpart for aligned cylindrical grains. We find percolation thresholds for each of the five platonic solids (i.e., tetrahedra, cubes, octahedra, dodecahedra, and icosahedra) as well as truncated icosahedra. For each polyhedron type, we consider aligned and randomly oriented grains, finding distinct percolation thresholds for the former versus the latter only for cubes. The anomalous diffusion exponents we find differ from those of the universality class for discrete models on 3D lattices.