Percolation Phase Transitions Involving Impermeable Torus Shaped Grains

Porous materials, often made up of impenetrable granular inclusions, admit the flow of fluid or charge through irregularly shaped voids among the grains. With large scale dynamical infiltration simulations, we calculate critical grain densities for randomly placed inter-penetrating torus-shaped barrier particles. A salient feature of the latter is their concave geometry. Moreover, we consider a range of torus shapes from barrier particles resembling spheres with small antipodal dimples to thin ring shaped toruses. This set of geometries straddles a topological transition as the dimples expand toward each other and join to form a central hole. We find a peak in the critical grain density coinciding with the increase in topological genus from zero to one, which we ascribe to the torus central holes serving as channels to increase the void network connectivity. We find that the percolation thresholds (in terms of percentage of void volume) for assemblies of randomly oriented very thin ring shapes with square and circular cross sections tend to the counterparts for square prisms and cyinders respectively in the high aspect ratio limit. The consistently lower critical concentrations for aligned grains may be due to a walling off of central holes from void volume networks.