Deterministic Infiltration of Voids among Impermeable Barrier Particles*

Among the simplest genuine second order phase transitions, percolation transitions are nonetheless relevant in a variety of physical settings. While percolation models on a discrete lattice have been extensively studied, in other systems the systematic identification of navigable networks of connected volumes has been elusive. A salient case is fluid flow through voids around impermeable barrier particles. In such systems, the percolation transition is marked by the coalescence of void volumes into a bulk level contiguous network as the concentration of barrier particles decreases below a critical density. With the geometry of void volumes being difficult to predict a priori, stochastic techniques have been used in which virtual tracer particles undergo specular reflection from impenetrable grains. While in principle geometrically exact since no discretization scheme is imposed, low dynamical exponents near the percolation transition mean that the tracer particles may visit and revisit portions of the void volume network many times instead of exploring the broader network. To circumvent this inefficiency, we instead use a deterministic method which may infiltrate connected volumes while avoiding redundant visitation of any region of the network containing the randomly selected point. Our method, valid for any system made up of impermeable polyhedral grains with a finite number of facets, begins at a randomly selected point in a void volume. We then reconstuct the concave void volumes by identifying faces bounding the void network as well as their connectivity to neighboring faces. The volume is expanded in this manner until the process terminates either in a finite void volume or a system spanning network corresponding to an "infinite cluster" in the context of percolation models on discrete lattices; disorder averaged fractions of the latter may be analyzed to calculate the critical concentration. To validate our technique, we calculate critical grain concentrartions for cases in which grains are platonic solids (i.e. tetrahedra, cubes, octahedra, dodecahedra, and icosahedra), and we consider both aligned and randomly oriented varrier particles.