A Sheath/Tunnel Topological Phase Transition of free surfaces in assemblies of randomly placed Inclusions

Percolation transition in systems comprised of randomly placed impermeable grains is often described either in terms of system-spanning clusters of inclusions appearing at a relatively low density per volume ρ_c1 or the much higher void percolation crtical concentration ρ_c2 beyond which empy tunnel-like interstitial spaces cease to exist on a macroscopic scale.  We interpret ρ_c1 and ρ_c2 as the low and high density boundaries of percolating free surfaces.  We show with direct large-scale Monte Carlo calculations that in the thermodynamic limit, the exposed surfaces either form sheaths about clusters of connected grains or instead line tunnel-shaped voids.  We identify and characterize (e.g. in terms of the critical concentration and associated critical exponents) a third phase boundary ρ_c* intermediate between ρ_c1 and ρ_c2 a third phase boundary in which, with increasing grain concentration, sheath-like surfaces abruptly become tunnels.  To locate this second order topological phase transition, we employ a technique which identifies free surfaces in a geometrically exact manner with a computational cost that scales only linearly in the system volume.  To locate ρ_c*, we calculate the fraction of free surfaces which are sheaths and the portion which are tunnels with a topological test that requires a detalied understanding of the geometry of an exposed surface, gleaned with our technique which operates in all grain density regimes.  We calculate ρ_c* (as well as ρ_c1 and ρ_c2) for the case of the Platonic solids as well as the semi-regular truncated icosahedron for both aligned and randomly oriented cases.