Worms and tunnels; Never shall the Twain meet: A new phase boundary in the percolation of free surfaces in systems of impermerable interpenetrating inclusions

Percolation phenomena in systems made up of impermeable interpenetratiions are often described either in terms of a phaese transition at lower densities in which particles begin to overlap and form system spanning clusters and at much high densities in where connected clusters of void volumes become closed off and cease to exist on a macroscopic scale with increasing concentration of the impenetrable grains. Both the former and latter exhibit standard critical behavior of percolation transitions in three dimensions. In this work, we regard these two transitions and low and high density boundaries of a phase of percolating exposed surfaces. This regime of grain densities typically exists over a range of particle concentrations spanning at least an order of magnitude in terms of density per unit volume. We present results of a geometrically exact method for finding and characterizing free surfaces. This approach provides an efficient (i.e. scaling only as the system volume) and accurate geometrically direct way to find the critical grain percolation for a wide range of convex faceted inclusions; as an example, we report results obtained in this way for the percolation of voids around randomly placed Platonic solids. In the present study we also characterize the free surfaces themselves in denstity regimes in which exposed surfaces percolate. Among the most basic questions one can ask is withether a surface formes a sheath about a cluster of inclusions or whether the exposed surface is instead tunnel-like, linining the interior of a void volume among impermeable grains. We find that in the large system size limit, free surfaces are either sheaths or tunnels for a particular density, but not both. We find and characterize a third phase boundary for intermediate grain densiteis which subdivides the percolation exposed surface phase into sheath-like surfaces which bound clusters of connected grain particles and tunnel-like surfaces which instead line void volumes. As the density of inclusions increases, we find that the free surfaces undergo an abrupt transition from sheaths to tunnels as surfaces cease to bound paricle clusters and instead define void regions. We find that this transition is a second order phase transition with critical exponents differeing from those of the standard three dimension percolation universality class.