A Lattice-Based Approach to Percolation in Penetrable Sphere Systems

An heuristic mapping is presented that enables analysis of continuum percolation by penetrable spheres in terms of an analogous lattice model. The resulting framework is based upon the well-known Bethe lattice, which is modified by the inclusion of an adjustable proportion of fully connected subgraphs that provide a particularly simple, albeit operational, vehicle for modeling the effect of inter-particle clustering. The volume fraction at the percolation threshold is calculated as a function of (i) the degree of interpenetrability amongst the spheres, and: (ii) the extent of particle clustering, and the results are compared with findings from Monte Carlo (MC) simulations of this problem. It is found that a suitably chosen ansatz that reflects a monotonic increase in the degree of clustering with increasing inter-sphere penetrability leads to close agreement between percolation thresholds calculated using the present approach and the results of MC simulations.