Clustering and percolation in multicomponent systems of randomly centered and permeable spheres

Abstract

For a multicomponent system of particles in equilibrium, an exact integral equation is derived for the pair connectedness function (which measures the probability that two particles, with centers a distance r apart, are connected). The pair connectedness function, mean cluster size, and percolation thresholds for mixtures of randomly centered (noninteracting) spheres and permeable spheres are then obtained analytically in the Percus–Yevick (PY) approximation. (The permeable-sphere model provides a one-parameter bridge from randomly centered sphere mixtures to hard-sphere mixtures.) For this family of models, connectedness is defined by particle overlap. It is found that, within the PY approximation, a multicomponent mixture of randomly centered spheres percolates at ξ3=π/6∑iρiR3i =1/2, independent of the concentration and size distributions of the particles. For the permeable-sphere case the percolation threshold depends on the relative densities, size, and interparticle permeability among the species. The loci of the percolation thresholds of a binary mixture of permeable spheres are explicitly determined.