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

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/6Σ_iρ_iR _i³, = 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.