Percolation-based mean field theory for disordered particle packings

Abstract

Dense disordered packings of hard particles are useful models for granular materials, composites, colloids, foams, and condensed matter. It is very challenging to devise predictive theories of random packings, due to their intrinsic non-equilibrium and non-local nature. Here, we develop a mean-field theory for disordered particle packings by exploiting a recently discovered percolation transition of the hard-particle contact network that precedes the jamming transition. Assuming that the salient structural correlations in the final jammed packing are mainly inherited from those emerging in the percolating particle network, we devise a mean-field formalism that maps the hard particle percolation to analytically solvable Bethe-lattice models. This allows us to establish an analytical relation connecting the packing fraction ϕ and average contact number Z for a wide spectrum of congruent non-spherical hard particles in three-dimensional Euclidean space ℝ³, where the particle shape factor is rescaled by the percolation critical exponent, reflecting the coupling of local and large scales in our mean-field formalism. The accuracy of our theory is ascertained by comparing its predictions of ϕ to the corresponding numerical values reported in literature. Our mean-field theory can be readily generalized to higher dimensions.