Thermal percolation in mixtures of monodisperse spheres

Abstract

The connectivity of individual species in a locally heterogeneous granular mixture strongly influences assembly-scale behavior. A behavioral transition is observed at the percolation threshold for a given constituent; that is, the mixing fraction at which the constituent has statistical connectivity between two opposing boundaries. This behavior is particularly evident in conductivity phenomena, e.g., the percolation of conductive particles (thermal, electric) or the relative degree of connectivity of the void space (hydraulic). Hard-core (first nearest-neighbor or lattice) percolation has been extensively studied experimentally, theoretically, and numerically. That hard-core percolation occurs in dense randomly packed bi-phasic mixtures of monodisperse spheres occurs at a mixing fraction of 0.15 v/v is well-accepted. Radiant conduction (e.g., heat), however, is influenced by hard-core “soft-shell” percolation, which is an nth-nearest neighbor problem and less well-studied. In the current work, we use discrete element method simulations coupled with a thermal network model that leverages a robust domain decomposition algorithm to simulate large assemblies of spheres to investigate soft-shell percolation numerically. Our results show that the thermal conductivity of a randomly packed assembly obeys a power law with respect to volume fraction of conductive particles while percolation threshold follows a power law with respect to coordination number. The ability of the pore fluid to transmit heat over a longer distance results in an increase of thermal conductivity and a decrease in thermal percolation threshold. Moreover, we observe that, contrary to previous findings, critical percolation density is not a dimensional invariant and depends on the microstructure of the assembly.