Percolation of disordered jammed sphere packings

TL;DR

This study determines the site and bond percolation thresholds for disordered jammed sphere packings, revealing thresholds comparable to simple cubic lattices and confirming the universality of certain ratios, with implications for various physical systems.

Contribution

First measurement of bond percolation threshold in disordered jammed sphere packings, confirming universality of ratios and providing thresholds relevant for multiple scientific applications.

Findings

Site percolation threshold p_c = 0.3116(3)

Bond percolation threshold p_c = 0.2424(3)

Universal ratios confirmed across different lattices

Abstract

We determine the site and bond percolation thresholds for a system of disordered jammed sphere packings in the maximally random jammed state, generated by the Torquato-Jiao algorithm. For the site threshold, which gives the fraction of conducting vs. non-conducting spheres necessary for percolation, we find $p_c = 0.3116(3)$, consistent with the 1979 value of Powell $0.310(5)$ and identical within errors to the threshold for the simple-cubic lattice, 0.311608, which shares the same average coordination number of 6. In terms of the volume fraction $\varphi$, the threshold corresponds to a critical value $\varphi_c = 0.199$. For the bond threshold, which apparently was not measured before, we find $p_c = 0.2424(3)$. To find these thresholds, we considered two shape-dependent universal ratios involving the size of the largest cluster, fluctuations in that size, and the second moment of the size distribution; we confirmed the ratios' universality by also studying the simple-cubic lattice with a similar cubic boundary. The results are applicable to many problems including conductivity in random mixtures, glass formation, and drug loading in pharmaceutical tablets.