Excluded Volume and the Continuum Percolation of Polyhedra

As nanoscience looks to manufacture designer materials from its ever-growing catalog of mesoscopic building blocks, a common goal has been the synthesis of connected structures. Percolation has long served as the foundational model for investigating such networked materials and has enjoyed a historied investigation by physicists and mathematicians alike. However, while universality implies that many of percolation's features can be gleaned from simplified simulations constrained to a lattice, many of its key properties including the location of the percolative phase transition are sensitive to the details of the model. There is therefore a need for simulations to calculate these non-universal properties and inform networked material design. Previous simulation work in the continuum has primarily focused on the percolation of two-dimensional objects or has been restricted to particles within a limited family of geometries. In this work, we implement the efficient Newman-Ziff algorithm to allow for the study of percolation for the more general class of convex polyhedra to investigate the role of shape on the position of the percolative transition and the robustness of the resultant network. Our results on regular polyhedra show strong agreement with the established theory of excluded volume and reinforce the notion that particle asphericity plays a primary role in the percolative behavior of particle systems.