Random geometric graph description of connectedness percolation in rod systems

TL;DR

This paper models continuum percolation in rod dispersions using weighted random geometric graphs, deriving percolation thresholds that incorporate rod interactions and orientations.

Contribution

It introduces a graph-based framework for analyzing rod percolation, connecting it to the second-virial approximation and enabling inclusion of interactions.

Findings

Percolation thresholds derived from the graph model match second-virial approximation results.

The model accounts for rod interactions and orientation distributions.

Neglecting closed loops simplifies the analysis for large aspect ratio rods.

Abstract

The problem of continuum percolation in dispersions of rods is reformulated in terms of weighted random geometric graphs. Nodes (or sites or vertices) in the graph represent spatial locations occupied by the centers of the rods. The probability that an edge (or link) connects any randomly selected pair of nodes depends upon the rod volume fraction as well as the distribution over their sizes and shapes, and also upon quantities that characterize their state of dispersion (such as the orientational distribution function). We employ the observation that contributions from closed loops of connected rods are negligible in the limit of large aspect ratios to obtain percolation thresholds that are fully equivalent to those calculated within the second-virial approximation of the connectedness Ornstein-Zernike equation. Our formulation can account for effects due to interactions between the rods, and many-body features can be partially addressed by suitable choices for the edge probabilities.