Nearest-neighbor connectedness theory: A general approach to continuum percolation

TL;DR

This paper introduces a new geometric method to estimate continuum percolation thresholds, providing accurate predictions for line segments and disks in two dimensions, outperforming traditional approaches.

Contribution

A novel nearest-neighbor connectedness theory that improves percolation threshold predictions in two-dimensional systems.

Findings

Predicted percolation threshold for segments: $ ho_c l^2 oughly 5.83$

Predicted percolation threshold for disks: $ ho_c a oughly 1.00$

Close agreement with Monte Carlo simulations for both models.

Abstract

We introduce a method to estimate continuum percolation thresholds and illustrate its usefulness by investigating geometric percolation of non-interacting line segments and disks in two spatial dimensions. These examples serve as models for electrical percolation of elongated and flat nanofillers in thin film composites. While the standard contact volume argument and extensions thereof in connectedness percolation theory yield accurate predictions for slender nanofillers in three dimensions, they fail to do so in two dimensions, making our test a stringent one. In fact, neither a systematic order-by-order correction to the standard argument nor invoking the connectedness version of the Percus-Yevick approximation yield significant improvements for either type of particle. Making use of simple geometric considerations, our new method predicts a percolation threshold of $\rho_c l^2 \approx 5.83$ for segments of length $l$, which is close to the $\rho_c l^2 \approx 5.64$ found in Monte Carlo simulations. For disks of area $a$ we find $\rho_c a \approx 1.00$, close to the Monte Carlo result of $\rho_c a \approx 1.13$. We discuss the shortcomings of the conventional approaches and explain how usage of the nearest-neighbor distribution in our new method bypasses those complications.