On multi-scale percolation behaviour of the effective conductivity for the lattice model with interacting particles

TL;DR

This paper investigates how particle interactions influence multi-scale percolation thresholds in lattice models, revealing that repulsive interactions limit the shift range and analytical formulas can predict thresholds at small scales.

Contribution

It extends previous multi-scale percolation models by incorporating interactions, deriving analytical formulas for thresholds at small scales, and analyzing the effects of repulsive interactions.

Findings

Interaction reduces the range of percolation threshold shifts.

Exact thresholds at scales 2 and 3 are derived analytically.

Threshold approaches 0.75 for large scales, with negligible odd-even differences.

Abstract

Recently, the effective medium approach using 2x2 basic cluster of model lattice sites to predict the conductivity of interacting droplets has been presented by Hattori et al. To make a step aside from pure applications, we have studied earlier a multi-scale percolation, employing any kxk basic cluster for non-interacting particles. Here, with interactions included, we examine in what way they alter the percolation threshold for any cluster case. We found that at a fixed length scale k the interaction reduces the range of shifts of the percolation threshold. To determine the critical concentrations, the simplified model is used. It diminishes the number of local conductivities into two main ones. In the presence of a dominance of the repulsive interaction over the thermal energy, the exact percolation thresholds at scales k=2 and 3 can be obtained from analytical formulas. Furthermore, by a simple reasoning, we obtain the limiting threshold formula for odd k. When k>>1, the odd-even difference becomes negligible. Hence, the 0.75 is the highest possible value of the threshold.