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

TL;DR

This paper investigates how the size of local clusters in lattice models affects the percolation threshold and effective conductivity in heterogeneous media, revealing scale-dependent behaviors and simplified modeling approaches.

Contribution

It extends lattice models to arbitrary cluster sizes and explores their impact on percolation thresholds and conductivity, including a simplified scalable model.

Findings

Increasing cluster size shifts percolation thresholds

Behavior is typical of materials with dominant linear conducting paths

Simplified model retains original thresholds

Abstract

Macroscopic properties of heterogeneous media are frequently modelled by regular lattice models, which are based on a relatively small basic cluster of lattice sites. Here, we extend one of such models to any cluster's size kxk. We also explore its modified form. The focus is on the percolation behaviour of the effective conductivity of random two- and three-phase systems. We consider only the influence of geometrical features of local configurations at different length scales k. At scales accessible numerically, we find that an increase in the size of the basic cluster leads to characteristic displacements of the percolation threshold. We argue that the behaviour is typical of materials, whose conductivity is dominated by a few linear, percolation-like, conducting paths. Such a system can be effectively treated as one-dimensional medium. We also develop a simplified model that permits of an analysis at any scale. It is worth mentioning that the latter approach keeps the same thresholds predicted by the former one. We also briefly discuss a three-phase system, where the double-thresholds paths appear on model surfaces.