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

Abstract

 We study the multi-scale site percolation in a system with interacting objects.  The interaction energy modifies the value of the percolation threshold in a limited range. When a repulsive interaction dominates thermal energy, the exact c at scales k = 2, 3 is found.  In turn, when k >> 1, the evaluated formula for c(k) implies the limit threshold value 0.75. A B S T R A C TRecently, the effective medium approach using 2  2 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 k  k 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.

Anisotropy in electrical conductivity of two-dimensional films containing aligned nonintersecting rodlike particles: Continuous and lattice models

“…Recently, the effect of filler alignment on the electrical conductivity of 2D composites has been simulated within a continuous approach for intersecting rodlike particles [26]. The multi-scale percolation behavior of the effective conductivity has been studied using a lattice model [27,28]. A lattice approach has also been applied to study the electrical conductivity of a monolayer produced by the random sequential adsorption (RSA) of non-overlapping conductive rodlike particles onto an insulating substrate [29].…”

Influence of defects on the effective electrical conductivity of a monolayer produced by random sequential adsorption of linear k-mers onto a square lattice

“…Recently, the percolation behaviour of the effective conductivity for a lattice model with interacting particles was reported [18]. Percolation and jamming phenomena have been investigated for the random sequential adsorption (RSA) of dimers on a square lattice, where the influence of dimer alignment on the electrical conductivity was examined [19].…”

“…The physical properties of monolayers produced by RSA have been widely studied and discussed with special attention being paid to the effects of defects [13][14][15] and particle size distribution [16,17]. Recently, the percolation behaviour of the effective conductivity for a lattice model with interacting particles was reported [18]. Per-colation and jamming phenomena have been investigated for the random sequential adsorption (RSA) of dimers on a square lattice, where the influence of dimer alignment on the electrical conductivity was examined [19].…”