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

Recently, the effective medium approach (EMA) using 2×2 basic cluster of model lattice sites to predict the conductivity of interacting microemulsion droplets has been presented by Hattori et al. To make a step aside from pure applications, we 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 EMA-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 two small scales can be revealed from analytical formulas. Furthermore, at large scales, the highest possible value of the estimated threshold can be obtained.