Effective conductivity in a lattice model for binary disordered media with complex distributions of grain sizes

Using numerical simulations and analytical approximations we study a modified version of the two-dimensional lattice model [R. Piasecki,phys. stat. sol. (b) 209, 403 (1998)] for random pH:(1-p)L systems consisting of grains of high (low) conductivity for H-(L-)phase, respectively. The modification reduces a spectrum of model bond conductivities to the two pure ones and the mixed one. The latter value explicitly depends on the average concentration gamma(p) of the H-component per model cell. The effective conductivity as a function of content p of the H-phase in such systems can be modelled making use of three model parameters that are sensitive to both grain size distributions, GSD(H) and GSD(L). However, to incorporate into the model information directly connected with a given GSD, a computer simulation of the geometrical arrangement of grains is necessary. By controlling the polydispersity in grain sizes and their relative area frequencies, the effective conductivity could be raised or decreased and correlated with gamma(p). When the phases are interchanged, a hysteresis-loop-like behaviour of the effective conductivity, characteristic of dual media, is found. We also show that the topological non-equivalence of the system's microstructure accompanies some GSDs, and it can be detected by the entropic measure of the spatial inhomogeneity of model cells.