Simulation of diffusion in 2-D heterogeneous systems: comparison with effective medium and percolation theories

We report Monte Carlo simulations of the conductivity of a two-dimensional static system of non-overlapping, non- or weakly conducting discs in a conducting medium. The conductivity is evaluated using the “blind ant” diffusion algorithm. We investigate the dependence of the conductivity on (i) the size, (ii) the area fraction and (iii) the conductivity of the discs. Aim of the investigation is to interpret the experimental data obtained by Tamada, Kim and Yu (Langmuir 9 (1993) 1545) for the lateral diffusion of tracer molecules in biphasic phospholipid monolayers (liquid expanded/gas coexisting phases). We discuss the simulation results in terms of the effective medium and the percolation theories, respectively. We show that the theoretical equation of Keller and Sachs for non-conducting discs is in excellent agreement with the simulation data for large obstacles and can also be scaled to describe the case of conducting discs. Application of the symmetrical Bruggeman-Landauer equation is restricted to the case of comparable conductivity of both phases.