Two-dimensional continuum percolation and conduction

In physical systems, we encounter phenomena well described by the continuum percolation as often as those characterized by the lattice percolation. However, the conductivity data of the continuum percolation are scarce compared to those of the lattice percolation. With this situation in mind, we consider here two types of two-dimensional continuum percolation problem and compute, by the finite element method, the critical volume fractions of the metallic elements for the metal-insulator transition. We also compute the conductivities. The geometrical configuration of the first (or second) type is formed by random deposits of square metals (or holes) of the same size and orientation on an insulator substrate (or on a metal). The critical volume fractions are calculated to be, respectively, 0.58 and 0.42. We compare the computed conductivities with results from the effective-medium approximation and from the extended Watson–Leath equation, with the measurement of Smith and Lobb and with the conductivities of composite materials having a regular structure.