Percolation and Conduction

TLDR

Resistor networks, from which resistors are removed at random, provide the natural generalization of the lattice models for which percolation thresholds and percolation probabilities have previously been considered. Extensions of percolation theory to treat transport are described. A “Green’s function” derivation of the effective medium theory, which makes contact with similar treatments of disordered alloys, is presented. The study shows that the normalized conductance of random resistor networks is sharply defined near the percolation threshold, depends only on dimensionality, and is accurately described by effective medium theory away from threshold, providing the first quantitative test of the theory and revealing that percolation channels above threshold are three‑dimensional.

Abstract

Extensions of percolation theory to treat transport are described. Resistor networks, from which resistors are removed at random, provide the natural generalization of the lattice models for which percolation thresholds and percolation probabilities have previously been considered. The normalized conductance, $G$, of such networks proves to be a sharply defined quantity with a characteristic concentration dependence near threshold which appears sensitive only to dimensionality. Numerical results are presented for several families of $3D$ and $2D$ network models. Except close to threshold, the models based on bond percolation are accurately described by a simple effective medium theory, which can also treat continuous media or situations less drastic than the percolation models, for example, materials in which local conductivity has a continuous distribution of values. The present calculations provide the first quantitative test of this theory. A "Green's function" derivation of the effective medium theory, which makes contact with similar treatments of disordered alloys, is presented. Finally, a general expression for the conductance of a percolation model is obtained which relates $G$ to the spin-stiffness coefficient, $D$, of an appropriately defined model dilute ferromagnet. We use this relationship to argue that the "percolation channels" through which the current flows above threshold must be regarded as three dimensional.