A two-bond theory of conductivity in bond-disordered resistor networks

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Abstract
A cluster theory is given to derive the effective (or macroscopic) conductivity of disordered square and simple cubic resistor networks in which bonds are broken at random (bond model). The cluster approximation that goes beyond single-bond effective medium approximations (single-bond EMA) can be obtained from the partial sum of two-bond terms by means of a diagrammatic representation of the perturbation series. The two-bond effective medium approximation is also derived from different points of view in order to clarify the meaning and applicability of the method. The thresholds pc for the bond percolation on a square lattice and on a simple cubic lattice are calculated by the two-bond EMA. The results give the exact value pc=1/2 for the square lattice and certainly show an improvement over the single-bond EMA for the simple cubic lattice. It is also found that the sum of two-bond terms gives the second-order term of the broken-bond concentration c on a non-self-consistent two-bond approximation. A comparison of the second-order term with that of the continuum model is made and the differences between the lattice and continuum models are demonstrated.

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