Universal conductivity curve for a plane containing random holes

Abstract

This paper examines the general percolation problem of cutting randomly centered insulating holes in a two-dimensional conducting sheet, and explores how the electrical conductivity σ decreases with the remaining area fraction. This problem has been studied in the past for circular, square, and needlelike holes, using both computer simulations and analog experiments. In this paper, we extend these studies by examining cases where the insulating hole is of arbitrary shape, using digital-image-based numerical techniques in conjunction with the Y-∇ algorithm. We find that, within computational uncertainty, the scaled percolation threshold, ${\mathit{x}}_{\mathit{c}}$=${\mathit{n}}_{\mathit{c}}$〈${\mathit{L}}_{\mathrm{eff}}^2$〉=5.9±0.4, is a universal quantity for all the cases studied, where ${\mathit{n}}_{\mathit{c}}$ is the critical value at percolation of the number of holes per unit area n, and 〈${\mathit{L}}_{\mathrm{eff}}^2$〉 is a measure of ${\mathit{n}}_{\mathit{I}}^{\mathrm{-}1}$, the initial slope of the σ(n) curve, calculated in the few-hole limit and averaged over the different shapes and sizes of the holes used. For elliptical holes, ${\mathit{L}}_{\mathrm{eff}}$=2(a+b), where a and b are the semimajor and semiminor axes, respectively. All results are well described by the universal conductivity curve: σ/${\mathrm{\sigma }}_0$=[(1-x/5.90)(1+x/5.90-${\mathit{x}}^2$/24.97)(1+x/3.31${)}^{\mathrm{-}1}$ ${]}^{1.3}$, where x=${\mathit{nL}}_{\mathrm{eff}}^2$, and ${\mathrm{\sigma }}_0$ is the conductivity of the sheet before any holes are introduced.