TY - JOUR

T1 - Universal conductivity curve for a plane containing random holes

AU - Garboczi, E. J.

AU - Thorpe, M. F.

AU - Devries, M. S.

AU - Day, A. R.

PY - 1991

Y1 - 1991

N2 - 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 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, xc=ncLeff2=5.90.4, is a universal quantity for all the cases studied, where nc is the critical value at percolation of the number of holes per unit area n, and Leff2 is a measure of nI-1, the initial slope of the (n) curve, calculated in the few-hole limit and averaged over the different shapes and sizes of the holes used. For elliptical holes, Leff=2(a+b), where a and b are the semimajor and semiminor axes, respectively. All results are well described by the universal conductivity curve: /0=[(1-x/5.90)(1+x/5.90-x2/24.97) (1+x/3.31)-1]1.3, where x=nLeff2, and 0 is the conductivity of the sheet before any holes are introduced.

AB - 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 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, xc=ncLeff2=5.90.4, is a universal quantity for all the cases studied, where nc is the critical value at percolation of the number of holes per unit area n, and Leff2 is a measure of nI-1, the initial slope of the (n) curve, calculated in the few-hole limit and averaged over the different shapes and sizes of the holes used. For elliptical holes, Leff=2(a+b), where a and b are the semimajor and semiminor axes, respectively. All results are well described by the universal conductivity curve: /0=[(1-x/5.90)(1+x/5.90-x2/24.97) (1+x/3.31)-1]1.3, where x=nLeff2, and 0 is the conductivity of the sheet before any holes are introduced.

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U2 - 10.1103/PhysRevA.43.6473

DO - 10.1103/PhysRevA.43.6473

M3 - Article

AN - SCOPUS:0001701899

SN - 1050-2947

VL - 43

SP - 6473

EP - 6482

JO - Physical Review A

JF - Physical Review A

IS - 12

ER -