Reference: E.J. Garboczi, M.F. Thorpe, M. DeVries, and A.R. Day, Physical Review A 43, 6473-6482 (1991).
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Universal conductivity curve for a plane containing random holes
E.J. Garboczi
National Institute of Standards and Technology
Building Materials Division, 226/B348
Gaithersburg, MD 20899
M.F. Thorpe and M.S. DeVries
Department of Physics and Astronomy
Michigan State University, East Lansing, MI 48824
A.R. Day
Department of Physics
Marquette University
Milwaukee, WI 53233
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
decreases with the remaining area fraction. This problem has been studied in the past
for circular, square, and needle-like holes, using 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 =
nc Leff2 = 5.9 ± 0.4, 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 semi-major and semi-minor axes, respectively. All results are well
described by the universal conductivity curve:
where x = n Leff2, and o is the conductivity of the sheet before any holes are
introduced.
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