The problem studied in this paper is that of the conductivity of a random continuum. We start with a uniform two-dimensional sheet of host conducting material with conductivity σo, and randomly introduce extended defects having conductivity σd and some given shape. These defects are allowed to freely overlap. When σd is non-zero, we have a composite material. When σd is zero, the defect is thought of as a hole, resulting in a continuum percolation problem. In this case, p denotes the area fraction of σo material remaining after a given number of defects have been introduced, or holes punched out.
In recent years, much attention has been given to the value of critical exponents at the percolation threshold. For the conductance case, the exponent of interest, usually denoted by t, determines how the conductance goes to zero at p = pc, the critical threshold. In this paper, we concentrate on two subjects that are arguably of more importance in the kind of continuum percolation problems that arise in real materials: 1) the overall behavior of the conductance as a function of p, and 2) the relationship between the effect of one defect on the overall conductivity and the many-defect critical threshold.
Previous work on which this present effort builds includes a joint experimental and computer simulation study of the continuum percolation problem of needle-shaped insulating defects [1], hereafter referred to as I, and a study of the dependence of the percolation threshold on the aspect ratio of the elliptical holes introduced [2], denoted hereafter as II.
In Section II we briefly review previous efforts to relate one-defect properties with percolation thresholds, which have been mainly cast as attempts to formulate a dimensionless, invariant percolation threshold using lengths or areas that are definable by one-defect properties only. In Section III, we present new computer simulations for the needle problem, using both continuum and lattice techniques. We define the initial slope and critical threshold of the needle problem in terms of the number of defects per unit area and use this to define a generalized dimensionless variable that is natural for this problem and that suggests the invariant defined in this paper. Ellipse percolation threshold data from II is used as a first test of the new invariant in Section IV. Section V presents digital-image simulations of percolation problems with arbitrarily-shaped defects. We show that the critical concentration of defects can be predicted from knowledge of the conductivity in the few-defect limit without even knowing the shape of the defects. In Section VI we define a universal conductance curve for the class of percolation problems described in this paper. It is shown that all previous conductivity data, both from simulations and experiments, falls on this universal conductivity curve. In the summary, we discuss how the universal conductivity curve should be used.