We have presented a new invariant derived from the percolation threshold in terms of the number of defects per unit area, nc, and a length Leff, which is defined by the change in electrical conductivity caused by a single insulating defect when placed in a conducting host. This new invariant has been tested only in problems where the defects are randomly-centered and overlapping. It is very important that the variable be the number of defects and not the area fraction remaining p. These are related in a non-linear way by p = exp(−nA), where A is the area of a single defect.
Using values of xI = nI Leff2 = 8/π = 2.55 and xc = nc Leff2 = 5.9, and optimizing a shape parameter α = 0.7, leads to the universal conductivity curve given in eq. (15), with x = n Leff2. This applies when the anisotropic holes are aligned horizontally and vertically, or isotropically, to make the conductivity isotropic. In general Leff is not known analytically and must be obtained by aligning the horizontal axes of the experimental and universal conductivity curves. This is a straight-forward procedure, as x is linearly proportional to the number of inclusions n. In this case, Leff is treated as a fitting parameter. In a few cases, we have more information about Leff from the dilute limit. For ellipses, Leff = 2(a + b) where a and b are the semi-major and semi-minor axes, respectively. This case has been fully discussed in I and also in this paper. For all other shapes, no exact solutions to the single inclusion problem exist. For squares, it appears as if there is an equivalence to circles of the same area , so that Leff = 4s / (π)½, where s is the edge length of the square. On the other hand, in I, it was shown that a long thin w x L rectangle behaved electrically as an ellipse with the same area, with major axis 2a = L, so that Leff = L + 4w/π. We have found no simple way to generalize these rules to arbitrary shapes, despite some effort. In general, the most we can say is that Leff = gL, where L is the maximum spanning diameter of the object, and g is a constant of order 1.