Estimation of percolation thresholds using shape functionals

A natural method for obtaining an estimation of the percolation point for conduction problems involving a second phase of randomly inserted objects can be developed by considering the change in the electrical conductivity of a material upon adding a small concentration of objects into the matrix which are "insulating" (much less conducting than the suspending matrix) or "superconducting" (much more conducting than the suspending matrix). The particles are positioned at random locations and with random orientation so that the effective conductivity of the medium, σ_eff, is a scalar that is invariant under rotation of the macroscopic material as a whole. An estimate of the conductive percolation threshold can then be obtained from a simple perturbative criterion [62]. A large variation of the composite conductivity, σ_eff, might be expected when the leading order perturbation in eq. (12) is on the order of unity,

This perturbative condition [62] defines an order of magnitude estimate of a critical concentration at which the property σ_eff, should become rapidly varying, indicative of some kind of "percolation threshold", φ^*, as was discussed in Section 1. For perfectly conducting or insulating particles this critical concentration can be simply identified with a geometrical percolation threshold [18,19]. Previous percolation studies in 2-D [29] for elliptical particles and particles of other shapes have indeed shown a nearly shape-independent relation between n_c for geometric percolation and [σ]_∞ that is more precise than estimates based on the excluded volume concept. For overlapping objects we recall that the average volume fraction occupied, φ, by objects of general shape is related to the particle volume V_p (or the area A_p in 2-D) and the number of particles per unit volume as [29]

as opposed to impenetrable particles where φ = nV_p. In 2-D it has been shown that the critical number density n_c for geometric percolation times [σ]_∞ equals

In the limit of highly anisotropic particles, where n_c A_p is very small, we have n_c A_p φ_c as in non-overlapping particles. Eq. (33) thus accords qualitatively with expectations from eq.(31). We note that recently, the product A_p [σ]_∞ has been shown to exactly equal [58]

for particles of general shape, where C_L is the "transfinite diameter" or "logarithmic capacity" [58]. The extension of eq. (31) to 3-D is not obvious [63]. Since [σ]_∞ and [σ]_o generally vary quite differently with particle anisotropy in 3-D, as opposed to being equal in magnitude in 2-D, we are led to expect different conductivity percolation transitions from eq. (31) depending on whether the anisotropic particles are insulating or superconducting. Such separate transitions are, in fact, obtained for the transition in conductivity. Consider the example of adding spherical overlapping objects to a conducting matrix. If the spheres are superconducting, the composite conductivity diverges when the spheres percolate at a volume fraction of 0.29 [64]. If the spheres are insulating, the resistivity of the composite diverges at a sphere volume fraction of 0.968, the point at which the matrix becomes disconnected [65].

We also remark that a relation similar to eq. (31) has been suggested involving < V_ex > that is applicable to 3-D. A virial expansion of the pair connectedness function for overlapping objects naturally leads to the low-density approximation [66] (compare with eq. (31))

which has been predicted to become asymptotically exact in the needle limit [66]. This approximation was introduced heuristically in earlier work by Balberg et al. [67]. More recently Balberg [38] has argued that the constant in eq. (35) is more generally on the order of unity, and he gives the rough bounds,

which include the estimate of eq. (28). Evidently, n_c < V_ex > is not a true invariant, but this ratio does seem to capture the main trends in the dependence of the percolation threshold on particle shape for overlapping particles. In 2-D this relation becomes [38]

These shape functionals and bounds on the percolation threshold provide a point of reference for discussing our computations of p_c.