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Dimensional Invariants for Percolation Thresholds

Looking at the percolation literature as a whole, there are two main bodies of work. The first is produced by those researchers who are interested in the percolation phase transition as a model second-order phase transition. The focus of their work is on the various critical exponents displayed as p approaches its critical value, p_c, either from above or below, and the interrelations between these critical exponents.

The second body of work is produced by those who are more interested in applications of percolation geometry to real materials problems. These researchers tend to focus on the value of pc as a function of material microstructure, as the mere existence of a percolation threshold is what tends to dominate material properties, rather than the exact values of the critical exponents.

Most theoretical progress seems to have been made in the first area above, that of critical exponents and their interrelationships. Good reviews of percolation critical properties exist [3]. Our work, however, deals with the latter category, that of relating p_c to microstructure via invariants constructed out of p_c and properties of microstructural constituents.

The first work in this area was done by Scher and Zallen [4,5], who proposed, for site percolation problems on a lattice, that p_c, the critical fraction of sites present, would be an invariant when expressed as a volume fraction, depending only on dimensionality. Their construction was to center a sphere, with radius equal to half the nearest-neighbor lattice spacing d, at each site. The critical volume fraction occupied, a_c = p_c π d² / (4 a_prim) in two dimensions, and v_c = p_c π d³ / (6 v_prim) in three dimensions, was then found to be invariant, within 10% or less, for regular and irregular lattices [5], where a_prim and v_prim are the primitive site volumes for the lattices considered. This invariant seems to hold only for percolation clusters built up of non-overlapping particles, since, for example, in two dimensions, overlapping circles with random centers require a higher area coverage of 0.68 in order to percolate [6]. Recently, in a lattice-based growth model for the reactive growth during curing of cement-based materials, percolation thresholds for solid and pore phases were found to roughly agree, in two and three dimensions, with the Scher and Zallen invariants [7,8]. Thus there are some complex percolation problems to which the Scher and Zallen conjecture is relevant, although the precise conditions necessary for this conjecture to hold are not yet clear.

For continuum percolation problems, with overlapping defects, for which the Scher and Zallen conjecture is definitely not applicable, the main work on predicting p_c has been by Balberg and co-workers [9]. The focus of their work has been on using the excluded volume or area to construct invariants with which to predict p_c, in problems where randomly-centered defects like circles and rods are introduced into a host material, gradually culminating in a percolation transition.

The excluded volume of an object is that volume of space around the object in which the center of another such object can be placed so as to guarantee an overlap. For example, for a circle in two dimensions, the excluded area a_ex is 4 π r², since two circles with radius r whose centers are less than 2r apart must overlap. In general, if the objects have a size or orientation distribution, one defines <a_ex> or <v_ex>, the local excluded area or volume averaged over these distributions, so that the total excluded area or volume at percolation is <A_ex> = n_c <a_ex> or <V_ex> = n_c <v_ex>, where n_c is the percolation threshold defined in terms of a critical number of objects per unit area.

Refs. [9] and [10] summarize in more detail what is known about the proposed invariants <V_ex> and <A_ex>. The following two results are known: 1) the critical excluded volume is a dimensional invariant for continuum systems where the defects are all of the same shape and orientation (circles or parallel squares with various size distributions), 2) when random orientations of the defects are allowed, then the critical excluded volume is much more variable. Even though the critical excluded volume is not the same for cases 1 and 2, it can be approximately bounded [10] by

3.2 <   <A_ex>   < 4.5 , and 0.7 <   <V_ex>   < 2.8.

In Section IV, a better invariant is described that is the same for both cases 1 and 2 above.