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Results and Discussion

Table 1 gives the geometric and percolation data for ellipsoids of revolution whose aspect ratio a/b spanned a range of six orders of magnitude (1/2000 to 500). Each result is the average of at least five realizations, although the number of particles at percolation did not vary more than a few percent between realizations. The number of particles at percolation was recorded (n_c), and the volume fraction of particles at percolation (p_c) was then calculated via eq. (32). For most of the shapes studied, the actual size of the particle, in relation to the unit edge length periodic computational cell used, was adjusted so that about 20,000 particles were present at percolation. The longest dimension of the particles was kept to less than one tenth of the box size to avoid size scaling problems [70]. For the very prolate particles, this forced the number of particles at percolation to increase sharply, thus practically limiting the computations to a maximum aspect ratio of 500.

Figure 3 shows the inverse of p_c plotted against the aspect ratio a/b. In the extreme oblate limit, it is clear that 1/p_c scales linearly in the inverse of the aspect ratio, as can also be seen in Table 1. In the extreme prolate limit, 1/p_c seems to also scale linearly in the aspect ratio, although it would be necessary to go an order of magnitude further in aspect ratio in order to rule out any small slowly-varying non-linear terms. In these limits, we can write the asymptotic forms approximately as:

It is interesting to note that the numerical prefactors differ by close to a factor of two in these very different limits. Qualitatively, we observe that 1/p_c is minimized for spherical particles. The data strongly suggests a new isoperimetric theorem: Of all objects of a given volume, the sphere has the maximum percolation threshold p_c for overlapping objects.

Figure 3: Inverse of the critical volume fraction for percolation (1/p_c) plotted vs. aspect ratio of ellipsoids of revolution. The solid lilne is a Pade-type approximant described in the text. It is fit to both asymptotic limits, the value of 1/p_c for the sphere, and is forced to have zero slope at a/b = 1.

The solid line in fig. 3 is a Pade-type approximant,

where x = a/b, the aspect ratio of the ellipsoids, and P(x) is fit to the reciprocal of p_c. Five of the six parameters are fit to the asymptotic behavior of the curve (two slopes, two intercepts) in the extreme prolate and oblate limits, and the known value of 1/p_c for the sphere. The sixth parameter is used to force the slope of P(x) to be zero for the sphere (a/b = 1), since the minimum value of 1/p_c was found at this point, and we suspect the above theorem to be true. We call the quantity P(x) a "Pade-type" approximant, since usually Pade approximants only contain integer powers of x [71]. We found that a fractional power was necessary in order to match the data around a/b = 1. The Pade approximant fits the data extremely well, using the following values: h = 7.742, ƒ = 14.61, g = 12.33, c = 1.763, d = 1.658, and s = 9.875.

After generating the basic percolation data, in light of the perturbative criterion summarized in eq. (31), it then seems natural to attempt to find some combination of shape functionals that approximates the variation in 1/p_c. Of course, we ultimately seek a relation that is not restricted to ellipsoidal particles, such as eq. (33), which appears to hold quite generally in 2-D. We search for this relation in terms of finding a shape functional whose product with p_c is a constant, since the value of p_c goes to zero in the oblate and prolate limits, while all the shape functionals displayed in Fig. 1 diverge in these limits, so that the product of the two has a chance of being invariant.

To connect with the extensive work of Balberg and co-workers, we first present a graph of n_c, normalized by < V_ex > for that particular shape so as to have a dimensionless quantity, vs. aspect ratio in Fig. 4. Eq. (11) must be multiplied by the appropriate excluded volume for a sphere of equal volume to the ellipsoid using the dimensions for the objects given in Table 1. We have also put back in the extra factor of 2 used by Balberg [28,38]. In Fig. 4, it is seen that the normalization with < V_ex > produces a clear invariant as the extreme oblate limit is approached, and what appears to be a different constant in the extreme prolate limit:

The linear behavior shown in Fig. 3 and eq. (41) puts restrictions on which shape factors have the potential to form an invariant with the percolation threshold.

Fig. 4 also shows the quantity n_c [σ]_∞ plotted vs. aspect ratio. This combination, analogous to that used successfully to give an invariant in 2-D, clearly fails in 3-D for prolate particles. This could have been predicted using the asymptotic relations of Sec. 2.7.

The result we have obtained in the disc limit disagrees with a previous result of Charlaix [70], who found n_c < V_ex > = 1.80. Using the same normalization of < V_ex >, we find a substantially different value for this product, n_c < V_ex >. Our value falls between the inequalities found by Balberg [28,38] while Charlaix's result does not. The percolation threshold for discs embedded in a three-dimensional space might intuitively be expected to have properties that span the range between two and three dimensions [58], and this is exactly what our numerical results imply. In Fig. 4, we see that as the ellipsoids change from discs (a / b → 0) to spheres (a/b = 1) and then to needles (a / b →∞), the value of n_c < V_ex > changes from 3.0 to 2.7 to 1.5, with 2.8 and below marking the "3-D" range found by Balberg [28,38]. Study of Charlaix's computation of n_c does not reveal any obvious mistakes, so we are puzzled by the reason for this disagreement. Figure 4 shows that the variation of n_c < V_ex > with aspect ratio is very small for all oblate shapes, and since n_c < V_ex > = 2.7 and p_c = 0.29 are well-known for spheres [64], the estimate n_c < V_ex > appears anomalous.

Studying the asymptotic limits given in eqs. (17)-(30), we see that there are only four possible choices of shape functions that have the correct or nearly correct prolate and oblate limits: the excluded volume, the radius of gyration R_g times the surface area A, the mean radius of curvature R times the surface area A, and Σ ≡ ( [σ]_∞ [σ]_o )^½. All these quantities have the correct asymptotic forms, except that Σ has a square root of the logarithm of the aspect ratio in its denominator in the prolate limit. The 1/p_c data, as mentioned above, cannot really rule out such a slight logarithmic dependence in the prolate limit. We have also tried the quantities p_c [σ]_∞ and n_c V_p [σ]_∞, because of their successful use in 2-D [29], but have found that [σ]_∞ and V_p [σ]_∞ increase much faster than does 1/p_c and n_c in the prolate limit. These quantities do work very well, however, in the oblate limit, as can be seen from eq. (18).

Figure 5 shows the result of multiplying 1/p_c by the above shape factors, plotted as a function of aspect ratio. All these quantities do an equally good job of reducing 1/p_c to a constant in the prolate and oblate limits, although these constants are different by about a factor of two, similar to the behavior seen in Fig. 4. Fig. 5 clearly shows that the excluded volume does no better than the other shape factors. None of these shape functionals, however, describe the data as well as the Pade approximant shown in Fig. 3.

Figure 5: The critical volume fraction from Fig. 3, normalized by four candidate shape factors that all give the correct scaling with aspect ratio in the extreme oblate and prolate limits, plotted vs. aspect ratio.

We therefore conclude that the dependence of the percolation threshold, even for the simple case of overlapping identical ellipsoidal objects, cannot be completely described by simple single- particle shape functionals, although the scaling of 1/p_c in the extreme prolate and oblate limits can be correctly predicted by several such functions and combination of functions. It is interesting to note that the intrinsic conductivity for superconducting particles, [σ]_∞, in 3-D fails in scaling p_c or n_c to an invariant, even though it worked extremely well in 2-D. The excluded volume, on the other hand, works about the same, to within a factor of two or so, in both 2-D and 3-D. It is hoped that the data in this paper will serve to test new mathematical theories for predicting percolation thresholds based on "microstructural shape" quantities.