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Intrinsic conductivity for general relative conductivity

The limiting values of [σ] for Δ → ∞ and Δ → 0 and the expansion of [σ (Δ)] to second order in (Δ − 1) provides 5 pieces of information about [σ (Δ)] corresponding to bodies of general shape. Explicit numerical and analytical calculations (see below) of [σ (Δ)] indicate that this function always seems to increase monotonically with Δ. Based on these results, we introduce a Pade approximant [41] to describe [σ (Δ)] for general shaped bodies and values of Δ,

Equation (13) was constructed by starting with a ratio of two quadratic polynomials in Δ. This ratio has five independent coefficients. These five coefficients are then determined from the five pieces of information mentioned above: the value of [σ (Δ)] and its first two derivatives at Δ = 1, and the values of [σ]_o and [σ]_∞. An approximation with a similar mathematical form was introduced previously by Eyges and Gianino [42] to summarize their numerical results for the polarizability of a cube (d = 3) as a function of the relative dielectric constant of the cube to the surrounding medium (see below). We also observe that Eqn. (13) in d = 2 reduces to the exact result for an ellipse, Eqn. (12). The Δ → ∞ limit for an ellipse is given by

Eqn. (13) also recovers [σ (Δ)] for hyperspheres in d dimensions given in (2). It is a good test of our approximate formula that it reduces to the exact equations in known limits.

As a further test of Eqn. (13), we consider [σ (Δ)] for ellipsoids of revolution where exact analytical results are known. In Figs. 1a and 1b we compare the exact results for [σ (Δ)] to the approximant Eqn. (13) for oblate and prolate ellipsoids of revolution where the ratio of the larger to the smaller axis lengths in each case equals 100. Each example reveals a non-trivial variation of [σ] with Δ, which is accurately described by the approximant (13). Deviations were less than 0.2 % for the prolate case, and were exact within computer roundoff error (10 significant digits) for the oblate case.

Figure 1: The intrinsic conductivity [σ (Δ)] vs. the relative conductivity Δ for an (a) oblate and a (b) prolate ellipsoid of revolution, with the ratio of the longest axis to the shortest axis equal to 100. The solid line is the Pade approximant Eqn. (13), the circles are the exact result for a selected number of values of Δ.

Some further insight into these crossover curves for [σ (Δ)] at high aspect ratios can be obtained from the exact d = 2 ellipse result. Fig. 2 shows this function covering six orders of magnitude in the aspect ratio x. The inflection points in Fig. 2 scale linearly with x on the log Δ scale (there are no inflection points on a regular Δ scale). Evidently, Δ must be increasingly large or small as the aspect ratio of the particles increases for these particle shapes to correspond to the limiting superconducting or insulating particle limits.

Figure 2: The intrinsic conductivity [σ (Δ)] divided by the superconducting limit, [σ]_∞, vs. relative conductivity Δ for a series of d = 2 ellipses with varying aspect ratio x.

We next consider some finite element calculations of [σ (Δ)] that generalize our previous calculations for insulating and highly conducting particles [1]. In these calculations particles are represented as digital images built up from cubic elements. A standard lattice of size 120³ was used, which is the largest that was practical considering both computer memory and running time. Because of the overall computational cell size limit, a compromise had to be taken between using enough pixels to give a good representation of the particle, and keeping the particle small compared to the overall unit cell, so as to keep the volume fraction small enough to be in the linear regime in concentration. The size and complexity of the objects that could be treated in this fashion is necessarily limited, but a good approximation to a wide range of physically interesting objects could still be obtained. The typical computational time for about 20 values of Δ and a particular shape was about 10 hours on a Convex 3820 supercomputer. Details of our computational procedure were given in our previous paper [1]. Comparisons of these calculations against exactly soluble examples showed a systematic overestimation of [σ]_∞ of about 5-6 % in d = 3, for higher values of Δ, and a smaller underestimation, of 2-3 %, in d = 2 [1].

Figure 3 shows the important case of a cube as a function of Δ where the solid line denotes the approximant, Eqn. (13), using the numerical estimates of Eyges and Gianino [42],

The results in Fig. 3 agree with the numerical results of Eyges and Gianino [42] to within 1%, and usually the agreement is much better. Eyges and Gianino evaluated [σ(Δ)] based on a numerical solution of an integral equation describing this problem.

Figure 3: The intrinsic conductivity [σ (Δ)] vs. the relative conductivity Δ for a cube, as computed by Eyges and Gianino [42] and the Pade approximant (13) of the text.

Finite element calculations were performed for rectangular parallelipipeds having dimension ratios 2:2:1 and 1:1:2. Eqn. (13) again gives an excellent description and thus we only summarize the limiting values of [σ] needed to reproduce these results:

We emphasize that these are finite element estimates of the intrinsic conductivity rather than the exact values.

The important physical example of a right circular cylinder is considered next. Precise analytical calculations of the polarizability in the superconducting particle limit have been made [43] and [σ]_∞ values based on these calculations are given in Table 1. In addition, [σ]_o has been calculated using finite element methods, and these results are also given in Table 1. This comparison shows a 5-6 % error in our numerical calculations for the superconducting limit. The error for the insulating limit, while we have no exact values against which to compare them, is probably in the range of 2-3%, based on experience with other shapes [1]. We have found that the insulating limit is usually computed more accurately with our finite element method than is the superconducting limit. Better accuracy is possible, of course, if sufficient computer memory is available, at the expense of much greater computational times. Accurate numerical data for [σ]_∞ and [σ]_o, for a wide range of diameter- height ratios, could be used to obtain approximants for these virials that then could be used in conjunction with Eqn. (13) to make more general analytical estimates of [σ(Δ)] for comparison with experiments on cylindrical conducting fibers.

In our next example we treat an idealized "sponge-like" body. Consider a cube of unit edge length, in which a square channel is cut through the center of each face, where each channel passes completely through the cube. A picture of an object of this kind is shown in Fig. 4a. The parameter m is taken to be the edge length of the cutout face in units of the total cube edge length. We obtain a rigid cubic wire frame when m approaches 1. Notice that cutting out the center, which makes the particle more sponge-like, has a very large effect on [σ]_∞, as can be seen in Table 2. It would be interesting to push the effect to the extreme in a different way by generating a Menger sponge [44] fractal by a repeated decimation of the cube at different scales so that [σ]_∞ would diverge in a fashion related to the fractal dimension of the sponge. The memory capacity of our computer was not large enough to allow us to consider more than one or two generations of such an iteratively constructed "diffuse" object, so we presently confine ourselves to the first generation wire frame structure shown in Fig. 4a.

Fig. 4b presents our numerical results as a function of Δ for the sponge with m = 23/27. Note that [σ(Δ)] is quite insensitive to Δ for relatively insulating particles (Δ < 1), which is typical for extended or diffuse objects. As in previous comparisons, the approximant Eqn. (13) describes the numerical data very well. Table 3 includes numerical data for [σ]_o corresponding to other representative shapes considered in our previous paper [1]. A "jack" is a sphere punctured by three intersecting rectangular parallelipipeds. The square ring and square hollow tube are self-explanatory. Values of [σ]_o were not given in our previous paper, which focused on comparisons between [σ]_∞ and the intrinsic viscosity [η].

Numerical calculations of [σ(Δ)] for inclusions having sharp corners give rise to subtle variations of this virial coefficient with Δ. Such cases provide a good test for our approximant Eqn. (13). In Fig. 5 we consider finite element calculations for a 12:1 rectangular region in d = 2 (the lattice size in d = 2 was 1800²). Observe the subtle feature of a "wavy" variation of [σ(Δ)] for this type of inclusion, which is followed by Eqn. (13) remarkably well. In the insulating limit, we found:

The exact value of [σ]_o for a 12 :1 rectangle is −6.38 [40], indicating an error of only 1.4% for the finite element method at this resolution in d = 2. The values of [σ] for finite Δ should have even less error. To save computer time, in all the d = 2 finite element computations we took advantage of the Keller-Mendelson inversion theorem [36] and only calculated [σ(Δ)] for Δ < 1.

As a final example, we considered a completely asymmetric inclusion to make sure that particle asymmetry did not invalidate our approximant. Two dimensions was chosen to allow a higher resolution to be used in order to obtain more accurate computational results. The L-shaped asymmetric particle, of no particular symmetry, chosen to illustrate this case is shown in the insert of Fig. 6. Equation (13) again provides a very good approximation to the intrinsic conductivity data (Fig. 6) where we found,

This agreement for many different kinds of particles, including a completely anisotropic one, suggests that Eqn. (13) should be a very reasonable approximation for a wide range of complex- shaped particles.