Analytical continuation methods

There are other methods for conductivity/permittivity analysis. The Bergman-Milton (Bergman, 1978; 1980; 1982; Milton, 1980; 1981; Korringa and LaTorraca, 1986; Stroud et al., 1986; Berryman, 1992) analytical approach to understanding some general effective transport coefficient or permittivity -- which we take for example to be $\sigma^*$ -- of two-component inhomogeneous media shows that  

$$\sigma^* = S(\sigma_1,\sigma_2) = \sigma_1S(1,0) + \sigma_2... ...rac{dy {\cal S}(y)}{\frac{1}{\sigma_1} + \frac{y}{\sigma_2}},$$ (40)

where S(1,0) and S(0,1) are constants depending only on the geometry and ${\cal S}(y) \ge 0$ is a resonance density functional also depending only on the geometry. The integral in (40) is known as a Stieltjes integral (Baker, 1975). This formula is typically derived and used for the case of complex constants: $\sigma_1$, $\sigma_2$, and $\sigma^*$.But we will restrict consideration here - as Bergman (1978) did in his early work - to pure conductors so that $\sigma_1$, $\sigma_2$, and $\sigma^*$ are all real and nonnegative.

A short derivation of (40) is instructive, so we will present one now.

Following (for example) Korringa and LaTorraca (1986) we consider the defining equation for the function Z(s)  

$$\sigma^*= \sigma_1Z(s),$$ (41)

where  

$$s \equiv \sigma_1/(\sigma_1 - \sigma_2).$$ (42)

Then, Milton (1981) shows [also see Korringa and LaTorraca, 1986] that  

$$Z(s) = 1 - \sum_{n=0}^N A_n(1-s_n)/(s-s_n),$$ (43)

where the s_n's are the locations of the poles, and are enumerated in increasing order. The A_n's are the residues. These real constants satisfy the following inequalties: 0 < A_n < 1, $0 \le s_n < 1$, and $\sum_n A_n \le 1$. Note that N might be a very large number in practice, so that it may then be more convenient to think of turning this sum into an integral. Define a density functional  

$${\cal A}(s) \equiv \sum_{n=1}^N A_n\delta(s-s_n),$$ (44)

where $\delta$ is the Dirac delta function. Then, (43) can be rewitten as  

$$Z(s) = 1 - A_0/s - \int_0^1\,dx{\cal A}(x)(1-x)/(s-x),$$ (45)

which is so far just a restatement of (43), assuming only that there exists a finite A₀ for which $s_0 \equiv 0$. Substituting (42) into (45) and rearranging, we find  

$$Z(s) = 1 - A_0 + A_0\frac{\sigma_2}{\sigma_1} - \int_0^1\,dx... ...(x) \frac{(1-x)(\sigma_1-\sigma_2)}{(1-x)\sigma_1+x\sigma_2}.$$ (46)

We can then symmetrize this expression by adding and subtracting the term $x\sigma_2$ in the numerator of the displayed ratio inside the integral. Then we can pull out another constant and finally have the form we want:  

$$Z(s) = [1 - A_0 - \int_0^1\,dx{\cal A}(x)] + A_0\frac{\sigma... ..._0^1\,dx{\cal A}(x) \frac{\sigma_2}{(1-x)\sigma_1+x\sigma_2}.$$ (47)

Substituting this back into the original definition (41), we find the symmetrical result  

$$\sigma^* = \frac{\sigma_1}{F_1} + \frac{\sigma_2}{F_2} + \int_0^1\,dx{\cal A}(x) \frac{1}{(1-x)/\sigma_2+x/\sigma_1},$$ (48)

where $1 \ge 1/F_2 = A_0 \gt 0$ and $1 \gt 1/F_1 = 1 - A_0 - \int_0^1\,dx{\cal A}(x) \ge 0$,since $\sum_{n=0}^\infty A_n = A_0 + \int_0^1dx\,{\cal A}(x) \le 1$.The F_i's are known as ``formation factors'' (Archie, 1942; Avellandea and Torquato, 1991).

This equation is not yet in the same form as (40), but it is nevertheless worthwhile to pause for a moment to consider this form on its own merits. In particular, the first two terms on the right hand side are exactly what is expected when conductors are connected in parallel inside a complex conducting medium. And the remaining integral looks like some sort of weighted average of conductors connected in series. The first physical analogy (conductors in parallel) is entirely appropriate. The second one is no doubt an oversimplification of what is happening in the medium, since the weights in the denominator (i.e., x and 1-x) are not really volume fractions (even though they do range from 0 to 1), and the density functional ${\cal A}$ in the numerator also contributes important numerical weights depending on the local shapes and interconnectedness of the microstructure of the conductors. This dependence on microstructure would correspond approximately to the network connectivity in a resistor network, but usually does not have a perfect analog for most 3D conducting composites.

To complete the derivation of (40), we now need only to make the further substitutions x = 1/(1+y) where y ranges from to $\infty$, and define ${\cal S}(y) \equiv {\cal A}(x)/(1+x)$. Then, we arrive finally at precisely (40), having found that S(1,0) = 1/F₁ and S(0,1) = 1/F₂. Furthermore, taking the limit $\sigma_1 = \sigma_2 = 1 = \sigma^*$, we find the useful sumrule  

$$\frac{1}{F_1} + \frac{1}{F_2} + \int_0^\infty\, dy\frac{{\cal S}(y)}{1+y} = 1.$$ (49)

Clearly, other choices of the integral transform in (48) may also be useful. In particular, taking instead x = 1/(1-y) is a good choice in preparation for analysis of the resonance density ${\cal S}(y)$ itself, as this transform places it most appropriately on the negative real axis. But for present purposes either (40) or (48) is a satisfactory choice for study.