Solid Mixture Permittivities

Abstract

A macroscopically homogeneous aggregate contains two materials A, B, with dielectric constants εA, εB, in volume ratio p: (1—p); what is the dielectric constant of the aggregate? By a method used by Yvon and others in molecular theory, this very old problem is formulated rigorously and solved in series form. One form of the solution is ε/ε′ = 1+⅓p(1—p)(δ′/ε′)2+..., where ε′ = pεA+(1—p)εB, δ′ = εA — εB. The cubic and higher terms depend on statistical properties of particle geometry: namely, on such functions as p(3)123, the probability that three specified points 1, 2, 3 are all in material A. The relation to earlier calculations is discussed.

Heterogeneity and disorder: Contributions of Rolf Landauer

“…Although major advances in the understanding of electrical conductivity of disordered and heterogeneous media were made by a number of his contemporaries, Landauer's approach was unique in this field: On one hand, people like William Fuller Brown, Jr. [9] or Zvi Hashin and Shmuel Shtrikman [10] confined themselves to a discussion of systems where a classical physics approach is valid, and described the local electrical response in terms of a position dependent conductivity. This lead, eventually, to concepts like percolation threshold, which determines the macroscopic response of a metal/insulator mixture [11].…”

Transport properties of heterogeneous materials derived from Gaussian random fields: Bounds and simulation

“…For a macroscopically homogeneous isotropic material p n only depends on the distances r ij = |r i − r j | between the points. Since volume and ensemble averages are equivalent in such a medium [40] we can use the latter to evaluate eqn. (9).…”

“…The effective conductivity will then depend on the σ i , their respective volume fractions, and the spatial distribution (microstructure) of each phase [40]. The first bounds on σ e were calculated by Wiener [41] who proved that < σ −1 > −1 ≤ σ e ≤< σ >.…”

“…Rigorous approximations for p 2 for the cases |α| ≪ 1 and |α| ≫ 1 are derived in appendix A. A useful non-rigorous approximation to p T 123 can be developed by requiring that the approximation have similar properties to the actual function for r ij ≫ 1 and satisfy the known consistency conditions in various limits [40]. Using the compact notation p…”

“…The so called microstructure parameter ζ 1 is given by a number of equivalent integrals [4,7,14], of which the formulation due to Brown [40] is the best for our purposes,…”