Trapping constant, thermal conductivity, and the microstructure of suspensions of oriented spheroids

The n-point probability function S_n (rⁿ) is fundamental to the study of the macroscopic properties of two-phase random heterogeneous media. This quantity gives the probability of finding n points with positions rⁿ ≡ {r₁,...,r_n} all in one of the phases, say phase 1. For media composed of distributions of oriented, possibly overlapping, spheriods of one material with aspect ratio ε in a "matrix" of another material, it is shown that there is a scaling relation that maps results for the S_n for sphere systems (ε = 1) into equivalent results for spheriod systems with arbitrary aspect ratio ε. Using this scaling relation it is then demonstrated that certain transport and microstructural properties of spheriodal systems generally depend upon purely shape-dependent functions and lower-order spatial moments of S ₂ (minus its long-range value) of the equivalent spherical system. Specifically, the following three distinct calculations are carried out for both hard, oriented spheroids and overlapping (i.e., spatially uncorrelated), oriented spheroids: (1) bounds on the diffusion-controlled trapping constant; (2) bounds on the effective conductivity tensor; and (3) fluctuations in the local volume fraction as measured by the "coarseness." These computations enable us to investigate the effects of statistical anisotropy (i.e., particle asymmetry) and particle exclusion volume on the aforementioned quantities.