Intrinsic conductivity of objects having arbitrary shape and conductivity

We study the electrical conductivity sigma of a dispersion of randomly oriented and positioned particle inclusions having common shape and conductivity sigma(p), suspended in an isotropic homogeneous matrix of conductivity sigma(0). For this problem, the mixture conductivity is a scalar and we concentrate on the leading order concentration virial coefficient, the ''intrinsic conductivity'' [sigma]. Results for [sigma] are summarized for limiting cases where there is a large mismatch between the conductivities of the inclusions and the suspending matrix. For a general particle shape, we then treat the more difficult case of arbitrary relative conductivity Delta=sigma(p)/sigma(0) through the introduction of a Pade approximant that incorporates (exact or numerical) information for [sigma(Delta)] in the Delta-->infinity,Delta-->0(+), and Delta approximate to 1 limits. Comparison of this approximation for [sigma(Delta)] to exact and finite element calculations for a variety of particle shapes in two and three dimensions shows excellent agreement over the entire range of Delta. This relation should be useful for inferring particle shape and property information from conductivity measurements on dilute particle dispersions. The leading order concentration virial coefficient for other mixture properties (thermal conductivity, dielectric constant, refractive index, shear modulus, bulk modulus, viscosity, etc.) are equally well described by a similar Pade approximant.