The conductivity of random two-phase media is a property that has traditionally been associated with percolative phenomena [56]. Exact relations are known between the percolation threshold of the conductivity and pc [18,19] in limiting situations (see below). The "virial expansion" for the conductivity can be developed much like the pressure of a non-ideal gas in statistical mechanics to yield the expansion in the particle volume fraction φ [57]
where σm is the conductivity of the pure medium without any particles and [σ] is defined as the intrinsic conductivity [58]. Two special limits of the intrinsic conductivity are [σ]∞ for superconducting particles and [σ]o for insulating inclusions [58]. The magnitude of these shape functionals is minimized by the sphere for all objects having a finite volume [40,59]. Notably, in 2-D, the relation −[σ]o = [σ]∞ holds for all shapes [60], but this relation does not extend to other dimensionalities. In a separate paper [58] we have shown that in 3-D [σ]∞ is proportional (to within 5%) to the intrinsic viscosity (exactly similar to the intrinsic conductivity but for the viscosity of suspensions of rigid particles in a fluid) for a wide range of particle shapes, so that this important shape functional is also implicitly considered in our discussion of [σ]∞ here. In 2-D [σ]∞ is conjectured to equal the intrinsic viscosity.
For ellipsoids of revolution, the intrinsic conductivity for superconducting inclusions, [σ]∞ , is given by
where L is a depolarization factor [61]:
and ε is the eccentricity. The intrinsic conductivity for insulating inclusions, [σ]o, can be similarly expressed in terms of L,
For a sphere, eqs. (13) and (16) reduce to the well-known results of Maxwell, [σ]∞ = 3 and [σ]o = −3/2 [58].