Comparison of the effects of a graded interphase and a homogeneous interphase on the macroscopic properties of a particulate composite

Two conceptual models are reviewed for interphase zones that surround spherical inclusions in a particulate composite: a thin-shell layer within which the local conductivity of the interphase is uniform, and a graded interphase within which the local conductivity varies with radius according to a power law. The analysis is carried out in the context of thermal conductivity, but the analysis applies to all transport properties that are governed by a linear relation that is analogous to Fourier’s law, such as electrical conductivity (Ohm’s law), fluid flow through a permeable medium (Darcy’s law), and solute diffusion (Fick’s law). The question is posed as to whether a graded interphase can be replaced by a homogeneous-shell interphase, by some rational a priori choice of the shell thickness, so that the homogeneous-shell model leads to the same macroscopic effective conductivity as does the graded interphase model. Examination of some specific cases, as well as analysis of the analytical solution for the homogeneous interphase model, reveals that, in general, no single a priori choice of the shell thickness can yield an exact equivalence between the effects of the graded interphase and the homogeneous interphase, for all possible parameter ranges. However, for interphases in which the variation of the local conductivity is not too extreme, an expression is derived for the shell thickness that leads to approximate agreement between the graded interphase and homogeneous-shell interphase models.