Thermal and Electrical Conductivity of Composites with Graded Interfaces

An analytical solution is derived for the problem of a homogeneous spherical inclusion that is surrounded by an interphase zone in which the conductivity varies according to a power law, and which perturbs and otherwise uniform flux. This solution is then used to derive an expression for the effective conductivity of a material that contains a dispersion of such inclusions. This solution reduces to the classical Maxwell result in the case when the interphase is homogeneous and has the same conductivity as the pure matrix. Results are shown for several important cases, such as the electrical conductivity of water-saturated concrete, in which case the inclusions are nonconductive and the interphase zone is more conductive than the pure matrix. Introduction The estimation of the effective conductivity of a composite material containing spherical inclusions is an important and classic problem that has been studied since the 19 th century, starting with Maxwell. The earliest models for the effective conductivity assumed that the two components are both homogeneous, and are perfectly bonded across a sharp and distinct interface [1,2]. However, in some materials, the components are well bonded to each other, but the interface is not sharp. In polymer-fiber composites, for example, as well as in some metal-matrix composites, diffusion of material between the matrix and fiber may create a conductivity profile that smoothly varies from that of the fiber to that of the matrix [3]. In some polymer composites, a binding agent is applied to the fibers to promote adhesion between the fiber and the matrix [4]. This binding agent may diffuse into the matrix during the curing process, leading to a gradient in resin concentration. This gradient, in turn, leads to a gradient in the conductivity. In other cases, such as the interfacial transition zone (ITZ) around inclusions in mortar or concrete [5], the conductivity of the matrix varies as the inclusion particle is approached, but the interface with the inclusion is still distinct, since the inhomogeneous region is restricted to the matrix phase. Recognition of the importance of modeling the “interphase zone” in composite materials has existed for some time. Kerner [1] explained how his approach could be modified to account for a coating on each inclusion, which could be represented as a thin, homogeneous interphase layer. Hashin [6] discussed this three-shell model in more detail, and also developed an approximation in which a thin interphase layer is replaced by a jump condition at the interface between the matrix and inclusion. Hervé [7] developed a scheme to extend the three-shell model to account for an arbitrary number of shells. Such models have been used to model the electrical conductivity [8] and chloride diffusivity [9] in mortar. In the context of the effective elastic moduli, Lutz and Zimmerman [10] modeled the elastic moduli outside of the inclusion with a constant term plus a power-law term, thereby allowing a smooth transition between the interphase layer and the matrix. They used the method of Frobenius series to derive a closed-form solution for a body containing such an inclusion, under hydrostatic far-field loading, and found an expression for the effective elastic bulk modulus of a material that contained a dispersion of such inclusions. This solution was successfully used in [5] to model the bulk modulus of concrete. Lutz and Zimmerman [11] showed that the conductivity problem is Advances in Science and Technology Online: 2006-10-10 ISSN: 1662-0356, Vol. 45, pp 1097-1102 doi:10.4028/www.scientific. net/AST.45. 1097 © 2006 Trans Tech Publications Ltd, Switzerland All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications Ltd, www.scientific. net. (Semanticscholar.org-19/03/20,08:44:06) exactly analogous to the elastic problem, if the Poisson ratios of all phases are set equal to zero. However, for simplicity and clarity, in the present paper we will not make use of this analogy, but directly use the mathematical approach of [10] to study the effective conductivity of a particulate composite in which the inclusions are surrounded by an interphase region in which the conductivity varies according to a power law. Single inclusion surrounded by graded interphase The effective conductivity (thermal, say) is investigated by starting with the problem of an inclusion perturbing an otherwise uniform temperature gradient. If the governing equation is expressed in a spherical co-ordinate system centered on the inclusion, with the z-axis aligned with the far-field temperature gradient of magnitude G (Fig. 1a), the problem is governed by 0 )] , , ( ) ( [ r T r k , (1) where T is the temperature and k(r) is the local conductivity. Using the standard expressions for the gradient operator in spherical co-ordinates, this equation can be written explicitly as . 0 ) ( sin 1 sin 1 ) ( sin sin 1 ) ( 1 2 2 T r r k r T r r k r r T r k r r r (2) The boundary conditions for the temperature are that as r , T To Gz To Gr cos , (3) where is the angle of inclination from the z-axis, along with the usual continuity conditions at the inclusion/matrix interface for the temperature and flux. Inserting a temperature field of the form T (r, , ) To f (r)cos into Eq. (2) yields the following equation for f (r) :