Abstract
The effective conductivity of composites containing simple or core-shell particles has been estimated in the literature using the Mean Field Approximation (MFA) and the Self-Consistent Approximation (SCA) among other techniques. It is shown here that for both simple and core-shell particles the two approximations agree to first order in the particle volume fraction but differ at the second order term. For simple particles the coefficient of the second order term calculated by SCA is at much better agreement with previous exact results than the coefficient calculated by MFA. For core-shell particles the results of the two approximations are almost identical up to particle volume fraction 0.20 but diverge with increasing volume fraction and particle-to-matrix conductivity ratio.
Highlights
There is a vast literature on effective properties of particulate composites, especially for mechanical and electrical/optical properties which concern the majority of applications
The present paper is limited to the simpler problem of electrical conductivity but the results apply to thermal conductivity and diffusivity when the constitutive equations are linear
In the present paper we extend Self-Consistent Approximation (SCA) to core-shell particles and compare the results with the corresponding Mean Field Approximation (MFA) results
Summary
There is a vast literature on effective properties of particulate composites, especially for mechanical and electrical/optical properties which concern the majority of applications. For spherical and uniformly sized dispersed particles Maxwell in Ref. 3 derived by an intuitive argument his often quoted equation for the effective electrical conductivity which is correct only to first order in the particle volume fraction. At volume fractions above the dilute limit Hashin, Ref. 8 derived simple approximate results for the effective conductivity using SCA. This version of SCA involves an additional unspecified parameter for a certain value of which the results are in good agreement with the results of Jeffrey (1973) to second order in the particle volume fraction, see Ref. 9. THE MFA AND SCA FOR SIMPLE SPHERICAL PARTICLES TO SECOND ORDER IN THE VOLUME FRACTION.
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