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Introduction

Small amounts of additives are often quite effective in modifying the properties of materials. The extent of the effect depends on the property involved, particle dispersion, concentration, and shape, and tends to be larger the more unlike the additive material properties are from those of the suspending matrix. In the common situation of low additive volume fraction, φ, the effective properties can be developed in a power series in φ. The leading order "virial coefficient," corresponding to the linear order concentration correction to the pure medium property, plays an important role in understanding the influence of particle shape and property mismatch on the effective property of the mixture [1]. The low additive concentration regime is also important in the inverse problem of inferring particle shape and/or properties from measurements of effective mixture properties over a range of low additive concentrations. This strategy is commonly followed in the polymer science literature to determine polymer molecular architecture [2].

In a previous paper [1], we made an extensive tabulation of the leading order transport property virial coefficients for a wide range of particle shapes and a large set of material properties (electrical and thermal conductivity, dielectric constant, refractive index, shear viscosity). Almost all of these previous calculations, analytical and numerical, were restricted to the case where the ratio of the additive property to the matrix property either vanished or diverged. Although the property mismatch between the additive and the matrix may be large, the limits considered previously are clearly idealizations in comparison with real systems where the transport property ratio is generally a finite, non-zero value. The present paper develops an approximate description of transport virial coefficients in terms of the relative conductivity Δ, the ratio of the inclusion conductivity to that of the matrix, and the limiting values of the transport virial coefficient for large property mismatch.

In Section 2 we review basic results about the conductivity virial expansion for suspensions of highly conducting and insulating particles, which was the subject of our previous paper [1]. A Pade approximant describing the intrinsic conductivity for general particle shapes and values of Δ is introduced in Section 3 and then compared to exact and numerical (finite element) calculations of [σ( Δ)] . Some approximations for the limiting values of the transport virial coefficient for large property mismatch are discussed in Section 4, which remain to be checked in future numerical studies.