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Discussion

The problem of calculating the effective properties of a medium containing dispersed particles having complex shape and different properties than the suspending matrix arises in many applications. For many properties (refractive index, dielectric constant, magnetic permeability, thermal and electrical conductivity, tracer diffusion constant) this general problem reduces to a common mathematical description [1,27,28,29]. In the present paper we have specialized the language to electrical conductivity to make our discussion more concrete. We have also confined our attention to the dilute regime where each inclusion independently influences the properties of the medium and where exact calculations for special shapes become possible, at least for limiting values of the relative conductivity and special shaped inclusions.

The computationally more difficult and practically important problem of the conductivity virial coefficient [σ (Δ)] for arbitrary Δ has been attacked by the pragmatic procedure of developing an approximant incorporating exact information for Δ ≈ 1 and numerical (and sometimes exact analytical) information for [σ (Δ)] in the insulating (Δ → 0) and the superconducting (Δ → ∞) limits. Numerical calculations of [σ (Δ)] for particles having a variety of shapes have shown very good agreement with this approximant. However, using Eqn. (13) for estimating [σ (Δ)] for generally shaped particles in d = 3 is still limited by the difficulty in calculating the virial coefficients in these limits.

Given these difficulties, it is useful to also develop some simple approximations for [σ]_o and [σ]_∞ appropriate for commonly encountered classes of inclusions. In Fig. 7 we give exact results for [σ]_o and [σ]_∞ for an ellipsoid of revolution as a function of the aspect ratio x (dimension of ellipsoid along axis of symmetry relative to dimension normal to axis of symmetry). This figure shows that for needle- like particles [σ]_o is insensitive to particle shape. For a sphere [σ]_o = − 3/2 and for an infinitely thin needle [σ]_o = − 5/3, so we expect that the average of these results,

should be a useful approximation for insulating particles modestly extended along one direction. Equation (19) should perhaps also apply to diffuse sponge-like structures as in Fig. 4 and random- coil polymers, which have a low density. A similar approximation is often employed in aerodynamic literature for the virtual mass of an object [45], which is directly related to [σ]_o [1]. The approximation Eqn. (19) is poor for sheet-like insulating structures, which strongly modify the conductivity of a conducting matrix (see Fig. 7), so the approximation should not be employed uncritically.

There are actually few previous calculations of [σ]_o for any shape, so some comment on other applications of this shape functional are worth mentioning. Wang [46] (with Onsager's advice) investigated the role of protein particle shape on the self-diffusion coefficient D_s of water in protein solutions, and found the concentration dependence,

where φ is the protein volume fraction. The notation of the present paper is adopted in Eqn. (20) (see also Appendix A of Ref. [1] for further discussion). Recent NMR measurements on the diffusion of water in swollen gels by Geissler and Hecht [47] have shown that [σ]_o ≈ −1.66, which is the expected result for slender particles (see Eqn. (19) and Fig. 7). Wang discusses the importance of virial expansions such as Eqn. (20) for inferring the particle shape and surface properties of suspended particles [46], and there are also many classic polymer science studies devoted to this general problem [48].

The [σ]_∞ virial exhibits a more complex shape dependence. In our previous paper we established a general approximate relation between [σ]_∞ and the intrinsic viscosity [η] of a suspension of rigid particles, [η] ≈ [(d + 2) / 2d] [σ]_∞. A slender body approximation of Debye [49] for [η], and an approximation relating the translational friction ƒ_T of a Brownian particle to the capacity of the particle C [50] (Newtonian capacity C rather than the logarithmic capacity C_L discussed above), suggests a general approximation for [σ]_∞ corresponding to linearly extended particles such as polymer chains, needle-like inclusions, etc. :

where R_g is the average particle radius of gyration and V_p is the particle volume. The scaling [σ]_∞ ≈ R_g² C / V_p is consistent with exact calculations for long ellipsoidal particles where [σ]_∞ scales with aspect ratio x as [1,51],

Equation (21) is potentially a useful approximation since R_g is readily calculated from simple geometry and accurate and efficient numerical methods have recently been developed for calculating C of general shaped objects [49,52]. Future numerical work should examine the ratio Γ,

to determine the degree of its universality, at least for bodies extended along one direction. The development of readily implemented and accurate approximations for [σ]_o and [σ]∞ should be very useful in estimating [σ(Δ)] of polymeric particles. For example, the relation Eqn. (21) implies a non-trivial molecular weight dependence of [σ] for suspensions of highly conducting polymers [1], and a correspondingly strong influence on the conductive properties of solutions of these polymers.

The problem of calculating transport virial coefficients as a function of the relative property Δ is encountered in a wide class of properties. The treatment of the conductivity Γ applies equally as well to thermal conductivity, dielectric constant, refractive index, magnetic permeabililty, and other properties [27,28,29]. There are also many applications involving the elastic constants of solid composites and the viscosity of suspensions that involve similar, but somewhat more complicated mathematics. For example, the virial coefficient [G] for the shear modulus of an elastic particle in an incompressible medium depends on the ratio Δ_G of the shear modulus of the particle to that of the embedding medium and the intrinsic shear viscosity [η] of a suspension of fluid particles depends on the relative viscosity Δ_η [1]. These virials can also depend on the surface boundary condition (partial slip) and on other parameters (Poisson ratio, surface shear viscosity or modulus, etc.) in addition to a general dependence on particle shape. It turns out to be possible to treat this general class of problems by basically the same Pade approximant method.

To illustrate this generality, we also consider the calculation of [G]. For this generalization we simply replace σ by G in Eqn. (9) and note the corresponding result for [G'' ], in d = 2 and d = 3,

which can be inferred from Refs. [29,53,54]. In Figs. 8a and 8b we compare exact results [55] to the approximant Eqn. (13) with G replacing σ for randomly oriented prolate and oblate ellipsoids of revolution with relative shear modulus Δ_G. As in the conductivity virial case, the agreement is excellent. The calculation of the intrinsic viscosity [η] is very similar to the calculation of [G] [1,56] so it should be possible to extend Eqn. (13) to these properties as well.

For the general elastic case, when particles with bulk and shear moduli K_p and G_p are embedded in a matrix with K₀ and G₀ (Δ_K = K_p /K₀ and Δ_G = G_p / G₀), the same Pade approximant again holds. The quantities [G''] and [K''] are given, in d = 3, by,

and in d =2 by,

where ν₀ is the Poisson's ratio of the matrix [54]. In the small (Δ_K − 1 ) and (Δ_G − 1 ) limits, the expansion for K depends only on (Δ_K − 1 ) and that for G only on (Δ_G − 1 ), to second order in these quantities [54]. These quantities can also be obtained from the Hashin elastic bounds [29,53] for the moduli, which are known to be exact to second order in (Δ_K − 1 ) and (Δ_G − 1 ). Using the exact solution for [G] and [K] for ellipsoids of revolution (d=3) [55], Eqn. (13) again agrees very well for randomly oriented prolate and oblate ellipsoids of revolution, as can be seen in Figs. 9a and 9b. Having worked well for both electric and elastic cases, for a wide variety of particle shapes, we then expect the Pade approximant may apply to other properties as well.