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Maxwell [3] first treated the conductivity σ of a particle suspension in which the suspended spherical particles have a different conductivity σp than the suspending medium σo. He recognized that the change in conductivity reflected the average dipole moment induced by the particles on the suspending medium in response to an applied field. For a dilute suspension of hard spheres the effect is the simple additive sum of the effects caused by the individual particle dipoles. The effective conductivity of the dilute mixture, for the case of hard spheres, then equals,
| σ/σ0 = 1 + [3 (Δ − 1) / (Δ + 2)]φ + O(φ2), | (1) |
where Δ = σp / σo is the "relative conductivity" and φ is the volume fraction of suspended spherical particles. Exact results that go beyond this classic result are limited, however. There are effective medium calculations that attempt to extend the "virial expansion" (1) to higher powers of φ [4], but only for the case of spherical inclusions. Sangani [5] recently generalized Maxwell's calculation for spherical particles to d dimensions,
Levine and McQuarrie [6] calculated the second order virial coefficient for highly conducting (Δ → ∞) spheres while Jefferey [7] treated this quantity for arbitrary Δ.
The virial expansion (1) has been verified experimentally for dilute suspensions of numerous substances. For example, Eqn. (1) implies that the leading order virial coefficient for highly conducting spheres equals 3. This value has been observed by Voet for nearly spherical iron particles (diameter = 10 µm;) in linseed and mineral oils [8], and has also been found for emulsions of salt water in fuel oil and mercury drops in different oils [9]. The corresponding prediction for insulating suspended spheres, −3/2 (Δ → 0), has been observed for suspensions of glass beads and sand particles in salt solutions [10] and for gas bubbles in salt solutions [11]. Good agreement with Eqn. (1) has also been observed in fluidized beds where the relative conductivity Δ was tuned over a range of values [12].
The practically important inverse problem of determining the volume fraction of a suspension of complicated shaped particles from electrical measurements motivated the generalization of Eqn. (1) to particles having arbitrary shape and conductivity. Fricke [13] treated the case of ellipsoidal particles and utilized a Clausius-Mosotti-style [14] effective medium theory to approximate the higher concentration regime. These effective medium calculations are exact in the dilute regime where they reduce to a virial expansion of the form (1). We avoid further discussion of the higher concentration regime where approximate methods must be employed.
The low concentration σ virial expansion of randomly oriented and arbitrarily shaped particles equals [1,15]:
where [σ (Δ)] is called the intrinsic conductivity. We adopt this notation by analogy with the leading order concentration virial for the suspension viscosity which is conventionally called the "intrinsic viscosity" [η] [2]. The magnitude of [σ (Δ)] can be a strong function of particle shape for extended or flat particles depending on the magnitude of the relative conductivity Δ, so that the effect of adding a given amount of material to a suspension can be greatly dependent on particle shape.
The polarizability α is a second rank tensor [16,17] that generally depends on particle orientation, shape, size, and Δ. The average polarizability, which is 1/d times the trace of the polarizability tensor, is an invariant under rotations [18,19] so that the virial coefficient [σ] = < α > / Vp is a functional of particle shape and Δ only. Calculation of the average polarizability is often easier than the full polarizability tensor, since any three orthogonal directions can be chosen for the field directions in the calculation of [σ]. Equivalently, we can angularly average the polarization tensor over all orientation angles with uniform probability [18,19]. In some applications it is useful to orient the suspended particles, in which case the effective conductivity of the composite is anisotropic and becomes explicitly dependent on the components of the polarizability tensor [20]. Historically, the anisotropic case was found to be very important in the design of microwave lenses and other artificial dielectrics where large scale conducting elements are arrayed in an insulating matrix [21,22,23,24]. The anisotropic situation is also encountered in the optical properties of sheared anisotropic particle suspensions [25]. In the present paper, we emphasize the average polarizability, < α >, which is relevant to suspensions in which the particle orientation is completely random.
In an electrostatic context the polarizability describes how the charges of a body of dielectric constant εp, embedded in a medium having a dielectric constant εo, are distorted in response to an applied electric field [14,26]. The distorted charge distribution gives rise to a dipolar field that reacts upon the applied field, thereby modifying the net effective field in the proximity of the body. This connection between conductivity and the dielectric constant is natural since Eqns. (1)-(4) also describe the dielectric constant of suspensions of particles with a relative dielectric constant Δ ε = εp / εo . Moreover, these equations also apply to the magnetic permeability, diffusion coefficient and the thermal conductivity of dilute suspensions, where the magnetic field, concentration gradient, and the temperature gradient are the corresponding "fields" [27,28,29].
Although simple in principle, calculations of the polarizability tensor for objects of general shape is a mathematical problem of notorious difficulty. Indeed, the ellipsoid [13,26] is the only shape in d = 3 for which exact analytic results have been obtained as a function of Δ. There have been recent numerical calculations of the polarizability tensor for other objects in relation to Rayleigh scattering (e.g. radar) applications [30,31]. The situation is better for limiting values of the relative conductivity Δ where the polarizability tensor α(Δ) simplifies. For highly conducting inclusions, the polarizability tensor reduces to the electric polarizability αe,
and [σ] for randomly oriented inclusions, having a much higher conductivity than the matrix, then equals,
As above, < αe > denotes the average electric polarizability tensor. The case of insulating inclusions in a conducting medium corresponds formally to Δ → 0+ , so that we similarly have
where αm is the magnetic polarizability (see Ref. 1 for a discussion of the magnetic- electric polarizability analogy). In the Δ → 0+ and Δ → ∞ limits, the intrinsic conductivity is simply a functional of particle shape and spatial dimension. In our previous paper we discussed the important exact relation of αm to the hydrodynamic virtual mass of the particle which yielded many exact results for this quantity [1].
The intrinsic conductivity is rather insensitive to particle shape when the conductivity of the particles is similar to the embedding medium (Δ ≈ 1) , and a formal Taylor expansion about this limit can be made. Explicit calculation shows that this expansion does not depend on particle shape at all to second order in (Δ − 1) [32,33],
where [σ''] = −1/d [28] for the electrical problem in d dimensions. Eqn. (9) is found to be very useful in the next section where an approximant for [σ (Δ)] is developed for particles of general shape.
More general results are possible for [σ (Δ)] in the superconducting and insulating particle limits in d = 2 based on general conformal mapping results. In particular, the intrinsic conductivity of an arbitrarily-shaped superconducting inclusion [σ]∞ can be exactly expressed [1] in terms of the "transfinite diameter" CL of the inclusion [34,35] (see below),
where A is the area of the inclusion. Moreover, the Keller-Mendelson inversion theorem [36] for d = 2 implies that [σ] for an insulating inclusion, [σ]o , is related to [σ]∞ by a change of sign
The transfinite diameter CL is a basic measure of the average size of a bounded plane set, and can be defined in a variety of equivalent ways [35,37]. CL, for example, is defined as the conformally invariant magnitude of Dirichlet's integral associated with the exterior of the region defining the particle [35]. The equivalent transfinite diameter can be expressed in terms of the Euclidean metric defining the distance between points in the set [37]. Perhaps the most useful definition of CL involves the purely geometrical construction of mapping the exterior of a region having an arbitrary but simply connected shape and finite area onto a circular region in such a fashion that the points at a large distance from the region are asymptotically unaffected by the transformation [38]. The radius of this uniquely defined transformed circular region equals CL. This transformation is basically the content of the Riemann mapping theorem [39]. Since CL is a central object of harmonic analysis in two dimensions, there exist extensive tabulations of CL [34,35]. We may combine this information with (10) and (11) to obtain exact results for [σ]o and [σ]∞ in d = 2. An extensive tabulation of exact analytic results for symmetrically shaped regions and numerical estimates of these virials for irregularly shaped regions have been given in our previous paper for d = 2 and also for d = 3 [1].
Unfortunately, real inclusions often do not correspond to the ideal limits of perfectly insulating (Δ = 0) or perfectly conducting (Δ → ∞) inclusions, and explicit calculation (see next section) shows that the extent to which these limits apply depends on particle asymmetry. There is a need for accurate estimates of [σ (Δ)] for variable Δ if accurate inference of particle shape from conductivity (or other related property) measurements is required.
Exact results for general values of Δ are very limited, however. Maxwell's [3] and Sangani's [4] result for spheres were mentioned in Eqns. (1) and (2). The ellipsoid in d = 3 is also analytically tractable, although no simple closed form analytic expression exists for [σ (Δ)] , even for this simple class of particle shapes (except for ellipsoids of revolution [1]). The final known [σ (Δ)] expression is for an elliptical inclusion in d = 2,
where x is the ratio of the semi-major to the semi-minor axis lengths. In the next section we provide an approximant for [σ] for particles of general shape based on previously tabulated values of [σ]o and [σ]∞ [1], the spatial dimension d, and Δ.