Next: Intrinsic Viscosity and Up: Main Previous: Introduction

Polarizability, Intrinsic Conductivity, and Virtual Mass

Maxwell [4] first considered the classic problem of the conductivity σ of a particle suspension in which the suspended particles have a different conductivity σ_p than the suspending medium σ_o. 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 then equals,

The virial expansion (1) has been verified experimentally for dilute suspensions of numerous substances. For example, the leading order virial coefficient for conducting spheres equals 3. This value has been observed by Voet for nearly spherical iron particles (diameter = 10 µm in linseed and mineral oils [31]. Emulsions of salt water in fuel oil and mercury drops in different oils have also been found to be consistent with eqn. (1) with Δ_σ large [32]. The corresponding prediction for insulating suspended spheres (Δ_σ → 0) has been observed for suspensions of glass beads and sand particles in salt solutions [33] and for gas bubbles in salt solutions [34]. The virial coefficient notably changes sign and equals -3/2 in the insulating spherical particle suspension. Good agreement with Eqn. (1) has also been observed in fluidized beds where Δ_σ was tuned over a range of values [35].

At higher concentrations the independent particle approximation of the dilute regime no longer holds, but the leading order virial coefficient still plays a primary role in theoretical estimates of the high concentration variation of transport properties. In the simple effective medium theories of Bruggeman [36] and Brinkman [37], for example, the resummation of the virial expansion for an arbitrary transport property P of a suspension is quite generally given by,

where P_o is the property for a pure suspending medium. Consistency at low concentration requires that the 'critical exponent' m has the same magnitude as the leading order virial coefficient. This simple prediction, which is derived on the basis of very simplistic reasoning, is often in remarkably good agreement with observations in physically important systems [31,32,33,38,39,40,41]. For example, 'Archie's law' for the conductivity of rocks saturated with salt water [39,41] follows directly from (1) and (2). The corresponding Brinkman-Roscoe [37,38] result has also been often cited as a useful description of the viscosity of concentrated suspensions. We mention these approximate calculations only to illustrate the primary role of the leading order virial coefficient in developing a theoretical description of mixture properties at higher concentrations. The development below is restricted to the low concentration regime, where such uncontrolled approximations are unnecessary.

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. Fricke [8] treated the case of ellipsoidal particles and utilized a Clausius-Mosotti-style [42] 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).

The low concentration σ virial expansion of randomly oriented and arbitrarily shaped particles equals [43]:

where [σ] is the 'intrinsic conductivity'. The magnitude of [σ] can be rapidly varying for extended or flat particles depending on the relative conductivity Δ_σ, so that the effect of adding a given amount of material to a suspension can be greatly dependent on particle shape and composition. It is often convenient to define virials such as [σ] in terms of a number concentration when the suspended particles have zero volume, as in the cases of needles, plates, and idealized random walk chains, for example.

The 'polarizability' α describes the average dipole moment induced on a particle in an applied field (electric or magnetic) and the calculation of the 'virial coefficient' [σ] therefore requires the determination of α or at least an average of the matrix elements defining α (see below). The quantity α is a second rank tensor [4,44] that depends on particle orientation, shape, size, and the ratio of the particle property to the matrix property for the property that is being considered (see (1), (2). The average polarizability < α >, which is 1/d times the trace of the polarizability tensor, is a scalar which is invariant under particle rotations [45,46]. The polarizability has the units of volume so that the ratio of < α > and the particle volume V_p is a scale invariant functional of particle shape and Δ_σ. Calculation of < α > is often easier than the full polarizability tensor α, since any three orthogonal directions can be chosen for the field directions in the calculation. Equivalently, we can angularly average α over all orientation angles with uniform probability [45,46] to obtain < α >. In some applications it is useful to orient the suspended particles, in which case the effective conductivity of the composite becomes explicitly dependent on α [9]. Historically, the anisotropic case was found to be important in the design of microwave lenses and other artificial dielectrics where large scale conducting elements are arrayed in an insulating matrix [47,48,49,50]. The anisotropic situation is also encountered in the optical properties of sheared anisotropic particle suspensions [51]. In the present paper, we emphasize the average polarizability , 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 ε_p, embedded in a medium having a dielectric constant ε_o, are distorted in response to an applied electric field [42,52]. 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. Equations (1)-(4) also describe the dielectric constant of suspensions of particles with a relative dielectric constant Δ _ε = ε_p / ε_o . Moreover, these equations apply equally as well to the magnetic permeability, diffusion coefficient (see Appendix A) and the thermal conductivity of dilute suspensions, where the magnetic field, concentration gradient, and the temperature gradient are the corresponding 'fields' [1,2,21].

Although simple in principle, the calculation of the polarizability tensor for objects of general shape is a mathematical problem of notorious difficulty. Indeed, the ellipsoid [8,52] is the only shape for which exact analytic results have been obtained as a function of Δ _σ. There have been recent numerical calculations of the polarizability tensor for other objects in relation to Rayleigh scattering (e.g. radar) applications [53,54]. The situation is better for limiting values of the relative conductivity where the polarizability tensor α( Δ_σ) simplifies. For highly conducting ('superconducting') inclusions, the polarizability tensor reduces to the 'electric polarizability' α_e,

and [σ] for randomly oriented inclusions, having a much higher conductivity than the matrix, equals,

where < α_e > denotes the average electric polarizability tensor. The case of insulating inclusions in a conducting medium corresponds formally to Δ_σ → 0⁺, so that we have

which is completely independent of particle shape in leading order.

The limiting relations (6) and (8), connecting the intrinsic conductivity to the electric and magnetic polarizabilities of a conductor, can be appreciated from a more general relation between the generalized electric E and magnetic H field polarizability tensors, α(E) and α (H), which allows a unified discussion of the response of complicated shaped objects to both electrostatic and magnetostatic fields. Senior [55] has proven the validity of the general relations,

where Δ_ε and Δ_µ are the relative dielectric constant and magnetic permeability,

and X is the same function for both electric and magnetic fields. A perfect conductor (Δ_ε → ∞) is magnetically 'impermeable' (Δ_µ → 0⁺) while a perfect insulator (Δ_ε → 0⁺) is formally a 'magnetic conductor' (Δ_µ → ∞) [50,56]. The relevance of the magnetic polarizability in descibing the insulating limit (Δ_ε → 0⁺) of the intrinsic conductivity is thus apparent.

Keller et al. [57] recently emphasized the equivalence of the magnetic polarizability tensor α_m and the hydrodynamic 'effective mass' tensor M describing a particle translating through an inviscid, irrotational, and incompressible ('perfect') liquid. M is equal the particle volume V_p plus the 'added mass' or 'virtual mass' W associated with the kinetic energy imparted to the fluid from the particle motion,

Almost all hydrodynamic calculations of M assume that the particle density is much higher than the surrounding fluid. Birkhoff [60,61,62,63] showed that the dipolar field disturbance induced by the motion of particles having comparable density to the fluid medium involve the general polarizability tensor X in eqn. (10) where the field permittivity parameter, corresponding to Δ_σ, is related to the relative density of the particle and the fluid. This generalized relation implies that M of low density objects (e.g. air bubbles in water) corresponds to α_e rather than α_m [60]. In the discussion below we restrict ourselves to the more conventional case in which the moving object is assumed to have a much higher density than the fluid. The terms 'effective mass' and 'virtual mass' will then always imply the Keller-Kelvin relation (12). A non-perturbative generalization of eqn. (12) to finite concentrations is discussed in Appendix A.

It should be mentioned that the hydrodynamic applications of M are not limited to transient hydrodynamic phenomena associated with the forces on accelerating particles in a fluid medium [64,65]. The presence of a body in a converging stream of an inviscid fluid, such as air in a wind tunnel impinging on an aircraft model, gives rise to a force on the body that is determined by the M tensor ( M = (3V_p/2) I for a sphere and the angular average of M for near-spherical and slender particles usually differs little from the sphere value; see below) and the pressure gradient in the channel. The force is in the direction of the pressure gradient. Kelvin [66] and Taylor [67] have shown this 'buoyancy drag force' is obtained even in non-simply connected spaces, like porous media. The Kelvin-Taylor result for drag forces in 'perfect' fluids is a natural counterpart to the Stokes drag force [64] on translating particles in viscous fluids, where a shape functional similar to M arises [see eqn. (22)].

The importance of eqns. (6), (8), and (12) derives from the extensive mathematical and technical literature relating to the calculation of α_e and M [6,26,46,52,68,69,70,71,72,73,74,75,76]. These functionals of object shape are naturally encountered in the solution of the Laplace equation on the exterior of regions of various shapes and consequently these shape functionals have attracted a mathematical interest quite apart from technical applications [26,46]. For example, it has rigorously been shown that < α_e > and <M> achieve their absolute minima for a circle and sphere in d = 2,3 dimensions [26,77] of all objects having a finite area or volume, respectively. ( Presumably, a hypersphere minimizes these functionals in d dimensions [78]. Numerical illustrations of this sphere minimization property are presented below. It is also known (implicitly from the work of Keller and Mendelson [78] that < α_e > = - < α_m > in d=2 and that these shape functionals have a general geometric interpretation in terms of conformal mapping, as will be discussed in Sec. 5 below.

Payne and Weinstein [79] proved that some components of α_e and M for certain regions (having reflection symmetry in d=2 or axisymmetric particles in d=3) are related to the capacity C of the region. The capacity C is another shape functional related to solving the Laplace equation on the exterior of a particle. Numerous applications of this quantity are summarized in Ref. [80], which also describes a probabilistic method for calculating C by hitting a region of arbitrary shape with Brownian paths launched from an enclosing surface. This development makes the Payne- Weinstein relation attractive for the calculation of α_e and α_m.

Finally, we mention that exact calculations of α_e and M can be made for certain object boundaries and associated coordinate systems for which the Laplace equation is separable [46] and for regions related to the separable boundary cases by Kelvin inversion [4,52]. This leads to quite a few intrinsic conductivity results for objects with interesting shapes that are useful in checking numerical methods applicable to more generally shaped regions. Some results of this kind are summarized below.

The technology literature is also a rich source of results for α_e, α_m, and M. Apart from the relation to transport coefficients like σ, ε, µ, and D, mentioned above, the magnetic and electric polarization tensors have fundamental interest because they completely determine [55,56,57] the scattering of electromagnetic waves having long wavelengths relative to the (metallic) scattering object size, i.e., Rayleigh scattering [81]. It is a crucial application to discriminate object shape to the maximum extent possible from long wavelength radiation like radar and, needless to say, the technical literature reflects a preoccupation with objects having the shapes of missiles and space vehicles [53,68]. Weather radar applications are also important [82]. We also note that scattering of long wavelength sound waves from hard obstacles is determined by M and the particle volume V_p [83,84,85,86] and that _e and M are also fundamental in the scattering of electromagnetic and sound waves through apertures [72,87,88,89,90,91]. These applications especially require the calculation of α_e and M for plate-shaped objects [92].

We next tabulate the components of the polarization tensors α_e and α_m , since this information is important in the comparisons below with intrinsic viscosity data. These tabulations should give the reader some feeling for the magnitudes involved and illustrate some of the general ideas stated above.

The ellipsoid provides the simplest example of an object having variable shape. Appendix B summarizes the necessary mathematical and numerical computations involved in calculating α_e (ellipsoid) for a range of principal axis radii ratios. In Figs. 1 and 2 we present the polarizability results for ellipsoids of revolution per unit particle volume (which actually have closed form analytical solutions--see Appendix B). Enough points have been calculated to make the graphs of these quantities appear as smooth curves. The abscissa x denotes the length of the ellipsoid along the symmetry direction relative to the axis length normal to the symmetry axis ('aspect ratio'). The component of α_e and α_m along the symmetry axis is denoted by L and the component normal to the symmetry axis is denoted by T. The average polarizabilities, < α_e > / V_p = [σ]_∞ and < α_m > / V_p = [σ]_o , are also shown. These quantities are invariant under particle rotations and are functionals of particle shape only. A tabulation of this numerical data for ellipsoids of revolution is given in Table 1. We give the data in the table in the dimensionless form α / V_p, since the results are then independent of the absolute particle size. A more general tabulation for the triaxial case is given separately for α_e and α_m in Table 2 and Table 3 for a range of the two principal axis ratios. All reported digits shown in Table 1, Table 2, and Table 3 are significant.

Figure 1: The longitudinal (L) and transverse (T) components of the dimensionless (normalized by the particle volume) electric polarizability tensor α_e for ellipsoids of revolution and the average of these components, the intrinsic conductivity of a conductor [σ]_∞.

Figure 2: The longitudinal (L) and transverse (T) components of the dimensionless (normalized by the particle volume) magnetic polarizability tensor α for ellipsoids of revolution and the average of these components, the intrinsic conductivity of an insulator [σ]_o.

We observe in Figs. 1 and 2 that the averages < α_e > / V_p = [σ]_∞ and < α_m > / V_p = [σ]_o obtain absolute minima for x = 1. This accords with the exact results [26,77] mentioned above, which indicate that this minimum is achieved in the case of a sphere for all objects having a given finite volume. These virial coefficients are observed to be quite sensitive to the aspect ratio x in the approach to the disc limit, but there are significant effects of particle asymmetry on the virial coefficients for highly conducting needle-like (x >> 1) particles. Needle-like non-conducting inclusions lead to remarkably little change in the intrinsic conductivity. Thus, we can understand the general experimental observation that non-conducting asymmetric inclusions in a conducting medium often lead to nearly the same intrinsic conductivity [93], except in the case of platelet shaped particles [94]. This implies that the most economical means of making a medium more insulating is through the introduction of a small concentration of non-conducting platelet-like particles. Fig. 2 also suggests that the polarizability of very irregular non-conducting objects, such as alkane chains and other non-conducting polymers, should increase in simple proportion to volume (molecular weight) for homologous molecular series, since the change in shape will not appreciably affect the intrinsic conductivity. This effect is observed in gas phase polarizability estimates on normal alkanes, based on dielectric constant and refractive index measurements [43]. Corresponding gas phase measurements on conjugated polymeric systems, on the other hand, exhibit a rapidly increasing polarizability with molecular weight [43], in accord with the calculations of α_e and the simplistic view of such polymers as 'conductors'. The variation of the colors of dyes with molecular weight [95] and certain attractive forces between long chain molecules [96] can be similarly understood using this kind of picture of 'insulating' and 'conducting' polymers and geometrical estimates of the polarizability.

There are other shapes for which α_e and α_m can be determined exactly. Most of these additional exact results are summarized by Schiffer and Szego [46], but are not well- known in the physical science literature. ( This is probably due to the rather complicated mathematical form of these exact analytical results.) As an example, we indicate exact α_e results, per unit particle volume, for a torus in Fig. 3, where the symbol L again denotes the axis of symmetry. The abscissa is the ratio of the overall torus radius a to the radius b of the tube forming the body of the torus itself. For example, the limit b → 0 for a fixed radius a gives an infinitely thin wire ring of radius a. Table 4 tabulates the corresponding numerical values of the polarizability components used in Fig. 3, based on previous tabulations of the equivalent of these numbers by Belovitch and Boersma [97]. The present tabulation is given in a dimensionless form that avoids the problem of the choice of units. Further tabulations of analytic results for the electric and magnetic polarizability tensors, corresponding to other shapes [46], will require careful numerical work.

Figure 3: The longitudinal (L) and transverse (T) components of the dimensionless (normalized by the particle volume) electric polarizability tensor α_e for tori and the average of these components, the intrinsic conductivity [σ]_∞, plotted versus the 'aspect ratio' a/b.