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Appendix C: Intrinsic Viscosity Formalism for Ellipsoids and Other Shapes

Haber and Brenner have worked out the isotropically averaged intrinsic viscosity for ellipsoidal particles [14]. This formalism is valid for any choice of the axis lengths, a₁, a₂, and a₃. Values of the A_i functions are required, as well as auxiliary functions that can be defined in terms of the A_i, denoted A_i'and A_i''. They are given as follows (i,j,k are permuted cyclically):

(The expression for A_i' is opposite in sign from that given in Haber and Brenner [14] and corrects a small typographical error in that paper.) The following quantities are then formed:

The intrinsic viscosity can be expressed in terms of these constants,

This equation is tabulated in Table 5 and Table 6.

The intrinsic viscosity for anisotropic particles is in general a fourth rank tensor, with the same symmetries as the elastic stiffness tensor [117] for the given symmetry of the particle. For example, the intrinsic viscosity for an aligned triaxial ellipsoid is a 4th rank tensor having the same symmetry as the elastic stiffness tensor for an orthorhombic crystal [117,139]. Formally, this can be stated using either the stress to define the effective viscosity of a single-particle suspension, or using the energy dissipation rate [14]. In the following, the discussion of Haber and Brenner is followed [14] in presenting both methods.

Consider an isolated particle, immersed at rest in a homogeneous, incompressible fluid with viscosity η_o. We restrict consideration to particles which have a center of symmetry. The traceless rate of strain field is denoted S, such that if the particle were not there, S = S_o would be a uniform traceless rate of strain tensor. Far away from the particle, the velocity fields must go to v = S_o r and the average of S over all space, denoted <S>, equals S_o [14]. Solving Stokes' equations for the fluid velocity and pressure fields due to the presence of the rigid particle and then averaging the viscous stress over the volume of the sample gives,

where φ is the volume fraction of the particles. This result is limited to a sufficiently dilute suspension so that each particle can be considered to be independent of the others. Ψ_ijkl is a 4th rank tensor of the same symmetry as the particle. If the calculation is carried out in a periodic array of particles, as was done in the finite element work explained in Appendix E, then Ψ_ijkl has whichever symmetry is lower, the particle or the array. For example, when considering a cubic array of spheres, Ψ_ijkl has cubic symmetry and is not isotropic like the sphere.

The isotropically averaged intrinsic viscosity that we have studied in this paper is obtained by calculating the isotropic average < Ψ_ijkl > , using the standard definition of rotational tensor averaging [140] and then using eqn. (14) to obtain

We are operating under the assumption of "overwhelming Brownian motion" [117], so that the applied shear is weak compared to the Brownian motion of the particle, so that all orientations are equally probable. In this case, the quantity Ψ_ijkl is calculated when holding the particle positionally and orientationally fixed in space. The rotational average of Ψ_ijkl then incorporates the fact that all orientations are equally probable. The opposite case would be the strong applied shear case, where Brownian motion can be ignored, and the particle has anisotropic orientation due to the applied shear field. For an anisotropic particle, that would correspond to the elastic case, since in the elastic case a rigid particle maintains its shape but can orient itself with the applied field.

Since the intrinsic viscosity tensor has the same symmetries as the equivalent elastic tensor, the averaging procedure discussed above is exactly the same as taking the Voigt average [141] of the elastic stiffness tensor for a polycrystalline sample. This procedure has been worked out for every crystal symmetry of interest and results are available in the literature. For example, all the particles considered in this paper have tetragonal or higher symmetry. Tetragonal symmetry means square symmetry in the cross-sectional plane, with the third direction being different (a rectangular parallelipiped, with a = b ≠ c, has tetragonal symmetry). The angularly averaged shear modulus is obtained from the elastic stiffness tensor C_ijkl (tetragonal symmetry) by,

To obtain the rotationally averaged intrinsic viscosity, simply substitute _ijkl for C_ijkl, and multiply by a factor of 2, because the elastic tensor averages are defined for a system of notation such that the shear strain is twice that used for the rate of shear strain in fluid mechanics [142,143]. One should note that in Refs. [14] and [117] a factor of 5/2 is often taken out of the definition of Ψ_ijkl so that < Ψ_ijkl > is then normalized to equal 1 for a spherical particle.

The intrinsic viscosity can also be defined via the energy-dissipation rate. Since this is the way that the finite-element simulations were done and since it offers an easy way to do the averages numerically, we present this method as well. We again follow the discussion of Ref. [14].

For the same case as considered above, it is found that the rotationally averaged rate of energy dissipation <E> is given by

For tetragonal symmetry and higher it is possible to select the terms of S_o so that the energy dissipation rate E [143]

gives the terms needed to form the average (87-88). One choice for S^o _ij which gives the correct combination of terms for tetragonal symmetry is the following:

Computing the energy for this applied rate of strain tensor exactly gives the combination of terms in eqns. (87-88). By subtracting the original energy, that found without a particle being present, the rotationally averaged intrinsic viscosity can then be read off from the numerical results.