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Appendix F: Ellipse Virial Coefficients in d=2

The intrinsic conductivity and viscosity for ellipses in a two dimensional fluid can be obtained from the triaxial ellipsoid results in three dimensions by taking appropriate limits and averaging. The average required to obtain virial coefficients in d=2 involves rotationally averaging around the z axis of the ellipsoid and letting a₃ (the semi-axis length in the z direction) approach infinity. We then need to evaluate A₁, A₂, and A₃ from Appendix B in the limit a₃ → ∞. From (71) and (72) along with the limit a₃ → ∞ we find,

and the A_i sum rule in d=2 becomes,

The rotational average in d=2 for the polarization is then,

since the 2nd-rank polarization tensor is diagonal for orthorhombic symmetry and higher. The intrinsic conductivity then equals,

which agrees with Garboczi et al. [155].

The intrinsic viscosity in d=2 is found in the same way as the shear modulus from an averaging of the elastic stiffness tensor G having rectangular symmetry and is given by,

After inserting the appropriate combinations of A₁, A₂, and A₃ into Haber and Brenner's formalism given in Appendix C and Ref. [14], the rotationally averaged intrinsic viscosity becomes

where the Q_i's are defined in Ref. [14]. This reduces to

which is exactly equal to the intrinsic conductivity given in eqn. (117).