Tensors for Physics

16.4 Viscosity and Alignment in Nematics 337

Miesowicz coefficients obey the inequalitiesη 2 >η 3 >η 1 andγ 2 <0. For oblate
particles, i.e. forQ<1, one hasη 2 <η 3 <η 1 andγ 2 >0. An adaption of this
model to partially ordered nematics and comparison with experimental data is found
in [148]. The alternative approach, where the dynamics of the alignment tensor is
taken into account, is discussed in Sects.16.4.4–16.4.6.
The anisotropy of the heat conductivity and of the diffusion tensor can also be
treated by the affine transformation approach. For diffusion, see the following exer-
cise. An amended affine transformation model which takes the partial alignment
into account and a comparison with molecular dynamics computer simulations is
presented in [156].

16.3 Exercise: Diffusion of Perfectly Oriented Ellipsoids
In the nematic phase, the fluxjof diffusing particles with number densityρobeys
the equation

jμ=−Dμν∇νρ, Dμν=D‖nμnν+D⊥(δμν−nμnν),

whereD‖andD⊥are the diffusion coefficients for the flux parallel and perpendicular
to the directorn.
Use the volume conserving affine transformation model for uniaxial particles, cf.
Sect.5.7.2, to derive

D‖=Q^4 /^3 D 0 , D⊥=Q−^2 /^3 D 0

for perfectly oriented ellipsoidal particles with axes ratioQ.HereD 0 is the reference
diffusion coefficient of spherical particles.
Furthermore, determine the anisotropy ratioR =(D‖−D⊥)/(D‖+ 2 D⊥),

the average diffusion coefficientD ̄ =^13 D‖+^23 D⊥and the geometric meanD ̃=

D^1 ‖/^3 D^2 ⊥/^3. Discuss the casesQ>1 andQ<1 for prolate and oblate particles.

16.4.3 Free Flow of Nematics, Flow Alignment and Tumbling

The antisymmetric part of the pressure tensor vanishes for a free flow, i.e. when no
orienting external field is applied. This implies

pμ=εμνλnν

[

γ 1

(

∂nλ

∂t

−ελκτωκnτ

)

+γ 2 nκ∇κvλ

]

= 0. (16.115)

For a spatially inhomogeneous situation and in the presence of external fields the
torque associated with (16.115) is not zero but balanced by the elastic and field-
induced torquesεμνλnν(KΔnλ+Fλκnκ)as described by (15.33).
Provided that|γ 2 |>γ 1 holds true, a stationary plane Couette flow leads to asta-
tionary flow alignmentwhere the director is in the plane spanned by the flow velocity