16.4 Viscosity and Alignment in Nematics 341
cf. Sect.12.2.2. Thus the moment equation for the second rank alignment tensor
becomes
∂
∂taμν− 2 εμλκωλaκν+ν 2 aμν− 4 ν 0 ζ 22(
1
5
Fμν+3
7
ζ 2 −^1 aμκFκν−ζ 4 −^1 aμναβFαβ)
= 0.
Use of the explicit expression (16.122) for the tensorF..leads to
∂
∂taμν− 2 εμλκωλaκν− 2 κ ∇μvκaκν+ν 2 (Aaμν−√
6 Baμκaκν)+ 5 ν 2 ζ 4 −^1 T−^1 T∗aμναβaαβ=R(
∇μvν− 5 ζ 4 −^1 aμναβ∇αvβ)
,
(16.124)
with
A= 1 −T∗
T
, B=
√
5
7
T∗
T
, (16.125)
and
κ=1
7
ζ 2 R=3
7
Q^2 − 1
Q^2 + 1
. (16.126)
The second equality in (16.126) pertains to the hydrodynamic expression (16.123)
the parameterR. For long, rod-like particles, corresponding toQ 1, the quantity
κapproaches 3/ 7 ≈ 0 .4.
In the isotropic phase and for small alignment where terms nonlinear in the second
rank alignment tensor and the fourth rank alignment tensor can be disregarded,
(16.124) reduces to an equation similar to (16.74), viz.
∂
∂taμν− 2 εμλκωλaκν− 2 κ∇μvκaκν+τa−^1 Aaμν=−τa−^1 τap√
2 ∇νvμ,
(16.127)
where the relaxation time coefficients are now related to the parameters occurring in
the Fokker-Planck approach by
τa=ν− 21 =( 6 ν 0 )−^1 ,√
2 τap=−Rτa. (16.128)The comparison of (16.127) with (16.74) shows two additional features. The first
is the term involving the parameterκ, which describes an effect of the deformation
rate on the alignment. The second is the occurrence of the factorA= 1 −T∗/T.
This implies thepre-transitional increaseof the relaxation timeτ =A−^1 τaand
of the flow birefringence∼A−^1 , when the temperatureTapproaches the transition