Tensors for Physics

16.4 Viscosity and Alignment in Nematics 347

pμ=εμνλnν

(

γ 1 Nλ+γ 2 ∇λvκnκ

)

, (16.145)

γ 1 = 3

ρ

m

kBTa^2 eqτa,γ 2 =

ρ

m

kBT

(

2

√

3 aeqτap−κa^2 eqτa

)

.

In this expression, the Ericksen-Leslie coefficientsγ 1 >0 andγ 2 , already introduced
in Sect.16.4.1, are related to the equilibrium alignmentaeq=

√

5 S, and to the model
parametersoccurringinthedynamicequationforthealignmenttensor.Noticethatthe
Fokker-Planck approach yieldsτap<0 for rod-like particles, cf. (16.126), (16.128),
and consequentlyγ 2 <0. For disc-like particles, on the other hand, one hasγ 2 >0.In
general, terms of higher power in the order parameteraeqcontribute to the Ericksen-
Leslie coefficients, when higher rank tensors are taken into account in the solution of
the Fokker-Planck equation. The expressions given here contain the leading terms.
In the weak flow approximation, the relation (16.136) for the symmetric traceless

part of the friction pressure tensors leads to an expression for pμν, as presented in
(16.101), now with the viscosity coefficientsη,η ̃ 1 ,η ̃ 2 ,η ̃ 3 given by

η=

ρ

m

kBT

(

τp+

1

6

κ^2 a^2 eqτa

)

, η ̃ 1 =−

ρ

m

kBTκaeq

(

2

√

3 τap+

1

2

κaeqτa

)

,

η ̃ 2 =

ρ

m

kBTκaeq

(√

3 τap−

1

2

κaeqτa

)

, η ̃ 3 =

1

2

ρ

m

kBTκ^2 a^2 eqτa. (16.146)

The Onsager symmetry relation 2η ̃ 2 =γ 2 is obeyed. Notice thatκ=0 implies
η ̃ 1 = ̃η 3 =0. In lowest order in the alignment, one hasη ̃ 1 =−κγ 2. This relation
can be used to obtain an estimate for the size of the parameterκfrom experimental
data.

16.4.7 Scaled Variables, Model Parameters

The relaxation term of the inhomogeneous equation (16.135) for the second rank
alignment tensor involves the derivative of the Landau-de Gennes potential, which in
turn contains the three parametersA,B,C. When the alignment is expressed in units
ofthenematicorderparameterani= 2 B/( 3 C),cf.Sect.15.2.2,thecoefficientsBand
Care replaced by specific numbers. More precisely, the alignment tensor is written as
aμν=ania∗μν, the derivative of the potential is expressed asΦμν(a)=ΦrefΦμν∗(a∗)

with the reference valueΦref =ani 2 B^2 /( 9 C)=aniδniA 0. As in Sect.15.2.2,the
scaled temperature variableθ = A(T)/A(Tni)= ( 1 −T∗/T)( 1 −T∗/Tni)is
used. The time is scaled in units of a reference time equal to the relaxation time, at
coexistence temperatureTni,viz.

t=τreft∗,τref=τa( 1 −T∗/Tni)−^1 A− 01 =τa

9 C

2 B^2

=τaaniΦref−^1. (16.147)