Tensors for Physics

348 16 Constitutive Relations

Shear rates, in units ofτref−^1 are now denoted byΓ, thus one hasΓ=γτref. Further-
more, the tumbling parameter, cf. Sect.16.4.3, is written as

λeq=λK

ani

aeq

+

1

3

κ, λK=−

2

3

√

3

τap

τa

a−ni^1. (16.148)

The equilibrium order parameter, in the nematic phase,aeqis given byaeq/ani=
3
4 +

1

4

√

9 − 8 θ,θ≤^98 ,see(15.23). Notice thatλeqdecreases with increasing order
aeq. For small shear rates,λeqdetermines the flow alignment angleχ, within the
Ericksen-Leslie theory, cf. Sect.16.4.3, according to cos( 2 χ)=−γ 1 /γ 2 = 1 /λeq,
provided thatλeq>1. Forλeq<1, no stable flow alignment exists. The actual
dynamics following from the alignment tensor theory, as discussed in Sect.17.3,is
more complex than the tumbling motion inferred from the Ericksen-Leslie director
approach. The quantityλKwhich is the tumbling parameter at the transition temper-
ature, forκ=0, is used as a model parameter in the scaled dynamic equation for
the alignment tensor.
In the following, when no danger of confusion exists, the scaled alignment tensor

a∗μνis denoted by the original symbolaμν.LetΩλ=ωλτrefandΓμν=∂vμ/∂rν
be the dimensionless vorticity and deformation rate tensor. The (16.135) governing
the dynamics of the alignment then is equivalent to

daμν

dt

− 2 εμλκΩλaκν− 2 κΓμκaκν+Φμν=

√

3

2

λKΓμν, (16.149)

where it is understood thatΦμνstands for the scaled derivative of the relevant poten-
tial, viz.
Φμν=θaμν− 3

√

6 aμκaκν+ 2 aμνaλκaλκ. (16.150)

This corresponds to the derivative of a Landau-de Gennes potential withA=θ,
B=3,C=2.
A scaled symmetric traceless stress tensorΣμνal, associated with the alignment is
introduced via

− pμν

align

=

√

2 GalΣμνal, Gal=

3

4

ρ

m

kBTλ^2 KδniA 0 ani^2 , (16.151)

whereδni= 1 −T∗/Tni, andGalis a shear modulus linked with the alignment. For

pμν

align

see (16.136). The scaled version of this equation corresponds to

Σμνal =

2

√

3

λ−K^1 Φ ̃μν, Φ ̃μν=Φμν+

2 κ

3 λK

√

6 aμκΦκν. (16.152)

The Fokker-Planck equation approach implies 2κ/( 3 λK)=

√

5 ani/7.