Tensors for Physics

17.2 Nonlinear Relaxation, Component Notation 361

In the absence of a flow, one hasaμνst =aeqTμν^0 , when thez-direction is put
parallel to the directorn. The equilibrium value of the order parameter isaeq=
3
4 +

3

4

√

9 − 8 θ,θ<^98. The relaxation equation forδaμνthen reduces to

∂

∂t

δaμν+(θ+ 2 aeq^2 )δaμν+ 4 a^2 eqTμν^0 δa 0 − 6

√

6 aeqTμλ^0 δaλν = 0. (17.34)

Due to (17.24), the last term in this equation is equivalent to

− 6

√

6 aeqTμλ^0 δaλν

=aeq

(

− 6 Tμν^0 δa 0 + 6 Tμν^1 δa 1 + 6 Tμν^2 δa 2 − 3 Tμν^3 δa 3 − 3 Tμν^4 δa 4

)

.

The resulting equations for theicomponents of the distortion can be written as

∂

∂t

δai+ν(i)δai= 0 , i= 0 ,..., 4 , (17.35)

with dimensionless relaxation frequenciesν(i)=ν(i)(aeq). The stationary solution
is stable against these different distortionsδaiwhenν(i)>0 holds true. The case
ν(i)=0 pertains to a marginal linear stability. Then terms nonlinear inδaihave to
be taken into account.
For a uniaxial distortion whereδaμν=Tμν^0 δa 0 applies, one obtains the relaxation
frequency

ν(^0 )=θ− 6 aeq+ 6 a^2 eq= 3 aeq− 2 θ.

The last equality follows from the equilibrium conditionθ− 3 aeq+ 2 aeq^2 =0.
At the phase transition temperatureTnione hasθ=1,aeq=1 and consequently
ν(^0 )=1. At lower temperaturesν(^0 )becomes larger and the exponential relaxation
of a uniaxial distortion is even faster. The highest temperature where a meta-stable
nematic phase exists, corresponds toθ= 9 /8, withaeq= 3 /4 and consequently
ν(^0 )=0. This is a marginal stability.
The linear stability analysis for biaxial distortions, in particular the determination
of the relaxation frequenciesν(^1 )=ν(^2 )andν(^3 )=ν(^4 ), are deferred to the next
exercise.

17.2 Exercise: Stability Against Biaxial Distortions
Compute the relaxation frequenciesν(^1 )andν(^3 )for biaxial distortionsδaμν =
Tμν^1 δa 1 andδaμν=Tμν^3 δa 3 from the relevant relations given in Sect.17.2.4.
Solve the full nonlinear relaxation equation fora 3 witha 1 =a 2 =a 4 =0 and
a 0 =aeq.