Tensors for Physics

342 16 Constitutive Relations

temperatureTni, from above. Notice thatTniis slightly larger than the pseudo-critical
temperatureT∗, cf. Sect.15.2.2.
In general, however, and in particular in the nematic phase, terms nonlinear in the
alignment matter and the fourth rank tensor has to be taken into account. Then an
equation is needed foraμναβ,in(16.124). Multiplication of the generalized Fokker-
Planck equation (16.119) by the fourth order expansion functionφμναβand subse-
quent integration leads to an equation for the fourth rank alignment tensor, that is
analogous to that of the second rank tensor. There, however, not only a coupling
with the second rank but also with the sixth rank tensor occurs. Clearly, the game
may be continued leading to a hierarchy of coupled equations for tensors with rank
= 2 , 4 , 6 ,.... The equation for the second rank tensor is the only one which con-
tains an inhomogeneous term. The tensors of rank>2 relax faster than the second
rank tensor, cf. (12.44). In [157], a closure of the set of equations was achieved by
disregarding the tensors of rank 6 and higher. When furthermore, the co-rotational
time derivative of the fourth rank tensor and terms nonlinear in the deformation rate
∇vare disregarded, this approximation amounts to puttingaμναβ∼aμνaαβ with
a proportionality coefficient analogous to that one occurring for equilibrium align-
ment, cf. Sect.12.2.4. Use of the relation (12.34) for uniaxial alignment in the high
temperature approximation, which is equivalent to

ζ 4 −^1 aμναβ=

5

7

ζ 2 −^2 aμνaαβ, (16.129)

leads to the closed equation governing the second rank alignment tensor

∂

∂t

aμν− 2 εμλκωλaκν− 2 κ ∇μvκaκν+ν 2 Φμν

=R

(

∇μvν−

10

21

aμνaλκ ∇αvβ

)

. (16.130)

Here

Φμν=Aaμν−B

√

6 aμκaκν+Caμνaλκaλκ,

is the derivative of the Landau-de Gennes potential, cf. Sect.(15.2.2), where now the
coefficientsAandBare given by (16.125) andCis found to be

C=

12

49

(

T∗

T

) 2

. (16.131)

Apart from the last term on the right hand side of (16.130), this equation is equivalent
to (16.127) where nowΦμνappears in the relaxation term instead ofAaμν.The
limiting case of a weak alignment in the isotropic phase was already discussed above.
The other limiting case corresponds to a weak flow in the nematic phase where the