Tensors for Physics

340 16 Constitutive Relations

for uniaxial ellipsoids with the axis ratioQ. ValuesQ>1 andQ<1 pertain to
prolate, i.e. rod-like and oblate, i.e. disc-like particles, respectively. One hasR>
0 andR <0 for these cases. The quantityRvanishes for spherical particles,
corresponding toQ=1.
The second term in (16.122) is associated with the internal field proportional to
the alignment tensor. The characteristic temperatureT∗is linked with the strength
of the alignment energy, just as in the Maier-Saupe theory, cf. Sect.15.2.3.Inthe
absence of a flow, the stationary solution of the kinetic equation (16.119)is

f=feq∼exp[T−^1 T∗aμνφμν],

which is essentially the Maier-Saupe distribution function.

Multiplication of the kinetic equation (16.119)byφμν=ζ 2 uμuνand integration
overd^2 uand use of (16.121) with (16.122) leads to a nonlinear, inhomogeneous
equation foraμν, which is, however, not yet a closed equation for the second rank
alignment tensor. More specifically, the moment equation foraμν, as inferred from
(16.119), is

∂

∂t

aμν− 2 εμλκωλaκν+ν 2 aμν−ν 0

〈

(Lλφμν)(Lλφαβ)

〉

Fαβ= 0 ,

withν 2 = 6 ν 0 ,cf.(12.44). Computation of the expression within the bracket〈...〉
of the last term yields

4 ζ 22 ελκμuκuνελσ αuσuβFαβ= 4 ζ 22

(

uνuβFμβ −uμuνuαuβFαβ

)

.

Use ofuνuβ=^13 δνβ+uαuβ in the first term on the right hand side and of the
relation

uμuνuαuβFαβ=

2

15

Fμν+

4

7

uμuκFκν+uμuνuαuβFαβ,

cf. (11.58), leads to

(Lλφμν)(Lλφαβ)= 4 ζ 22

(

1

5

Fμν+

3

7

uμuκFκν−uμuνuαuβFαβ

)

.

The orientational average of this expression involves the fourth rank alignment tensor

aμναβ=〈φμναβ〉,φμναβ=ζ 4 uμuνuαuβ,ζ 4 =

3

4

√

70 ,