278 15 Liquid Crystals and Other Anisotropic Fluids
The alignment tensor is uniaxial fora 1 =0, with its symmetry axis parallel toe(^3 ).
The tensor is planar biaxial fora 0 =0. For this reason,a 0 anda 1 are referred to
asuniaxial order parameterandbiaxial order parameter. Notice, however, that the
alignment tensor is also uniaxial whena 1 =±
√
3 a 0 holds true. In these cases, the
symmetry axis is parallel toe(^2 )ande(^1 ), respectively.
In terms ofa 0 anda 1 , the third scalar invariantI 3 ∼aμνaνκaκμ, which is essen-
tially the determinant, cf. (5.44), is determined by
I 3 =√
6 aμνaνκaκμ=a 0 (a 02 − 3 a^21 ). (15.4)The factor
√
6, which was not included in (5.44), is inserted here for convenience.
The biaxiality parameterb, cf. Sect.5.5.2,isnowgivenby
b^2 = 1 −I 32 /I 23. (15.5)With
a 0 =acosα, a 1 =asinα, (15.6)whereadetermines the magnitude of the alignment and the angleαis a measure for
the biaxiality, the scalar invariants and the biaxiality parameter are given by
I 2 =a 02 +a 12 =a^2 , I 3 =a^3 cosα(cos^2 α−3sin^2 α)=a^3 cos 3α, b=sin 3α.
(15.7)The argument 3αreflect the fact that the roles of the three principal axes can be
interchanged without changing the physics described.
Ordinary nematic liquid crystals are uniaxial in thermal equilibrium and when no
distortions are imposed. Then one hasα=0,a 1 =0 anda 0 =a. Furthermore,
the unit vector parallel to the space-fixed symmetry direction is denoted byn, rather
thane(^3 ), and calleddirector. The alignment tensor is written as
aμν=√
3
2
anμnν. (15.8)Due to
nμnνaμν=√
2
3
a,and (15.1), the order parameterais determined by
a=√
3
2
ζ 2 〈uμuν〉nμnν=√
5 S 2 , S 2 ≡〈P 2 (u·n)〉, (15.9)