290 15 Liquid Crystals and Other Anisotropic Fluids
The local orientation of liquid crystals, as observed optically via its birefringence,
i.e. between crossed polarizer and analyzer, may show defects. In the theoretical
description based on the director field, the defects are treated as mathematical singu-
larities. The alignment tensor theory, like (15.38) is closer to the physical reality. It
takes into account, that the defect is not a point, but rather a spatial region, where the
alignment tensor is no longer uniaxial, as assumed by the director theory. Further-
more, the magnitude of the order parameters are spatially dependent and in the core
of a defect, the alignment can even vanish, i.e. locally, within a small volume, the
fluid is isotropic [87]. For a specific example, the comparison between the alignment
tensor theory and the director description is presented in [88]. An example for the
alignment tensor field, in the vicinity of a point, which would be treated as a defect
in a director description, is shown in Fig.7.8. Notice that the sides of the bricks are
the eigenvalues of a second rank tensor which is the sum of the alignment tensor and
a constant isotropic tensor, chosen such that all eigenvalues are positive.
Incholestericsandblue phase liquid crystalsan additional term linear in the
spatial derivative of the alignment tensor has to be included in the free energy density
(15.35). The contribution to the free energy density associated with the chirality is
fachol=1
2
(ρ/m)ε 0 ξ 0 σchενλκaμν∇λaκμ, (15.39)where the coefficientσchis a pseudo-scalar. For a uniaxial alignmentaμν =
√
3 / 2 aeqnμnνwith a spatially constant order parameteraeq=
√
5 S, the expression
(15.39) reduces to
fachol=1
2
(ρ/m)15
2
S^2 ε 0 ξ 0 σchnνενλκ∇λnκ.Comparison with (15.34) shows that the coefficientsq 0 andσchcharacterizing
the chiral behavior are linked according to 2K 2 q 0 =(ρ/m)^152 S^2 ε 0 ξ 0 σch,forK 2
see (15.36).
The general structure of the free energy constructed from the spatial derivatives of
the alignment tensor up to second order and of all orders in the second rank alignment
tensor, with special emphasis on chiral terms, was studied in [89].
15.4 Cubatics and Tetradics.
Some anisotropic fluids are composed of particles or have a local structures which
cannot be described by a tensor of rank two, but where higher rank tensors, e.g.
tensors of rank three or four are needed. These substances are referred to astetradics