288 15 Liquid Crystals and Other Anisotropic Fluids
The free energy density has a minimum whenα=q 0 z. The quantityP= 2 π/q 0
is referred to as thepitchof the helix. Due to the fact thatnand−nare physically
equivalent, the periodicity of the cholesteric helix isπ/q 0.
15.3.3 Alignment Tensor Elasticity
The alignment tensor approach used for the description of the phase transition
isotropic↔nematic, cf. Sect.15.2.2, can be generalized to spatially inhomogeneous
systems. This allows to treat the elastic behavior and to derive expressions for the
Frank elasticity coefficients which are closer to a microscopic interpretation. The
free energyFais written as an integral over a free energy densityfaassociated with
the alignment and its spatial derivatives, viz.Fa=
∫
fad^3 rwithfa=(ρ/m)kBTΦ+fainhom.Hereρ/mis the number density, of a fluid with the mass densityρ, composed of
particles with massm. As in Sect.15.2.2,Φstands for the dimensionless free energy
functionaldependingonthealignment,e.g.theLandaudeGennesexpression(15.11),
(15.12) involving the second rank tensora, which now depends on the positionr.
The additional contributionfainhomcharacterizes the ‘energy cost’ associated with
spatial derivatives of the alignment.
First, the ansatz (15.12) is used with
fainhom=(ρ/m)ε 0 ξ 02[
1
2
σ 1 (∇νaνμ)(∇λaλμ)+1
2
σ 2 (∇λaνμ)(∇λaνμ)]
, (15.35)
whereε 0 andξ 0 are a reference energy and a reference length. The energy scale can
be associated with the transition temperatureTni,viz.ε 0 =kBTni. The length scale
is of the order of a molecular length and can be linked with the average inter-particle
distance according toξ 03 =(ρ/m)−^1. The dimensionless characteristic coefficients
σ 1 andσ 2 can be expressed in terms of integrals involving the anisotropic interaction
potential and the pair correlation function [83]. Here, these quantities are treated as
phenomenological coefficients.
In equilibrium, and in the absence of external orienting fields, the order parameter
tensor is uniaxial, cf. (15.8), thus one hasaμν =
√
3
2 aeqnμnν, where the order
parameteraeq =
√
5 Seqis essentially the equilibrium value of the Maier-Saupe
order parameter. Assuming thatSeqis constant, the free energy (15.35) reduces to
the form (15.34) with the Frank elasticity coefficients given by
K 1 =K 3 =K 0
(
1
2
k 1 +k 2)
, K 2 =K 0 k 2 , (15.36)K 0 =(ρ/m)ε 0 ξ 02 =kBTniξ 0 −^1 , k 1 , 2 = 3 aeq^2 σ 1 , 2 = 15 Seq^2 σ 1 , 2.