15.3 Elastic Behavior of Nematics 289
Clearly, fork 1 =0, all three elasticity coefficients are equal, corresponding the
isotropic case (15.31). On the other hand, to obtain the full anisotropy of the elastic
energy with three different values for the Frank coefficients, the ansatz (15.35) has
to be extended. An approach discussed in [83] is the inclusion of the fourth rank
alignment tensoraμνλκin the theoretical description. This means, instead of (15.12),
the potential
Φ=ΦLdG−√
70
6
Daμνaλκaμνλκ+1
2
E 0 aμνλκaμνλκ,is used with two additional coefficientsDandE 0. In equilibrium, this implies
aμνλκ=√
70
6
D
E 0
aμνaλκ.As a side remark, from a lowest order expansion of the entropy associated with the
single particle distribution function followsA 0 =E 0 =1,B=
√
5 /7,C= 5 /7,
see the Exercise15.1, but alsoD= 3 /7.
Furthermore, additional terms are included in the expression for the free energy
density involving the spatial derivatives of the type
(∇λaμν)(∇κaμνλκ), (∇κaμνλκ)(∇τaμνλτ), (∇τaμνλκ)(∇τaμνλκ).The mixed term with the product of the spatial derivatives of the second and fourth
rank tensors provides the desired full anisotropy. The experimentally observed tem-
perature dependence of the elasticity coefficients of ten liquid crystals can be fitted
rather well when all terms are included infainhom, for details see the 1982 article
of [83]. An alternative approach for the computation of the elasticity coefficients is
presented in [84]. There a local perfect order is assumed which can be treated by
an affine transformation model. A microscopic method for calculations of the twist
elasticity coefficientK 2 is derived and tested in [85].
The variational principle applied to a free energy density f = f(aμν,∇λaμν,
∇λaλμ)leads to the differential equation
∂f
∂aμν−∇λ∂f
∂∇λaμν−∇μ∂f
∂∇λaλν= 0. (15.37)
In the simple case corresponding to the isotropic elasticity, this equation is
Φμν−Fμν−ξ^2 Δaμν= 0. (15.38)HereΦμνis the derivative of the potentialΦ, with respect toaμν, e.g. the Landau
de Gennes expression (15.13), the tensorFμνcharacterizes the influence of an orient-
ing electric or magnetic field, and the lengthξis linked with the quantities occurring
in (15.35)viaξ^2 =kεB^0 Tξ 02 σ 2.