Tensors for Physics

286 15 Liquid Crystals and Other Anisotropic Fluids

For the derivation of this expression from (15.27), the condition (15.28) equations
(4.8), (4.10) for the product of two epsilon-tensors have been used. With help of the
relation

∇μ(nν∇νnμ−nμ∇νnν)=(∇νnμ)(∇μnν)−(∇μnμ)(∇νnν),

Equation(15.29) can be cast into the form

felast=

1

2

K 2 (∇νnμ)(∇νnμ)+

1

2

(K 1 −K 2 )(∇μnμ)(∇νnν) (15.30)

+

1

2

(K 3 −K 2 )nν(∇νnλ)nμ(∇μnλ)−

1

2

K 2 ∇μ(nν∇νnμ−nμ∇νnν).

The last term, being a total spatial derivative, contributes at the surface only, when
the free energy density is integrated over a volume. For typical low molecular weight
nematic liquid crystal, one hasK 2 <K 1 <K 3. Some qualitative features of the
nematic elasticity can be treated theoretically in the “isotropic” approximationK 1 =
K 2 =K 3 =K. Then (15.30), with the surface term disregarded, reduces to

felast=felastiso ≡

1

2

K(∇νnμ)(∇νnμ). (15.31)

The Frank elasticity coefficients have the dimension ofenergy density times length
squared. On the other hand, ordinary elastic coefficients, like the shear modulus, as
treated in Sect.16.2, have the dimension of anenergy densityor equivalently, of a
pressure.
In the presence of external electric or magnetic fields, the free energy density con-
tains additional contributions. For substances without permanent dipole moments,

but with anisotropic electric and magnetic susceptibilitiesχμνel =χaelnμnν and

χ

mag

μν =χ

mag

a nμnν,cf.(5.34), one has

ffield=−

1

2

(ε 0 χaelEμEν+μ− 01 χamagBμBν)nμnν=−

1

2

Fμνnμnν, (15.32)

whereχa=χ‖−χ⊥is the difference between the relevant susceptibilities parallel
and perpendicular to the directorn. The symmetric traceless field tensorFμνis
defined by (15.32).
The stationary director field is the solution of a spatial differential equation which
follows from a variational principle, viz. the spatial integral

F=

∫

fd^3 r=

∫

(felast+ffield)d^3 r