15.3 Elastic Behavior of Nematics 285
15.3.1 Director Elasticity, Frank Coefficients
Standard nematic liquid crystals, in thermal equilibrium, are uniaxial, cf. (15.8),
(15.9), and their order parameterS=S 2 is constant. The local directorn, however,
depends on the spatial positionr. The spatial variation, in general, is influenced
by boundary conditions and by orienting magnetic or electric fields. The functional
dependencen=n(r)of the director field is governed by an equation, which follows
from a variational principle for the relevant free energy densityfelast. The pertaining
free energyFelast=
∫
felastd^3 ris the spatial integral of the energy density. The
“elasticity” of the director field discussed here is of a different character as compared
with the elastic behavior of solids under deformations, cf. Sect.16.2. The standard
ansatz for the free energy density associated with the ‘elasticity’ of the director field is
felast=1
2
[
K 1 (∇·n)^2 +K 2 (n·(∇×n)^2 +K 3 (n×∇×n)^2]
, (15.27)
with theFrank elasticity coefficients K 1 ,K 2 ,K 3. The distortions of the director field
described by the divergence∇·n, by a rotation parallel to the directorn·(∇×n),
and by a rotation perpendicular ton,viz.n×∇×n), are referred to assplay,twist,
andbend deformations, as indicated in the sketch Fig.15.6.
The undistorted, spatially homogeneous state has the lower free energy, provided
thatKi>0,i= 1 , 2 ,3 holds true. Expressions like (15.27) were first introduced
by Oseen and Zocher, later refined by Frank [82]. In the literature, the coefficientKi
are calledFrank-Oseen elasticity, or mostly,Frank-elasticitycoefficients.
The directorn=n(r)is a unit vector, thusn·n=1, and consequently one has
nν∇λnν= 0. (15.28)In Cartesian component notation, the free energy density (15.27) reads
felast=1
2
K 1 (∇μnμ)(∇νnν) (15.29)+1
2
K 2
[
(∇νnμ)(∇νnμ)−(∇νnμ)(∇μnν)−nν(∇νnλ)nμ(∇μnλ)]
+
1
2
K 3 nν(∇νnλ)nμ(∇μnλ).Fig. 15.6The splay, twist
and bend deformation of a
director field