15.3 Elastic Behavior of Nematics 287
has to be a minimum. This impliesδF=
∫
f(n+δn)d^3 r−∫
f(n)d^3 r=0. The
changeδnmust conserve the length of the director, thusn·δn=0 has to hold true.
In the expression forfelast, an integration by parts removes the spatial derivative
acting onδn. For the simple case of the isotropic elastic energy (15.31), the resulting
differential equation is
−K∇ν∇νnμ−Fμνnν= 0.The constraint∇νnν=1 can be taken care of by a cross product of this equation
withn,
ελκμnκ(KΔnμ+Fμνnν)= 0. (15.33)
This is essentially a torque balance, cf. (5.34). Of course, the differential equation
has to be supplemented by boundary conditions. An example for the director field
is shown in Fig.7.7. The defect seen in the lower left corner of the right figure with
a ‘half integer winding number’ is typical for a tensor field. The winding number
counts the number of 360◦turns of the director when one follows its direction on
a closed path, for 360◦, around the defect. The head-tail symmetry of the director
allows half integer winding numbers, viz. the turn of the director for 180◦only.
Defects with integer winding number, the only ones allowed for a vector field, are
also possible for a director field.
15.3.2 The Cholesteric Helix
Incholesteric liquid crystals, the director field has a screw-like spatial behavior.
The cholesteric phase is essentially a spontaneously twisted nematic state with a
characteristic pitchPof the helix. For cholesterics, the twist-part of the free energy
density (15.27), involving the elasticity coefficientK 2 , contains a term linear in the
spatial derivative,
felastchol=1
2
[
K 1 (∇·n)^2 +K 2 [n·(∇×n)+q 0 ]^2 +K 3 (n×∇×n)^2]
. (15.34)
Conservation of parity requires that the quantityq 0 is a pseudo scalar. It is non-zero
only for substances containing chiral particles or which possess a helical short range
structure.
Let the spatial dependence ofnbe of the twist type
n=cosαex+sinαey,α=α(z).Since∇νnμ=α′ezν(−sinαexμ+cosαeyμ), whereα′stands fordα/dz, one has
nλελνμ∇νnμ=−α′, and (15.34) reduces to
felastchol=1
2
K 2 (α′−q 0 )^2.