Question 5.2.1: Determine the equilibrium field of directors for the ground ......
Determine the equilibrium field of directors for the ground state of a cholesteric liquid crystal (or simply cholesteric).
The difference between the cholesterics and nematics is that molecules of the former have no center of inversion. Therefore, they can be also characterized by pseudoscalar quantities, and their elastic free energy, \mathcal{F}, has an extra term of the form α(\vec{n} ·(\vec{∇} ×\vec{n})), where α is a pseudoscalar. Thus, the respective free energy takes the form
\mathcal{F} = K_{1}(\vec{∇} · \vec{n})^{2} + K_{2}(\vec{n} · (\vec{∇} ×\vec{n}))^{2}
+K_{3}(\vec{n} × (\vec{∇} × \vec{n}))^{2} + α(\vec{n} · (\vec{∇} × \vec{n})) (5.23)
The equilibrium ground state corresponds to a minimum of \mathcal{F}, and, unlike the nematics, for the cholesterics this is not a uniform field of directors. Indeed, it follows from expression (5.23) that the extremum conditions for the free energy now read
\vec{∇} · \vec{n} = 0, \vec{n} × (\vec{∇} × \vec{n}) = 0,
(\vec{n} · (\vec{∇} × \vec{n})) = −α/2K_{2} ≡ −t_{0} = const (5.24)
It means the presence of a uniform twist deformation in the ground state of the cholesteric. For example, consider the solution of equations (5.24) that depends only on one coordinate, say z. Then, the condition \vec{∇} · \vec{n} = 0 results in n_{z} = const, which, together with the \vec{n} × (\vec{∇} × \vec{n}) = 0, yields n_{z} = 0.
This means a plane field of directors, (n_{x}, n_{y}), which can be represented as: n_{x} = \cos \phi(z), n_{y} = \sin \phi(z). Since in this case (\vec{∇} × \vec{n})_{x} = −\cos \phi d\phi /dz, and (\vec{∇} ×\vec{n})_{y} = −\sin \phi d\phi /dz, it follows from the last of equations (5.24) that (\vec{n} · (\vec{∇} × \vec{n})) = −d \phi /dz = −t_{0}, that is \phi(z) = \phi_{0} + t_{0}z. Thus, the ground state of the cholesteric liquid crystal has a helicoidal structure, in which \vec{n} is uniform in each of the family of parallel planes, and it is uniformly twisted about the axis that is normal to these planes. This structure has a full pitch equal to 2π/t_{0}, but since \vec{n} and −\vec{n} are equivalent, the physically significant period is equal to π/t_{0}.
