Cholesteric

A cholesteric liquid crystal, in contrast to a nematic liquid crystal as was considered in the tutorial in Chapter X, favors a twisted state. The liquid crystal elastic energy is modified to include a preferred chiral wavevector $q_{0}$, $$ \begin{equation} F=\frac{1}{2}\int_{C}K_{11}\left(\nabla\cdot\mathbf{n}\right)^{2}+K_{22}(\mathbf{n}\cdot\nabla\times\mathbf{n}-q_{0})^{2}+K_{33}\left|\mathbf{n}\times\nabla\times\mathbf{n}\right|^{2}dA.\label{eq:CholestericFreeEnergy} \end{equation} $$ The cholesteric example minimizes the above equation in a square domain $(x,y)\in[-L,L]$, with $L=1/2$, together with an anchoring energy, $$W\int(\mathbf{n}\cdot\mathbf{\hat{y}})^{2}dl,$$ imposed on the top and bottom boundaries to promote planar degenerate alignment, i.e. $\mathbf{n}$ prefers to lie any direction in the $x-z$ plane. The optimized structure with $q_{0}=\pi/2$ is displayed in Fig. (7.2).