Effect of curvature on cholesteric liquid crystals in toroidal geometries

The confinement of liquid crystals inside curved geometries leads to exotic structures, with applications ranging from biosensors to optical switches and privacy windows. Here we study how curvature affects the alignment of a cholesteric liquid crystal. We model the system on the mesoscale using the Landau-de Gennes model. Our study was performed in three stages, analyzing different curved geometries from cylindrical walls and pores, to toroidal domains, in order to isolate the curvature effects. Our results show that the stresses introduced by the curvature influence the orientation of the liquid crystal molecules, and cause distortions in the natural periodicity of the cholesteric that depend on the radius of curvature, on the pitch, and on the dimensions of the system. In particular, the cholesteric layers of toroidal droplets exhibit a symmetry breaking not seen in cylindrical pores and that is driven by the additional curvature.

Figure 1

Schematic of a cholesteric in contact with a wall (dark bar on the left) of curvature radius $R$. The pitch $P$ is defined here as the distance needed for the orientational field to rotate by $2\pi$. The configuration presented corresponds to the limit of a flat surface ($R\to \infty$). The color indicates the off-plane (angular, $n_{\theta }$) component of the orientational field. White: the director points out of the plane; Black: the director is in-plane.

Figure 2

(a) Square of the off-plane (angular) component of the director field $n_{\theta }^2$ and (b) cholesteric pitch $P$ as functions of the distance to the wall $d$ with curvature radii $R/P_0=0.001,0.01,0.1,1,\infty$. $P_0$ is the natural pitch of the system. In the inset: maxima of $P$ as a function of the distance. The dashed lines correspond to $P/P_0-1\sim {(d/P_0)}^{-\alpha }$ with $\alpha =0.01,0.05$ and are represented for comparison.

Figure 3

Comparison between (a) the pitch deviations $P/P_0$ and (b) the splay-bend parameter $S_{sb}$, as functions of the distance to the wall $d$ with curvature radii $R/P_0=0.001,0.01,0.1$.

Figure 4

Comparison between the cholesteric configurations for the no curvature limit [(a) and (b)] and for a curvature radius $R=50\xi$ [(c) and (d)]. Systems (a) and (c) have boundary conditions with $\varphi =0$ and systems (b) and (d) have $\varphi =\frac{\pi }{2}$. The color represents $n_{\theta }^2$, the square of the component of the director along ${\mathbf{e}}_{\theta }$. The gray bars represent the director field.

Figure 5

Equilibrium configurations for a cholesteric liquid crystal inside an infinite cylinder with radius $r=500\xi$ and $P_0/r=2,0.5,0.2$. (a–c) Fixed boundary conditions with director parallel to the long axis of the cylinder. (d–f) Fixed boundary conditions with director tangent to the limiting circumference. The panels (g) and (h), (i) and (j) correspond to metastable states obtained, respectively, for off-plane and in-plane boundary conditions. The color represents $n_{\theta }^2$, the off-plane component of the director. The gray bars represent the director field. In the bottom row (k–o) we represent the splay-bend parameter $S_{\text{sb}}$ for the configurations in the top row. For better viewing and comparison we have restricted the splay-bend parameter to $-15<S_{\text{sb}}{P}_0^2<15$.

Figure 6

Cholesteric configurations for a system with radius of curvature $R=50\xi$ and cross-section $r=500\xi$. Panels (a)–(e) correspond to boundary conditions with $\varphi =0$, and (f)–(j) have boundary conditions with $\varphi =\frac{\pi }{2}$. The color represents $n_{\theta }^2$, the square of the component of the director along ${\mathbf{e}}_{\theta }$. The gray bars represent the director field.

Figure 7

Configurations of a cholesteric inside a torus, with multiple layers. (a) $\frac{r}{R}=0.8,\frac{P_0}{r}=0.25$, (b) $\frac{r}{R}=0.5,\frac{P_0}{r}=0.5$, (c) $\frac{r}{R}=0.25,\frac{P_0}{r}=0.5$. The color scale refers to $n_{\theta }^2$ and the gray bars represent the director field.