Twist transition of nematic hyperbolic hedgehogs

Stability of an idealized hyperbolic hedgehog in a nematic liquid crystal against a twist transition is investigated by extending the methodology of Rüdinger and Stark [Liq. Cryst. 26, 753 (1999)], where the hedgehog is confined between two concentric spheres. In the ideal hyperbolic-hedgehog the molecular orientation is assumed to rotate proportionally with respect to the inclination angle, $\theta$ (and in the opposite sense). However, when splay, $k_{11}$, and bend, $k_{33}$, moduli differ this proportionality is lost and the liquid crystal deforms relative to the ideal with bend and splay. Although slight, these deformations are shown to significantly shift the transition if $k_{11}/k_{33}$ is small. By increasing the degree of confinement the twist transition can be inhibited, a characteristic both hyperbolic and radial hedgehogs have in common. The twist transition of a hyperbolic defect that accompanies a particle is found to be well predicted by the earlier stability analysis of a thick shell.

Figure 1

Geometry of an idealized hyperbolic-hedgehog confined within a thick shell extending from $r_{\min }$ to $r_{\max }$, where the gray bars represent the fixed surface director orientation. After the twist transition, the director rotates in the direction of ${\widehat{e}}_{\varphi }$.

Figure 2

Angular dependence of $\overline{f}(\alpha ,x)$, $h\left(x\right)$, (minimum eigenvalue solution) when ${\alpha }_{\max }-{\alpha }_{\min }\to \infty$ for 5CB, with $k_{11}/k_{33}=0.79$ and $k_{22}/k_{33}=0.43$. Solid line is the full numerical solution [normalized to conform to the form of Eq. (14)] and the dotted line is the approximation given by first-order perturbation analysis.

Figure 3

White region indicates where the untwisted hyperbolic configuration is stable as (i) ${\alpha }_{\max }-{\alpha }_{\min }\to \infty$, calculated numerically. Dark-gray region indicates the stability region of the twisted hyperbolic configuration with (ii) ${\alpha }_{\max }-{\alpha }_{\min }=ln\left(50\right)$. Solid and dotted lines correspond to the analytic approximation of Eq. (16) for (i) and (ii). ($+$) Threshold value of $k_{22}/k_{33}$ below which twist occurs with $r_{\max }/r_{\min }=50$ given by the nonlinear analysis of Sec. 3.

Figure 4

(a) Twist, $\chi (\alpha ,\theta )$, and (b) tilt relative to the ideal hedgehog, $\psi (\alpha ,\theta )-\theta$, of a hyperbolic hedgehog given by the nonlinear analysis assuming the elastic constants of 5CB and ${\alpha }_{\max }-{\alpha }_{\min }=ln\left(50\right)$. Minimum is shaded white and the maximum black. Peak twist is $28.3^{\circ }$ and deviation in tilt is at most $\pm 2.4^{\circ }$.

Figure 5

Twist transition threshold given by the nonlinear analysis can be fitted by $k_{22}/k_{33}<m{k}_{11}/k_{33}+c$ for a given value of ${\lambda }_{\alpha }$, where $m$ and $c$ are evaluated at $k_{11}/k_{33}=1$. ($+$) Numerical values of (a) $1/m$ and (b) $c{\lambda }_{\alpha }$ as functions of ${\lambda }_{\alpha }$, well approximated by the linear fits ($-$) given by (a) $1/m=1.159{\lambda }_{\alpha }+0.994$, (b) $c{\lambda }_{\alpha }=0.141{\lambda }_{\alpha }-0.027$.

Figure 6

Twist, $\chi (\zeta ,\mu )$, about the dipole state of a spherical particle assuming the elastic constants of 5CB and $r_{\max }/r_{\min }=50$. Peak twist is $30.7^{\circ }$.

Figure 7

Threshold value of $k_{22}/k_{33}$ beneath which twist occurs for ($-$) a companion defect about a particle and (- -) in a hyperbolic shell, using the nonlinear analysis of Sec. 3. In (a) $k_{11}/k_{33}$ is varied with $r_{\max }/r_{\min }=50$, whereas in (b) $r_{\max }/r_{\min }$ is varied with $k_{11}/k_{33}=1$.