Conically degenerate anchoring effect in planar nematic-liquid-crystal shells

We study the defect texture in symmetric and asymmetric states of a nematic-liquid-crystal shell with conic and planar degenerate surface anchorings on the inner and outer spherical boundaries, respectively. To achieve the equilibrium nematic orientation, we numerically minimize the Landau–de Gennes free energy by employing surface potentials on the shell walls. The symmetric nematic shells energetically have stable configurations independent of thickness. In thick shells, the director field satisfies bipolar and hexadecapolar configurations between boundaries. In thin shells, the boojums transform into two stable disclination curves.

Figure 1

(a) Schematic defect structure inside a nematic spherical shell. $R=0.5\mu \text{m}$ and $a$ are the radii of outer and inner spheres, respectively, $\vec{r}$ shows the center-to-center distance vector of the two spheres, and $\theta$ expresses the polar angle of $\vec{r}$ with respect to the $z$ axis. Axis $z$ is determined by the normal vector of the surface containing the saturn ring defect around the inner sphere shown by red color. Boojum defects on both surfaces are also shown by red dots. (b) A schematic of the conically degenerate anchoring. Equilibrium direction ${\widehat{n}}^e$ freely can rotate on a cone with the vertex angle ${\psi }_{{}_{\mathrm{e}}}={45}^{\circ }$. Surface normal vector $\widehat{\nu }(={\widehat{e}}^{\left(1\right)}\times {\widehat{e}}^{\left(2\right)})$ defines a right-handed coordinate system in which the ${\widehat{e}}^{\left(2\right)}$ axis lies in the plane surface of $\widehat{\nu }$ and $\widehat{n}$.

Figure 2

General director field and defect structure of the shell with heterogeneous thickness.

Figure 3

Total free energy ${\mathcal{F}}_a\left(\vec{r}\right)$ dependence on off-center displacement vector $\vec{r}$ of the inner sphere for different amounts of radius $a$.

Figure 4

Free energy dependence on the off-center displacement at various amounts of $a$ with (a) $\theta =0^{\circ }$, (b) $\theta ={30}^{\circ }$, (c) $\theta ={60}^{\circ }$, and (d) $\theta ={90}^{\circ }$. Free energy difference is defined as $\mathrm{\Delta }\mathcal{F}={\mathcal{F}}_a\left(\vec{r}\right)-{\mathcal{F}}_a\left(0\right)$.

Figure 5

Maximum free energy values at maximum amounts of $r$ varying with $\theta$. Inner sphere is nearly touching the wall of the outer sphere.

Figure 6

Possible defect structures of NLC shells with conically degenerate and degenerate planar anchorings on inner and outer surfaces, respectively: (a)–(c) boojums bipolar structure on a symmetric thick shell with $a=0.7R$, (d)–(f) triangular defect texture on a highly asymmetric shell with $a=0.7R$, $r=0.2R$, and $\theta =0$, and (g)–(i) tetrahedral defect structure in a very thin symmetric shell where $a=0.9R$. The first row shows defect structures and planar director field on the outer surface. The second and third rows show a cross section of the director field inside the bulk and on the shell surfaces, respectively.