Geodesic defect anchoring on nematic shells

Nematic shells are colloidal particles coated with nematic liquid crystal molecules, which may freely glide and rotate on the colloid's surface while keeping their long axis on the local tangent plane. Molecular dynamics simulations on a nanoscopic spherical shell indicate that under appropriate adhesion conditions for the molecules on the equator, the equilibrium nematic texture exhibits at each pole a pair of $+\frac{1}{2}$ defects so close to one another to be treated as one $+1$ defect. Spirals connect the polar defects, though the continuum limit of the interaction potential would not feature any elastic anisotropy. A molecular averaging justifies an anchoring defect energy that feels the geodesics emanating from the defect. All our observations are explained by such a geodesic anchoring, which vanishes on flat manifolds.

Figure 1

Equilibrium vortex at the north pole.

Figure 2

Mercator map of $\mathit{n}$ in the $(\phi ,\theta )$ plane. For all directors $\beta \approx \frac{\pi }{2}$. Those frozen along the equator (for which $\beta =\frac{\pi }{2}$) are thicker than the others.

Figure 3

Loxodromes for $\nu =\frac{1}{2}$, ${\theta }_{\mathrm{c}}=1/50$, and ${\alpha }_{\mathrm{c}}\doteq 0.45\pi$.