Topological Defects in Spherical Nematics

We study the organization of topological defects in a system of nematogens confined to the two-dimensional sphere ($S^2$). We first perform Monte Carlo simulations of a fluid system of hard rods (spherocylinders) living in the tangent plane of $S^2$. The sphere is adiabatically compressed until we reach a jammed nematic state with maximum packing density. The nematic state exhibits four $+1/2$ disclinations arrayed on a great circle. This arises from the high elastic anisotropy of the system in which splay ($K_1$) is far softer than bending ($K_3$). We also introduce and study a lattice nematic model on $S^2$ with tunable elastic constants and map out the preferred defect locations as a function of elastic anisotropy. We find a one-parameter family of degenerate ground states in the extreme splay-dominated limit $K_3/K_1\to \infty$. Thus the global defect geometry is controllable by tuning the relative splay to bend modulus.

Figure 1

The ground-state configuration for spherical nematic ordering of 1082 rods with $L/D=15$ is shown as the central lines of rods for clarity, together with maps of the local nematic order parameter. The two views are: (a) near defects and (b) nematic bulk. The radius of the compressed sphere is $R_f/D=37.15$. The measurement of the order parameter along the great circle connecting defects is plotted as a function of angular distance in radians in (c). The exact locations of the four defects lying near one great circle are indicated in (d).

Figure 2

(a) In the limit of infinite $K_3/K_1$, the separation of a $+1/2$ disclination pair in the plane costs no energy. (b) On the 2-sphere, an infinite number of states with four $+1/2$ disclinations on a great circle can be generated from a state with two $+1$ disclinations by cut-and-rotate surgery.

Figure 3

The effect of the anisotropy $J_3/J_2$ on the configuration of defects. As the anisotropy is turned on the defects shift from a tetrahedral geometry to a coplanar great-circle alignment. The angular deviation of the defect positions from the great circle are measured as a function of the anisotropy.

Figure 4

At $J_3/J_2=1.2$, the energy, as a function of the rotation angle $\alpha$, is flat within $\pm 2\%$.