Nematic textures in spherical shells

The equilibrium texture of nematic shells is studied as a function of their thickness. For ultrathin shells the ground state has four short $\frac{1}{2}$ disclination lines but, as the thickness of the film increases, a three-dimensional escaped configuration composed of two pairs of half-hedgehogs becomes energetically favorable. We derive an exact solution for the nematic ground state in the one Frank constant approximation and study the stability of the corresponding texture against thermal fluctuations.

Figure 1

(Top panel) Two-dimensional texture characterized by four short disclination lines at the vertices of a tetrahedron inscribed in the sphere. The surface texture shown (inscribed on the surface of a baseball) is invariant throughout the thickness of the shell. (Bottom panel) Cut view of the escaped three-dimensional texture given by two pairs of half hedgehogs located at the north and south poles of the sphere.

Figure 2

Schematic illustration of the phase diagram of the texture of tilted molecules on a sphere as a function of the anisotropy parameter $ϵ$ superimposed on a plot of the free energy, $F$, stored in the texture versus $ϵ$. The two competing ground states (top view) are characterized by either pure bend (lines of longitude configuration at left) or pure splay (lines of latitude configuration at right).

Figure 3

(Color online) Schematic illustration of the baseball texture of a thin nematic shell. The same texture is reproduced from a different perspective in the top panel of Fig. 1.

Figure 4

Graphical construction of the stereographic projection. Regions close to the north pole have larger images in the conformal plane than regions of equal areas close to the south pole. The stereographic projection preserves the topology of the surface provided all points at infinity are identified with the north pole.

Figure 5

Illustration of the change in the branch cut structures of the complex function $\Omega \left(v\right)$ describing the nematic texture after performing a series of conformal transformations. In the top panel we have simply performed a stereographic projection from the sphere. The middle panel shows the “fold up” transformation of Eq. (18) whereas the bottom panel corresponds to the transformation in Eq. (22) that symmetrizes the positions of the cuts.

Figure 6

(Color online) Different views of a track of parallel nematogens which partitions the sphere into two equal areas, each containing two $s=\frac{1}{2}$ disclination defects. It resembles a “fattened” version of the seam of a baseball.

Figure 7

(Color online) Illustration of the nematic texture in stereographic projection. The red and black field lines correspond to two energetically degenerate (in the one Frank constant approximation) families of bend and splay rich textures. The dots indicate the four tetrahedral $s=\frac{1}{2}$ disclination defects.

Figure 8

(Color online) Side view of the 2D nematic texture ansatz solution in Eq. (55). The cylindrical slab has height $h$ and radius $R$. The arrows can be interpreted as flow lines of a fluid entering a narrow and long channel from a point source located on the top plate. To obtain the nematic texture the vectors need to be normalized and viewed as rods with both ends identified as shown in Fig. 9.

Figure 9

Nematic texture in a thin spherical shell. The nematic director near the pair of half-hedgehogs indicated in the figure is well described by the one calculated for the slab in Fig. 8.

Figure 10

(Color online) Splay rich (hyperbolic) and bend-rich (ellipsoidal) families of nematic “flow” lines generated by two $s=\frac{1}{2}$ disclinations. The two families of flow lines are the equipotential and field lines of a complex function $\Omega \left(z\right)$ whose branch cut is a horizontal line connecting the two defects.

Figure 11

(Color online) (a) The ground state of a tetratic phase exhibits eight short disclination lines located at the vertices of a square antiprism inscribed in the sphere. (b) The 12 disclinations that characterize hexatic order on a sphere lie at the vertices of an icosahedron.

Figure 12

(Color online) Schematic illustration of the inversion through an equatorial plane of symmetry (shaded circle bisecting a sphere) for a reflection $\sigma$. The longitudinal displacement of the defect is unchanged while the direction of the latitudinal one is inverted.