Spherical nematic shells with a threefold valence

We present a theoretical study of the energetics of thin nematic shells with two charge-one-half defects and one charge-one defect. We determine the optimal arrangement: the defects are located on a great circle at the vertices of an isosceles triangle with angles of ${66}^{\circ }$ at the charge-one-half defects and a distinct angle of ${48}^{\circ }$, consistent with experimental findings. We also analyze thermal fluctuations around this ground state and estimate the energy as a function of thickness. We find that the energy of the three-defect shell is close to the energy of other known configurations having two charge-one and four charge-one-half defects. This finding, together with the large energy barriers separating one configuration from the others, explains their observation in experiments as well as their long-time stability.

Figure 1

(a) In a two-dimensional nematic, a $s=1$ topological defect (black dot in left panel) can lower its elastic energy by splitting into two $s=1/2$ defects (purple dots in right panel). (b) A singular line (left panel) spanning the shell with a winding number of one at the boundaries is topologically and energetically unstable. The singular core is indicated by a black dot in the top view shown in the top panel and by the vertical bold line in a cut shown in the bottom panel. One way of reducing the elastic energy escaping in the third (vertical) dimension (right panel), thereby leaving a point defect (green dot), called boojum, on each boundary.

Figure 2

Four views on the bend texture of the director field on the sphere containing a $+1$ defect and two $+1/2$ defects arranged in an isosceles triangle with ${\beta }_{12}={\beta }_{13}\approx {132}^{\circ },{\beta }_{23}\approx 96.4^{\circ },{\alpha }_1\approx {48}^{\circ }$, and ${\alpha }_2={\alpha }_3\approx {66}^{\circ }$. The defects lie on a great circle.

Figure 3

Four views on the splay texture of the director field on the sphere containing a $+1$ defect and two $+1/2$ defects arranged in an isosceles triangle with ${\beta }_{12}={\beta }_{13}\approx {132}^{\circ },{\beta }_{23}\approx 96.4^{\circ },{\alpha }_1\approx {48}^{\circ }$, and ${\alpha }_2={\alpha }_3\approx {66}^{\circ }$. The defects lie on a great circle:

Figure 4

(a–c) Cross-polarized images of a shell at different stages of a swelling process, which makes the shell more homogeneous in thickness. Each bounding surface of the shell has three topological defects: one defect of charge $+1$, characterized by four black brushes, and two defects of charge $+1/2$, characterized each one by two black brushes. The three defects depict a triangle whose angles, ${\alpha }_1,{\alpha }_2$, and ${\alpha }_3$, change as the shell swells, see evolution from left to right. The exact values of ${\alpha }_1,{\alpha }_2$, and ${\alpha }_3$ as a function of the shell average thickness $h/R$ are shown in (d). The triangle depicted by the defects is initially oriented perpendicularly to the gravitational direction, but it tilts off as the shell swells, as shown in (e), where ${\theta }_z$ stands for the angle between gravity $\overline{g}$ and the flat triangle normal $\overline{N}$.

Figure 5

Defect configuration of a very thin and almost homogeneous trivalent nematic shell. (a–c) Cross polarised images of the shell at different focal planes: the $+1$ defect is in an upper plane shown in (a), while the $+1/2$ defects are at lower planes; see arrows in (b) and (c). (d, e) Histograms of the angular distances between defects. The three defects depict an isosceles triangle where the two equal sides correspond to the distance between the +1 defect and each of the +1/2 defects, ${\beta }_{12}={(118\pm 19)}^{\circ }$, and the unequal side corresponds to the distance between the two $+1/2$ defects, ${\beta }_{23}={(77\pm 15)}^{\circ }$.

Figure 6

Elastic energy as a function of shell thickness for divalent (red, dashed), trivalent (green, solid), and tetravalent (blue, dotted) defect configuration for $R/a={10}^5$ and $E_c=0$. Either the divalent or tetravalent configuration, but not the trivalent configuration, is lowest in energy.

Figure 7

Coexistence of divalent, trivalent, and tetravalent configurations in shells with (a) $h/R$ in the range $[0.20,0.35]$ and (b) $h/R\simeq 0.1$. The histogram shows the different populations in a sample of 60 shells.

Figure 8

Schematics of the three nontrivial eigenmodes corresponding to (a) ${\mathbf{u}}_4$, (b) ${\mathbf{u}}_5$, and (c) ${\mathbf{u}}_6$. The defects (represented by dots) continue to lie on a great circle in (a) and (c), but not in (b). The defects continue to form an isosceles triangle in (a) and (b), but not in (c).