Ordered materials exhibit fascinating behaviors when they are confined to curved spaces. They can display completely new structures and include topological defects in their ground states.

Nematic liquid crystals are typically composed by rod-like molecules that tend to align parallel to each other along a given direction called the director, n. When the nematic lies on a flat surface, its ground state is characterized by a uniform director field, as shown in Figure 1a. However, when the nematic is forced to live on the surface of a sphere, topological constraints prevent occurrence of a uniform director field. No matter how we arrange the rods, there will always be regions in space where the director is undefined. These unavoidable singularities or defects are characterized by a topological charge, s, which is a measure of how much the director rotates around the defect. According to a theorem due to Poincaré and Hopf, every possible configuration must have a total topological charge of +2. Figures 1b and 1c show two possible configurations in which the rods are aligned along the meridians and parallels of the sphere. In both cases, two defects of charge s = +1 appear at the north and south poles. Although there are many other topologically equivalent configurations, these are different from an energetic point of view. In fact, the minimum energy configuration for a system with nematic order on a sphere is not predicted to have two s = +1 defects, but rather four s= +1/2 defects arranged at the vertices of a tetrahedron, as shown in Figure 1d. Topology establishes unavoidable restrictions that must be fulfilled, but it is the energy landscape that determines the ground state of the system. It is this fascinating interplay between topology and energy what we study experimentally in our laboratory.

To make spherical shells of liquid crystal we generate double emulsions using an axi-symmetric microfluidic device, see Video 1. Basically, we encapsulate a drop of water inside a drop of nematic which is in turn dispersed in water. When the radius of the inner drop (a) is comparable to the radius of the outer one (R), the nematic gets confined to a thin shell. Both the inner and outer fluids contain a surfactant that stabilizes the double emulsion and enforces boundary conditions for the nematic. By playing with the flow rates of the three fluids, we can control the overall radius of the double emulsion and the shell thickness.

Despite nematic shells generated with microfluidic techniques can be quite thin, there is always a remaining thickness. In addition, due to the different densities of the middle and inner fluids, the inner drop typically floats inside the outer one; as a result, the experimental shells are heterogeneous in thickness. These two features have important consequences in the final defect structure of the shell. While bidimensional shells are predicted to have four s = +1/2 defects arranged in a tetrahedral fashion, a richer phenomenology is observed in our experimental shells. Figure 2 shows cross-polarized images of the four different defect structures that arise when the nematic molecules are tengentially anchored to both confining surfaces. We observe shells with 4 s = +1/2 disclination lines, shells with 2 s = +1/2 disclination lines and an s = +1 point defect, and shells with 2 s = +1 point defects, as shown in Figure 2.

We have recently investigated the consequences of changing the boundary conditions for n at the outer surface from planar to perpendicular in shells with 2 s = +1 point defects on each bounding surface. By adding the surfactant sodium dodecyl sulphate (SDS) to the continuous phase, which induces perpendicular anchoring of the nematic molecules at the outer surface, a hybrid shell is formed, and the initial defect structure evolves to a final state that possesses two s = +1 defects on the inner surface only. When the initial state is a thin bipolar shell the transition involves the disappearance of the two outermost boojums of the bipolar shell, the formation of a disclination line that shrinks and eventually vanishes, and the relocation of the two innermost boojums. This transformation is reminiscent to the one reported for the bipolar to radial drop transition; however, for shells, the evolution of the defect line can proceed via two different routes in which the two remaining boojums play a key role. For thinner shells, we typically see that the two defects on the inner surface remain below the ring at around a ring diameter away, as shown in Figure 3, while for thicker shells we typically see that one of these two boojums remains below the center of the defect ring, as shown in Figure 4. We believe that the repulsion between the boojums, which lie on the inner surface, can qualitatively account for these different pathways.

We are currently addressing other fundamental questions. If we were able to make an extremely thin shell, would the defects de-confine to the tetrahedral configuration? Could we induce transitions between the different defect structures by changing the shell thickness? And in case that we could, how would these transitions be? Continuous or discontinuous? Reversible or irreversible? Are all the transitions allowed? To answer these questions, we have developed a method based on osmotic effects to swell and de-swell the inner droplet in a controlled way. In this way we can change shell thickness to explore its influence over the resultant defect structures.

References

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[2] Nelson, D. R., Toward a tetravalent chemistry of colloids. Nano Letters 2002, 2, (10), 1125-1129.

[3] Bates, M. A., Nematic ordering and defects on the surface of a sphere: A Monte Carlo simulation study. Journal of Chemical Physics 2008, 128, (10), 4.

[4] Shin, H.; Bowick, M. J.; Xing, X. J., Topological defects in spherical nematics. Physical Review Letters 2008, 101, (3), 4.

[5] Fernandez-Nieves, A.; Vitelli, V.; Utada, A. S.; Link, D. R.; Marquez, M.; Nelson, D. R.; Weitz, D. A., Novel defect structures in nematic liquid crystal shells. Physical Review Letters 2007, 99, (15), 4.

[6] Lopez-Leon, T.; Fernandez-Nieves, A., Topological transformations in bipolar shells of nematic liquid crystals. Physical Review E 2009, 79, (2), 5.