# Chiral Chromonics Confined in Spherical Geometries
## 3 Theoretical Modeling of Defects in Thermotropic Microspheres  
Droplets are the simplest and most symmetric confining container for liquid crystals. When a liquid crystalline volume is confined to a spherical geometry, the anchoring conditions at the interface impose topological constraints on the director field that inevitably causes the topological defects to form [32]. These defects are interesting not only from a scientific point of view but, as mentioned in the introduction, they can also be useful for technological applications [33; 34]. The confinement of thermotropic liquid crystals in the microspheres is easily obtained by dispersing them in an immiscible fluid (for example, water-based polymers or glycerol). Due to their immiscibility, an interface is formed between the liquid crystal and the surrounding fluid. Since the interface formation costs energy, the liquid crystal droplets tend to minimize the surface free energy by assuming a perfectly spherical shape. Inside the microsphere, the nematic director vector field depends on three factors: the anchoring of the liquid crystal at the interface, the elastic properties of the liquid crystal and the size of the sphere. In a liquid crystal microsphere, the balance between the volume and surface energy is complex. The volume elastic energy \(F_{V}\) scales linearly with the radius of the drop \(R\), whereas the surface energy \(F_{S}\) is proportional to \(WR^{2}\) where \(W\) is the anchoring energy. From this, it is evident that as the radius \(R\) of the liquid crystalline droplet increases, the interface contribution predominates over the volume contribution and, therefore, the influence of the topological constraints imposed by the interface vanishes for the small droplet sizes. Comparing the two energy contributions, it is possible to define a critical radius as the ratio of the elastic energy to the anchoring energy \(K\)/\(W\)[32]. For the droplets of radius smaller than the critical radius, it will be energetically favorable to change the anchoring at the surface [35] and, thus, relax the elastic deformation leading to a uniform orientation of the director (Figure 4). This happens for the droplets with a radius in the order of a micron. If the droplets radius is larger than the critical value, topological defects will appear. In an article from 2011, [PERSON] and [PERSON] exhaustively collected the phenomenology of the observed textures identifying them as a function of their elastic properties and anchoring conditions [36]. Among them, for the planar anchoring with \(K_{11}<K_{33}\), a bipolar configuration is typically observed (Figure 5a). In this configuration there are two defects, called boojums, at the poles. The bipolar configuration shows splay distortions near each defect and bend distortions in the rest of the drop. Adding a twist to this configuration, a twisted bipolar configuration is obtained (Figure 5b). The resulting structure still maintains the two boojums at the poles, but the surface director field is at an angle relative to the axis that connects the two point defects. The bipolar twisted configuration forms when \(K_{11}\geq K_{22}+0.431K_{33}\). Instead, when \(K_{11}<K_{22}+0.431K_{33}\) and \(K_{11}>K_{33}\) and \(K_{11}<K_{33}\), the director organizes itself into a series of concentric circles, forming a structure that has a defect line along the diameter of the drop (Figure 5c). This structure is unstable. Figure 4: Schematization of a nematic liquid crystal drop with homeotropic surface anchoring conditions and varying size. For homeotropic anchoring, the frequently observed configuration for the nematics, which favors a splay to bend distortion, is the radial drop (Figure 6a). A pure splay is obtained with a localized defect, called a hedgehog, in the center of the drop. A variant of this structure results from the presence of twist distortions to form a twisted radial drop (Figure 6b).