# Chiral Chromonics Confined in Spherical Geometries
## 3 Theoretical Modeling of Defects in Thermotropic Microspheres
For nematics that favor a bend to splay distortion, the director is organized around a ring defect located on the surface of the drop along one of the two equatorial lines (Figure 6c). In this case, the director has a cylindrical symmetry, and there is a defect line perpendicular to the director's preferred orientation (on the drop's equator). For the cholesteric phases, more complex configurations are formed. As it is well known, in a cholesteric liquid crystal, the director describes a helix in the volume and, due to the cylindrical symmetry of this phase, the whole structure is periodic after a half rotation. Two parameters control the configuration of the director within a cholesteric drop: the anchoring conditions at the interface and the twisting power \(qR\), where \(R\) is the radius of the drop and \(q\) is the inverse of half the pitch. For planar anchoring and a low twisting power (\(qR<1\)), the resulting drop configuration is a twisted bipolar (Figure 7a). For planar anchoring and a high twisting power (\(qR>1\)), the most frequently observed configuration is the Frank-Pryce structure, which consists of a central hedgehog with a defect line along the radius (Figure 7e). This line can be explained considering the liquid crystal as a set of nematic planes. The line is the result of the concentric stacking of these nematic layers inside the drop, with each layer slightly rotated with respect to the other. The defect line can be seen as a linear stacking of defect points in each cholesteric layer. By varying the droplet size with respect to the cholesteric pitch, the configuration of the drop progressively changes from a twisted bipolar to the Frank-Pryce structure (Figure 7b-d). The case for the homeotropic alignment was studied in detail by [PERSON] and co-workers [37]. The authors reported on the orientational structures observed in cholesteric droplets as a function of the helicity parameter N\({}_{0}\) = 2d/p (d is the droplet diameter). A sequence of structures were identified comparing the director configurations and topological defect-simulated textures with polarized light optical microscopy images [38; 39]. Figure 5: Director’s field schematization for a spherical droplet in planar anchoring conditions: (**a**) bipolar configuration, (**b**) twisted bipolar configuration (**c**) defect line along the diameter of the drop. Reprinted with permission from ref. [36]. Copyright 2011, Springer Nature. Figure 6: Director’s field schematization for a spherical droplet in homeotropic anchoring conditions: (**a**) radial drop, (**b**) twisted radial drop (**c**) ring defect located on the surface of the drop along one of the two equatorial lines. Reprinted with permission from ref. [36]. Copyright 2011, Springer Nature. When N\({}_{0}\) = 1.1, the radial structure was observed, the point defect was located in the center of the drop and the director distribution was not twisted. In the drops where N\({}_{0}\) > 1.1, the twisted radial structure was observed (Figure 8a). When 2.9 \(\leq\) N\({}_{0}\)\(\leq\) 5.8, the toron-like structure is observed. This structure presents the formation of an equatorial circular defect line near the drop surface (Figure 8b). A further increase in the droplet diameter leads to a layer-like L structure with a bipolar distribution of helical axes and with a double twist defect line near the surface (Figure 8d). There is also another structure called the intermediate layer-like structure L\({}_{0}\) that shows a number of twists and does not vary as a function of the drop size, having both a fixed geometry and a layered nature (Figure 8c). In some intervals, the different textures are observed simultaneously (Figure 9). Figure 8: Optical textures taken between the crossed polarizers (double arrows). The symmetry axis was oriented parallel (top row) and perpendicular (bottom row) to the viewing plane.