# Chiral Chromonics Confined in Spherical Geometries
## 4 Confinement of Nematic Chromonics and Native Chiral Chromonics in Microspheres
The concentric spherical packing of the cholesteric layers began at the interface between the microsphere and the immiscible matrix and gradually propagated towards the center of the drop. The center appeared black, highlighting the presence of an isotropic core. The image also showed the presence of a disclination, indicated by a white arrow, and an occasional dislocation, indicated by a red arrow. This configuration can be explained by introducing another term in the [PERSON] theory of the elastic energy that it is usually neglected for thermotropic LCs: the saddle-splay term. The saddle-splay term, denoted \(K_{24}\), refers to a deformation of the director field such that n(r) is normal to a saddle surface [43; 44]. Such surfaces have a negative Gaussian curvature, i.e., two principal curvatures of opposite signs and a saddle curve upwards along the crotch but downwards through the crotch. The corresponding energy density is a pure divergence term. \[f_{24}=-\frac{K_{24}}{2}\
abla\cdot(n\
abla\cdot n+n\times\
abla\times n). \tag{2}\]  
When considering strongly confined LC samples, such as droplets, the total free energy is the sum of three contributions: the volume, anchoring and saddle-splay. The latter is the only contribution that has a negative sign. A saddle-splay deformation of the boundary interface decreases the energy if \(K_{24}>0\). When the sample boundaries are flat, the contribution of \(f_{24}\) disappears totally. In contrast, on small microspheres with strongly  
Figure 11: Polarized light microscope image of a CNC cholesteric microdroplet. The scale bar is 50 mm. The white arrow indicates the defect running perpendicular to the cholesteric layers. The red arrow indicates an edge dislocation parallel to the cholesteric layers. Reprinted from [41]. curved boundaries that impose elastic distortions within the LC, this contribution can play an important role. A very important difference between chromonic and thermotropic chiral liquid crystals is the size of the isotropic core, which is much larger in chromonics. To understand the mechanism that leads to the creation of such a large isotropic core in the CNC droplets, the balance between the elastic, condensation and surface energies must be considered. The free energy associated with the formation of an isotropic core with a radius \(r_{i}\), in a microsphere in which the radius is larger than the cholesteric helical pitch, is given by the following formula [41]. \[F=8\pi(K_{11}-K_{24})(R-r_{i})+4\pi\sigma_{chi}r_{i}^{2}+\frac{4}{3}\pi(f_{i}- f_{ch})r_{i}^{3}, \tag{3}\]  
The first term is related to the distortion of the cholesteric layers, the second is related to the energy at the cholesteric-isotropic interface and the third is the difference in the free energy between the isotropic and cholesteric phases. \(\Sigma_{chi}\) is the tension at the interface between the chiral and isotropic phases and \(f_{i}\) and \(f_{ch}\) are the free energy densities of the isotropic and chiral phases, respectively. The only term that favors the existence of an isotropic core is \(-8\pi\) (\(K_{11}\)-\(K_{24}\)) \(r_{i}\), for which the elastic energy is reduced when the curved chiral layers are replaced in the center of the drop with an isotropic spherical region. This implies that a fraction of the CNC is transferred from the core into the cholesteric shell. The fraction transferred in the volume is very small and no significant change in the measured pitch is observed in the shell. For the values typical of chromonic liquid crystals, Equation (3) predicts a radius \(r_{i}\) in the order of 1 um, which is close to the experimentally measured value. This radius is two or three orders of magnitude larger than the one recorded for a thermotropic liquid crystal, usually between 2 nm and 50 nm [43; 45].