# Topological Defect Evolutions Guided by Varying the Initial Azimuthal Orientation
## 3 Results and Discussions  
When the infilled 8 CB is cooled from the isotropic state to the N phase at a rate of \(-0.3\,^{\circ}\)C/min, the LC directors follow the local alignment of SD1 on the substrate and gradually turn to a uniform vertical alignment toward the superstrate. Figure 2 reveals the generated N textures observed under polarization optical microscopy (POM). When \(\alpha_{0}=0^{\circ}\), a square lattice of Maltese crosses is exhibited (Figure 2a). When the polarizers are rotated, the Maltese crosses around \(+1\) point defects rotate in the same direction, whereas those around \(-1\) point defects rotate in the opposite direction. The above phenomena indicate that \(+1\) and \(-1\) point defects are located exactly on the predesigned \(+1\) and \(-1\) alignment singularities. Similar textures are observed in samples with \(\alpha_{0}=30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\), except for obvious rotations of Maltese crosses at the corresponding anchoring angles. The bright field images shown in the insets of Figure 2 further confirm the consistency of point defects on alignment singularities. Cyan and magenta interference colors are simultaneously observed in Maltese crosses around adjacent \(+1\) point defects. These differences are caused by the distinct phase retardations induced by the different tilt angles of the adjacent domains. To more precisely determine the director field around the point defects, a first-order retardation plate (\(\lambda=530\) nm) is inserted between the sample and the analyzers. Magenta is observed when \(n\) is parallel and vertical to the polarizers, and cyan blue and yellow colors appear when \(n\) is parallel and vertical to the slow axis of the \(\lambda\)-plate, respectively [27]. As shown in the bottom images of Figure 2a, cyan blue and magenta emerge twice alternatively in each unit, suggesting \(n\) rotates accordingly. The bottom images of Figure 2b-d show the cases with \(\alpha_{0}=30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\), respectively. The square lattices are maintained, while the interference colors change with varying \(\alpha_{0}\). The interference color variations are consistent with the presetting local alignments \(\alpha(x,y)\). Figure 1: (**a**) Schematic illustration of the hybrid cell. (**b**) Molecular structures of SD1, PDMS, and 8 CB. The square lattice of alignment singularities defined on the substrate with (**c**) \(\alpha_{0}=0^{\circ}\), (**d**) \(\alpha_{0}=30^{\circ}\), (**e**) \(\alpha_{0}=60^{\circ}\), and (**f**) \(\alpha_{0}=90^{\circ}\). Each lattice contains four singularities—two \(+1\) (red dots) and two \(-1\) (blue dots). The Landau-de Gennes theory is further utilized to reproduce the NLC director distribution and investigate the point defect evolution [28]. A traceless and symmetric tensor is adopted to describe the director field; this is expressed as \(Q_{ij}=S(n_{i}n_{j}-\frac{1}{3}\delta_{ij})\), where \(S\) is the uniaxial scalar order parameter. The total free energy \(F\) of the NLC is defined as follows:  
\[ \begin{split} F=&\int\limits_{v}f\mathrm{d}v+\int \limits_{s}f_{\text{surface}}\mathrm{d}s\\ &=\int\limits_{v}\left(f_{\text{bulk}}+f_{\text{elastic}}\right) \mathrm{d}v+\int\limits_{s}f_{\text{surface}}\mathrm{d}s\end{split}  \tag{2}\]  
The bulk energy \(f_{\text{bulk}}\) describes the degree of nematic order, which is given as follows:  
\[f_{\text{bulk}}=\frac{1}{2}AtrQ^{2}-\frac{1}{3}BtrQ^{3}+\frac{1}{4}\mathrm{C} \left(trQ^{2}\right)^{2} \tag{3}\]  
Here, A = A\({}_{0}\)(\(T-T\)*), where \(T\)* defines the supercooling temperature of the nematic phase; A\({}_{0}\), B, and C are defined as constants.