# Topological Defect Evolutions Guided by Varying the Initial Azimuthal Orientation
## 3 Results and Discussions
The director \(n\) distributions of three typical cross sections are further selected to study the defect evolution along with the variation in \(\alpha_{0}\). They are the middle layers in the \(x\)-\(y\) cross section (top images in Figure 3), and the \(x\)-\(z\) cross section passing through \(-1\) (\(y\) = 0) and \(+1\) (\(y\) = \(l/2\)) point defects with \(y\) = 0 (middle images in Figure 3) and \(y\) = \(l/2\) (bottom images in Figure 3), respectively. Here, \(\varphi\) is defined as the azimuthal angle between the director \(n\) and the \(x\)-axis. Figure 3a shows the \(\varphi\) distributions and director fields in the three cross sections for the sample with \(\alpha_{0}\) = 0\({}^{\circ}\). The top image reveals that the director \(n\) is radially distributed within each unit due to the guidance of the preset alignment. The middle image in Figure 3a reveals the variation of directors in the \(z\) dimension. According to the simulation, a \(+1\) point defect is classified into two types: (1) \(+1\) converging type (\(+1_{\text{c}}\)), in which the director field will converge toward the \(z\) axis; (2) \(+1\) diverging type (\(+1_{\text{d}}\)), in which the director field will diverge from the \(z\) axis. Moreover, for the \(-1\) defect point, the converging and diverging director fields coexist in orthogonal planes, resulting in only one type. The cross section passing through the center of the \(+1\) point defects (bottom image in Figure 3a) proves that there is no azimuth rotation for the \(+1\) point defects, differing from the case of the \(-1\) point defects; the calculated director distribution is consistent with the interference color in the POM image (Figure 2a). The director fields and \(\varphi\) distributions of samples with \(\alpha_{0}\) = 30\({}^{\circ}\) (Figure 3b), \(\alpha_{0}\) = 60\({}^{\circ}\) (Figure 3c), and \(\alpha_{0}\) = 90\({}^{\circ}\) (Figure 3d) are also simulated, indicating that the directors rotate with increasing \(\alpha_{0}\), which is consistent with the experimental interference colors. In the \(x\)-\(z\) cross sections with \(y\) = 0 (middle images), the variation in the director field astride the \(-1\) point defect indicates a transformation from the convergent type to the divergent type. The bottom images indicate that the type of \(+1\) point defect remains unchanged, whereas slight deformation occurs due to the increase in the azimuthal component of the surrounding directors as \(\alpha_{0}\) increases. Thus, the variation of \(\alpha_{0}\) leads to an azimuthal rotation of the director fields. When the sample is further cooled to the SmA phase, the LC molecules locally arrange into parallel layers and form TFCDs. In each domain, the LC layers are wrapped around a defect pair (red) of a straight line and a circle (the inset in Figure 4a). As reported previously [24], when \(\alpha_{0}\) = 0\({}^{\circ}\), TFCDs are generated only at \(+1_{\text{c}}\) point defects due to the conforming converging director fields. Moreover, \(+1_{\text{d}}\) point defects change to domains with positive Gaussian curvatures and are alternatively embedded into the TFCDs, resulting in a square TFCD array with a lattice constant of \(\sqrt{2}l\) (top image in Figure 4a). In this case, the TFCD exhibits infinite rotational symmetry, which is further verified by the circular contour in the bright field image of each TFCD (bottom image in Figure 4a). For the case of \(\alpha_{0}\) = 30\({}^{\circ}\), TFCDs are still formed, but are rotationally distorted. The distortion is attributed to the azimuthal twist of the LC director fields guided by the anchoring (top image in Figure 4b). Due to the inheritance of orientational order across the N-SmA phase transition, distortion induces the deformation of the LC layers. As a result, TFCDs are split into four partial lobes, whose boundaries are clearly observed as dark curves in bright field images (bottom image in Figure 4b). In this case, spiral TFCDs exhibiting chirality are formed due to symmetry breaking.