# Topological Defect Evolutions Guided by Varying the Initial Azimuthal Orientation
## 3 Results and Discussions
For Equation (3), the uniaxial scalar order parameter of the bulk equilibrium can be calculated as \(S_{\text{eq}}=\frac{B}{4\mathrm{C}}\left(1+\sqrt{1-\frac{24 AC}{B^{2}}}\right)\), and it depends on the temperature \(T\).\(f_{\text{elastic}}\) is used to describe the contribution of anisotropic elastic free energy, including the all-spatial gradient of tensor \(Q\), which is given as follows:  
\[ \begin{split} f_{\text{elsastic}}=&\frac{1}{2}L_{1 }\frac{\partial Q_{ij}}{\partial x_{k}}\frac{\partial Q_{ij}}{\partial x_{k} }+\frac{1}{2}L_{2}\frac{\partial Q_{ij}}{\partial x_{j}}\frac{\partial Q_{ik }}{\partial x_{k}}+\frac{1}{2}L_{4}Q_{ij}\frac{\partial Q_{kl}}{\partial x_{i} }\frac{\partial Q_{kl}}{\partial x_{j}}\\ +&\frac{1}{2}L_{24}\left(\frac{\partial Q_{jk}}{ \partial x_{i}}\frac{\partial Q_{ik}}{\partial x_{j}}-\frac{\partial Q_{ij}}{ \partial x_{j}}\frac{\partial Q_{ik}}{\partial x_{k}}\right)\end{split}  \tag{4}\]  
where the coefficients \(L_{i}\) (\(i\) = 1, 2, 4, 24) represent the splay (\(k_{11}\)), twist (\(k_{22}\)), bend (\(k_{33}\)), and saddle-splay (\(k_{24}\)) elastic constants of [PERSON] theory, and the relationships of them can be defined as follows:  
\[L_{1}=\frac{3k_{22}-k_{11}+k_{33}}{6S_{eq}^{2}},L_{2}=\frac{k_{11}-k_{22}}{S_ {eq}^{2}},L_{4}=\frac{k_{33}-k_{11}}{2S_{eq}^{2}},L_{24}=\frac{k_{24}}{S_{eq}^ {2}} \tag{5}\]  
Figure 2: POM micrographs of the samples in the N phase at 34.0 \({}^{\circ}\)C with (**a**) \(\alpha_{0}=0^{\circ}\), (**b**) \(\alpha_{0}=30^{\circ}\), (**c**) \(\alpha_{0}=60^{\circ}\), and (**d**) \(\alpha_{0}=90^{\circ}\) (top row), as well as corresponding micrographs observed with a \(\lambda\) plate (bottom row). Insets show bright field images. The white arrows denote the directions of a pair of crossed polarizers. The green arrow denotes the direction of the \(\lambda\)-plate. The scale bar indicates 30 \(\upmu\)m for all images. The second integral in Equation (2) is the anchoring energy, which includes the homeotropic anchoring energy of the superstrate covered with a PDMS layer and the planar anchoring contribution of the substrate covered with an SD1 layer. The calculation of homeotropic anchoring energy is defined by the Rapini-Papoule expression: \(f_{\text{surface}}=\frac{1}{2}W_{\text{h}}tr(Q-Q_{\text{s}})^{2}\). \(W_{\text{h}}\) represents the anchoring strength and \(Q_{\text{s}}\) is the priority tensor of the surface. The planar anchoring energy is calculated using the following equation: \(f_{\text{surface}}=\frac{1}{2}W_{\text{p}}(\widetilde{Q}_{ij}-\widetilde{Q}_{ ij}^{\perp})(\widetilde{Q}_{ij}-\widetilde{Q}_{ij}^{\perp})\), where \(\widetilde{Q}_{ij}=Q_{ij}+\frac{1}{2}S_{\text{eq}}\delta_{ij}\), using the projection operator: \(\widetilde{Q}_{ij}^{\perp}=P_{ik}\widetilde{Q}_{kl}P_{lj}\), \(P_{lj}=\delta_{ij}-v_{i}v_{j}\). Here, the tensor \(P_{ij}\) will subtract the normal of the \(Q\) tensor, and \(v\) represents the normal of the substrate. The total free energy of the LCs is further minimized based on the Euler-Lagrange equation, and is solved by a finite difference method. Here, the parameters in the simulations are selected as A = \(-0.172\times 10^{6}\) J/m\({}^{3}\), B = 2.12 \(\times\)\(10^{6}\) J/m\({}^{3}\), and C = 1.73 \(\times\)\(10^{6}\) J/m\({}^{3}\); thus, the scalar order parameter is \(S_{\text{eq}}\) = 0.533 [28]. The homeotropic anchoring strength of the PDMS superstrate is \(W_{\text{h}}\) = 6 \(\times\)\(10^{-6}\) J/m\({}^{2}\) and the planar anchoring strength of the SD1 substrate is \(W_{\text{p}}\) = 1 \(\times\)\(10^{-4}\) J/m\({}^{2}\)[29]. For the elastic constants, we choose \(k_{11}\) = \(k_{33}\) = 2\(k_{22}\) = 9.5 \(\times\)\(10^{-12}\) N and \(k_{24}\) = \(k_{22}\). Finally, our simulation is conducted on a cubic lattice with periodic boundary conditions along two lateral directions. Figure 3 reveals the simulated LC directors of samples with different \(\alpha_{0}\).