Selection rules and a new model for stable topological defect arrays in nematic liquid crystal

Orientation of nematic liquid crystal (NLC) around a topological defect is expressed by the equation, (Formula presented.), where (Formula presented.), (Formula presented.), (Formula presented.) and (Formula presented.) are the director spatial phase, topological charge, azimuthal angle and a spatial phase shift, respectively. Conventionally, defect array is modelled by simply adding the (Formula presented.) fields of all the defects. The resultant array is a combination of the defects imposed by design ((Formula presented.)) and the defects due to the lattice geometry ((Formula presented.)). However, defect arrays generated in homeotropic confinement show hyperbolic hedgehogs ((Formula presented.)) between the (Formula presented.) s, which are out of the scope of the conventional model. In this research, two-dimensional defect ‘crystals’ with various lattice structures (square and hexagonal), various defect shapes (radial or circular) and various lattice constants (pixel size) are generated by pixelated patterned electrodes in homeotropic NLC cells. The results verify the two selection rules, 1) The total (Formula presented.) must be zero, 2) (Formula presented.) must be a constant throughout the array, and a new model composed of (Formula presented.), (Formula presented.) and (Formula presented.) is established. Calculation proves that the selected modelled array has minimum free energy. The (Formula presented.) - (Formula presented.) - (Formula presented.) model is versatile and easy to apply, which benefits the design of new topological defect arrays.