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, ψ=sϕ+ϕ0, where ψ, s, ϕ and ϕ0 are the director spatial phase, topological charge, azimuthal angle and a spatial phase shift, respectively. Conventionally, defect array is modelled by simply adding the ψ fields of all the defects. The resultant array is a combination of the defects imposed by design (ψs) and the defects due to the lattice geometry (ψg). However, defect arrays generated in homeotropic confinement show hyperbolic hedgehogs (ψh) between the ψss, 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 s must be zero, 2) ϕ0 must be a constant throughout the array, and a new model composed of ψs, ψg and ψh is established. Calculation proves that the selected modelled array has minimum free energy. The ψs-ψg-ψh model is versatile and easy to apply, which benefits the design of new topological defect arrays.