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

ABSTRACT

Orientation of nematic liquid crystal (NLC) around a topological defect is expressed by the equation, $\psi =s\varphi +{\varphi }_0$, where $\psi$, $s$, $\varphi$ and ${\varphi }_0$ are the director spatial phase, topological charge, azimuthal angle and a spatial phase shift, respectively. Conventionally, defect array is modelled by simply adding the $\psi$ fields of all the defects. The resultant array is a combination of the defects imposed by design (${\psi }_s$) and the defects due to the lattice geometry (${\psi }_g$). However, defect arrays generated in homeotropic confinement show hyperbolic hedgehogs (${\psi }_h$) between the ${\psi }_s$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 $s$ must be zero, 2) ${\varphi }_0$ must be a constant throughout the array, and a new model composed of ${\psi }_s$, ${\psi }_g$ and ${\psi }_h$ is established. Calculation proves that the selected modelled array has minimum free energy. The ${\psi }_s$-${\psi }_g$-${\psi }_h$ model is versatile and easy to apply, which benefits the design of new topological defect arrays.

Graphical abstract

KEYWORDS:

• Nematic liquid crystal
• topological defect
• long-range order
• homeotropic alignment
• topological defect array

Acknowledgments

I am thankful to our colleagues in the Department of Electrophysics of National Chiao-Tung University for all the kind support and assistance while this research was funded in a start-up laboratory. I wish to extend my special thanks to Prof. Tsung-Hsien Lin, Prof. Chun-Da Wang, Dr. Joe Chun-Wei Chen, Oliver Heng-Yi Tseng and Cheng-Chang Li for their heartwarming support and encouragement.

Disclosure statement

There are no competing interests or conflicts to declare.

Supplementary material

Supplemental data for this article can be accessed here.

Additional information

Funding

This research was supported and sponsored by the Ministry of Science and Technology, Taiwan, R.O.C. under Grant no. MOST 108-2112-M-009-001-MY2.