Structures in Nematic Liquid Crystals

Liquid crystals are a phase of matter with properties somewhere between liquids and crystalline solids, with applications in display technologies, optical devices, and biosensors. This project studies theoretical frameworks for liquid crystal systems which require the development of advanced mathematical techniques. The research focuses on two areas: ferroelectric nematic liquid crystals, which exhibit switchable electric polarization and show promise for improved electro-optical devices, and liquid crystals with very high symmetries that require mathematical tools like tensors for an accurate description. The mathematical methods proposed in this project should improve computational simulations and help understand structural ordering and defects in these materials. By developing new theoretical models and computational methods, this work supports advances in sensing and simulation technologies. This project also contributes to education and workforce development in science and engineering, involving the training of both graduate and undergraduate students in this research area. This proposal presents two research directions investigating novel structures in nematic liquid crystals. The first focuses on ferroelectric nematic liquid crystals, a recently observed phase that was long predicted theoretically and enables polarity switching. These materials develop singular structures similar to domain walls in ferromagnetic systems, occurring in regions where polarity transitions sharply between states. The second research area examines nematic liquid crystals with symmetries beyond the typical head-tail symmetry, including tetrahedral and cubic symmetries that can be characterized using appropriate order parameters. For these high-order symmetries, higher-order tensors provide a more natural description than the second-order tensors used in classical Landau-de Gennes theory. Both research topics have practical relevance for sensor development and computational mesh generation for scientific simulations. The work requires developing new mathematical tools from tensor analysis, calculus of variations, and partial differential equations to characterize the complex structures observed in physical experiments and numerical simulations. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.