The principal investigator considers analytical questions related to nonlinear problems describing liquid crystalline phase transitions, with an emphasis on models based on complex order parameters. The main focus is the study of magnetic- and electric-field-induced phase transitions in smectic materials in general, and chiral tilted smectic systems in particular. The approach is to study smectic liquid crystals as described by generalized phenomenological Landau-de Gennes energies, which take into account the complete electromechanical coupling, finite surface anchoring and added interaction coupling terms. The investigator studies effects observed in experiments: reentrant phases under electric fields, shrinkage of smectic layers that lead to folding instabilities at smectic C* transitions, undulation induced by applied fields, field-induced phase transformations, and the emergent display of macroscopic chiral properties in liquid crystals that are composed of achiral molecules. These effects arise in considering both the behavior of liquid crystals and issues associated with manufacturing smectic liquid crystal devices. The approach uses a diverse collection of mathematical tools coming from nonlinear partial differential equations, including variational methods, asymptotic analysis and partial differential equation regularity theory. Part of the effort is to validate and refine the models that contain the essential features of the examined phenomena.
Smectic liquid crystals are less studied than nematic liquid crystals. As in nematic liquid crystals, the long molecules in smectic liquid crystals point in approximately the same direction, called the director; unlike nematic liquid crystals, the long molecules of smectic liquid crystals also are arranged approximately in planar layers. The tilt between the director and the normal to the layer, and the rotation of this tilt as one moves between layers, are involved in the switching properties of smectic liquid crystals. Liquid crystals are well-known for their use in liquid crystal display technology, where devices at the moment are mostly based on nematic mesophases and operate very close to physical limits. The future of this important application resides in tilted chiral smectic mesophases, which due to their ferroelectric or antiferroelectric properties have superior operational speed and resolution. Commercial applications of ferroelectric liquid crystals include spatial light modulators, optical memory, optical computers and high-resolution microdisplays. Due to the importance of current and future applications of smectic materials, there is a need for validation and extension of the available theoretical models through mathematical analysis. By theoretically studying the properties of the solutions of the proposed models, and comparing them to experimental results, the project adds to the understanding of the nature of the liquid crystalline phase transitions, and the electrical and optical properties of soft condensed matter. This in turn leads to better ways of building liquid crystal based devices.