NSF Award
2106551
Liquid crystals are composed of rod-like or disc-like molecules that enable these materials to behave in some ways like a liquid and in others like a solid. This project is an exploration of the fundamental problem of analyzing the configurations observed in phase transitions in liquid crystals. Principally, the project will elucidate how the shape of the interface between different phases is determined by a balance of elastic forces and surface tension. The results will be of importance in mathematics, physics, and materials science since development of various morphologies in coexisting phases is a ubiquitous natural phenomenon. The acquired knowledge will bring about new and powerful tools to experimentally control these morphologies, leading to new and exciting applications. This interdisciplinary program further provides opportunities to educate students in the mathematics and physics of liquid crystals. <br/><br/>The project will focus on the characterization and description of the overall shape and internal structure of soft condensates representing nuclei of the orientationally ordered phases shaped by bulk elasticity with disparate elastic moduli and anisotropic surface tension. The proposed work will consider a broad spectrum of liquid crystals, embracing nematic and cholesteric liquid crystals that are described by non-orientable director line fields and experience transitions into partially positionally ordered phases, and newly discovered ferroelectric nematic liquid crystals that are described by the director and a vector-valued electric polarization. The mathematical emphasis is on the analysis of stationary variational problems for energies defined over vector- and tensor-valued fields in two and three dimensions and associated gradient flows, so as to describe the equilibrium configurations of liquid crystals and dynamics of interfaces in a plethora of their phase transitions. The principal novelty of the proposed work is that it will consider singular limits of liquid crystal energies that impose severe geometric rigidity on admissible classes of functions. The primary challenges to the analysis arise due to high dimensionality of a target space and the need to consider singular asymptotic limits that result in highly nonlinear constraints. The combined analytical/experimental approach will be essential in that it will provide a clear check of success for the analytical predictions; these predictions, in turn, will guide the experiments.<br/><br/>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.
