The goal of the project is the analysis of mathematical models described by nonlinear partial differential equations describing liquid crystal and superconducting materials, focusing on thin liquid crystal films, ferro-electric liquid crystal materials, polar-modulated liquid crystal materials such as bent-core fibers, and high-temperature superconducting materials. The investigators identify qualitative features of solutions to the mathematical models describing these phenomena, including pattern formation of various phases of the materials and the location, nature, and number of defects. The models are highly nonlinear and expressed in terms of nonconvex, second-order energies. Developing methods in partial differential equations to analyze these features is part of the project.
The project is related to applications in materials. For example, physicists have proposed to build lattices of colloidal particles by coating each with a thin film of liquid crystal material in such a way that the film's defects act as natural bonding sites (i.e., chemical linking locations) between the particles. This self-assembly allows the creation of functionalized micron-sized objects similar to the molecules characteristic of organic chemistry, which can be used for particle-based bioassays and catalysis, and for photonic band-gap materials. Ferro-electric liquid crystals are used to make optical switches, nano-devices, and displays. Liquid crystal bent-core fibers have been proposed by physicists to model artificial muscles and other biological applications. High-temperature superconductors are used for small-scale sensors such as squids (superconducting quantum interference detectors) and to make powerful magnets. The project takes up the mathematical investigation of models for these applications.