Many materials designed for high-tech applications today are engineered at the micro or nano scale. The physics of how these materials behave in applied electric fields or boundary treatments is still being worked out through experiments and simulations. An important challenge in this area is that the small-scale structure of these materials inevitably contains defects that make their responses even more complex. Physicists have proposed mathematical models to explain this behavior. It is important to investigate the nature of solutions in these models to better understand their structure under outside influences, to obtain greater predictability of their behavior, and to develop effective applications for them. That is the goal of this project. The project includes the training and participation of graduate students.
The investigator and his colleagues study the formation of defects in soft matter systems, including liquid crystal materials and layered superconductors. Physicists describe stable states of these materials using vector fields and scalar or tensor-valued order parameters that minimize a free energy. Pattern formations under external influences, as well as defects and other topological characteristics, are studied using the calculus of variations and nonlinear partial differential equations associated with the free energy. In nematic liquid crystals, the Landau-de Gennes and Maier-Saupe Q-tensor models form the basis of the investigation. Of special interest is the nature of configurations obtained in the limit of vanishing elasticity. Geometric influences, such as flat or curved, thick or thin liquid crystal materials, are considered. In layered high-temperature superconductors, the Lawrence-Doniach model is the basis for this investigation. The nature of defects and behavior of solutions in applied magnetic fields of various intensities and orientations is investigated. The above work includes the training and participation of graduate students.
