The investigator and her colleague study mathematically the interaction of smectic liquid crystals with external electric or magnetic fields. Typical liquid crystals are nematic and smectic liquid crystals, consisting of rod-like molecules that tend to align themselves lengthwise along a common axis, called the director. The local orientation of a nematic liquid crystal is described by the director field. However, in smectic phases the molecules of the liquid crystal also tend to lie in layers. To describe the local orientation, a smectic order parameter must be introduced in addition to the director field. The free energy of smectic liquid crystals has a resemblance to the Ginzburg-Landau energy for superconductivity, and it is a nonlinear, nonconvex second order energy. While nematic phases are rather well understood, the mathematical theory of smectic phases is comparatively underdeveloped. This study consists of two main projects. In the first project, the investigators advance the mathematical understanding of smectic liquid crystals in the area of instabilities driven by external magnetic fields, by means of mathematical analysis and numerical simulations. Partial differential equation methods for phase transition, calculus of variations, Gamma convergence theory, and bifurcation theory are employed. In the second project, the investigators study ferroelectric liquid crystals such as smectic C* and bent-core phases, which are named after the bent or banana-shaped molecules. The response time of bent-core liquid crystals is only in the range of microseconds and thus this material offers huge potential for many applications. The investigator studies an electrodynamic model of the molecular reorientation of ferroelectric liquid crystals.
Liquid crystals are used in many different applications such as optical switching devices and temperature sensors. Nematic liquid crystal materials have been widely studied because of their applications to flat panel displays. However, it is known that ferroelectric liquid crystals such as smectic C phases with chiral molecules exhibit much faster switching times. Thus understanding the features of smectic liquid crystals such as defect structures and switching dynamics is a very important issue in practical applications.
