Mathematical Study of Smectic Liquid Crystals

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JooDMS-0908538 The investigator and her colleague study mathematically theinteraction of smectic liquid crystals with external electric ormagnetic fields. Typical liquid crystals are nematic and smecticliquid crystals, consisting of rod-like molecules that tend toalign themselves lengthwise along a common axis, called thedirector. The local orientation of a nematic liquid crystal isdescribed by the director field. However, in smectic phases themolecules of the liquid crystal also tend to lie in layers. Todescribe the local orientation, a smectic order parameter must beintroduced in addition to the director field. The free energy ofsmectic liquid crystals has a resemblance to the Ginzburg-Landauenergy for superconductivity, and it is a nonlinear, nonconvexsecond order energy. While nematic phases are rather wellunderstood, the mathematical theory of smectic phases iscomparatively underdeveloped. This study consists of two mainprojects. In the first project, the investigators advance themathematical understanding of smectic liquid crystals in the areaof instabilities driven by external magnetic fields, by means ofmathematical analysis and numerical simulations. Partialdifferential equation methods for phase transition, calculus ofvariations, Gamma convergence theory, and bifurcation theory areemployed. In the second project, the investigators studyferroelectric liquid crystals such as smectic C* and bent-corephases, which are named after the bent or banana-shapedmolecules. The response time of bent-core liquid crystals isonly in the range of microseconds and thus this material offershuge potential for many applications. The investigator studiesan electrodynamic model of the molecular reorientation offerroelectric liquid crystals. Liquid crystals are used in many different applications suchas optical switching devices and temperature sensors. Nematicliquid crystal materials have been widely studied because oftheir applications to flat panel displays. However, it is knownthat ferroelectric liquid crystals such as smectic C phases withchiral molecules exhibit much faster switching times. Thusunderstanding the features of smectic liquid crystals such asdefect structures and switching dynamics is a very importantissue in practical applications.