Mathematical Study of Smectic Liquid Crystals

JooDMS-0908538 这位研究者和她的同事用数学方法研究了近晶液晶与外部电场或磁场的相互作用。 典型的液晶是液晶和近晶液晶，由棒状分子组成，这些分子倾向于沿着一个共同的轴纵向排列，称为指向矢。 液晶的局域取向由指向矢场描述。 然而，在近晶相中，液晶的分子也倾向于位于层中。 为了描述局域取向，除了指向矢场外，还必须引入近晶序参量。 近晶相液晶的自由能类似于超导的Ginzburg-Landau能，是一个非线性、非凸的二阶能。 虽然近晶相已被相当好地理解，但近晶相的数学理论还相对不发达。 本研究包括两个主要项目。 在第一个项目中，研究人员通过数学分析和数值模拟，推进了近晶液晶在外部磁场驱动的不稳定性领域的数学理解。 相变的偏微分方程方法，变分法，伽玛收敛理论，分歧理论。 在第二个项目中，研究人员研究铁电液晶，如近晶C* 和β-核相，它们以弯曲或香蕉形状的分子命名。 双核液晶的响应时间仅在微秒范围内，因此这种材料具有很大的应用潜力。 研究了铁电液晶分子重取向的电动力学模型。 液晶被用于许多不同的应用中，例如光学开关器件和温度传感器。 向列型液晶材料由于其在平板显示器中的应用而得到了广泛的研究。 然而，众所周知，铁电液晶如具有手性分子的近晶C相表现出快得多的开关时间。 因此，了解近晶相液晶的缺陷结构和开关动力学等特性在实际应用中具有十分重要的意义.

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.