Mathematical Study of Smectic Liquid Crystals

研究人员JooDMS-0908538和她的同事对近晶液晶与外加电场或磁场的相互作用进行了数学研究。典型的液晶是向列型和近晶型液晶，由棒状分子组成，这些分子倾向于沿一个共同的轴垂直排列，称为指向矢。向列型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层状存在。为了描述局域取向，除了导向场之外，还必须引入近晶有序参数。近晶液晶的自由能类似于超导的金兹堡-朗道能，它是一种非线性的、非凸的二阶能。虽然向列相被很好地理解，但近晶相的数学理论相对不发达。本研究由两个主要项目组成。在第一个项目中，研究人员通过数学分析和数值模拟的方法，推进了近晶液晶在外加磁场驱动的不稳定性领域的数学理解。相变的部分微分方程法、变分法、Gamma收敛理论和分叉理论被用到。在第二个项目中，研究人员研究铁电液晶，如近晶C*相和弯曲芯相，它们是以弯曲或香蕉形状的分子命名的。弯芯液晶的响应时间仅在微秒范围内，因此这种材料在许多方面具有潜在的应用前景。研究人员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，如光学开关设备和温度传感器。纳米晶体材料因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，如近晶C相与手性分子显示出更快的开关时间。因此，了解近晶液晶的缺陷结构和开关动力学等特性在实际应用中是非常重要的。

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.