Mathematical Study of Smectic Liquid Crystals

近晶液晶的数学研究

• 批准号: 1120637
• 负责人: Sookyung Joo
• 金额: $ 8.59万
• 依托单位: Old Dominion University Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1120637&HistoricalAwards=false
• 关键词: Mathematical; Study; Smectic; Liquid; Crystals

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.

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。典型的液晶是向列型和微晶液晶，由杆状分子组成，它们倾向于沿着一个叫做导向的公共轴纵向排列。向列液晶的局部取向由指向场描述。然而，在同晶相中，液晶的分子也倾向于分层分布。为了描述局部方向，除了director字段外，还必须引入一个微序参数。同晶液晶的自由能类似于超导的金兹堡-朗道能，是一种非线性的非凸二阶能。虽然向列相已被很好地理解，但近晶相的数学理论却相对不发达。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，推进了对近晶液晶在外部磁场驱动下不稳定性领域的数学理解。采用了相变的偏微分方程方法、变分法、伽玛收敛理论和分岔理论。在第二个项目中，研究人员研究了铁电液晶，如近晶C*和弯曲核相，它们以弯曲或香蕉形状的分子命名。弯核液晶的响应时间仅在微秒范围内，因此这种材料具有许多应用潜力。研究了电液晶分子重定向的电动力学模型。液晶用于许多不同的应用，如光开关器件和温度传感器。向列液晶材料因其在平板显示器上的应用而受到广泛的研究。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一个非常重要的问题。