Modeling, Computation, and Analysis of Complex Liquid Crystal Systems and Transitions

Gartland DMS-0608670研究人员对液晶物理的两个重要新领域进行了数值模拟：(1)双轴和弯曲晶体材料相的平均场理论；(2)液晶体系(无流体或有流体时)点缺陷动力学和静力学的宏观模型的数值解。这位研究人员利用现代数值分叉分析技术，结合奇点和对称性分叉理论，研究了最近的双轴材料平均场模型的完全分叉和相行为，以及他和他的合作者正在为重要的新一类弯芯材料开发的模型。研究人员对具有运动和变形物体的计算流体的方法进行了改编和扩展，以提供有效的技术来准确地模拟具有挑战性的问题，包括宏观液晶导向器模型中点缺陷的动力学和静力学，并应用于最近关于填充液晶材料的毛细管中缺陷对湮灭的令人惊讶的物理实验。液晶是具有部分分子取向有序的复杂流体。它们可以用来控制灯光，这一特点构成了它们许多重要的技术应用的基础。近年来，人们合成了具有某些特殊相的新型液晶分子。这些新的相被称为“双轴”，因为它们除了具有摩尔酮“单轴”材料的特性外，还具有二次有序特性。液晶界对这些新材料的潜在新应用寄予厚望。相对早期的双轴液晶理论和实验研究。在这个项目中，研究人员分析了这类材料的数学模型，该模型描述了系统的整体平衡状态和相关的转变，作为控制参数(如温度)的函数。这项研究需要数学和数值两种技术。此外，研究人员和合作者还为当前非常感兴趣的“弯曲核”液晶这一特殊子类开发了新的模型。这项工作的优点有两个：(1)验证模型和(2)开发预测工具，以帮助设计实验和确定技术应用的参数范围。该项目的第二个主要主题涉及液晶中缺陷动力学的数值模拟(表征介质取向特性的场中孤立奇点)。缺陷在液晶系统中无处不在，从数学和数值的角度来看，缺陷都造成了相当大的困难。研究人员和同事应用了计算流体力学其他领域的数值程序，都对一个目标问题进行了数值模拟，该目标问题涉及到填充液体晶体的毛细管中一对点缺陷的湮灭，并将这些数值结果与最近公布的实验数据进行了比较。这些技术(液晶领域的新技术)应该会为其他一些数值建模问题打开大门，这些问题涉及以前受到非物理数值假象阻碍的缺陷。

GartlandDMS-0608670 The investigator carries out numerical modeling of problemsin two important new areas of the physics of liquid crystals: (1)mean-field theories for phases of biaxial and bent-coreliquid-crystal materials, and (2) numerical solution ofmacroscopic models for the dynamics and statics of point defectsin liquid-crystal systems (in the absence or presence of fluidflow). The investigator uses techniques from modern numericalbifurcation analysis together with the theories of singularitiesand bifurcations in the presence of symmetry to study the fullbifurcation and phase behavior of recent mean-field models forbiaxial materials, along with models that he and collaboratorsare developing for the important new class of bent-corematerials. The investigator adapts and extends methods fromcomputational fluids with moving and deforming bodies ("movingoverset grid methods") to provide effective techniques toaccurately model challenging problems involving the dynamics andstatics of point defects in macroscopic liquid-crystal directormodels, with application to surprising recent physicalexperiments on defect-pair annihilation in capillaries filledwith a liquid-crystal material. Liquid crystals are complex fluids that possess partialmolecular orientational order. They can be used to controllight, and this feature underlies many of their significanttechnological applications. In recent years, new liquid-crystalmolecules have been synthesized that exhibit certain specialphases. These new phases are termed "biaxial," because they havesecondary ordering properties in addition to those of the morecommon "uniaxial" materials. The liquid-crystal community hashigh hopes for potential new applications of these new materials. The theoretical and experimental study of biaxial liquid crystalsis at a relatively early stage. In this project the investigatoranalyzes a mathematical model for such materials thatcharacterizes the bulk equilibrium state of the system andassociated transitions, as a function of control parameters suchas temperature. This study requires both mathematical andnumerical techniques. In addition, the investigator andcollaborators develop new models for the special sub-class of"bent core" liquid crystals, which are of high current interest. The merits of this work are two-fold: (1) validation of themodels and (2) the development of predictive tools to aid in thedesign of experiments and in the identification of parameterranges for technological applications. The second main topic ofthe project concerns the numerical modeling of the dynamics ofdefects in liquid crystals (isolated singularities in the fieldsthat characterize the orientational properties of the medium). Defects are ubiquitous in liquid-crystal systems, and they causeconsiderable difficulty both from mathematical and numericalpoints of view. The investigator and co-workers apply numericalprocedures from other areas of computational fluid dynamics, bothto model numerically a target problem involving the annihilationof a pair of point defects in a capillary tube filled with liquidcrystal and to compare these numerical results to recentlypublished experimental data. These techniques (new to the areaof liquid crystals) should open the door to a number of othernumerical-modeling problems involving defects that havepreviously been hampered by unphysical numerical artifacts.