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

研究者在液晶物理的两个重要的新领域中进行问题的数值模拟：(1)双轴和弯曲的共液晶材料相的平均场理论，(2)液晶系统中点缺陷的动力学和静力学宏观模型的数值解（在没有或存在流体流动的情况下）。研究者利用现代数值分岔分析技术，结合对称性存在下的奇点和分岔理论，研究了双轴材料的全分岔和相行为的最新平均场模型，以及他和合作者正在为重要的新型弯曲核材料开发的模型。研究者适应并扩展了具有运动和变形体的计算流体方法（“移动偏移网格方法”），以提供有效的技术来准确地模拟涉及宏观液晶定向模型中点缺陷的动力学和静力学的挑战性问题，并应用于最近令人惊讶的关于充满液晶材料的毛细血管中缺陷对湮灭的物理实验。液晶是具有部分分子取向秩序的复杂流体。它们可以用来控制光线，这一特性是它们许多重要技术应用的基础。近年来，合成了具有某些特殊相的新型液晶分子。这些新相被称为“双轴”，因为除了更常见的“单轴”材料外，它们还具有二级有序性质。液晶界对这些新材料的潜在新应用寄予厚望。双轴液晶的理论和实验研究还处于比较早期的阶段。在这个项目中，研究者分析了这些材料的数学模型，该模型表征了系统的总体平衡状态和相关的转变，作为温度等控制参数的函数。这项研究需要数学和数值技术。此外，研究者和合作者开发了“弯核”液晶的特殊亚类的新模型，这是当前的高兴趣。这项工作的优点有两个方面：(1)模型的验证和(2)预测工具的开发，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缺陷动力学的数值模拟（表征介质取向特性的场中的孤立奇点）。缺陷在液晶系统中是普遍存在的，从数学和数值的角度来看，它们都造成了相当大的困难。研究者和同事们应用了计算流体动力学其他领域的数值程序，对一个目标问题进行了数值模拟，该问题涉及一个充满液晶的毛细管中一对点缺陷的湮灭，并将这些数值结果与最近发表的实验数据进行了比较。这些技术（对于液晶以外的领域来说是新技术）应该为许多涉及缺陷的其他数值模拟问题打开了大门，这些问题以前被非物理数值伪影所阻碍。

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.