DMREF/Collaborative Research: Materials engineering of chromonic and colloidal liquid crystals via mathematical modeling and simulation

该研究项目是在非线性现象的物理学，应用数学，非线性分析和计算领域的交叉点。 基于液晶的悬浮液的知识目前进展相当迅速，受到材料科学以及生物系统中的应用的推动。 在一个基本的水平上，与正常的悬浮液的无序性质相反，液晶基质中的无序导致胶体颗粒之间或与边界壁之间的长程弹性相互作用，从而导致各种意想不到的现象。 此外，矩阵中的顺序被悬浮的粒子扭曲，导致不可避免的拓扑缺陷，必须与粒子一起移动。 一方面，液体基质中结构的存在为流动控制、加工和悬浮液稳定性提供了新的机会。 同时，出于同样的原因，这些系统的有效工程需要我们目前对简单流体胶体的理解取得重大进展。 从新的显示技术和纳米流体设备的建议，以更基本的问题，在粒子系统中的聚类和去聚类的机制，新的实验结果需要主要的建模和分析工作。 例如，结构化介质中的电泳研究可以促进生物学中的相关努力，以建模和控制纳米流体传输，并有助于理解癌细胞的运动及其在肿瘤转移中的聚集。 该项目通过制定，分析和模拟允许存在缺陷的液晶胶体的变分模型来解决这些重要的挑战。 技术转让是拟议研究的另一个组成部分。对液晶锚定和缺陷动力学的更好理解将允许更高分辨率、更快的显示设备。该研究旨在开发一种各向异性液晶基质中悬浮液传输的预测理论，包括静电荷粒子和离子。 这些粒子可以具有任意形状，可以是刚性的或柔性的，可以是带电的或电中性的，或者是各向同性-非磁性相变的域（有色）。 分析和计算将用于探索静态和时间相关的问题。 初级变分方法将被采用，无论是在能量最小化的静态问题或时间相关的问题的最小耗散原理。 新的理论方面包括结构化介质中的胶体系统的综合模型，该模型结合了弹性矩阵，表面锚定，电场，离子和流动及其相互作用。 这些影响的相对重要性将通过该项目的建模和实验组件之间的反馈来确定。 决定双折射液晶胶体行为的一个重要特征是悬浮颗粒伴随着双折射基质中的拓扑缺陷。 作为奇异结构，从数学的角度来看，缺陷本质上是难以处理的，但是它们必须被纳入任何物理正确的模型中。 所提出的工作的主要挑战和贡献是制定，分析和模拟的液晶胶体，允许存在的缺陷的变分模型。 其中要解决的问题是所观察到的非线性电泳和粒子悬浮，在各向同性溶致变色液晶的微结构域的调查建模，和建模的实验观察到的运动的向错曲线伴随着可忽略不计或没有流量。

This research project is at the intersection of the fields of physics of nonlinear phenomena, applied mathematics, nonlinear analysis, and computation. Knowledge of liquid-crystal-based suspensions is currently advancing quite rapidly, motivated by applications in materials science as well as in biological systems. At a fundamental level, and in contrast with the disordered nature of normal suspending fluids, nematic order in a liquid crystalline matrix leads to long range elastic interactions either among colloidal particles or with bounding walls, resulting in a variety of unexpected phenomena. Furthermore, order in the matrix is distorted by the suspended particles, resulting in unavoidable topological defects that must move with the particles. On the one hand, the existence of structure in the liquid matrix affords new opportunities for flow control, processing, and suspension stability. At the same time, and for the same reasons, efficient engineering of these systems requires major advances to our current understanding of simple fluid colloids. From proposals for new display technologies and nanofluidic devices to more fundamental questions about the mechanisms of clustering and de-clustering in systems of particles, new experimental findings call for major modeling and analysis efforts. For example, studies of electrophoresis in structured media can facilitate related efforts in biology to model and control nano-fluidic transport as well as contribute towards understanding of motion of cancer cells and their clustering in tumor metastasis. This project addresses these important challenges through the formulation, analysis, and simulation of variational models of liquid crystalline colloids that allow for the presence of defects. Technology transfer is another component of the proposed research. Improved understanding of liquid crystal anchoring and defect dynamics will allow for higher resolution, faster display devices. The research aims to develop a predictive theory of transport in suspensions within an anisotropic liquid crystalline matrix, including electrostatically charged particles and ions. The particles can have arbitrary shapes, be rigid or soft, charged or electrically neutral, or be domains of the isotropic-nematic phase transition (chromonics). Analysis and computation will be used to explore both static and time dependent problems. Primarily variational methods will be employed, either within energy minimization for static problems or within the minimum dissipation principle for time dependent problems. Novel theoretical aspects include comprehensive models of colloidal systems in structured media that incorporate elasticity of the nematic matrix, surface anchoring, electric field, ions, and flow and their interplay. The relative importance of these effects will be established via the feedback between the modeling and experimental components of this project. A significant feature that determines the behavior of nematic liquid crystalline colloids is that the suspended particles are accompanied by topological defects in the nematic matrix. As singular structures, defects are inherently difficult to handle from a mathematical point of view, however they must be incorporated into any physically correct model. The principal challenge and contribution of the proposed work is formulation, analysis, and simulation of variational models of liquid crystalline colloids that allow for the presence of defects. Among the questions to be addressed are those of modeling of observed nonlinear electrophoresis and particle levitation, investigation of nematic domains in isotropic lyotropic chromonic liquid crystal, and modeling of the experimentally observed motion of disclination curves accompanied by negligible or no flow.