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

这项研究项目涉及非线性现象物理、应用数学、非线性分析和计算等多个领域。在材料科学和生物系统应用的推动下，基于液晶的悬浮液的知识目前发展得相当快。在基本水平上，与正常悬浮流体的无序性质不同，液晶基质中的向列相有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，导致各种意想不到的现象。此外，悬浮粒子扭曲了基质中的有序，导致不可避免的拓扑缺陷，这些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在我们目前对简单流体胶体的理解上取得重大进展。从新的显示技术和纳米流体设备的提议，到关于粒子系统中聚集和解聚机制的更基本的问题，新的实验发现需要进行重大的建模和分析工作。例如，对结构化介质中的电泳的研究可以促进生物学中的相关工作，以模拟和控制纳米流体的传输，并有助于了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、分析和模拟允许缺陷存在的液晶胶体的变化模型来解决这些重要挑战。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。这项研究旨在发展一种预测悬浮液在各向异性液晶基质中传输的理论，包括带静电的粒子和离子。粒子可以具有任意形状，刚性或软性，带电或电中性，或者是各向同性-向列型相变(色调)的区域。分析和计算将用于探索静态和时间相关问题。主要将采用变分方法，对于静态问题，在能量最小化范围内；对于时间相关问题，在最小耗散原理范围内。新颖的理论方面包括结构化介质中胶体系统的综合模型，其中包括向列相基质的弹性、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本项目的建模组件和实验组件之间的反馈来确定。决定向列型液晶胶体行为的一个重要特征是悬浮粒子伴随着向列型基质中的拓扑缺陷。作为奇异结构，缺陷本质上很难从数学的角度来处理，然而它们必须被合并到任何物理上正确的模型中。这项拟议工作的主要挑战和贡献是建立、分析和模拟允许缺陷存在的液晶胶体的变化模型。其中需要解决的问题包括观察到的非线性电泳法和粒子悬浮法的模拟，各向同性溶致变色液晶向列相的研究，以及实验观察到的倾斜曲线运动的模拟以及可以忽略或没有流动。

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.