Vorticity driven dynamics in orientationally ordered systems

液晶（Liquid Crystals，LC）是一种取向有序的流体。在更广泛的意义上，LC可以被认为是具有取向有序的软物质，这是一类新兴的复杂流体。有许多实验表明液晶中有趣的流动现象。 我们专注于其中两个，涡，驱动外部来源，创造物质流。 在Janossy效应中，在液晶主体中的光激发下的染料分子充当分子马达的转子，并且产生涡旋和随后的流动。在Yokoyama-Tabe实验中，朗缪尔单分子层中的手性分子被水蒸气的横向流动所引起，旋转并产生流动。该奖项将支持旨在理解这种现象的工作，其中局部扭矩，基本上作为涡量的点源，引起宏观平移和流动。 这项工作将通过涉及应用数学和物理学的跨学科合作进行，目标是开发一个健全的数学模型，能够预测流动起重要作用的定向有序系统的动力学行为;并建立强大的计算工具，以捕捉流动和微观结构之间相互作用的基本特征。我们提出的数学模型有两个重要组成部分。首先，它是使用全取向概率密度函数，而不是它的一些时刻来表征取向顺序。 第二，在模型中包括了流动动力学。非均匀流动对瞬态微结构的形成起着重要的作用，因此，取向状态必须与宏观流场耦合。这将为研究和设计其他复杂的软物质系统奠定框架，其中流动和局部涡度起着重要作用。一个新的方面是探索同行指导的有效性，从应用数学和物理的研究生合作这个项目。 在快速发展的材料科学领域，复杂流体正在成为越来越重要的一类材料。 它们能够表现出非常广泛的行为，以响应激励，导致各种各样的应用。 这种丰富的响应性起源于复杂的内部结构使之成为可能的物理过程。 复杂流体的范围从液晶到微米和纳米胶体，病毒悬浮液和活性生物流体。许多这些系统的一个关键组成部分是成分的取向顺序。取向顺序和平移之间的耦合可能导致显着和意想不到的行为;这包括橡胶激光器和光驱动塑料马达。该奖项将支持旨在理解这些现象的工作，目标是建立计算工具来描述并帮助理解和利用负责任的机制。拟议工作的更广泛的影响是开发的工具在设计光响应软物质，超材料转换光学和生物运输战略的更大的社区的有用性。 它们包括对研究生阶段科学和数学教育的贡献。两名研究生，一个在应用数学和化学物理，参加这个项目将获得在这个多方面的工作的各个方面的专业知识，并将受益于与研究团队的更高级成员和彼此的互动。将大力鼓励妇女和代表性不足的群体成员参与这一项目。

Liquid Crystals (LCs) are orientationally ordered fluids. In a broader sense, LCs can be thought of as soft matter with orientational order, an emerging class of complex fluids. There are many experiments which show interesting flow phenomena in liquid crystals. We focus on two of these, where vorticity, driven by external sources, creates material flow. In the Janossy effect, dye molecules under photoexcitation in a liquid crystal host act as rotors of molecular motors, and generate vorticity and subsequent flow. In the Yokoyama-Tabe experiment, chiral molecules in a Langmuir monolayer are caused, by a transverse current of water vapor, to rotate and give rise to flow. This award will support work aimed at understanding such phenomena, where local torques, acting essentially as point sources of vorticity, give rise to macroscropic translation and flow. The work will be carried out through an interdisciplinary collaboration involving applied mathematics and physics, with the goal of developing a sound mathematical model which is capable of predicting the dynamical behavior of orientationally ordered systems where flow plays an essential role; and building robust computational tools that will capture the essential features of the interplay between flow and microstructure. The mathematical model we propose has two important components. First, it is the use of the full orientational probability density function, rather than a few of its moments to characterize orientational order. Second, the flow dynamics is included in the model. It is known that inhomogeneous flows play important roles in forming transient microstructures, thus the orientational state must be coupled with the macroscopic flow field. This will lay the framework for the study and design of other complex soft matter systems where flow and local vorticity play important roles. A novel aspect is the exploration of the effectiveness of peer mentoring by graduate students from applied mathematics and physics collaborating on this project. In the rapidly advancing field of materials science, complex fluids are emerging as an increasingly important class of materials. They are capable of exhibiting a remarkably broad range of behavior in response to excitations, leading to a wide variety of applications. This rich responsivity originates in physical processes which are made possible by complex internal structure. Complex fluids range today from liquid crystals to micro- and nanocolloids, virus suspensions and active biofluids. A key component of many of these systems is orientational order of the constituents. Remarkable and unexpected behavior can result from the coupling between orientational order and translation; this includes rubber lasers and light driven plastic motors. This award will support work aimed at understanding such phenomena, with the goal of building computational tools to describe and to help to understand and utilize the responsible mechanisms. The broader impacts of the proposed work are the usefulness of the developed tools to the larger community in the design of photo-responsive soft matter, metamaterials for transformation optics and strategies for biological transport. They include contributions to science and mathematics education at the graduate level. The two graduate students, one in applied mathematics and one in chemical physics, participating in this project will gain expertise in various aspects of this multifaceted work and will benefit from interactions with more senior members of the research team and each other. A vigorous effort will be made to encourage the participation of women and members of underrepresented groups in this project.