Nonlinear Dynamics of Colloidal Rotors: Chaos and Order

被称为胶体的微小、微米大小的颗粒在工程和生物医学应用中随处可见，例如微流体和药物输送，以及油漆等消费品。近年来，利用电场和磁场将胶体组装成具有设计性结构的新型功能材料的工作取得了很大进展。特别令人感兴趣的是旋转的胶体(转子)，因为在这样的系统中，旋转的颗粒搅拌的流动增加了静电相互作用，从而极大地扩展了可能的结构。本项目将研究放置在两个平面电极之间的胶体的动力学理论，以了解场驱动粒子组装的机制。该项目整合了应用数学、流体力学和软物质等领域的知识，这将对与该项目相关的学生的培训非常有益。视觉上吸引人的实验将有助于教育公众有关数学和流体动力学的知识。这一建议涉及由Quincke效应驱动的旋转胶体动力学的理论研究。Quincke效应是介质球在外加均匀电场中的自发自旋。自由空间中的Quincke转子动力学由Lorenz方程描述，该系统作为Lorenz混沌的物理实现之一引起了人们的关注：在高电场下，球体绕固定的轴不规则地自转。然而，昆克转子在受限空间中的动力学与在自由空间中的动力学有很大的不同。单个转子显示出周期性的旋转轴重新定向的三维运动。人们发现，盘旋的Quincke转子集体自组织成稳定的星团或蛇形链条。该项目将研究放置在两个平面电极之间的粒子的电流体动力学。新的数学挑战很多，包括边界附近单粒子动力学的解析解，在约束条件下仔细考虑成对的流体动力和静电相互作用，以及将这些分析结果整合到一个针对大量粒子的基于斯托克斯动力学的算法中。建议进行基准实验来补充数学研究，以证明和验证数学模型，并确保研究在应用数学界内外都有影响。教育影响包括为西北大学的应用数学本科生和研究生带来流体力学和软物质研究的直接实验经验。Quincke转子的紧急行为可能会在不同领域开辟新的研究方向，例如非平衡软物质、材料工程和流体动力学。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Tiny, micron-sized particles called colloids find ubiquitous uses in engineering and biomedical applications, for example, microfluidics and drug delivery, and consumer products, like paint. In recent years, there has been great effort to assemble colloids into new functional materials with designer architecture using electric and magnetic fields. Of particular interest are colloids that spin (rotors), because in such systems the flows stirred by the rotating particles add to the electrostatic interactions thereby vastly expanding the possible structures. This project will study the theory of dynamics of colloids placed between two planar electrodes in order to understand the mechanisms of the field-driven particle assembly. The project integrates knowledge across the fields of applied math, fluid mechanics and soft matter, which will be very beneficial for the training of the students associated with the project. Visually appealing experiments will help educate the public about mathematics and fluid dynamics. This proposal is concerned with a theoretical investigation of the dynamics of rotating colloids powered by the Quincke effect, which is the spontaneous spinning of a dielectric sphere in an applied uniform electric field. The Quincke rotor dynamics in free space is described by the Lorenz equations and the system has gathered attention as one of the physical realizations of Lorenz chaos: at high electric fields the sphere spins irregularly around a fixed axis. However, the dynamics of Quincke rotors in confinement is strikingly different than that in free space. An individual rotor displays three-dimensional motion with periodic reorientation of the axis of rotation. Collectives of hovering Quincke rotors are found to self-organize into stable clusters or snaking chains. This project will investigate the electrohydrodynamics of particles placed between two planar electrodes. The novel mathematical challenges are many and include analytical solutions of single particle dynamics near boundaries, careful consideration of pair-wise hydrodynamic and electrostatic interactions in confinement, and integration of these analytical results into a Stokesian-dynamics-based algorithm for a very large numbers of particles. Benchmark experiments to complement the mathematical research are proposed to justify and validate the mathematical models and to ensure that the research has impact both in and beyond the applied mathematics community. Educational impact includes bringing direct experimental experience in fluid dynamics and soft matter research to applied mathematics undergraduate and graduate students at Northwestern University. The emergent behavior of the Quincke rotors will likely open new research directions across various fields, e.g., non-equilibrium soft matter, materials engineering, and fluid dynamics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.