Nonlinear Dynamics of Colloidal Rotors: Chaos and Order

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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.