Fast Electrostatics and Brownian Hydrodynamics in Doubly-Periodic Geometries

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New computational tools are required for large-scale modeling of a broad range of biological and engineered systems such as lipid bilayer membranes in biological cells and engineered vesicles, thin liquid crystal films in displays, confined electrolytes in batteries, and colloidal monolayers in materials science. These systems are all quasi two-dimensional (2D) because the system is bounded in the third (normal) direction, they all involve particles (lipid molecules, liquid crystal molecules, ions, colloids) suspended in a solvent fluid, and in all of these systems diffusion via Brownian motion needs to be modeled accurately. This project will develop novel mathematical techniques and computational codes for computational modeling of these types of systems. The new algorithms will allow the research team to study collective diffusion in quasi-planar systems over unprecedented length and time scale, vastly expanding our ability to answer fundamental science questions about diffusion in quasi-2D materials, as well as helping us engineer better devices and materials such as battery electrodes. The PI will involve and train several undergraduate, Ph.D students and a postdoctoral fellow in the project.The project team will develop novel computational techniques for modeling collective diffusion in a broad range of physical systems such as electrolyte solutions, lipid bilayer membranes, thin liquid crystal films, colloidal monolayers on a liquid-liquid interface or sedimented on a bottom wall, and colloidal clusters. The team will develop numerical techniques based on the Fast Fourier Transform and Chebyshev polynomials that can compute electrostatic and hydrodynamic interactions in doubly-periodic geometries in linear time in the number of particles. The team will develop tools for long-time Brownian HydroDynamics simulations of many-particle systems by designing algorithms to efficiently generate Brownian velocities in the presence of hydrodynamic interactions. The team will develop efficient public-domain parallel codes for electrostatics and Brownian dynamics, capable of handling hundreds of thousands of particles over long time scales. The team will also use the developed techniques to illuminate physical phenomena such as electro-hydrodynamic flows, collective diffusion in colloidal monolayers, and collective dynamics in driven colloidal layers. In particular, the team will characterize diffusion of ions in electrolytes, as well as lipids and protein inclusions in bilayers, over a broad range of time scales. This will elucidate the range of applicability of the widely-used Poisson-Nernst-Planck equations for electric double layers, as well as the Saffman model of membrane hydrodynamics. This project will support one graduate student in the second and third years.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.