Fast Electrostatics and Brownian Hydrodynamics in Doubly-Periodic Geometries

需要新的计算工具来对广泛的生物和工程系统进行大规模建模，例如生物细胞和工程囊泡中的脂双层膜、显示器中的薄液晶膜、电池中的受限电解液以及材料科学中的胶体单分子膜。这些系统都是准二维的，因为系统在第三(法)方向上是有界的，它们都涉及悬浮在溶剂流体中的粒子(脂类分子、液晶分子、离子、胶体)，并且在所有这些系统中，需要精确地模拟布朗运动的扩散。该项目将为这些类型的系统的计算建模开发新的数学技术和计算代码。新的算法将允许研究团队在前所未有的长度和时间尺度上研究准平面系统中的集体扩散，极大地扩展我们回答有关准2D材料中扩散的基础科学问题的能力，并帮助我们设计更好的设备和材料，如电池电极。该项目将涉及并培训几名本科生、博士后和一名博士后研究员。该项目团队将开发新的计算技术，用于对广泛的物理系统中的集体扩散进行建模，这些物理系统包括电解液溶液、脂类双层膜、薄液晶膜、液-液界面上的胶体单层或沉积在底壁上的胶体单层，以及胶体团簇。该团队将开发基于快速傅立叶变换和切比雪夫多项式的数值技术，这些技术可以在粒子数量的线性时间内计算双周期几何图形中的静电和流体相互作用。该团队将开发用于对多粒子系统进行长期布朗流体动力学模拟的工具，方法是设计算法，在存在流体动力学相互作用的情况下有效地生成布朗速度。该团队将为静电学和布朗动力学开发高效的公共领域并行代码，能够在长时间尺度上处理数十万粒子。该团队还将使用开发的技术来阐明物理现象，如电流体动力学流动、胶体单分子层中的集体扩散以及驱动胶体层中的集体动力学。特别是，该团队将在广泛的时间范围内表征离子在电解液中的扩散，以及在双层中的脂类和蛋白质包裹体。这将阐明广泛使用的电双层Poisson-Nernst-Planck方程以及膜流体动力学的Saffman模型的适用范围。该项目将在第二年和第三年支持一名研究生。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。

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.