RUI: Computational Methods for Measuring Topological Entanglement in Polymers

本项目旨在利用计算和数学技术研究聚合物纠缠和结构对材料性能的影响。自爱德华兹在60年代提出S的原始模型以来，纠缠聚合物物理一直是人们研究的对象，该模型目前仍在研究之中。在某些条件下，我们可以将高分子链看作空间中的数学曲线，并度量它们的拓扑复杂性。然而，将拓扑纠缠用于聚合物纠缠的研究还没有得到充分的探索，这是因为将这两个概念联系起来是困难的，需要来自拓扑学和聚合物物理和工程学的背景。该项目包括与来自数学和化学工程的研究人员的跨学科努力，以解决量化拓扑纠缠和聚合物结构对聚合物材料性质的影响的问题。我们的贡献是一种创新的方法，它集成了分析、计算和实验方法来解决聚合物物理、拓扑和几何界面上的问题。了解微观性质是如何影响材料性质的，不仅将导致新材料的智能制造，而且还将有助于理解生物。该奖项将在该项目的三年中每年资助一名研究生。为了理解和量化聚合物的微观结构和宏观性质之间的相互作用，我们建议使用拓扑学中的数学概念，通过计算机模拟来研究不同长度尺度下的聚合物的性质。我们的结果将得到实验数据的补充和验证。所做的工作可概括如下：(1)在自洽场理论(SCFT)模拟中创建新的方法来解释聚合物纠缠；(2)开发新的分割算法来模拟纠缠聚合物的流固相互作用；(3)开发用于测量开放曲线的拓扑纠缠的新的计算用户程序包；(4)通过模拟和实验来综合应用上述所有工具来理解不同结构的聚合物熔体的自组装、组织和粘弹性性质。这项研究通过定义和研究用于测量空间中开放曲线的几何/拓扑复杂性的新工具，促进了拓扑学和几何学领域的知识，并通过设计可重用代码中的算法并将其原型化，特别是研究缠绕聚合物模拟的各个方面(例如，此类系统的流体-结构相互作用和拓扑相互作用)，促进了计算基础设施的发展。这项工作还扩展了SCFT模拟，以在计算上可行的方式解释聚合物的拓扑方面，这是目前SCFT模拟中所没有的。这种整体的方法将彻底研究不同结构的聚合物中的纠缠，这些聚合物目前对材料、制造和纳米技术非常感兴趣，我们的结果有可能立即影响到实际制造。我们的结果也为研究具有潜在生物技术影响的生物聚合物提供了有价值的工具。该项目具有教育目标，包括对本科生研究产生重大影响，致力于促进STEM中代表性不足的群体。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。

This project aims to investigate the effects of polymer entanglement and architecture on material properties using computational and mathematical techniques. Entangled polymer physics are a subject of study since Edwards' original model in the 60's which is still under examination. Under some conditions we can see polymer chains as mathematical curves in space and measure their topological complexity. However, the use of topological entanglement for the study of polymer entanglement has not been fully explored, due to the difficulty of bridging the two notions, requiring background from topology and polymer physics and engineering. The project consists of an inter-disciplinary effort with researchers from Mathematics and Chemical Engineering to solve the problem of quantifying the effects of topological entanglement and polymer architecture to material properties of polymers. Our contribution is an innovative approach that integrates analytical, computational and experimental methods to solve a problem at the interface of polymer physics, topology and geometry. Understanding how microscopic properties affect material properties will lead not only to the smart manufacturing of new materials, but also to the understanding of living matter. This award will support 1 graduate student for each of the three years of the project. In order to understand and quantify the interplay between microstructure and macroscopic properties of polymers, we propose to use mathematical concepts from topology and investigate properties of polymers at different length-scales through computer simulations. Our results will be complemented and validated by experimental data. The proposed works can be summarized as follows: (1) the creation of new methods to account for polymer entanglement in Self-Consistent Field Theory (SCFT) simulations, (2) the development of new partitioned algorithms to simulate the fluid-structure interaction for entangled polymers (3) the development of new computational user-packages for measuring topological entanglement of open curves and (4) the combined application of all the above mentioned tools to understand the self-assembly, organization and viscoelastic properties of polymer melts of varying architecture using simulations and experiments. This study advances knowledge at the area of topology and geometry, by defining and studying new tools for measuring the geometrical/topological complexity of open curves in space and also advances computational infrastructure, by designing and prototyping algorithms in reusable code that in particular studies aspects in entangled polymer simulations (such as fluid-structure interactions for such systems and topological interactions). This work also extends SCFT simulations to account for topological aspects of polymers in a way that it is computationally feasible, which is presently absent in SCFT simulations. This holistic approach will thoroughly study entanglement in polymers of varying architecture that are currently of great interest in materials and manufacturing and nanotechnology, with the potential of immediate impact of our results to practical manufacturing. Our results also provide valuable tools for studying biopolymers with potential impact in biotechnology. This project has educational objectives including strong impact on undergraduate research with a commitment in promoting underrepresented groups in STEM.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.