RUI: Computational Methods for Measuring Topological Entanglement in Polymers

该项目旨在利用计算和数学技术研究聚合物缠结和结构对材料性能的影响。纠缠聚合物物理是自爱德华兹在60年代提出的原始模型以来的一个研究课题，该模型仍在研究中。在一定条件下，我们可以把高分子链看作空间中的数学曲线，并度量其拓扑复杂性。然而，使用拓扑缠结的研究聚合物缠结尚未得到充分的探讨，由于难以桥接这两个概念，需要从拓扑和聚合物物理和工程的背景。该项目由数学和化学工程研究人员的跨学科努力组成，以解决量化拓扑纠缠和聚合物结构对聚合物材料性能的影响的问题。我们的贡献是一个创新的方法，集成了分析，计算和实验方法，以解决问题的高分子物理，拓扑和几何的接口。了解微观特性如何影响材料特性不仅会导致新材料的智能制造，而且会导致对生命物质的理解。该奖项将支持1研究生为该项目的三年中的每一年。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的数学概念，并通过计算机模拟研究不同长度尺度下聚合物的性质。我们的研究结果将得到补充和实验数据的验证。 拟议工程可概述如下：（1）在自洽场理论（SCFT）模拟中创建新方法来解释聚合物纠缠，（2）开发新的分区算法来模拟缠结聚合物的流体-结构相互作用（3）开发新的计算用户包来测量开放曲线的拓扑缠结（4）所有上述工具的组合应用，以理解使用模拟和实验的不同结构的聚合物熔体的自组装、组织和粘弹性。 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，并通过在可重复使用的代码中设计和原型化算法，特别是研究纠缠聚合物模拟方面（如此类系统的流体-结构相互作用和拓扑相互作用），推进了拓扑和几何领域的知识。这项工作还扩展了SCFT模拟，以计算可行的方式考虑聚合物的拓扑方面，这是目前SCFT模拟中缺乏的。这种整体的方法将彻底研究不同结构的聚合物中的纠缠，这些聚合物目前在材料和制造以及纳米技术中引起了极大的兴趣，我们的结果可能会对实际制造产生直接影响。我们的研究结果也为研究生物聚合物在生物技术中的潜在影响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，并致力于促进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知识价值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.