The Statistical Mechanics of Lattice Models of Polymers

聚合物晶格模型的统计力学

• 批准号: RGPIN-2019-06303
• 负责人: JansevanRensburg, Esaias
• 金额: $ 1.24万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=715993
• 关键词: Statistical; Mechanics; Lattice; Models; Polymers

Polymers appear in many different forms, namely as soft or hard solids, as liquids, glues, or melts, or as networks of covalently bonded monomers, or adsorbed onto a surface, or twisted and entangled in knotted conformations. Underlying this rich set of forms and thermodynamic phases is the notion of polymer entropy, which is related to the number of conformations of a polymer molecule. In this application I shall give an overview of my ongoing work in this field and explain several avenues for continuing my research program. I will also explain the current involvement of students in my research program and their contributions, as well as future projects which are suitable for training students in Monte Carlo simulations and statistical mechanics. Polymer entropy was studied by Nobel prize winners Flory in the 1940s, and de Gennes in the 1970s. A rich mathematical theory to model the entropy of string-like objects was built on their work. This includes the self-avoiding walk and related models, directed path models in combinatorial mathematics, percolation, networks, as well as numerical methods including Monte Carlo methods. These models are ubiquitous in the statistical mechanics of random clusters and in the theory of phase transitions and are related to classical models of spin system models of magnetic materials. This area of research straddles rigorous and applied statistical mechanics, combinatorial mathematics, probability theory, and mathematical physics. There is also a connection to experimental and theoretical polymer physics and chemistry. Research in this area is important because it adds to the understanding of phase behaviour and scaling in models of interacting and dense polymers and on networks. Over the last cycle my students and I have worked on mean field scaling for networks in molecular biology and on a self-avoiding walk model of compressed dense polymers. With other collaborators I have worked on the phase diagrams of linear and branched polymers and on partition function zeros of adsorbing self-avoiding walks. My short term goals are to expand my research into the partition function zeros of models of self-avoiding walks and directed lattice paths, to apply Flory-Huggins theory (a theory of dense polymer solutions) to models of copolymer melts, and to perform simulations of lattice spin systems using the GARM algorithm. In addition, I am investigating the phase diagram of pulled adsorbing models of branched polymers using self-avoiding walk models. Studies on partition function zeros and models of dense polymers will be done with graduate students. The longer term goals are to consider the usefulness of Flory-Huggins theory in creating a framing for understanding the phase diagram of dense polymer systems on the one hand, and on the other hand to examine the mathematical properties of partition function zeros and the role they play in creating critical points in self-avoiding walk models of interacting polymers.

聚合物出现在 许多不同的形式，即软的或硬的固体，如液体， 胶水或熔体，或作为共价键合的单体网络，或 吸附在表面上，或扭曲并缠绕在一起 构象。在这种丰富的形式和热力学相的基础上， 聚合物熵的概念，它与分子的数量有关 聚合物分子的构象。在这份申请书中，我将给出 概述我在这一领域正在进行的工作，并解释几个 继续我的研究计划的途径。我还将解释学生目前在我的研究项目中的参与情况和他们的贡献，以及未来适合培训学生蒙特卡洛模拟和统计力学的项目。 聚合物的熵是 诺贝尔奖得主弗洛里在20世纪40年代和德根恩在20世纪40年代进行了研究 20世纪70年代。丰富的数学理论，用来模拟 弦一样的物体是建立在他们的工作基础上的。这包括 自回避行走和相关模型，有向路径模型 组合数学、渗流、网络以及 数值方法包括蒙特卡罗方法。这些型号是 普遍存在于随机星系团的统计机制和 相变理论，并与经典的 磁性材料的自旋系统模型。这一研究领域既严谨又实用。 统计力学，组合数学，概率论， 和数学物理。还有一种与实验有关的联系 以及理论聚合物物理和化学。这一领域的研究很重要，因为它增加了对相互作用的致密聚合物模型和网络中的相行为和比例的理解。 在上一个周期中 我的学生和我一直致力于网络的平均场缩放 分子生物学和压缩致密聚合物的自回避行走模型。我与其他合作者一起参与了该阶段的工作 线型和支化聚合物图及配分函数 零点吸收自我回避的散步。我的短期目标是 将我的研究扩展到模型的配分函数零点 自回避漫游和有向格子路径，以应用 Flory-Huggins理论(稠密聚合物溶液理论)到模型 以及执行晶格自旋的模拟 使用GARM算法的系统。此外，我还使用自回避行走模型研究了支化聚合物的拉式吸附模型的相图。配分函数的研究 密集聚合物的零点和模型将在研究生中完成。 较长期的目标是考虑Flory-Huggins理论的有用性，一方面为理解稠密聚合物体系的相图创建一个框架，另一方面检查 配分函数零点及其在相互作用聚合物的自回避行走模型中创建临界点的作用。