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，即作为软或硬固体，作为液体， 胶或熔体，或作为共价键合的单体的网络，或 吸附在一个表面上，或扭曲和纠缠在打结 构象在这丰富的形式和热力学相的基础上， 聚合物熵的概念，它与聚合物的数量有关。 聚合物分子的构象。在本申请中，我将 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参与我的研究计划和他们的贡献，以及未来的项目，这是适合培训学生在蒙特卡洛模拟和统计力学。 聚合物熵为 20世纪40年代诺贝尔奖得主弗洛里和20世纪60年代的德·热内研究了这一问题。 70年代一个丰富的数学理论来模拟 像绳子一样的物体是在他们的工作上建造的。 这包括 自我回避步行和相关模型，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。这些模型 普遍存在于随机簇的统计力学和 相变理论，并与经典的模型， 磁性材料自旋系统模型 这一研究领域跨越了严谨性和应用性 统计力学，组合数学，概率论， 和数学物理。也有一个连接到实验 以及理论聚合物物理和化学。 该领域的研究很重要，因为它增加了对相互作用和致密聚合物模型以及网络中的相行为和缩放的理解。 在上一个周期中 我和我的学生研究了网络的平均场缩放， 分子生物学和压缩致密聚合物的自避免行走模型。 与其他合作者，我曾在阶段， 线性和支化聚合物的图解和配分函数 零的吸附自我回避行走。 我的短期目标是 将我的研究扩展到模型的配分函数零点， 自避免遍历和定向格点路径，以应用 Flory-Huggins理论（一种稠密聚合物溶液理论） 的共聚物熔体，并进行模拟的晶格自旋 使用GARM算法的系统。 此外，我正在研究的相图拉吸附模型的支化聚合物使用自回避行走模型。 配分函数的研究 零和模型的密集聚合物将与研究生。 长期目标是一方面考虑Flory-Huggins理论在创建理解致密聚合物体系相图的框架中的有用性，另一方面检查 配分函数零点和它们在相互作用聚合物的自避免行走模型中创建临界点时所起的作用。