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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759008
• 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.

聚合物以许多不同的形式出现，即作为软或硬固体，作为液体、胶水或熔体，或作为共价键合单体的网络，或吸附在表面上，或以打结构象扭曲和缠结。在这组丰富的形式和热力学相的基础上是聚合物熵的概念，它与聚合物分子的构象数量有关。在本申请中，我将概述我在该领域正在进行的工作，并解释继续我的研究计划的几种途径。  我还将解释学生目前参与我的研究项目的情况和他们的贡献，以及适合培训学生蒙特卡罗模拟和统计力学的未来项目。诺贝尔奖获得者弗洛里 (Flory) 在 20 世纪 40 年代和德热尼斯 (de Gennes) 在 1970 年代研究了聚合物熵。他们的工作建立了丰富的数学理论来模拟弦状物体的熵。这包括自回避行走和相关模型、组合数学中的有向路径模型、渗滤、网络以及包括蒙特卡罗方法在内的数值方法。这些模型在随机簇统计力学和相变理论中普遍存在，并且与磁性材料自旋系统模型的经典模型相关。  该研究领域涵盖严格的应用统计力学、组合数学、概率论和数学物理学。它还与实验和理论聚合物物理和化学有关。  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分子生物学网络的平均场缩放以及压缩致密聚合物的自回避行走模型。我与其他合作者一起研究了线性和支化聚合物的相图以及吸附自回避游走的配分函数零点。我的短期目标是将我的研究扩展到自回避行走和有向晶格路径模型的配分函数零点，将 Flory-Huggins 理论（一种致密聚合物溶液理论）应用于共聚物熔体模型，并使用 GARM 算法对晶格自旋系统进行模拟。   此外，我正在研究使用自回避行走模型的支化聚合物的拉吸附模型的相图。  配分函数零点和致密聚合物模型的研究将由研究生完成。长期目标是一方面考虑弗洛里-哈金斯理论在创建理解致密聚合物系统相图的框架方面的有用性，另一方面检查配分函数零点的数学性质以及它们在创建相互作用聚合物的自回避行走模型中的临界点中所发挥的作用。