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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675855
• 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，或熔融物，或作为共价键合单体的网络，或 * 吸附在表面上，或扭曲和缠结在打结 * 构象。在这些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有关。在这份申请中，我将概述我在这一领域正在进行的工作，并解释继续我的研究计划的几个途径。 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。聚合物熵在20世纪40年代由诺贝尔奖获得者弗洛里（Flory）研究，在20世纪70年代由德·热内（de Gennes）研究。在他们的工作基础上，建立了一个丰富的数学理论来模拟弦状物体的熵。 这包括 * 自避免行走和相关模型，* 组合数学中的有向路径模型，渗透，网络，以及 * 数值方法，包括蒙特卡洛方法。这些模型在随机团簇的统计力学和相变理论中无处不在，并且与磁性材料的自旋系统模型的经典模型有关。 这个研究领域横跨严格和应用 * 统计力学，组合数学，概率论，* 和数学物理。还与实验 * 和理论聚合物物理和化学有关。 这一领域的研究很重要，因为它有助于理解相互作用和致密聚合物模型以及网络中的相行为和缩放。在上一个周期中，我和我的学生研究了分子生物学中网络的平均场标度，以及压缩致密聚合物的自回避行走模型。 与其他合作者，我曾在相图的线性和支化的聚合物和分区功能 * 零吸附自回避行走。 我的短期目标是 * 扩展我的研究到 * 自避免行走和定向晶格路径模型的配分函数零点，将 *Flory-Huggins理论（一种稠密聚合物溶液理论）应用于共聚物熔体模型 *，并使用GARM算法模拟晶格自旋 * 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。 将与研究生一起研究致密聚合物的配分函数 * 零点和模型。长期目标是一方面考虑Flory-Huggins理论在创建用于理解致密聚合物系统的相图的框架中的有用性，另一方面研究 * 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避免行走模型中的临界点中所起的作用。