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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=738827
• 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，即像柔软或硬固体一样，如液体，Glyces或融化或共价键合的单体的网络，或吸附在表面上，或在打结的会议上扭曲并纠缠在一起。这组丰富的形式和热力学相的基础是聚合物熵的概念，这与聚合物分子的会议数量有关。在此应用程序中，我将概述我在该领域正在进行的工作，并解释继续进行研究计划的几种途径。我还将解释学生当前参与我的研究计划及其贡献，以及未来的项目，这些项目适合培训蒙特卡洛模拟和统计力学的学生。 1940年代，诺贝尔奖获奖者Flory和1970年代的De Gennes对聚合物熵进行了研究。一个丰富的数学理论来建模类似字符串的对象的熵的作品。这包括避免自我的步行和相关模型，组合数学中的定向路径模型，渗透，网络以及包括蒙特卡洛方法在内的数值方法。这些模型在随机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。研究领域跨越了严格和应用的统计力学，组合数学，概率理论和数学物理学。还与实验和理论聚合物物理和化学有联系。在该领域的研究很重要，因为它增加了对相互作用和密集聚合物和网络模型中相位行为和缩放的理解。在最后一个周期中，我和我的学生研究了分子生物学网络的平均田间缩放，并为压缩致密聚合物的自我避免行走模型。与其他合作者，我曾在线性和分支聚合物的相图上以及吸附自我避免自我的步行的分区功能零。我的短期目标是将我的研究扩展到自我避免步行和定向晶格路径模型的分区功能零，以将Flory-Huggins理论（一种密集聚合物溶液理论）应用于共聚物融化的模型，并使用Garm Algorithm进行晶格旋转系统的模拟。此外，我正在研究使用自我避免行走模型的分支聚合物吸附模型的相图。有关分区功能零和密集聚合物模型的研究将与研究生进行。长期目标是考虑一方面，一方面，考虑创建框架来理解密集聚合物系统的相位图，另一方面，以研究分区功能零的数学属性及其在创建中所扮演的作用。相互作用聚合物的自我避免步行模型的关键点。