Statistical Physics of Polymers

Transcription

1 Statistical Physics of Polymers

2 Advanced Texts in Physics This program of advanced texts covers a broad spectrum of topics which are of current and emerging interest in physics. Each book provides a comprehensive and yet accessible introduction to a field at the forefront of modern research. As such, these texts are intended for senior undergraduate and graduate students at the MS and PhD level; however, research scientists seeking an introduction to particular areas of physics will also benefit from the titles in this collection. Springer-Verlag Berlin Heidelberg GmbH

3 Toshihiro Kawakatsu Statistical Physics of Polymers An Introduction With 129 Figures t Springer

4 Prof. Toshihiro Kawakatsu Tohoku University Department of Physics Aoba, Aramaki, Aoba-ku Sendai , Japan Cataloging-in-Publication Data. Kawakatsu, Toshihiro. (Kobunshi butsuri no kiso. English) Statistical physics of polymers: an introduction / Toshihiro Kawakatsu. p.cm. - (Advanced texts in physics, ISSN ) Includes bibliographical references and index. ISBN ISBN (ebook) DOI / Polymers. 2. Statistical physics. I. Title. II. Series. QC173.4.P65K '29-dc Original Japanese edition published by Saiensu-Sha Co., Ltd, Tokyo, 2001 ISSN ISBN This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations are liable to prosecution under the German Copyright Law. springeronline.com Springer-Verlag Berlin Heidelberg 2004 Originally published by Springer-Verlag Berlin Heidelberg New York in 2004 Softcover reprint of tlie hardcover I st edition 2004 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data prepared by the author using a Springer TEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN /3141/tr

5 Preface This book is an introductory textbook on the statistical mechanics of polymers and complex fluids aimed at senior undergraduate and graduate students and non-specialist researchers who are starting research in this field. Modern statistical mechanics on polymers and complex fluids is based on many fields, such as chemical physics, statistical mechanics, quantum mechanics, stochastic processes, theory of phase transitions, hydrodynamics, rheology, and so on. This book provides an overview of the basic concepts and methods used in current research on the physics of polymers and complex fluids. Using simple but essential examples, we describe how to derive the physical properties of polymers theoretically, focusing on the structure and dynamics on mesoscopic scales. Here, the term 'mesoscopic scales' means intermediate lengths and time scales between the microscopic atomic scale and the macroscopic scale. Properties on mesoscopic scales are the central issue of the physics of polymers and complex fluids, because these materials are well characterized by spatiotemporal structures on these scales, where we can extract universal properties that are independent of the microscopic details of the system. In order to understand universal mesoscopic properties, it is important to eliminate microscopic degrees of freedom irrelevant to these universal properties. Although such an approach has been very successful in describing macroscopic properties of simple liquids, it is not so simple to find out whether this powerful framework can also be applied to complex fluids, because of the coexistence of structures over a wide range of length and time scales. The complexity of complex fluids usually leads to theoretical models described by nonlinear non-equilibrium equations. In order to extract useful predictions from such theoretical models, we are compelled to rely on analytic approximations or direct numerical calculations. Using polymeric systems as the main target, we show how these numerical calculations and analytical theories can be constructed.

6 VI Preface This book is organized as follows: Chapter 1 contains a brief overview of various mesoscopic phenomena in complex fluids, together with a brief introduction to the modeling of these phenomena. Chapter 2 focuses on polymers, i.e., chain-like macromolecules, as a typical example of complex fluids. We discuss microscopic modeling of the statistical mechanics of such polymer chains. We introduce a molecular model called the Gaussian chain model, which serves as a useful starting point for theoretical studies on the equilibrium and dynamic properties of polymers. In the derivation of this Gaussian chain model, we extensively utilize the so-called coarse-graining procedure, where unimportant microscopic degrees of freedom are eliminated. To demonstrate the efficiency and usefulness of the Gaussian chain model, we compare the theoretical results of this model with recent experimental results concerning biological polymers. Chapter 3 further applies the coarse-graining procedure to produce a continuum description of the polymer chain in terms of the density fields and the probability distribution of the shape of the chain. One such theoretical framework is the so-called self-consistent field theory, where many-body problems of dense polymeric liquids and blends are formulated using the mean-field approximation and the probability distributions of the Gaussian chain model. Theoretical as well as numerical analysis using the selfconsistent field theory has been widely applied to predict the mesoscopic structures of inhomogeneous polymer systems, such as polymer blends or polymers near surfaces. We will exhibit examples of recent research based on this approach. Although the self-consistent field theory is useful and quantitatively reliable, it often requires a heavy numerical calculation. To avoid such a numerical task, we develop another coarse-grained model called the Ginzburg-Landau model in Chap. 4. This model was originally proposed for the critical phenomena associated with second order phase transitions to explain their universal properties. The same technique is applicable to polymer solutions and polymer blends. The model contains several parameters that reflect microscopic information concerning the structure of the polymer chain. Viscoelasticity is one of the most important macroscopic properties of dense polymer solutions. It is instructive to see how such macroscopic viscoelastic properties are derived from microscopic molecular models. For this purpose, we discuss reptation theory and its recent advances in Chap. 5. This book owes much to the kind help of many colleagues. First, I would like to thank Professor Seiji Miyashita (Tokyo University), Mr. Nobuhiko Tajima and Mr. Kosuke Hirase (Saiensu-Sha Co. Ltd.) for their continuous help, encouragement and advice during the preparation of the original

7 f>reface \TIl Japanese edition of this book. The publication of the English edition came into being thanks to Dr. Claus Ascheron and Dr. Angela Lahee (Springer \Terlag, Berlin). I also thank Dr. Hiroshi Morita (JST, CREST), Mr. Kenji Furuichi, Mr. Yohei Yamaguchi, Ms. Yuko Mizuta (Nagoya University), Mr. Rihei Endo (Tohoku University), Mr. Yoshikatsu Hayashi (Lund University), Professor Takao Ohta (Kyoto University), Professor Tohru Okuzono (ERATO, JST), and Mr. Takashi Honda (Nippon Zeon Co. Ltd.) for their useful comments on the original Japanese edition. This book is based on a series of lectures that I gave for graduate students at several universities in Japan. A note on one such lecture at Kyoto University in June 1998 was edited by Dr. Yoshihiro Yamazaki (Waseda University) and Mr. Masataka Imura (Nara Advanced Institute of Science and Technology), and was published in Bussei Kenkyuu (in Japanese). This lecture note was useful in preparing this text book. I would also thank authors and publishers for their kind permission to reproduce figures. Last but not least, discussions with Professor Masao Doi (Nagoya University) and members of the Doi Laboratory and those of the Doi Project sponsored by the Ministry of Economy, Technology and Industry (Japan) are gratefully acknowledged. Sendai, January 2004 Toshihiro K awakatsu

8 Contents 1 Introduction Complex Fluids and Polymers Complex Systems Complex Fluids Mesoscopic Structures in Complex Fluids Polymers: A Typical Example of Complex Fluids Molecular Structure of Polymers Mesoscopic and Macroscopic Properties of Polymers Modeling the Physical Phenomena of Polymers Modeling Mesoscopic Structures Static Properties Dynamic Properties Physical Properties and Gaussian Chain Statistics Gaussian Chain Model and Statistics of Polymers A Simple Model of a Polymer Chain: The Lattice Model Definition of the Lattice Model of Polymers Ideal Chain Statistics of Lattice Models Bead-Spring Model of Polymer Chain and Gaussian Chain Statistics Coarse-Graining Procedure and Bead-Spring Model Statistical Mechanical Theory of Equilibrium Conformations of a Gaussian Chain Ideal Chain Statistics Correlation Functions and Scattering Functions of an Ideal Chain Statistical Mechanics of Chains with Interactions and Approximate Theories Statistical Properties of Many-Chain Systems Dynamical Models of a Polymer Chain Based on a Molecular Description Formulation of Brownian Motion Rouse Model of a Single Polymer Chain in a Solvent Hydrodynamic Effects in Dilute Polymer Solutions... 73

9 X Contents 2.5 Justification of the Gaussian Chain Model from a Microscopic Point of View Full Atomistic Model of Polymer Chains United Atom Model Statistical Theories and Experiments on Semi-Flexible Chains Worm-Like Chain Model Statistical Properties of a Stretched Worm-Like Chain Experiments on Worm-Like Chains Using Biopolymers Molecular Simulations of Polymer Dynamics Molecular Simulation Methods Models of Interaction Potentials for Coarse-Grained Chains Examples of Molecular Simulations Exercises Mesoscopic Structures and Self-Consistent Field Theory Mesoscopic Phenomena in Polymer Systems Formulation and Simple Examples of the Self-Consistent Field Theory of Polymers Mean Field Approximation and Self-Consistent Field Path Integral Formalism for Polymers Classical Approximation for Self-Consistent Field Theory Numerical Methods for the Self-Consistent Field Theory of Polymers Functionals, Functional Derivatives and Functional Integrals General Expression for the Free Energy Numerical Solutions of Self-Consistent Field Equations Examples of Numerical Simulations U sing Self-Consistent Field Theory Exercises Ginzburg-Landau Theory Formulation of the Ginzburg-Landau Theory Ginzburg-Landau Model Expansion of the Free Energy Evaluation of Expansion Coefficients Using the Random Phase Approximation Applications of the Ginzburg-Landau Theory Phase Diagram of Block Copolymer Melts Extensions to Dynamical Processes Exercises

10 Contents XI 5 Macroscopic Viscoelastic Theory of Polymers Viscoelastic Properties of Polymeric Liquids Polymers and Viscoelasticity Hydrodynamic Descriptions of Viscoelasticity Reptation Theory for Linear Polymers Concept of Reptation Theory Stress Relaxation Function Extensions of Reptation Theory and Nonlinear Viscoelasticity Contour Length Fluctuations Chain Retraction Constraint Release Contribution to Viscoelasticity from Phase-Separated Domains Exercises References Index