Scattering and Dynamics of Polymers E-Book

Charles C. Han

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 387

Veröffentlichungsjahr: 2011

Beschreibung

Scattering is a very powerful tool to study the structure of polymers. Written by highly regarded and respected scientists in the field, this book presents the latest developments in the field of scattering in a uniform, systematic manner. This volume arms readers with both theoretical and experimental aspects of the intended area, offering much simplified theoretical explanations on the physics of scattering. The authors provide discussion on applications of experimental techniques. Han and Akcasu begin with a traditional treatment of light scattering from plane waves, followed by consistent application of density (in both real and Fourier space) correlation functions in both space and time. The authors do not distinguish among light, X-ray, and neutron, excepting their scattering length, q-range, coherence and detection differences. Readers can therefore concentrate on exactly the scattering tools they need to use, while theoretical explanation on the physics of scattering can be made much more simplified and uniform. * Presents the latest development in the field of scattering in a uniform, systematic manner * Arms readers with both theoretical and experimental aspects * Gives a much simpler theoretical explanation on the physics of scattering * Demonstrates application of experimental techniques

Leseprobe

Contents

Cover

Title Page

Copyright

Foreword by Professor Timothy P. Lodge

Foreword by Professor Hyuk Yu

Preface

Chapter 1: Plane Waves, Scattering, and Polymers

1.1 Single-Particle Scattering/Multi-Particle Scattering

1.2 Molecular Weight of Particles with Thermodynamic Interactions

1.3 Scattering Structure Factor of a Polymer/Point Scattering Approximation

Appendix 1.A: Thermodynamics [14]

Chapter 2: Fluctuations, Correlation, and Static/Dynamic Scattering

2.1 Space-Time Correlation Function

2.2 Density in and Space

2.3 Some Properties of and Dynamic Scattering

2.4 Examples of Dynamic Light Scattering in Polymer Solutions

2.5 Light, X-Ray, and Neutron Scattering

Appendix 2.A: Gaussian Stochastic Variable Approximation [74]

Appendix 2.B: Spin Incoherence [75]

Appendix 2.C: The Basic Scattering Laws for Incompressible Systems [76]

Chapter 3: Dynamics and Kinetics of Phase Separation in Polymer Systems

3.1 Thermodynamics of Polymer Blends

3.2 The Theory of Kinetics of Phase Separation

3.3 Spinodal Decomposition in Normal Binary Homopolymer Systems

3.4 Nucleation Phase Separation

3.5 Phase Separation and Phase Behavior under Shear Flow

3.6 Spinodal Decomposition in Complicated Systems

Appendix 3.A: Nonlinear Langevin Equation Approach to the Kinetics of Polymer Mixtures

Chapter 4: Statistical Mechanical Approach to the Theory of Dynamic Scattering

4.1 Introduction

4.2 A Brief History of Brownian Movement

4.3 Einstein's Explanation of Brownian Movement

4.4 Langevin Equation Approach

4.5 Scattering from Non-interacting Brownian Particles

4.6 Zwanzig-Mori Projection Operator Technique

4.7 Molecular Theory of Brownian Movement

4.8 Markov Processes and Fokker-Planck Equation

4.9 Stochastic Differential Equation and Fokker-Planck Equation

4.10 Rouse Dynamics

4.11 Hydrodynamic Interaction

4.12 Kirkwood-Risemann Equation

4.13 Diffusion Coefficient

4.14 Molecular Weight Dependence of the -Ratio and a Method for Measuring the Draining Parameter

4.15 Calculation of the Dynamic Scattering Function

Appendix 4.A: Radius of Gyration

Appendix 4.B: Diagonalization of the Rouse Matrix

Appendix 4.C: Solution of the Diffusion Equation without Hydrodynamic Interaction

Appendix 4.D: Solution of

Appendix 4.E: Some Trigonometric Formulae [23]

Index

This edition first published 2011

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Library of Congress Cataloging-in-Publication Data

Han, Charles C.

Scattering and dynamics of polymers: seeking order in disordered systems / Charles C. Han, A. Ziya Akcasu.

p. cm.

Summary: ‘In this book, we will start from the traditional light scattering for dilute solution, and then bring in the concepts of fluctuations, correlations, and also space-time correlation function’ – Provided by publisher.

Includes bibliographical references and index.

ISBN 978-0-470-82482-5 (hardback)

1. Light–Scattering. 2. Polymers–Spectra. 3. Polymers–Rheology. I. Akcasu, Ziyaeddin A. II. Title.

QC427.4.H36 2011

547'.70455–dc22

2011009939

Print ISBN: 978-0-470-82482-5

ePDF ISBN: 978-0-470-82483-2

oBook ISBN: 978-0-470-82484-9

ePub ISBN: 978-0-470-82825-0

Mobi ISBN: 978-1-118-08041-2

Foreword

By Professor Timothy P. Lodge

This monograph is unique in both scope and focus, and addresses a fascinating set of interrelated topics in modern polymer physics. In particular, the use of a suite of radiation scattering techniques – light, X-ray, and neutron – to explore the structure and dynamics of polymer-solvent and polymer-polymer mixtures is explored in detail. The underlying principles that govern the scattering techniques themselves, and the kinds of information that can be obtained, are developed. Then, the interpretation of the results via modern statistical thermodynamic and hydrodynamic theories is presented. In addition, representative experimental results are described and discussed, and promising directions for future research are identified. The choice of topics is informed by the particular expertise and experience of the authors, both of whom are world leaders in the field. Despite this personal flavoring, however, the topics covered capture a broad swath of the most exciting areas of polymer physics from recent decades.

Polymer materials are ubiquitous in modern technology, for example as plastics, fibers, adhesives, rubbers, coatings, and foams. For a variety of reasons (low density, flexibility, toughness, high specific strength, cost effectiveness), polymer materials continue to make substantial inroads into areas traditionally dominated by “hard materials.” Some commercial polymer applications date back to the nineteenth century, although the modern “polymer age” really began with the development of synthetic fibers and the search for synthetic rubber in the middle of the twentieth century. Not until the last part of the twentieth century, however, has a thorough understanding of the fundamental relationships between long-chain molecular structure and the properties of polymer liquids been developed. With this understanding has come the realization that polymers, and especially polymer blends, can also serve as model systems for studying the detailed structure, thermodynamics, kinetics, and dynamics of molecular mixtures.

From a structural point of view, polymers are built up of many repeat units, each less than a nanometer in length; the polymers themselves, depending on degree of polymerization (N) and chain conformation, are typically 10–100 nm in size. Aggregation of polymers, whether driven by self-assembly or by phase separation, leads to domain dimensions from 10 nm up to many microns. Thus experimental tools need to cover a wide range of length scales, extending over at least four orders of magnitude. As most of the systems of interest are amorphous, classical diffraction analysis is not applicable. However, the related tools of small angle light, X-ray, and neutron scattering are essential, as collectively they can access most of this range. In addition, neutron scattering is particularly powerful, because deuterium labeling permits access to the conformations of particular components in a dense mixture, with minimal (but not negligible!) perturbations to the thermodynamics.

From a dynamics perspective, the demands on experimental technique are even more stringent. The motions of small chain segments typically occur in the sub-microsecond range, whereas the relaxation of the entire chain, or the flow of a molten polymer, can occur on time scales of seconds, hours, or even weeks. This vast spread of time scale has two origins. One is the strong dependence of chain mobility on N; an increase in N by a factor of 103 can increase the viscosity of the molten polymer by more than a factor of 1010. The other is the inherent proximity to the glass transition temperature, and the associated highly non-Arrhenius temperature dependence of chain dynamics. For example, the viscosity of a particular polymer melt can vary by as many as 12 orders of magnitude over a 200-degree temperature window. These considerations must be borne in mind when adopting a model polymer system for study. In fact, this allows a great deal of experimental optimization, in the sense that a given dynamic process, whether relaxation of fluctuations at equilibrium or the evolution of a system toward equilibrium after a phase transition, can be tuned to an appropriate time scale.

Given these experimental challenges, it is fair to ask whether dense polymer phases are amenable to any kind of tractable analytic theory. However, the fortunate fact is that one can understand many features of polymer behavior in a relatively straightforward way; this is particular true of the structural and thermodynamic aspects on longer length scales. The primary reason for this surprising simplicity lies in the effect of the large values of N. For all flexible polymers in the melt, the chain conformational distribution is closely approximated by the Gaussian form, and the dimensional scaling is that of the random walk. Then, the strength of the thermodynamic interactions between chemically dissimilar repeat units is typically very small, on the order of 0.001 to 0.1 kT. When two polymers are blended into a homogeneous mixture, the combination of weak interactions and huge coordination numbers (of order 10 times N) makes mean-field theory an excellent approximation.

One example of how polymer systems can serve as an excellent model for a more general class of mixtures is the kinetics of liquid-liquid phase separation. This topic plays a central role in Chapter 3 of this book. For a binary blend of two polymers that exhibits a substantial one-phase region in the composition-temperature plane, mean-field theory (known as Flory–Huggins theory in the polymer literature) can describe the phase diagram rather well (i.e., the critical point, coexistence curve, and stability limit or spinodal). A quench into the two-phase region leads to phase separation, by nucleation and growth in the metastable regime, or by spinodal decomposition in the unstable regime. These processes are fascinating in their own right, but are also of huge technological importance for many different classes of materials. The classic Cahn–Hilliard theory of spinodal decomposition was not formulated with polymers in mind, yet, as detailed in this book, the first thorough experimental tests were conducted on polymers. The reasons for this have been alluded to above, namely by appropriate choice of polymer system, the chain dynamics could be matched to the experimental window, so that early, middle, and late stages of the process could be followed conveniently. Furthermore, the associated length scales fall neatly in the range accessible by a combination of light and neutron scattering. Another example that is featured in this book builds upon this quiescent case, namely the role of strong flow on both the phase boundaries and the kinetic pathways of structural evolution.

Although excellent monographs exist that focus on each of the scattering techniques discussed in this text, this work is unique in its choice of applications, and in its blend of theory and experiment. The book will serve as an excellent introduction to a rather large and disparate primary literature, and provide a coherent framing of the issues that cannot be easily extracted from any prior reference.

Timothy P. Lodge Department of Chemical Engineering and Materials Science Department of Chemistry University of Minnesota Minneapolis, USA

Foreword

By Professor Hyuk Yu

This is a unique monograph in polymer science. Unity is its singular distinction. Disparate subjects, commonly so perceived, are unified with an elegant introduction of a theoretical framework, and the book proceeds to cover a wide range of modern scattering experiments in the structure and dynamics of polymers and those of phase separation. It will undoubtedly be a constant companion and ready reference for those in polymers and related areas, novices and experts alike.

It starts with the scattering of a single particle and multiple particles, thermodynamic interactions of the particles, and the structure factor of an isolated polymer chain. Then, it develops the fluctuations in space and time, the van Hove space-time correlation function, and eventually the dynamic structure factor. In so doing, it makes contact with a number of critical experiments by X-ray, light, and neutron scattering over the past three decades. The pivotal chapter of the book is Chapter 3, on the structure and kinetics of polymer phase separation. This is the area of polymer science to which the authors have made premier contributions since 1976, with the collaborations of their respective colleagues at the University of Michigan and NIST. In my view, this is the chapter that will be widely accepted by experts in the field as a tour de force, in terms of its foundational starting point in statistical mechanics, the range of theoretical tools for linear and nonlinear systems, and the broad extent of relevant phenomena being addressed, such as shear effect on phase behavior. It concludes with a uniquely global perspective on quasielastic light scattering of polymers in solution to deduce the diffusive dynamics. Not so parenthetically, I must note that the appendices at the end of each chapter are pedagogical masterpieces of distinction.

I wish I had owned this book when I first started to include the subject of dynamic light scattering in a graduate polymer course at Wisconsin. It is my firm prognosis that this book will stand the test of time along with Dynamic Light Scattering by Berne and Pecora (1976), the two editions of Laser Light Scattering by Ben Chu (1991 and 2007), and Polymers and Neutron Scattering by Higgins and Benoit (1997).

Hyuk Yu Department of Chemistry University of Wisconsin Madison, Wisconsin, USA

Preface

An idealistic way of understanding the nature of polymers or macromolecules probably starts from the electronic and chemical structure of “parts” or “monomers” of the polymer/macromolecule, then goes into some possible secondary or even higher order molecular structures, such as chain folding or helix formation. Meanwhile, due to the long-chain nature of polymers, many static conformational distributions and dynamic responses of these molecules can be generalized and may even be “scaled.” One may think that this generalizable nature probably has made the physics of “high polymers” simple and straightforward. However, due to the endless possibilities of chemical and structural variations in polymers, such as copolymerization, tacticities, isomers, branching, and so on, and various short-range and long-range monomer-monomer interactions and monomer-solvent interactions, such as van der Waals force, hydrogen bonding, and electrostatic interactions, the situation is actually very complicated and complex. Especially if we increase the concentration of polymers or even bring in more than one component, then we will have to deal with many-body and multi-component interactions and possibly phase transition or transitions. Therefore, some “generalized order” in the static structure and in the dynamic responses is still what we are “chasing” and trying to understand in polymer physics. More importantly, the field of polymer physics is not only interconnected with chemistry, processing, and applications, it is also inter-crossed and overlapped with biomedical science, environmental science, optoelectronics, and even information science.

All these give us more reasons to search deeper into polymer physics and look for the fundamentals and to understand these multi-scale, multi-component systems better and more wisely. This should include bringing tools and knowledge, both theoretical and experimental, to and from all the related areas and research fields.

As we look back into the simple cases of polymers, we definitely need to start from dilute solutions of polymers where the single-chain conformation and dynamics can be studied. As the number of polymer chains in a given space (or concentration) becomes higher, depending on the molecular weights and dimensions of the chains, contacts between chains become inevitable. The concentration regions of semidilute, concentrated, and bulk are naturally entered. Again, the corresponding static and dynamic properties are of concern and need to be understood. The major differences and also difficulties in these regions compared to the single-chain (or dilute solution) case is that the inter-chain (and multi-chain) interactions will not only affect the static and dynamic properties of individual chains, but also the macroscopic properties, including possible phase transition (or transitions) of the system.

In this book, we will start from the traditional light scattering for dilute solutions, and then bring in the concepts of fluctuations, correlations, and also the space-time correlation function. From there, we will introduce the intermediate structure factor and the dynamic scattering technique. We will also introduce the static structure factor and related experimental techniques such as light scattering, X-ray scattering, and neutron scattering. We will discuss the isotope labeling technique in the neutron scattering, which is particularly useful in the areas of polymers, biology, and colloids. This technique can also be used in the areas of semidilute and concentrated solutions and even in bulk systems to study the single-chain behavior under the environment of multi-chain interactions.

We will then go into multi-component systems. Again, we have to deal with the equilibrium phase behavior as well as the phase separation kinetics. We will also try to introduce the effect due to an external perturbation, such as shear flow, on the phase separation, both in the steady state of phase separation and during the separation process (kinetics). We will also touch upon some nonlinear concentration fluctuation behavior in theory as well as in experiments. Finally, we will present a more formal approach to the dynamics and dynamic scattering of polymer solutions in the last chapter.

The authors, A. Ziya Akcasu from the University of Michigan and Charles C. Han (formerly of the National Bureau of Standards/National Institute of Standards and Technology, NBS/NIST, and now at the Institute of Chemistry, Chinese Academy of Sciences, ICCAS), have worked together since 1976. A. Ziya Akcasu spent many summers during the earlier period at NBS/NIST, and both authors worked together on the problems of polymer scatterings and dynamics. During that period, many stimulating discussions were carried out with many colleagues, including Professors/Drs Issac Sanchez, Charles Guttman, Frank McCrackin, Ed DiMarzio, Eric Amis, Tim Lodge, Boualem Hammouda, George Summerfield, John King, and many others.

The most frequently asked questions are “what is the structure?”, “what is the time dependence of the structure?”, and “what are the dynamics of the polymers?” We know we are most interested in the structure (also the time evaluation of structures during some transitions) and dynamics of the scale (dimensional) element defined by the scattering wave vector, q. We also realize that we are not dealing with a top-down, deterministic process (such as the lithographic process); we are dealing with systems controlled by thermal dynamics and kinetics/dynamics, with inevitably built in (thermal) noises and fluctuations. To us scattering provided a realistic and powerful tool in measuring and finding the “order” (only a correlated order perhaps) for the system of interest without artificially discriminating against noises, defects, and uncontrollable variables. We know there are drawbacks from scattering measurements. The two major problems with scattering measurements are that: (i) by definition, the scattering measurements are carried out in the reciprocal space: it is difficult for any experimenter to visualize the structure and order and compare them with the image from the real space where we live; and (ii) this problem is compounded by another problem that all detectors we normally use in scattering experiments are “square law” detectors, which detect only energy deposited at the detector surface. By doing so, we have already lost the phase angle of the carrier wave; consequently, we cannot invert the scattering data back to the corresponding “image” in real space. Still, we believe, the “space” and “time” correlated order from the scatting measurements have a lot to offer in the understanding of the static structures of polymeric systems and materials, their kinetics (time dependence) of transitions, and the dynamics (motions and responses) of individual molecules in the system and/or the systems as a whole. This measurement of “correlated order” in space and time often provides information that is holistic and undistorted and often cannot be “captured” by any real space measurements.

One of the authors, Charles C. Han, would like to thank his assistants, Guangcui Yuan, Ruoyu Zhang, and He Cheng, especially Guangcui Yuan, for their help in putting this book together. Also thanks go to Tsing Hua University, where Charles Han's part of this book has been used and refined as teaching material in a graduate course for the past three years.

Charles C. Han and A. Ziya Akcasu

Chapter 1

Plane Waves, Scattering, and Polymers

Traditionally the description of light scattering often starts from the electromagnetic plane wave approach, and X-ray and neutron scattering often start from the approach of scattering cross-section and then the Fourier transform of the real space correlation function. Sometimes they merge together, and sometimes they remain as if they were two branches of the field of scattering. It is the intention of this book to start from the traditional approach of scattering of a plane wave, and then bring out the correlation functional equivalent. We will show the equivalence and convenience of using the second approach, meanwhile illustrating the generality of the correlation approach without losing sight of the wave nature of light, X-ray, and neutron.

1.1 Single-Particle Scattering/Multi-Particle Scattering

When a particle of a transparent, non-absorbing substance lies in a light beam, it will scatter energy from the beam, behaving as a source of secondary radiation. The secondary or scattered light depends in character on both the incident light and on the particle itself, and so serves as a source of information about both. X-rays and neutrons also have wave-particle duality; their wave property can also be used to explain the scattering phenomenon.

Polarized, monochromatic light waves propagating in free space consist of an electromagnetic disturbance, having equal electric and magnetic fields, E and H, directed at right angles to one another varying in intensity sinusoidally with time; the wave travels perpendicularly to both of these components. The general equations for the electric field and magnetic field of a plane wave can be written as:

(1.1)

(1.2)

where is the maximum amplitude of ; the maximum amplitude of ; the frequency; t the time; C the velocity of light in the medium of propagation; and x the distance in the direction of propagation.

For the present we will limit our discussion to small isotropic, spherical particles, introducing the complicating factors of size and internal structure later. When such a particle is placed in an electric field, such as a light wave, an electric dipole moment may be induced. It is generally found, for fields of the intensities and frequencies we are concerned with here, that this induced dipole moment is proportional to the field strength; if is the dipole moment and represents the magnitude of the electric field,

(1.3)

where is the proportionality constant, called the polarizability.

An oscillating dipole is itself a source of electromagnetic radiation. This new radiation is what we mean by scattered radiation. According to classical electromagnetic theory, the scattered radiation is a spherical wave, extending in all directions, but the field strength depends on the direction. The field strength is proportional to , varies as where is the distance from the observer to the dipole, and at any given value of , is proportional to , where is the angle between the dipole axis and the line joining the point of observation to the dipole. Differentiating Equation (1.3) to obtain , introducing the factor , and dividing by for dimensional correctness, we thus obtain the field strength of scattered radiation at position ,

(1.4)

Equation (1.4) shows that the scattered radiation has the same frequency as the incident light, the amplitude varying with both the distance and angle between the observer and the scattering point.

The experimental measure of the energy in a light wave is the intensity, that is, the energy that falls on to unit area (such as 1 cm2) per second. According to Poynting's theorem [1], the intensity of scattering in any direction is given by times the square of the amplitude of the vibration. By use of Equation (1.1) we thus obtain the intensity of the incident light; by use of we obtain the intensity of the scattered light. The ratio of these two intensities is