Structural Science of Crystalline Polymers: Basic Concepts and Practices 9811595607, 9789811595608

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Kohji Tashiro

Structural Science of Crystalline Polymers Basic Concepts and Practices

Structural Science of Crystalline Polymers

Kohji Tashiro

Structural Science of Crystalline Polymers Basic Concepts and Practices

Kohji Tashiro Toyota Technological Institute Nagoya, Japan

ISBN 978-981-15-9560-8 ISBN 978-981-15-9562-2 (eBook) https://doi.org/10.1007/978-981-15-9562-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

When we open the review article of Dr. C. S. Fuller written in 1939, we can find his very impressive words in the conclusion part, “The ultimate aim of research on high polymeric substances is to explain the properties of these substances in terms of their inner colloidal and molecular nature. X-ray and electron diffraction investigations, as we have seen, have furnished considerable data on this inner structure. Nevertheless, it is of the most importance that the investigator in this field does not obtain a onesided view from the past work. Nearly always a given study relates to only one phase of the general problem. For example, it is wrong to assume that, since the unit cell for cellulose has been well established, we therefore know its entire structure. In reality we have determined simply that parts of the cellulose have this structure. …” [1]. Only about 20 years after the establishment of polymer science by Dr. H. Staudinger [2], polymer scientists realized the usefulness of diffraction techniques for the study of the relationship between structure and property of polymer substances [3–6]. It is the present author’s surprise to notice that the polymer scientists in the earlier stage of polymer history had such a wonderful and ultimate purpose to understand the structure-property relation from the various hierarchical levels! In fact, as Dr. Fuller pointed out above [1], the diffraction method has been playing important roles in the progress of polymer science and industry. For example, Dr. Staudinger utilized the X-ray diffraction method for the study of monomeric sequences in polymer chains [2, 3]. The X-ray diffraction pattern of a fibrous DNA taken by Dr. Franklin [7] revealed the doubly stranded helical conformation as analyzed by Dr. Watson and Dr. Crick [8, 9]. This was in 1953. About 5 years later, stereoregular polymers were synthesized by Dr. Ziegler [10] and clarified to take the beautiful helical forms by Dr. Natta with an X-ray structure analysis technique [11]. The electron diffraction pattern from a solution-grown single crystal of gutta percha, which was measured by Dr. Storks in 1938 [6], revealed the possibility of chain fold in the polymer crystal. An image of ideal random coil in the amorphous region of a polymer solid as well as in the molten state, which was predicted theoretically by Dr. Flory [12], was experimentally proved by Dr. Schelten et al. using a small angle neutron scattering technique [13]. In this way, the contribution of the diffraction (or scattering) methods to the progress of polymer science is greatly remarkable. Exponential progress in the X-ray diffraction analysis of crystalline polymers has been v

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attained by many scientists, including Dr. Natta, Dr. Corradini, Dr. Bunn [14], Dr. Tadokoro (my teacher), and his group members [15]. Dr. Fuller’s words are not limited to such a diffraction (scattering) method only. He suggested the importance of the structural study of polymer substances from the variously different points of view. Actually, in addition to these diffraction methods, the vibrational spectroscopy or the IR and Raman spectroscopy has been playing also an important role in the structural study of polymers [16]. In particular, we can see such brilliant spectroscopic works as the discovery of trans and gauche or the internal rotations around the CC bonds by Dr. Kohlrausch [17] and Dr. Mizushima [18], the discovery of the so-called accordion mode or the longitudinal acoustic mode of n-alkanes with finite chain lengths (by Dr. Shimanouchi et al. [19, 20]), the progression bands of alkanes (by Dr. Snyder [21–24]), the theoretical calculation of the vibrational frequency-phase angle dispersion curves of polyethylene chain and crystal (by Tasumi, Miyazawa, Krimm et al. [25–27]), and so on. Of course, we cannot forget the remarkable contribution of NMR spectroscopy opened the wide windows for the characterization of stereoregular polymers and the study of conformation and thermal mobility of polymer chains in the solution as well as in the solid state [28, 29]. The tremendously large contribution can be seen also for the study of thermal behaviors of polymer solids using the thermal analysis [30], which are combined now with the observation of morphological change by the optical microscopy, atomic force microscopy, and electron microscopy [31]. The great acceleration in the computation speed and memory size has led us to the remarkable progress in the computer science for the structural study of polymers. In 1970s, the prediction of energetically stable conformations was succeeded for the polymer solutions and also for the solid state composed of aggregated polymer chains. Following it the energy calculation of the chain packing structure was developed [32]. The computer simulation is now one of the most powerful and indispensable techniques for the structural study of solid polymer substances as well as in the solutions. The computer simulation was started from the classical mechanical calculation of a simple and small molecule, but it is now applicable to the study of time-dependent molecular motions (molecular dynamics), to the prediction of aggregation state of polymer blends from the mesoscopic viewpoint, and furthermore to the quantum mechanical prediction of polymer system of a large number of atoms [33, 34]. In this way, the structural study of polymer substances needs to use the various kinds of characterization techniques in order to obtain the detailed and concrete structural concepts of polymer molecules in the solid state. The thus-clarified structure information helps us to interpret the physical property of polymer solids from the various hierarchical levels. Of course, we cannot stay at the level of the qualitative interpretation but we should understand the properties quantitatively, which may lead us to the prediction of the more excellent properties or the so-called molecular design of novel polymer compounds with the higher functionality. In order to do so, we need to master the theoretical background for the calculation of physical properties from the atomistic level including the lattice dynamics, classical mechanics, and quantum mechanics.

Preface

vii

Another key point is seen in such a remarkable progress in computer science, as mentioned above, which might lead us to a misunderstanding about the significance of the structural study. It becomes now quite easy to know the three-dimensional structural models of polymer and even its physical property. We can click a mouse only once and then the computer kindly gives us the information needed for the research of the user. However, this is exactly a black box! How can we know the processes occurring in the computation? We have to understand the principles of these processes deeply and thoroughly. If not, polymer science may become a simple computer game and no new progress will be born out. Besides, more terribly, how can we believe the reasonableness of the thus-predicted structure models? In many cases, these models cannot reproduce the experimental data enough satisfactorily. The harmonic combination between the experimental analysis and the computer simulation technique is quite important for getting the trustable structure [35]. Recently, the population of structural scientists of synthetic polymer substances is becoming lower as a global trend. Sometimes we hear such a voice that the structural analysis of polymers [15] has been already completed and so we do not need to analyze the structure anymore. This is perfectly wrong! One of the reasons might come from the abovementioned situation. Even now we do not know the correct crystal structure of many important multi-purpose polymers. We cannot answer such a naive question of how it transforms to the other crystalline forms by heating or stretching. People cannot predict the physical properties of a polymer if they do not have any detailed and concrete knowledge about its structural characteristics. We have to check how many number of crystal structures were analyzed accurately without any questions among the polymer substances utilized for industrial usages. In spite of this serious situation, when we visit a polymer company and discuss with them, we notice immediately that many industries are frustrating how to improve the property of their polymer products and how to reduce the troubles encountered in the usage of polymers. It is impossible to predict the accurate physical properties without any such reliable structure information. As typically observed in the world-wide spreading of the synchrotron facilities, the recent progress in the characterization tools has been making it possible to collect more accurate data than before, giving us a chance to perform more reliable structure analysis. It is now necessary to recheck the previously analyzed structures by using more highly qualified experimental data collected from various points of view. In order to do so, we have to absorb the basic knowledge and the techniques necessary for it. This book has been written with the purpose for many polymer scientists and young researchers to understand the basic principles necessary for the study of the hierarchical structure of synthetic polymers by using the various types of analytical instruments including the X-ray, electron, and neutron diffraction methods, the IR and Raman vibrational spectroscopies and the NMR spectroscopy in combination with the computer science and also to understand the relation between structure and physical properties viewed from the atomistic level. But, the simple and brief description of the basic principles of the individual analytical instruments cannot satisfy the readers well. It is requested by them to know the physical meanings of the equations as well as to know the detailed routes to derive these equations in a

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concrete manner. Besides, the readers may want to know how to get the experimental data, how to analyze the data, and how to interpret them in order to get the concrete images of structures. The present author has made an effort to satisfy these requests as much as possible by showing the detailed derivation processes of the equations, by describing the concrete methods for the preparation of samples necessary for the experiments, by explaining the processes of the experimental data analysis, and by talking about the utilization of these data to the structural analysis of many kinds of polymers. The readers might feel that too many case studies are introduced in this book. But, they need to understand the reasons as mentioned above. This book consists of two volumes. The first volume is to describe the principles and applications of the analytical instruments. Since the author’s territory is not very wide, the description has been limited to the fields of X-ray, electron and neutron diffractions, IR and Raman spectroscopy, computer science, and NMR spectroscopy. If the readers want to know the techniques of how to measure the molecular weight, how to observe the optical microscopic images, or how to collect and analyze the thermal data in the heating and cooling process, please refer to the corresponding professional textbooks [27, 28]. The second volume of this book is about the relation between structure and physical properties of polymer crystals as well as the bulk samples. Polymer materials are in general too complicated to clarify the relation between the bulk physical properties and the macroscopic morphology. In some parts of the volume II, the experimental and theoretical methods on how to estimate the physical properties of the crystalline region will be described in detail. Some other parts are about the relation between the macroscopic structure and the bulk property by utilizing the complex models between the crystalline and amorphous phases. By understanding the contents concretely, the reader can realize the importance of the structure analysis described in the volume I of this book. The level of this book is for the graduate course students and researchers who want to enter the research field of structural science of polymers. The structural science of synthetic polymers needs a wide range of basic and important concepts about the behaviors of polymer chains in the processes of crystallization, phase transition under an external condition, and even in the chemical reactions. Once they master these contents, it may be easier for them to extend the research target to the species of biopolymers and biomacromolecules. It is my great pleasure if the readers would master this book and widen the gate of structure science of soft materials including both synthetic polymers and biopolymers. Nagoya, Japan

Kohji Tashiro

References 1. 2.

C. S. Fuller, The Investigation of Synthetic Linear Polymers by X-rays, Chem. Rev., 26, 143– 167 (1940). H. Staudinger, Der Aufbau der Hochmolecularen Organishcen Verbindung, Hirschwaldsche Buchhandlung, Berling (1932).

Preface 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

24. 25. 26. 27.

28. 29. 30. 31.

ix

H. Staudinger, H. Johner, P. Signer, G. Mie, H. Hengstenberg, Der Polymere Formaldehyd, ein Modell der Zellulose, Z. Physik. Chem. A126, 425–488 (1927). H. Mark, Über die röntgenographische Ermittlung der Struktur organischer besonders hochmolekularer Substanzen, Ber. Dtsch. Chem. Ges., 59, 2982–3000 (1926). H. Mark, Die Mechanischen Eigenschaften der Cellulose und ihrer Derivate im Festen Zustand, Physik und Chemie der Cellulose, p. 1–66, J. Springer, Berlin (1932). K. H. Storks, An Electron Diffraction Examination of Some Linear High Polymers, J. Am. Chem. Soc., 60, 1753–1761 (1938). R. E. Franklin, R. G. Gosling, Molecular Configuration in Sodium, Nature, 171, 740–741 (1953). W. Cochran, F. H. C. Crick, V. Vand, The structure of synthetic polypeptides. I. The transform of atoms on a helix, Acta Crystallogr., 5, 581–586 (1952). J. D. Watson, F. H. C. Crick, Molecular Structure of Nucleic Acids: A Structure for Deoxyribose Nucleic Acid, Nature, 171, 737–738 (1953). K. Ziegler, E. Holzkamp, H. Breil & H. Martin, Polymerisation voniithylen und anderen Olefinen (Zuschrift 7) , Angew Chem, 67, 426 (1955). G. Natta, P. Corradini, Kristallstruktur des isotaktischen Polystyrols, Makromol. Chem., 16, 77–80 (1955). P. J. Flory, The Configuration of Real Polymer Chains, J. Chem. Phys., 17, 303–310 (1949). R. G. Kriste, W. A. Kruse, J. Schelten, Die bestimmung des trägheitsradius von polymethylmethacrylat im glaszustand durch neutronenbeugung, Makromol. Chem., 162, 299– 303 (1973). C. W. Bunn, Chemical Crystallography, An Introduction to Optical and X-ray Methods, Oxford University Press (London) (1958). H. Tadokoro, Structure of Crystalline Polymers, John Wiley & Sons (New York) (1979). Handbook of Vibrational Spectroscopy, J.M. Chalmers and P. R. Griffiths (Eds), John Wiley & Sons, Ltd, Volumes 1–6 (2002). K. W. F. Kohlrausch, Raman effect and free rotation, Z. Phys. Chem., B18, 61–72 (1932). S. Mizushima, Y. Morino, K. Higashi, Raman effect and dipole moment in relation to free rotation. I, Sci. Pap. Inst. Phys. Chem. Res. (Tokyo), 25, 159–221 (1934). S. Mizushima, T. Shimanouchi, Raman Frequencies of n-Paraffin Milecules, J. Am. Chem. Soc., 71, 1320–1324 (1949). R. F. Schaufele, T. Shimanouchi, Longitudinal Acoustical Vibrations of Finite Polymethylene Chains, J. Chem. Phys., 47, 3605–3610 (1967). R. G. Snyder, Vibrational spectra of crystalline n-paraffins: Part I. Methylene rocking and wagging modes, J. Mol. Spectr., 4, 411–434 (1960). R. G. Snyder, Vibrational spectra of crystalline n-paraffins: II. Intermolecular effects, J. Mol. Spectr., 7, 116–144 (1961). R. G. Snyder, J. H. Schachtschneider, Vibrational analysis of the n-paraffins—I: Assignments of infrared bands in the spectra of C3H8 through n-C19H40, Spectrochim. Acta, 19, 85–116 (1963). J. H. Schachtschneider, R. G. Snyder, Vibrational analysis of the n-paraffins-II. Normal coordinate calculations, Spectrochim. Acta, 19, 117–168 (1963). M. Tasumi, T. Shimanouchi, Crystal Vibrations and Intermolecular Forces of Polymethylene Crystals, J. Chem. Phys., 43, 1245–1258 (1965). M. Tasumi, S. Krimm, Crystal Vibrations of Polyethylene, J. Chem. Phys., 46, 755–766 (1967). H. Sugeta, T. Miyazawa, General Method for Calculating Helical Parameters of Polymer Chains from Bond Lengths, Bond Angles, and Internal-Rotation Angles, Biopolymers, 5, 673–679 (1967). T. Kitayama, K. Hatada, NMR Spectroscopy of Polymers (Springer Laboratory) (2004). F. Bovey, P. A. Mirau, NMR of Polymers, Acdemic Press (1996) J. D. Menczel, R. B. Prime, Thermal Analysis of Polymers: Fundamentals and Applications, John Wiley and Sons, Inc. (2008). G. H. Michler, Microscopy of Polymers, Springer-Verlag Berlin Heidelberg (2008).

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32. Computer simulation of polymers, ed. E. A. Colbourn, (Polymer science and technology series), Longman Scientific and Technical (1994). 33. T. H. Pham, R. Ramprasad, and H-V. Nguyen, Density-functional description of polymer crystals: A comparative study of recent van der Waals functionals, J. Chem. Phs., 144, 214905 (2016). 34. T. E. Gartner, lll, A. Jayaraman, Modeling and Simulations of Polymers: A Roadmap, Macromolecules, 52, 755–786 (2019). 35. K. Tashiro, Molecular Theory of Mechanical Properties of Crystalline Polymers, Prog. Polym. Sci., 18, 377–435, Pergamon Press (1993).

Acknowledgements

The author wishes to acknowledge many people who support his research life, in particular, the late Dr. Hiroyuki Tadokoro (an Emeritus Professor of Osaka University), the late Dr. Masamichi Kobayashi (an Emeritus Professor of Osaka University), the late Dr. Yozo Chatani (an Emeritus Professor of Tokyo University of Agriculture and Engineering), and the late Yasuhiro Takahashi (Professor of Osaka University) for their heartfelt guiding when the author was the doctor-course student and postdoctoral fellow of Osaka University. The author wishes to thank all the students and all the post-doctoral fellows in his laboratory and also the researchers in many companies and countries, who have collaborated with him in the various types of researches. He wishes to express his sincere acknowledgement to Drs. Shinichi Koizumi, Taeko Sato and Nobuko Hirota of Springer Japan Co. Ltd. and Bharath Kumar Dhamodharan and Muruga Prashanth Rajendran of Springer Nature, Scientific Publishing Services Ltd. India, for their kind care of the manuscript and encouragement. Finally the author thanks his family, Sawako and Masashi Tashiro and his late parents for their kind and heartfelt support for long years. As long as people have ideas… … … … Dr. Paul J. Flory

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1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Brief Description of X-ray Structure Analysis of Polymer Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Real Space and Reciprocal Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Crystal Lattice and Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Symmetry and Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Symmetry of Isolated Molecules (Point Groups) . . . . . . . . 1.3.2 Symmetry of Crystals (Space Groups) . . . . . . . . . . . . . . . . 1.4 Principle of Diffraction Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Scattering and Diffraction of X-ray Beam . . . . . . . . . . . . . 1.4.2 Diffraction Intensity and Symmetry . . . . . . . . . . . . . . . . . . 1.4.3 Fourier Transform and Phase Problem . . . . . . . . . . . . . . . . 1.4.4 Introduction of Thermal Factor . . . . . . . . . . . . . . . . . . . . . . 1.4.5 Anomalous Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Generation and Detection of X-ray . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 X-ray Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 X-ray Diffraction Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Setting of Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Sample Setting and Corrections . . . . . . . . . . . . . . . . . . . . . . 1.7 Sample Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Unoriented Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2 Oriented Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.3 Equipments Surrounding Samples . . . . . . . . . . . . . . . . . . . . 1.8 Diffraction Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 2D X-ray Diffraction Patterns of Variously Oriented Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.2 Ewald Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.3 Multiplicity of Reciprocal Lattice Points . . . . . . . . . . . . . . 1.8.4 Polarization Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 5 5 17 17 20 40 40 46 49 54 59 63 63 65 69 75 75 76 81 81 86 95 102 102 104 109 111 xiii

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1.8.5 1.8.6

1.9

1.10

1.11

1.12

1.13

1.14 1.15

1.16 1.17

Effects of Laue Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation of Integrated Intensity and Structure Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crystal Structure Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.1 Determination of Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.2 Estimation of Unit Cell and Space Group (Orthorhombic Cell) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.3 Estimation of Unit Cell and Space Group (Monoclinic Cell) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.4 Modeling of Crystal Structure and Calculation of Diffraction Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure Refinement and Least Squares Method . . . . . . . . . . . . . . . 1.10.1 Least Squares Method (General) . . . . . . . . . . . . . . . . . . . . . 1.10.2 Constrained Least Squares Method . . . . . . . . . . . . . . . . . . . Calculation of 2D X-ray Diffraction Patterns . . . . . . . . . . . . . . . . . . 1.11.1 Using Atomic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.2 Finding Cross-points with Ewald Sphere . . . . . . . . . . . . . . Characteristic Diffraction Pattern of Helical Chains . . . . . . . . . . . . 1.12.1 X-ray Diffraction Pattern of Helical Chain . . . . . . . . . . . . . 1.12.2 Several Examples of X-ray Analysis of Helical Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12.3 Prediction of Chain Conformation by Energy Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12.4 Construction of Helical Chain Models . . . . . . . . . . . . . . . . Examples of Actual Crystal Structure Analysis . . . . . . . . . . . . . . . . 1.13.1 Orthorhombic Polyethylene . . . . . . . . . . . . . . . . . . . . . . . . . 1.13.2 Trigonal Polyoxymethylene . . . . . . . . . . . . . . . . . . . . . . . . . 1.13.3 atactic Poly(vinyl Alcohol) . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13.4 Giant Single Crystal of Polymer . . . . . . . . . . . . . . . . . . . . . Characteristic Structural Features of Polymer Crystals . . . . . . . . . . 1.14.1 Tilting Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diffraction and Structure Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15.1 Thermally Induced Disordering . . . . . . . . . . . . . . . . . . . . . . 1.15.2 Disorder in Relative Height of Chains . . . . . . . . . . . . . . . . . 1.15.3 Packing Disorder of Helical Chains . . . . . . . . . . . . . . . . . . . 1.15.4 Disorder in Copolymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15.5 Kink and Streaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15.6 Faults in Stacked Layer Structure . . . . . . . . . . . . . . . . . . . . 1.15.7 Domain Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15.8 Disorder in Polymer Blends . . . . . . . . . . . . . . . . . . . . . . . . . 1.15.9 Disorder of the Second Kind (Paracrystal) . . . . . . . . . . . . . Crystal Structure Analysis Using Powder Diffraction Data . . . . . . . X-ray Analysis of Amorphous and Liquid Structures . . . . . . . . . . . 1.17.1 Randomly Oriented Gas Molecule . . . . . . . . . . . . . . . . . . . . 1.17.2 Aggregation of Randomly Oriented Gas Molecules . . . . .

113 126 128 128 131 135 140 145 145 149 160 160 161 163 163 170 177 183 191 192 197 201 206 214 214 224 224 231 235 237 241 243 244 245 246 253 257 257 259

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1.17.3 X-ray Diffraction from Liquid . . . . . . . . . . . . . . . . . . . . . . . 1.17.4 Amorphous Solid State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.17.5 Actual Calculation of g(r) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.18 Degree of Crystallinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.19 Degree of Crystallite Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.19.1 Simple Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.19.2 Meridional Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.19.3 Orientation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.19.4 Higher Order Parameters of Crystal Orientation . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

262 265 265 267 270 270 272 273 274 278

2 Structure Analysis by Wide-Angle Neutron Diffraction Method . . . . 2.1 Neutron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Neutron as Wave and Particle . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Cross-Sectional Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Collection of Neutron Diffraction Data . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Atomic Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Crystal Structure Analysis by Neutron Diffraction Data . . . . . . . . . 2.3.1 High-Density Polyethylene . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Polyoxymethylene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Atactic-PVA and Its Iodine Complex . . . . . . . . . . . . . . . . . . 2.3.4 Deformed Electron Density Distribution . . . . . . . . . . . . . . 2.4 Structure Analysis by TOF Method . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Principle of TOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Data Collection by TOF Method . . . . . . . . . . . . . . . . . . . . . 2.4.3 Fiber Diffraction Pattern by TOF Method . . . . . . . . . . . . . 2.4.4 TOF Neutron Diffraction Data Analysis of Single Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

287 287 287 288 291 291 293 294 295 299 301 305 309 309 313 314

3 Structure Analysis by Electron Diffraction Method . . . . . . . . . . . . . . . . 3.1 Principle of Electron Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Electron Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Electron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Electron Microscopy (TEM, SEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Transmission Electron Microscope (TEM) . . . . . . . . . . . . . 3.2.2 Scanning Electron Microscope . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Sample Preparation for TEM . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Crystal Structure Analysis by ED . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Polyethylene Single Crystal . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Polyoxymethylene Whisker . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

321 321 321 323 326 326 331 332 332 332 337 339

318 319

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4 Small-angle X-ray Scattering Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Small-angle Scatterings and Hierarchical Structure . . . . . . . . . . . . . 4.2 Density Difference and SAXS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 X-ray Scattering and Density Difference . . . . . . . . . . . . . . 4.2.2 SAXS of Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Aggregated Particle Systems . . . . . . . . . . . . . . . . . . . . . . . . 4.3 2D-SAXS Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Simulation of SAXS Pattern . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 SAXS Simulation by MC Method . . . . . . . . . . . . . . . . . . . . 4.4 SAXS Measurement System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 SAXS Instruments and Slit System . . . . . . . . . . . . . . . . . . . 4.4.2 Setting of WAXD Detector on the SAXS Instrument . . . . 4.4.3 Samples and SAXS Data Treatments . . . . . . . . . . . . . . . . . 4.4.4 Contrast Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Examples of SAXS Data Analysis of Polymers . . . . . . . . . . . . . . . . 4.5.1 Lamellar Insertion in Melt-Isothermal Crystallization of POM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Higher Order Structural Change and Phase Transition of it-Polybutene-1 . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Change of Rg in the Crystallization from the Melt . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

341 341 342 342 343 352 366 368 374 375 375 380 380 385 387

5 Structure Analysis by Vibrational Spectroscopy . . . . . . . . . . . . . . . . . . . 5.1 Molecular Vibrations and Vibrational Spectra . . . . . . . . . . . . . . . . . 5.1.1 Energy of Molecule and Infrared Spectra . . . . . . . . . . . . . . 5.1.2 Transition (Perturbation Theory) . . . . . . . . . . . . . . . . . . . . . 5.1.3 Rotational Spectrum of Diatomic Molecule . . . . . . . . . . . . 5.1.4 Vibration–Rotation Spectra of Diatomic Molecule . . . . . . 5.1.5 Vibrational Spectra of Polyatomic Molecules . . . . . . . . . . 5.1.6 Raman Spectral Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.7 Group Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Spectrometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 IR Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 How to Control FTIR Spectrometer . . . . . . . . . . . . . . . . . . . 5.2.3 Raman Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Sample Preparation for IR Spectroscopy . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Oriented Film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Powder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Key Points for Spectral Measurements . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 IR Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Raman Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Various IR Spectral Measurement Methods . . . . . . . . . . . . . . . . . . . 5.5.1 Transmission Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

399 400 400 405 409 413 419 421 427 428 428 435 443 447 447 449 449 449 452 452 454 455 455

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5.5.2 Polarized IR Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Reflection Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Diffusion Refection Spectrum . . . . . . . . . . . . . . . . . . . . . . . 5.5.5 IR-RAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.6 IR Microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Various Methods of Raman Spectral Measurements . . . . . . . . . . . . 5.6.1 Polarized Raman Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Liquid Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Oriented Solid Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.4 Resonance Raman Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.5 Raman Spectra of Surface (SERS) . . . . . . . . . . . . . . . . . . . . 5.6.6 Raman Microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Normal Modes and Symmetry (Factor Group Analysis) . . . . . . . . . 5.7.1 Factor Group Analysis of Molecules . . . . . . . . . . . . . . . . . . 5.7.2 Factor Group Analysis of 1D Crystal Lattice . . . . . . . . . . . 5.7.3 Factor Group Analysis of 3D Crystal Lattice . . . . . . . . . . . 5.7.4 Symmetry Relation Between the Isolated Chain and the 3D Crystal Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.5 External Lattice Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Normal Coordinates Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Infinitely Repeated 1D Crystal Lattice . . . . . . . . . . . . . . . . 5.8.2 Finite 1D Crystal Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.3 GF Matrix Method (Isolated Molecule) . . . . . . . . . . . . . . . 5.8.4 Lattice Dynamics of Crystals . . . . . . . . . . . . . . . . . . . . . . . . 5.8.5 Application of Lattice Dynamics to Polyethylene Single Chain (1D Crystal) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.6 Application of Lattice Dynamics to Orthorhombic PE 3D Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.7 How to Adjust the Calculated Frequencies to the Observed Ones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Vibrational Frequency-Phase Angle Dispersion Curves . . . . . . . . . 5.9.1 Application to 1D Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.2 Application to Polyethylene Single Chain . . . . . . . . . . . . . 5.9.3 How to Use the Dispersion Curves . . . . . . . . . . . . . . . . . . . 5.10 Analysis of Vibrational Spectra Characteristic of Polymers . . . . . . 5.10.1 Disorder and Vibrational Spectra . . . . . . . . . . . . . . . . . . . . . 5.10.2 Band Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.3 Band Width and Molecular Motion . . . . . . . . . . . . . . . . . . . 5.10.4 Progression Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.5 LAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.6 Critical Sequence Length . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.7 Evaluation of Orientation of Polymer Crystals . . . . . . . . . 5.11 Vibrational Anharmonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11.1 Anharmonicity and Spectra . . . . . . . . . . . . . . . . . . . . . . . . . .

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456 458 465 466 469 472 472 472 474 481 483 486 487 487 495 499 500 507 508 508 510 514 529 532 544 548 549 550 551 556 560 560 566 572 575 589 597 602 611 611

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5.11.2 Fermi Resonance and Raman Spectra of Polyethylene Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Circularly Polarized Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12.1 Circularly Polarized Raman Spectra . . . . . . . . . . . . . . . . . . 5.12.2 Circularly Polarized IR Spectra . . . . . . . . . . . . . . . . . . . . . . 5.13 Brillouin Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13.1 Principle of Brillouin Scattering . . . . . . . . . . . . . . . . . . . . . . 5.13.2 Analysis of Brillouin Spectra . . . . . . . . . . . . . . . . . . . . . . . . 5.14 Practical Analysis of IR/Raman Spectra of Polymer Crystals . . . . . 5.14.1 Polyethylene (H/D) Blends . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14.2 Structural Regularization of Polyethylene in Melt-Crystallization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14.3 Vibrational Spectra of Poly(Vinylidene Fluoride) and its Copolymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.15 Simultaneous Measurement System . . . . . . . . . . . . . . . . . . . . . . . . . . 5.15.1 WAXD/SAXS Simultaneous Measurement . . . . . . . . . . . . 5.15.2 WAXD/SAXS/Raman Simultaneous Measurement . . . . . 5.15.3 WAXD/SAXS/IR Simultaneous Measurement . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Computer Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Significance of Computer Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Molecular Mechanics (MM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Potential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Energy Calculation of Crystal Lattice . . . . . . . . . . . . . . . . . 6.2.3 Minimization Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Prediction of Chain Conformations . . . . . . . . . . . . . . . . . . . 6.2.5 Prediction of Packing Structure . . . . . . . . . . . . . . . . . . . . . . 6.3 Molecular Dynamics (MD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Principle of MD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Statistical Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Physical Parameters Obtained by MD . . . . . . . . . . . . . . . . . 6.3.4 Actual MD Calculation Process . . . . . . . . . . . . . . . . . . . . . . 6.4 Monte Carlo Method (MC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Examples of MC Calculation . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Simulation of 2D SAXS Pattern . . . . . . . . . . . . . . . . . . . . . . 6.5 Quantum Mechanical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Valence Bond Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Molecular Orbital Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.4 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.5 Application of MO Theory to Polymer System . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

612 617 617 619 623 623 625 629 629 634 635 647 647 648 650 652 661 661 663 663 667 670 673 675 676 676 680 683 684 692 694 696 696 696 698 703 720 725 730

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7 NMR Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Chemical Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Spin-Spin Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 FT-NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Interpretation of NMR Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 1 H NMR Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 NMR Spectra of n-Alkanes . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 13 C NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Relaxation Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 2D NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Solid-State NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Broad Line NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 CP-MAS NMR Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.3 Solid-State 13 C NMR Spectra of Chain Molecules . . . . . . 7.7 NMR Spectra of the Other Spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 NMR Spectral Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

733 733 734 735 737 742 742 743 744 749 753 753 753 756 757 764 765 766

8 Phase Transition Behavior of Polymer Crystals . . . . . . . . . . . . . . . . . . . 8.1 Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Phase Transition of Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Isotactic Polybutene-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Polyoxymethylene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 PVDF and VDF Copolymers . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 Poly(Ethylene Imine)-Polymer Complexes . . . . . . . . . . . . 8.2.5 Structure and Morphological Change of Poly(3-Hydroxybutyrate) . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.6 Two-Stage Transitions of Aliphatic Nylons . . . . . . . . . . . . 8.2.7 Water-Induced Phase Transitions of Poly(Ethylene Imine) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.8 PEO-PE Diblock Copolymers . . . . . . . . . . . . . . . . . . . . . . . 8.2.9 Stress-Induced Phase Transition of Poly(Tetramethylene Terephthalate) . . . . . . . . . . . . . . . . 8.2.10 Photo-Induced Solid-State Polymerization Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

769 769 773 773 780 783 786 789 792 794 796 802 804 808

9 Character Tables Useful for Structural Science of Crystalline Polymers and Their Related Compounds . . . . . . . . . . . . . . . . . . . . . . . . . 813

Chapter 1

Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Abstract Crystal structure is one of the most important basic information for understanding the physical behavior of a crystalline polymer from the atomic level. The crystal structure has a high or low symmetry, which systematically connects the units in the crystal lattice. The X-ray diffraction occurs from the aggregation of atoms under such a symmetry relation. For solving the crystal structure based on the X-ray diffraction data, we need to master the various techniques how to prepare the samples, how to measure the accurate X-ray diffraction data, how to analyze the thus-collected X-ray diffraction data, and how to interpret the analyzed results. Since the crystalline polymers give quite limited number of diffraction spots of rather broad shapes, we need to know the special methods to index the observed diffraction spots, to speculate the initial structure models, and to refine the crystal structure. Concrete methods to understand the X-ray diffraction patterns characteristic of polymer crystals, including helical conformation, tilting phenomenon, structure disorder, crystallinity, orientation, amorphous phase, etc. will be learned concretely. Keywords Symmetry · X-ray diffraction · Crystal structure analysis · Helix · Disorder

1.1 Brief Description of X-ray Structure Analysis of Polymer Crystals Different from the process of the modern X-ray crystallographic method (Fig. 1.1), most of the X-ray structure analysis of polymer crystals follows the classical way shown in a flowchart of Fig. 1.2 [1–5]. The two-dimensional (2D) X-ray diffraction data collected for the uniaxially oriented sample is usually used for the crystal structure analysis of the polymer crystal. Of course, the one- or three-dimensional data are also useful for this purpose. The X-ray diffraction data are the Fourier transform of the crystal structure, as will be explained later, and the information of the reciprocal lattice is obtained by analyzing the diffraction data. At first, the (x, y) positions of the observed spots on the 2D diffraction image are read out manually or using a software. These (x, y) coordinates are converted to the (ξ , ζ ) coordinates or the cylindrical coordinates of the reciprocal lattice (Fig. 1.3) [1]. The ζ values give the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 K. Tashiro, Structural Science of Crystalline Polymers, https://doi.org/10.1007/978-981-15-9562-2_1

1

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1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.1 Flow chart showing the general X-ray structure analysis

repeating period c along the chain axis (or the drawing axis): m/c = ζ, where m is the number of the layer line. Using the ξ values on the equatorial line (ζ = 0), the indices hk0 of the observed reflections can be assigned by assuming the plausible reciprocal cell parameters a*, b*, and γ*. By repeating this process for all the layer lines of the constant ζ values, all the unit cell parameters in the reciprocal lattice (a*, b*, c*, α*, β* and γ *) can be estimated with the indices hkl assigned to all the observed diffraction spots. The next stage is to evaluate the integrated intensity of the observed spots, which can be estimated using the various methods. The intensities of the hkl spots I(hkl) are converted to the absolute values of the structure factors |F(hkl)|obsd after the various corrections. By investigating some systematic rules of the observed diffraction intensities (for example, only the spots of even h index are observed for a series of h00 diffraction data), the space group symmetry can be estimated for the unit cell. By positioning the atoms in the unit cell properly but by following the space group symmetry, the initial structure model is built up (Fig. 1.4). The structure factors |F(hkl)|calc are calculated using this initial model and compared with the observed values. The model is modified so that the observed structure factors can be reproduced as well as possible. The degree of the agreement between the observed and calculated structure factors is expressed using a reliability factor (R), which is defined as R = ||F(hkl)|obsd −|F(hkl)|calc |/ |F(hkl)|obsd , where the summation  is made for all the observed diffraction spots. The perfect agreement gives R = 0. In the polymer case, the structural model can be accepted for the model giving the R value lower than 20 % though the model of the lower R value is better of course. In some cases, even the model with the low R value cannot perfectly reproduce the observed diffraction profiles along the layer lines of the 2D X-ray diffraction image. This is

1.1 Brief Description of X-ray Structure Analysis of Polymer Crystals

3

Fiber Diagram

Conversion of Diffraction Data to - Cylindrical Coordinate System

Indexing of Reflections Lp correction Evaluation of Integrated Intensity (Sepa ration of Overlapped Peaks)

p,

Observed Structure Factors (Multiplicity,Absorption correction)

p

of Peaks

Refinement of Unit Cell

Space Group Determination

Initial Model s # Direct Method # Energy Calculation

No

Structural Refinement ?

No

Yes Output of Molecular and Crystal Structures

Fig. 1.2 Flowchart showing the X-ray structure analysis of oriented polymer sample

because the R value is calculated only for the observed diffraction spots. Even when the calculated diffraction peaks of unexpectedly strong intensity are present for the wrong model, they are not taken into account in the calculation of the R factor and so the R is still low. Since the total number of the observed diffraction spots is overwhelmingly small (several tens to several hundreds at most) compared with those of the single crystals of the low-molecular-weight compounds (several thousands to ten thousands), the relatively good agreement between the observed and calculated X-ray diffraction data might not necessarily give the “reliable” structure model. The reasonableness of the X-ray-derived structure model must be checked whether it can reproduce the other experimental data such as IR and Raman spectra, for example.

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1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.3 a X-ray diffraction experiment using a flat camera, b the measured 2D X-ray diffraction pattern, c the X-ray diffraction pattern transformed from the (x, y) coordinate system (b) to the (ξ, ζ ) reciprocal cylindrical coordinate system shown in (d), e the diffraction profile along the equatorial line, and f the indexing of diffraction peaks on the equatorial line

1.2 Real Space and Reciprocal Space Fig. 1.4 a The reciprocal unit cell, b the derived unit cell of the real lattice with the symmetries, and c the packing of the polymer chains in the real lattice

5

(a) reciprocal unit cell

(b) unit cell in the real lattice

(c) packing structure

c* b* c a*

b a

1.2 Real Space and Reciprocal Space 1.2.1 Crystal Lattice and Unit Cell As illustrated in Fig. 1.5, the regularly repeated array of molecules form the complicated structure as a whole. However, even when the molecular packing is complicated, the extraction of the representative points from all the repeated units gives the simpler array of these representative points. The smallest translational repetition of these representative points can be found more clearly by drawing the unit cell frames in various ways. Among many possibilities, the unit cell of the smallest size with the mutually perpendicular axes should be chosen preferably as the most suitable candidate (of course, the consideration of the higher symmetry is important, as will be mentioned in a later section). Figure 1.6 and Table 1.1 summarize the various types of unit cell [6]. The unit cell consisting of only one representative point is called the primitive cell (P). Sometimes one representative point corresponds to the plural molecules with the different orientations. If these molecules are not equivalent to each other, even when they are quite similar in shape but not perfectly equal in geometry, one representative point does not have any symmetry in this case. If the molecules in the primitive unit cell are perfectly equivalent in geometry, they are related to each other by the symmetries. These symmetries form the space group, as will be described in a later section. In this case, a set of symmetry operations belonging to one representative point is isomorphous to the corresponding point group, which is called the factor group (see Sect. 1.3.2.3). This situation can be understood by seeing one example; the crystal structure of biphenyl shown in Fig. 1.7. Two biphenyl molecules are included in the unit cell, but they are different in orientation. Therefore, one representative point consists of these two molecules. The unit cell is of the P type. We should notice that the basic units related by the symmetries are not the molecules, but benzene rings. The four benzene rings are related by 21 helical axis, the glide plane and the point of symmetry. The basic unit related by these symmetries (one benzene ring) is called

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1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

(a) Primitive unit cell (1) one molecule = one representative point representative points

b

b a

a

(2) two molecules = one representative point

b

b a a

Fig. 1.5 a The examples of the primitive lattice. (1) One representative point contains one molecule. (2) One representative point contains two molecules. b (page 7) The example of complex unit cell. The same aggregation structure of molecules is expressed using the two types of cell. (upper) the C-centered lattice and (lower) the primitive cell

the crystallographically asymmetric unit. The more detailed explanations and some exercises are given in the next section. Figure 1.8 shows the aggregations of the representative points. In addition to the primitive lattice (P), we can define the unit cells containing the plural number of representative points. These representative points can be expressed using the primitive cell as shown in Fig. 1.5, but the unit cell consisting of the several representative points is used because of its higher symmetry. They are named the Bravais lattices

1.2 Real Space and Reciprocal Space

7

(b) Complex unit cell

C-centered cell

b

b

a a

Primitive cell

a’

b’

a’

b’

Fig. 1.5 (continued)

(including the P lattice) as shown in Fig. 1.8. The symbols A, B, or C indicate the A-, B-, or C-centered lattices, respectively. The lattice points are located at the corner and the center of the corresponding lattice plane (for example, the bc-plane in the A-centered lattice). The crystal structure of aspartic acid is one example of the Ccentered lattice (Fig. 1.9). The F or face-centered lattice contains the four lattice points at the corner and centers of the cell. One example is shown in Fig. 1.10a, ¯ The 4 points are located at crystal structure of Aluminum. The space group is Fm3m. the corner and the 3 mutually perpendicular planes (ab, bc, and ac planes). In total, the 4 Al atoms are included, each of which corresponds to the representative point. ¯ Diamond crystal belongs also to the F lattice (Fd 3m). In this case, two C atoms correspond to one representative point. Another example is NaCl single crystal as shown in Fig. 1.10c. One representative point contains a pair of one Na+ and one Cl- ions. We can distinguish these 3 cases. Once when only the representative points are drawn, these three crystals become the same. As seen in Fig. 1.11a the CsCl single crystal belongs to the primitive lattice [not the body-centered lattice of Cs (Fig. 1.11b)]. In Fig. 1.8, the symbol R indicates the rhombohedral unit cell, which will be explained in a later section. The unit cell can be defined using the six independent parameters (unit cell parameters), a, b, c, α, β, and γ as shown in Fig. 1.6. Depending on the relation between these parameters we can classify the crystal unit cells into seven classes: cubic,

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Fig. 1.6 Crystal lattices and unit cells. Refer to Table 1.1

tetragonal, orthorhombic, monoclinic, triclinic, trigonal, and hexagonal systems as listed in Table 1.1. A lattice plane (hkl) is defined as the plane which cuts the unit cell at the positions of a/h, b/k, and c/l from the origin of the cell (Fig. 1.12). The lattice plane (hkl) is called also the Miller plane. There are infinite numbers of the hkl planes arrayed in parallel in the infinitely large crystal. The perpendicular distance between the neighboring (hkl) planes is d hkl . A vector phkl is defined as the vector directing toward the normal to the (hkl) plane, and the vector size is 1/d hkl . That is to say, this vector represents the direction of the repeated lattice planes with the spacing 1/d hkl . If all these different vectors are collected together at an origin of the three-dimensional space, then the

1.2 Real Space and Reciprocal Space

9

Table 1.1 Unit cell parameters Crystal system

Unit cell

Cubic

a = b = c, α= β = γ = 90°

Tetragonal

a = b = c, α = β = γ = 90°

Orthorhombic

a = b = c, α = β = γ = 90°

Monoclinic

a = b = c, principal axis (// 21 ) = c, α = β = 90°, γ = 90° a = b = c, principal axis (// 21 ) = b, α = γ = 90°, β = 90° a = b = c, principal axis (// 21 ) = a, β = γ = 90°, α = 90°

Triclinic

a = b = c, α = β = γ = 90°

Trigonal

rhombohedral type a = b = c, α = β = γ = 90° Hexagonal type a = b = c, α = β =90°, γ = 120°

Hexagonal

a = b = c, α = β =90°, γ = 120°

(a)

(b)

(c) #2

#1

b

1

1

1

1

#3

#4

a’

Fig. 1.7 The unit cell of biphenyl crystal. The unit cell parameters are a = 8.12 Å, b =5.63 Å, c = 9.51 Å, and β = 99.1°. a Packing mode of 2 biphenyl molecules. b Four benzene rings are related by the symmetry operations. c The representative point is only one, which consists of two biphenyl molecules. The space group is P21 /a. (lower)

end points of the vectors form a kind of three-dimensional lattice as illustrated in Fig. 1.13. The thus-constructed lattice is called the reciprocal lattice. The reciprocal unit cell can be defined for this reciprocal lattice using the parameters a*, b*, c*, α*, β*, and γ *. The three basic reciprocal lattice vectors a*, b*, and c* are defined as below using the basic vectors of the real space a, b, and c. (Fig. 1.14) a∗ = b × c/V, b∗ = c × a/V, and c∗ = a × b/V

(1.1)

a = b∗ × c∗ /V ∗ , b = c∗ × a∗ /V ∗ , and c = a∗ × b∗ /V ∗

(1.2)

where V = a·(b×c) = b·(c×a) = c·(a×b) is a volume of the real unit cell and V * = a*·(b*×c*) = b*·(c*×a*) = c*·(a*×b*) is a volume of the reciprocal unit cell; V

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1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method Primitive (P)

C-centered (C)

B-centered (B)

c b

a A-centered (A)

Trigonal Rhombohedraltype (R)

Body-centered (I)

Face-centered (F)

Trigonal Hexagonal-type(P)

Fig. 1.8 Bravais lattices. Gray circles express the representative points

= 1/V *. The vector a* is perpendicular to the vectors b and c. The similar relation is seen also for the other vectors. The vector phkl can be expressed using these basic vectors as phkl = ha∗ + kb∗ + lc∗

(1.3)

The indices h, k, and l are the integers. In more general, these indices may be nonintegers. The reciprocal space can be defined as a continuous space composed of the infinite numbers of reciprocal lattice points. The reciprocal lattice points with the integer coordinates (h, k, l) are important as the points giving the X-ray diffractions from the infinitely large crystal.

1.2 Real Space and Reciprocal Space

11

Fig. 1.9 Crystal structure of aspartic acid with a monoclinic C-centered lattice (a = 18.943 Å, 4 1 ∼ b = 7.433 Å, c = 9.184 Å, and β = 123.750°). The space group C12/c1. a 4 molecules  1 → 1 are contained in one representative point, which are related by the symmetry operations;  1 →  2 (2 or C 2 // b),  1 →  3 (σg (ac|c)), and  1 →  4 (i). b The two representative (E),  points (molecular groups) are included in the unit cell, as indicated by the blue broken circles. c The structure viewed along the b axis

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1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

(a)

(c) c

c

b

b

a

a

(b)

c

b

a

¯ , b diamond (a = b = Fig. 1.10 Crystal structure of a Aluminum (a = b = c = 4.050 Å, Fm3m) ¯ and c NaCl (a = b = c = 5.620 Å, Fm3m) ¯ c = 3.556 Å, Fd 3m) Fig. 1.11 a Crystal structure of CsCl (a = b = c = 4.123 ¯ Å, Pm3m), b Cs (a = b = ¯ c = 6.140 Å, Im3m))

(b)

(a)

Cs

Cl

c

Fig. 1.12 Lattice planes (hkl) and the corresponding vector phkl

phkl

phkl

c/l

d a/h

a

b b/k

d

1.2 Real Space and Reciprocal Space

13

Fig. 1.13 Reciprocal lattice constructed with the collection of all the vectors phkl . The coordinates at the reciprocal lattice points correspond to the directions of the lattice planes in the real lattice

Now let us see some examples about the relation between the real space and the reciprocal space. The orthorhombic unit cell is considered here, which has the real space vectors a, b, and c as follows.

Fig. 1.14 Real unit cell and reciprocal unit cell a the orthorhombic cell and b the monoclinic cell with the angle γ (the c-axis is perpendicular to the sheet plane)

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1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ a 0 0 a = ⎝0⎠ b = ⎝b⎠ c = ⎝0⎠ 0 0 c

(1.4)

The volume of the unit cell is V = abc. The reciprocal lattice vectors are calculated as ⎛

⎛ ⎛ ⎞ ⎞ ⎞ 1/a 0 0 a∗ = ⎝ 0 ⎠ b∗ = ⎝ 1/b ⎠ c∗ = ⎝ 0 ⎠ 0 0 1/c

(1.5)

It is easily checked that the vector a is parallel to a* and perpendicular to b* and c* in the case of orthorhombic cell. Using these basic vectors, the reciprocal lattice vector p111 can be expressed as ⎛

p111

⎞ 1/a = 1a∗ + 1b∗ + 1c∗ = ⎝ 1/b ⎠ 1/c

(1.6)

The magnitude of the vector p111 is     p111  = 1/d111 = a∗ + b∗ + c∗      1/2 = a∗2 + b∗2 + c∗2 + 2a∗ b∗ cos γ ∗ + 2b∗ c∗ cos α ∗ + 2c∗ a∗ cos β ∗

1/2  = (1/a)2 + (1/b)2 + (1/c)2 (1.7) Let us consider an example of a monoclinic unit cell where the a-axis is parallel to the X-axis and the angle between the b- and a-axes is γ . The Y-axis is perpendicular to the a-axis in the ab-plane. The c-axis is perpendicular to a- and b-axes. As shown in Fig. 1.14b, these vectors can be expressed as given below: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ a b cos(γ ) 0 a = ⎝ 0 ⎠ b = ⎝ b sin(γ ) ⎠ c = ⎝ 0 ⎠ 0 0 c

(1.8)

The volume of the unit cell V = abc·sin(γ ). Then the reciprocal lattice vectors are given as ⎛

⎛ ⎛ ⎞ ⎞ ⎞ 1/(a · sin(γ )) 0 0 a∗ = ⎝ cos(γ )/(a · sin(γ )) ⎠ b∗ = ⎝ 1/(b · sin(γ )) ⎠ c∗ = ⎝ 0 ⎠ 0 0 1/c The reciprocal lattice vector p111 can be calculated as

(1.9)

1.2 Real Space and Reciprocal Space

15

⎛

p111

⎞ 1/(a · sin(γ )) = a∗ + b∗ + c∗ = ⎝ cos(γ )/(a · sin(γ )) + 1/(b · sin(γ )) ⎠ 1/c

(1.10)

The length of p111 is   p111  = 1/d111

1/2  = 1/(a · sin(γ ))2 + {cos(γ )/(a · sin(γ )) + 1/(b · sin(γ ))}2 + 1/(c)2 (1.11) Similarly, p110 and p110 ¯ are given below: ⎛

p110

p110 ¯

⎞ 1/(a · sin(γ )) = a∗ + b∗ = ⎝ cos(γ )/(a · sin(γ )) + 1/(b · sin(γ )) ⎠ 0 ⎛ ⎞ 1/(a · sin(γ )) = a∗ − b∗ = ⎝ cos(γ )/(a · sin(γ )) − 1/(b · sin(γ )) ⎠ 0

(1.12)

(1.13)

It must be noted here that the size of these two vectors p110 and p110 ¯ is not equal to each other because the angle γ is not equal to 90°, different from the case of the orthorhombic cell. In the case of the trigonal lattice, two types of Bravais lattice are possible; hexagonal type (H) and rhombohedral type (R). As shown in Fig. 1.15a, for the primitive rhombohedral-type (hR) Bravais lattice, the unit cell parameters show the following relations: a = b = c and α = β = γ

(1.14)

The a-, b-, and c-axial vectors are rotated around the body-diagonal axis. Another expression of the rhombohedral lattice is that of hexagonal type (hP, Fig. 1.15b). The vectors of hR and hP lattices can be converted to each other using the following relation as seen in Fig. 1.15c: ⎞ ⎛ ⎞⎛ H ⎞ −1/3 1/3 1/3 a aR R = ⎝ bR ⎠ = ⎝ 2/3 1/3 1/3 ⎠⎝ bH ⎠ = MH , H = M −1 R cR cH −1/3 −2/3 1/3 ⎛

(1.15)

The symmetric relation of these two lattices will be seen in a later section. The unit cell parameters of the hexagonal unit cell are a little special. We define the four axes (a1 , a2 , a3 , and c) as shown in Fig. 1.15b. The lattice plane is indicated as (hkml). ¯ Similarly, the planes For example, the plane A in Fig. 1.15b is expressed as (1120). ¯ B and C are (1100) and (0001), respectively.

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1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

(a) Trigonal

(b) Trigonal

Rhombohedral-type (R)

Hexagonal-type (P) plane C

plane B

plane A

a4 a3

bR

a2

a1 cR

aR

(c) cH

bR cR

aR aH

bH

Fig. 1.15 Trigonal lattices. a Rhombohedral lattice and b Hexagonal lattice. In the (b) case, the ¯ (1100), ¯ planes A, B, and C are expressed as (1120), and (0001), respectively. c The spatial relation between (a) and (b)

Here we have to note the symbols to represent the various information: the reciprocal lattice vector phkl is parallel to the direction [hkl] of the group of (hkl) planes, {hkl}, with the inversed value of the lattice spacing (= 1/d hkl ). The index to show the X-ray diffraction coming from the reflection on the {hkl} plane group is named hkl.

1.3 Symmetry and Crystal Structure

17

1.3 Symmetry and Crystal Structure 1.3.1 Symmetry of Isolated Molecules (Point Groups) The shape of a molecule can be described using a set of symmetry [7]. The aggregation of molecules or the crystal structure is also governed by a collection of the threedimensional symmetry [6]. The symmetric consideration of the structure is useful for the interpretation of the X-ray diffraction and vibrational spectral data. In this section, we start from the symmetry of an isolated molecule (point group) and proceed to the symmetry of the one-dimensional polymer chain (1D space group or line group) and then the spatial symmetry of the 2D and 3D structures of the crystal lattice (plane group and space group, respectively). Figure 1.16 shows the shapes of the various types of molecules. The molecule (a) does not have any symmetry. We can say the symmetry of this molecule belongs to the point group C 1 , which means that this molecule has only an identity symmetry element (E). The second molecule shown in (b) has a mirror plane symmetry which passes the center of the molecule to divide the molecule into the right and left half parts. The corresponding point group is C s , which consists of an identity symmetry element (E) and a mirror plane (σ ). The chloroform molecule shown in Fig. 1.16d has a threefold rotation axis which passes through the central axis of the molecule. The molecule is separated into the three parts, which are related by the 120° rotation around this axis. The rotation can be made by 120° and 240° or −120°, which transforms the first part to the second part and to the third part, respectively. As a result, the molecules generated after these rotations cannot be distinguished from each other, meaning that the molecular shape does not change at all before and after the rotation operations. The rotation of 360° returns the molecule to the original orientation. These rotational actions are called the symmetry operations and expressed as C 13 (120°), C 23 (240°), and C 33 (360°). If the coordinate of one point on the molecule is (x 1 , y1 , z1 ), then the coordinate of the second equivalent point generated by the C 13 symmetry operation can be expressed as follows: X 2 = C 13 X 1 ⎛ ⎞ ⎛ ⎞⎛ ⎞ x2 cos φ − sin φ 0 x1 ⎝ y2 ⎠ = ⎝ sin φ cos φ 0 ⎠⎝ y1 ⎠ , φ = 120◦ z2 z1 0 0 1

(1.16)

Similarly, the coordinate of the third point can be expressed as follows: X 3 = C 13 X 2 = C 23 X 1 ⎛ ⎞ ⎛ ⎞⎛ ⎞ x3 cos 2φ − sin 2φ 0 x1 ⎝ y3 ⎠ = ⎝ sin 2φ cos 2φ 0 ⎠⎝ y1 ⎠ , φ = 120◦ z3 z1 0 0 1

(1.17)

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1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.16 Point groups and symmetry elements of the various molecules. a alanine, b thionyl chloride, c trans-1,2-dichloroethylene, d chloroform, and e tetrafluoroethylene

⎛

⎞⎛ ⎞ ⎛ ⎞ cos φ − sin φ 0 cos φ − sin φ 0 cos 2φ − sin 2φ 0 C 23 = (C 13 )2 = ⎝ sin φ cos φ 0 ⎠ ⎝ sin φ cos φ 0 ⎠ = ⎝ sin 2φ cos 2φ 0 ⎠ 0 0 1 0 0 1 0 0 1 The identity operation is equal to the C 33 operation or 360° rotation. ⎛

⎞ 100 C 33 = C 1 = ⎝ 0 1 0 ⎠ 001

(1.18)

1.3 Symmetry and Crystal Structure

19

The operation matrices, C 13 , C 23 etc., are called the representations of the corresponding symmetry operations. The chloroform possesses the additional symmetry element or three mirror planes which pass through the C 3 -axis. The combination of C 3 and mirror planes gives the point group C 3v , where “v” means the mirror plane (σv ) passing though the vertical C 3 -axis. Another case is seen for trans-1,2dichloroethylene (Fig. 1.16c). This molecule possesses the C 2 rotational axis. In addition, it has a mirror plane in the horizontal direction (σh ). The point group is C 2h , the symmetry elements of which are E, C 2 , σh , and i. The “i” is a point of symmetry. By using the following operation matrices we can understand easily why the point of symmetry i is existent: ⎛

⎛ ⎞ ⎞ −1 0 0 10 0 C 2 = ⎝ 0 −1 0 ⎠ σ h = ⎝ 0 1 0 ⎠ 0 0 1 0 0 −1

(1.19)

The product of C 2 and σ h is ⎛

⎞⎛ ⎞ ⎛ ⎞ −1 0 0 10 0 −1 0 0 C 2 ∗ σ h = ⎝ 0 −1 0 ⎠⎝ 0 1 0 ⎠ = ⎝ 0 −1 0 ⎠ = i 0 0 1 0 0 −1 0 0 −1

(1.20)

The case (e) is the symmetry of tetrafluoroethylene. The three mutually perpendicular C 2 -axes and the three mutually perpendicular mirror planes are detected. A point of symmetry is also existent at the center of the molecule. The point group consisting of a main C n -axis with the n perpendicular C 2 -axes is named Dn . In the case (e), a horizontal mirror plane and a point of symmetry are added to D2 , resulting in the point group D2h . The other mirror planes are automatically generated. The combination of C n and horizontal mirror σ h operations gives another symmetry operation or the rotation-reflection operation: Sn = σ h* C n . The S2 is equal to the point of symmetry i because ⎛

⎞⎛ ⎞ ⎛ ⎞ 10 0 −1 0 0 −1 0 0 S2 = σ h ∗ C 2 = ⎝ 0 1 0 ⎠⎝ 0 −1 0 ⎠ = ⎝ 0 −1 0 ⎠ = i 0 0 −1 0 0 1 0 0 −1 It must be noted that S 24 is the twice repetition of S 4 operations, not the combination of 90° × 2 + σ h .

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1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Table 1.2 Point groups

⎛

⎞⎛ ⎞⎛ ⎞⎛ ⎞ 10 0 0 −1 0 10 0 0 −1 0 ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ S24 = S4 ∗ S4 = (σ h ∗ C 4 )(σ h ∗ C 4 ) = ⎝ 0 1 0 ⎠⎝ 1 0 0 ⎠⎝ 0 1 0 ⎠⎝ 1 0 0 ⎠ 0 0 −1 0 0 1 0 0 −1 0 0 1 ⎛ ⎞ −1 0 0 ⎜ ⎟ = ⎝ 0 −1 0 ⎠ = C 2 0 0 1

The combination of rotation and inversion is also there. Table 1.2 shows the typical point groups and the corresponding symmetry operations with their symbols.

1.3.2 Symmetry of Crystals (Space Groups) Before we consider the space group symmetry, we need to know the helical symmetry and glide-plane symmetry.

1.3.2.1

Helical Symmetry

As shown in Fig. 1.17, one unit (#1) is positioned at the coordinate (x 1 , y1 , z1 ). This unit is rotated around the z-axis by an angle θ, followed by a translation along the z-axis with a pitch p. The thus-obtained unit is named the unit #2, the coordinate of which is expressed as follows:

1.3 Symmetry and Crystal Structure

21

Fig. 1.17 Helix. a Unit #1 is rotated by θ around the c-axis and translated by p along the chain axis, resulting in the generation of the unit #2. b An example of polyoxymethylene (9/5) helix. One monomeric unit is rotated by 200o around the c-axis to generate the second monomeric unit. Do not misunderstand in such a way that the rotation angle between the neighboring monomeric units is only 40o (= 360o /9)

⎛

⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ x2 cos(θ ) − sin(θ ) 0 0 x1 ⎝ y2 ⎠ = ⎝ sin(θ ) cos(θ ) 0 ⎠⎝ y1 ⎠ + ⎝ 0 ⎠ z2 z1 0 0 1 p X 2 = R(θ )X 1 + T

(1.21)

By repeating this process, we have a series of units along the chain axis and make a helical chain form. X i = R(θ )X i−1 + T = R(θ )2 X i−2 + R(θ )T + T = . . .

(1.22)

For an isolated chain, it is possible to have a helical form or helical conformation with any values of θ and p. If the m units are included in the repeating period and they rotate by the total angle m θ (= 360° × n, n: integer), this chain is called the (m/n) helix (θ = 360° × n/m). For example, polytetrafluoroethylene takes the (13/6) helix, meaning that the 13 CF2 units are included in the repeating period 16.88 Å and the six turns are made around the chain axis [8, 9]. That is to say, the rotation angle is 6 × 360°/13 = 166° per one CF2 unit around the helical chain axis. This description of helix assumes that the helical chain is uniform. In the crystal lattice, such a uniform helical symmetry disappears sometimes and the several limited helical

22

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

symmetries are possible; 21 , 31 , 32 , 41 , 42 , 43 , 61 , 62 , 63 , 64 , and 65 helical symmetries. For example, polyoxymethylene takes the (9/5) helix in an isolated state: 9 CH2 O units are included in the 5 × 360° rotation around the chain axis. But the chain possesses only 31 helical symmetry in the crystal lattice, meaning that the 3 CH2 O units form one crystallographically asymmetric unit, and the three asymmetric units are included in the repeating period of 17.39 Å. The 3 CH2 O units are not equivalent in geometry and so the bond lengths, bond angles, and torsional angles are different more or less among them. Such a lower symmetry of the molecule causes the various phenomena, for example, the band splitting of the vibrational spectra, the appearance of forbidden reflections in the X-ray diffraction, and so on.

1.3.2.2

Glide Symmetry

For example, as shown in Fig. 1.18a, let us imagine one upward right (Ru) hand. Using the mirror reflection operation and translating by b/2 along the b-axis, the original Ru hand is changed to the downward left (Ld) hand at the b/2 different position. By repeating this process, we have a series of Ru and Ld hands along the b-axis. The mirror plane sandwiched between these two different hands is called a glide plane (σg ). In the actual crystal lattice, the mirror is needed to be defined using the normal vector perpendicular to the mirror plane. For example, in Fig. 1.18b, the mirror plane is equal to the ab-plane, and its normal vector n is parallel to the c-axis. The Ru hand is positioned above the plane and converted to the Ld hand located at the a/2 shifted position along the a-axis. This glide plane is named the a glide [σg (ab|a)]. The case of Fig. 1.18a corresponds to the b-glide [σg (ab|b)]. Since the glide plane is parallel to the paper sheet plane, the symbol of 90°-bent short line with an arrow indicating the translational direction is used as a symbol. On the other hand, in the cases of (c) and (d), the mirror plane is the ac-plane. The Ru hand changes the handedness by a mirror operation and the two types of translation are possible, along the c-direction [σg (ac|c)] or along the a-direction [σg (ac|a)]. The mirror plane is perpendicular to the paper sheet plane; the in-plane translation (along the a-axis) is expressed by a broken line and the out-of-plane translation (along the c-axis) by a dotted line. The cases (b) and (d) are the same expression “the a glide” which confuses us, and so the more exact expressions σg (ab|a) and σg (ac|a) are better. When the translation occurs in the diagonal direction, the glide plane is named the n-glide: the translation is expressed as a/2 + b/2, for example. The symbols are one-dotted broken lines and the 90°-bent line with an arrow directing to the diagonal direction. See Table 1.3.

1.3.2.3

1D Space Group (Polymer Chain)

An infinitely long fully extended polymer chain can be assumed as the onedimensional lattice along the chain axis (the “fully extended” does not mean the perfectly linear wire but the energetically stable conformation). Let us see the symmetry elements for the planar-zigzag polyethylene chain as illustrated in

1.3 Symmetry and Crystal Structure

23

Fig. 1.18 Glide planes. The normal vector n perpendicular to the mirror plane is parallel to the c-axis. a A right upward (Ru) hand is changed to a left downward (Ld) hand by a mirror (process 1). This Ld hand is translated by b/2 into the b-direction (process 2). By repeating this process (1, 2, 1’, 2’,…) a series of Ru and Ld is obtained. This glide plane is called “b-glide”. b An Ru hand is changed to Ld hand by a mirror symmetry. This Ld hand is translated along the a-direction by a/2. This glide plane is named “a-glide”. The symbol of the glide plane is expressed using the 90°-bent line with an arrow directing to the translational direction. c The Ru and Lu hands are generated by applying the mirror plane (ac-plane) and the translation toward the c-axis by c/2 shift. This is the “c-glide”. The symbol is a dotted line. d The mirror ac-plane combined with the a/2 translation along the a-axis, which is the “a-glide” with the symbol of a broken line along the a-axis. In order to express the mirror plane and the translational direction, the symbol σg (ac|a) is easier to understand, which is different from the case (c), the “c-glide” with the symbol σg (ac|c)

24

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Table 1.3 Glide planes Ta/2

Tb/2

Tc/2

(Tb+Ta)/2

Tb/2 Ta/2

(Tc+Ta)/2

Tc/2

(Tb+Tc)/2

Glide plane // sheet plane (translation direction is indicated by an arrow)

Glide plane ⊥ sheet plane (In-plane translation is indicated by a broken line. The translation toward the perpendicular direction to the sheet is expressed by a dotted line)

n-glide plane with the diagonal translation

Fig. 1.19. The z-axis is parallel to the chain axis. The x-axis is perpendicular to the z-axis and included in the zigzag plane. The y-axis is perpendicular to the xz-plane or the zigzag plane. The chain axis is equal to the 21 screw helical axis [C 2 (z)s ], by which CH2 (1) unit is moved to CH2 (2), and CH2 (2) to CH2 (3) and so on to give the zigzag chain equivalent to the original one. The zigzag chain plane is equal to the mirror plane [σ(xz)], by which the two H atoms of CH2 unit or H11 and H12 are related to each other. A plane perpendicular to the mirror plane σ(xz) is a glide plane [σg (yz|z)]. The CH2 unit is moved to the next CH2 unit by the continuous operations of mirror operation σ(yz) and translation by c/2 along the z-axis. In this case, the H atoms are moved in a different way from the case of C 2 (z)s operation. The neighboring two CH2 units can be related to each other by a C 2 rotation axis [C 2 (y)]. These two units are connected by a point of symmetry [i]. It must be noted again that the symmetrically related H atoms are different in these two cases of C 2 (y) and i. The CH2 unit itself possesses a mirror plane [σ(yx)] and a C 2 (x)-axis, which relate the H atoms in a different way. In summary, a polyethylene zigzag chain has three C 2 -axes, three mirror (glide) planes, and a center of symmetry. This set of symmetry elements is repeated by translating the units along the chain axis. In other words, the whole symmetries are expressed as a direct product between the set of translational vectors and the set of point symmetry group. {Symmetry operations of PE chain} = {translations} ⊗ {E, 3C2 , 3σ, i}

(1.23)

The latter or the set of the symmetry elements without translational operations is called the factor group. The factor group of a PE chain is isomorphous to the point group D2h . Another example of an isolated chain is polyoxymethylene (POM) (Fig. 1.17). This polymer takes a (9/5) helical conformation as an approximation [10]. [Strictly speaking, the (25/16) conformation is more correct [11, 12].] The nine CH2 O monomeric units are included in a repeating period and in five turns. If this helical

1.3 Symmetry and Crystal Structure

25

(a)

σ (xy )

σ (xz )

C 2(x)

C 2s(z )

σ g(yz )

i C 2(y)

x z

c y

(b)

Fig. 1.19 a Symmetry of infinitely long planar-zigzag polyethylene chain. One repeating period includes the 2 CH2 units. The symmetry elements contained in the fiber period form the factor group. The whole symmetry group is a product of the factor group and the translational group (T z , T 2z , T 3z ,…). b Symmetry of infinitely long planar-zigzag poly(vinylidene fluroride) form II. (left) glide-plane symmetry. (right) 21 screw symmetry

form is assumed to be uniform, these units are rotated by 5 × 360°/9 monomeric units = 200° per one monomeric unit. The chain axis is equal to the ninefold helical axis C s9 . The nine C 2 rotation axes pass through the O atoms, which are perpendicular to the C s9 -axis. The factor group is isomorphous to the point group D9 . In the crystal lattice, the C s9 helix symmetry disappears and the three monomeric units become one crystallographically asymmetric unit and the POM chain takes a 31 helical axis (C s3 ). The factor group is now C 3 . Poly(vinylidene fluoride) form II takes a glide-plane symmetry, as shown in Fig. 1.19b [13, 14]. The CH2 CF2 unit is reflected by a mirror plane followed by

26

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

a translation by c/2 along the chain axis. You cannot confuse the glide symmetry and the 21 screw helix. In the latter case, one CH2 CF2 unit rotates by 180° around the chain axis followed with the translation of c/2 along the chain axis. These two different symmetries give the quite different conformations. The C atom of the R configuration is transformed to the C atom of the S configuration if the glide symmetry exists in the chain. If the 21 screw helix is applied, the carbon atom takes always only the R (or S) configuration along the chain axis. The torsional angles should be as follows: ¯ Glide plane symmetry T G T G 21 helical symmetry T C T C where T (trans) = 180°, G (gauche) = 60°, and C (cis) = 0°. (The torsional angles are automatically fixed to G and C, respectively, as long as one CC torsional angle is perfectly T (=180°). Refer to the equations of helical parameters. (Sect. 1.12.4))

1.3.2.4

3D Space Group Symmetry

We have the 1D, 2D, and 3D space group symmetries. The 2D space group is about the symmetry relation of the 2D structure, which is a projected structure of the 3D structure along a particular axis. The 1D space group was already discussed just above. Let us see the 3D space group symmetries here. In Fig. 1.6, we have learned the unit cells of the various crystal systems. The unit cell shape does not necessarily correspond to the unit cell symmetry. In the case of the monoclinic system, for example, the monoclinic lattice with β = 90°, similar to the orthorhombic lattice, is possible. The same situation can be seen for the triclinic system. An important point to distinguish these crystal systems is the difference in the symmetry of the lattice, as will be described later. As an example, let us see the space group P21 /c of the monoclinic lattice (Fig. 1.20) [6]. Figure 1.20a is the structure projected along the b-axis. The b-axis is a unique axis, and the 21 screw axis exists along the b-axis. The a- and c-axes do not have any symmetries except E (identity). The full name of the space group can be written as P 1 21 /c 1, where the three terms “1”, “21 /c”, and “1” indicate the symmetry elements along the a-, b-, and c-axes, respectively. The origin is a left upper corner. The a-axis is along the horizontal direction and the c-axis is oriented downward. The angle between a- and c-axes is β. The small open circles show the centers of symmetry located at (0, 0, 0), (0.5, 0, 0), and so on. The 21 screw axes are positioned at the (0, 0, 0.25), (0.5, 0, 0.25), and so on as indicated by a solid ellipsoidal shape with short tails. The normal to the glide plane σ g is parallel to the b-axis. One crystallographically asymmetric unit is copied to the other asymmetric units at the different positions by applying the symmetry operations using the abovementioned symmetry elements. The unit #2 is generated from the unit #1 by applying a point of symmetry. If the unit #1 is an optically active right-handed upward (Ru) molecule, the #2 should be a left-handed downward (Ld) molecule. The mirror image is indicated by a comma “,”

1.3 Symmetry and Crystal Structure

27

Fig. 1.20 a The unit cell projected along the b-axis. The open and sold circles with number are the crystallographically asymmetric units. b The unit cell projected along the a-axis. c The unit cell along the c-axis. d The relative heights of the units #1–#4 along the b-axis. Modified from Ref. [6] with permission of the International Union of crystallography, 1983

inside the circle [see (d)]. The unit #3 is generated by the 21 screw symmetry operation from the unit #2. The unit #3 is also the left-handed downward (Ld) molecule, which locates at the different height by b/2. The unit #4 is a right-handed upward molecule, which is generated from the #3 unit by a point of symmetry. The point #4 is generated also from the #2 by applying a glide-plane symmetry with the c/2 translation along the c-axial direction. The #4 point is generated from #1 by the 21 screw symmetry. Figure 1.20b shows the projected cell along the a-axis. The cp is equal to c·sin β. The center of symmetry is located at the origin. The 21 screw axis is parallel to the b-axis as shown by a horizontal arrow with one-side wing. The units #1 and #4 and the units #2 and #3 are related by the 21 helices. The normal to the glide plane is parallel to the b-axis and the translation is made on the plane along the c-axis. The broken line indicates the translation on the sheet plane. The units #1 ad #4 are related to the units #2 and #3 by the glide planes. Figure 1.20c is the unit cell projected to the c-axis. The symbol of the glide plane is now changed to the dotted line, indicating that the translation must be made to the direction normal to the sheet plane. Concretely, the units #2 and #4 (#1 and #3, also) are related by a dotted line (translation into the vertical direction from the sheet). It must be noted that the glide plane is at the height 1/4 from the origin along the b-axis (as indicated by the symbol “1/4” in (a)). As a result, the height of #2 is –y and that of #4 is y + 1/2.

28

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

In this way, the four units in the unit cell are generated using the different four types of symmetry operations. The fractional coordinates of these four units are given below: #1 (x, y, z), #2 (-x, -y, -z), #3 (x, -y + 1/2, z + 1/2), #4 (-x, y + 1/2, -z + 1/2) These information are given in the International Tables for Crystallography, volume A [6]. Here we need to notice that these coordinates are expressed using the fractional coordinates, which are different from the orthogonal Cartesian coordinates. ********************************************

As illustrated in Fig. 1.21, the axes x, y, and z of the fractional coordinate system are defined along the a-, b-, and c-axes of the unit cell. In the case of an orthogonal unit cell, the Cartesian coordinate axes and the fractional coordinate axes are parallel. The coordinate of a central blue-color point is expressed as follows: Cartesian coordinate [a/2, b/2] Fractional coordinate [1/2, 1/2]. In the case of a monoclinic cell with the principal c-axis, the angle γ is not necessarily 90°. As a result, the XY-coordinate axis and the xy-coordinate axis are not parallel to each other. The fractional coordinate of the central point is still the same as the case of the orthogonal cell. But the Cartesian coordinates are now changed as

Fig. 1.21 Relation between the Cartesian and fractional coordinate systems. a the orthogonal cell, b the monoclinic cell, and c the general case

1.3 Symmetry and Crystal Structure

29

given below: Cartesian coordinate [a/2 + (b/2) cos(γ ), (b/2) sin(γ )] Fractional coordinate [1/2, 1/2]. Including the c-axial component, the general relation between the fractional coordinate (x, y, z) and the Cartesian coordinate (X, Y, Z) of the blue particle is expressed as follows. The a-axis and the X-axis are assumed parallel. The b-axis is in the XYplane. On the basis of the Cartesian coordinate, the unit cell vector a is expressed as ⎛ ⎞ ⎛ ⎞ a aX a = ⎝ aY ⎠ = ⎝ 0 ⎠ aZ 0 Similarly, the b vector is given as ⎞ ⎛ ⎞ b cos(γ ) bX b = ⎝ bY ⎠ = ⎝ b sin(γ ) ⎠ bZ 0 ⎛

As for the c vector, there are the following three relations: a · c = ac · cos(β) = aX cX b · c = bc · cos(α) = bX cX + bY cY c · c = c2 = cX2 + cY2 + cZ2 From them, we have ⎛

⎞ ⎛ ⎞ cX c cos(β) c = ⎝ cY ⎠ = ⎝ c[cos(α) − cos(β) cos(γ )]/ sin(γ ) ⎠ cZ V /(ab · sin(γ )) The whole volume V = c·(a × b) = cZ ab · sin(γ ) = abc(A)1/2 where A = 1 − cos2 (α) − cos2 (β) − cos2 (γ ) + 2 cos(α) cos(β) cos(γ )

30

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

The Cartesian coordinate vector R and the fractional coordinate vector r of one point in the unit cell are related to each other in the following way: ⎞ RX R = ⎝ RY ⎠ = rx a + ry b + rz c RZ ⎛

=



⎛ ⎞ rx a b c ⎝ ry ⎠ = M r rz

⎞ ⎛ ⎞ a b cos(γ ) c · cos(β) aX bX cX ⎟ ⎜ ⎟ ⎜ M = a b c = ⎝ aY bY cY ⎠ = ⎝ 0 b sin(γ ) c[cos(α) − cos(β) cos(γ )]/ sin(γ ) ⎠ 0 0 V /[ab sin(γ )] aZ bZ cZ



⎛

By calculating the inversed matrix M −1 , we have another relation r = M −1 R ⎛

⎞

⎛

⎞ ⎞⎛ rx (1/a) − cos(γ )/[a · sin(γ )] bc[cos(α) cos(γ ) − cos(β)]/[V sin(γ )] RX ⎜ ⎟ ⎜ ⎟ ⎟⎜ 1/[b sin(γ )] ac[cos(β) cos(γ ) − cos(α)]/[V sin(γ )] ⎠⎝ RY ⎠ ⎝ ry ⎠ = ⎝ 0 rz RZ 0 0 ab · sin(γ )/V

*************************************************** Now let us pack the polymer chains in the monoclinic unit cell of the space group P21 /c. We treat here, as an example, the case of isotactic polypropylene α2 phase with this space group symmetry (see Fig. 1.22) [15]. The energetically most stable chain conformation takes a 31 helical symmetry in the isolated state (not in the crystal lattice) [16]. The right-handed chain is named R and the left-handed chain L. Fig. 1.23 shows the illustration of the R and L chains. A ribbon is attached on the chain. In the case of the R chain, the ribbon proceeds with a right-upward slope or a right-handed screw. For the L chain, the ribbon goes upward to the left direction. The R and L chains might have the side groups. Depending on the orientation of the side group, we can define the upward and downward R chains, Ru and Rd, respectively. Similarly, the Lu and Ld can be defined. Be careful that the helical handedness of the skeletal chain is the same between the Ru and Rd. Another indication of the upward and downward directions of a helical chain is to use the direction of a vector CH2 →CH(CH3 ) along the skeletal chain.

Fig. 1.22 Crystal structure of isotactic polypropylene (α 2 phase) with the space group P21 /c

1.3 Symmetry and Crystal Structure

31

Fig. 1.23 Helical chains of isotactic polypropylene. The right-handed and left-handed chains are denoted as R and L, respectively. The direction of side group (methyl) is used to denote the direction of chain: upward (u) and downward (d)

When an Ru chain is input into the unit cell, then we have a crystal structure consisting of the four chains as shown in Fig. 1.22. These four chains are Ru, Rd, Lu, and Ld chains created from the Ru chain using the different symmetry operations. In this example, we have to notice that the 31 helical chains do not have any symmetry (except E) in the unit cell, although they have the 31 helical symmetry in the isolated state. In other words, all the monomeric units contained in one chain are symmetrically independent in the crystal lattice. The bond angle C–C–C, for example, can take the various values independently, which are different among the originally equivalent three monomeric units. The symmetry at a crystal lattice site is named a site group symmetry. In this case, it is C 1 , because no symmetry element (except E) is located at the helical chain. Therefore, we have a relation of the symmetry groups among the isolated chain, a helix on a lattice site and the crystal lattice as shown below:

We check again the meaning of factor group. As already mentioned, the factor group is the group of symmetry operations included in one translational unit. For example, the crystal lattice symmetry is expressed as a direct product between the point group C 2h = {E, C2 (b), i and σ(ac)} and the group of translational operations T = {T x , T 2x , T 3x ,…, T y , T 2y , T 3y ,…, and T z , T 2z , T 3z ,…}, where T x is the translation by a pitch of a-axial length along the x (or a)-direction and T 2x is the translation by 2a pitch. The symmetry group of the crystal = C 2h ⊗T. This equation means that the group of symmetry elements is copied by the translational symmetries along all the unit cell axes. The symmetry group without the translational symmetries T is named the factor group. The correlation between the isolated chain symmetry,

32

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.24 The space group Pbca. By putting the unit #1, you can generate the other seven units in the unit cell. Modified from Ref. [6] with permission of the International Union of Crystallography, 1983

site symmetry, and space group symmetry is important in the interpretation of the behaviors of the chains in the data analyses of X-ray, IR, Raman spectra, and so on. Now, let us check another higher symmetry case. Figure 1.24 shows a unit cell of the orthorhombic system with the space group Pbca. The full name is P21 /b 21 /c 21 /a, the three terms “21 /b”, “21 /c”, and “21 /a” are about the symmetry elements along the a-, b-, and c-axes, respectively. We have the 21 screw rotation axes along the a-, b-, and c-axes. At the same time, the glide planes σg (bc), σg (ac), and σg (ab) are existent perpendicularly to the a-, b-, and c-axes, respectively. Similarly to the case of P21 /c, the combination of 21 (or 2) axis with a perpendicular mirror plane (or a glide plane) generates automatically a center of symmetry. The symbols of the glide planes must be checked carefully in the three projected figures. The “1/4” added to the 21 screw axis symbol means the height of this axis along the projection direction. It must be careful when you calculate the coordinate of a point by applying this 21

1.3 Symmetry and Crystal Structure

33

screw axis locating at the 1/4 height. The 21 screw axis consists of the two matrices: one is a rotation matrix and another is a translational vector. For example, about the 21 screw axis along the a-axis, these two matrices are given below. If the 21 screw axis is located at y = 0 position, then we have X = RX + T ⎛ ⎞ ⎛ ⎞ 1 0 0 1/2  s

R C2 (a) = ⎝ 0 −1 0 ⎠ T = ⎝ 0 ⎠ 0 0 −1 0 ⎛ ⎞ ⎛ ⎞ x + 1/2 x X = ⎝ y ⎠ → 21 (at y = 0) → X = ⎝ −y ⎠ −z z

(1.25)

(1.26)

(1.27)

Since the position of the 21 screw axis is shifted to the 1/4 along the b-axis, then the y-position must be corrected as shown below. ⎛ ⎛ ⎞ ⎞ x + 1/2 x X = ⎝ y ⎠ → 21 (at y = 1/4) → X = ⎝ −y + 1/2 ⎠ −z z

(1.28)

In this way, we have the eight points with the following coordinates in total: Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8

(x, y, z) (−x, −y, −z) (x + 1/2, y, −z + 1/2) (−x + 1/2, −y, z + 1/2) (−x, y + 1/2, −z + 1/2) (x + 1/2, −y + 1/2, −z) (−x + 1/2, y + 1/2, z) (x, −y + 1/2, z + 1/2)

The operation matrix is useful for the consideration of the symmetry elements. As pointed out before, the coexistence of a twofold rotation axis along the a-axis and a horizontal mirror plane σ(bc) gives a center of symmetry i. ⎛

⎞ ⎛ ⎞ 1 0 0 −1 0 0 C 2 (a) = ⎝ 0 −1 0 ⎠ σ(bc) = ⎝ 0 1 0 ⎠ 0 0 −1 0 01

34

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.25 The space group Pnam. The #1 and #1’ CH2 units are overlapped on the mirror plane at 1/4 height since the (1/2)CH2 unit is one asymmetric unit. The #2 and 2’ units are generated from #1 and #1’ by 21 screw axis along the c-axis. The σg (bc|n) gives the #3 and 3’ units. The #4 and 4’ units are generated by 21 screw axis from #3 or by 21 screw axis along the a axis from #1. Modified from Ref. [6] with permission of the International Union of Crystallography, 1983

⎛

⎞⎛ ⎞ ⎛ ⎞ 1 0 0 −1 0 0 −1 0 0 C 2 (a)σ(bc) = ⎝ 0 −1 0 ⎠⎝ 0 1 0 ⎠ = ⎝ 0 −1 0 ⎠ = i 0 0 −1 0 01 0 0 −1

(1.29)

As an exercise, let us build up the crystal structure of polyethylene-like polymer by applying the space group. The unit cell parameters are not known at this stage, but we know that the unit cell belongs to the orthorhombic system of the space group symmetry Pnam as illustrated in Fig. 1.25 [17]. From the observed density value (about 1 g/cm3 for PE), the 4 CH2 units are speculated to be included in the cell. At first, one CH2 unit is input at an arbitrary position. Since the mirror plane exists on the ab-plane, a half of CH2 unit is enough for building up the structure if this unit is cut into two pieces by the mirror (1 and 1’). This half-cut CH2 unit (1) is a crystallographically asymmetric unit. Applying the eight symmetry operations to this asymmetric unit, the other seven units are generated: #1 →E→ #1, #1 → σ(ab)→ #1’, #1 →21 (c)→ #2, #1 →i→ #2’, #1 → σg (bc|b)→ #3, #1 → σg (ac|a)→ #4, #1 →21 (a)→ #4’, #1 →21 (b)→ #3’. As a result, the packing structure of 4 CH2 units is obtained, which is similar to that of the orthorhombic polyethylene. The zigzag conformation is also spontaneously built up. The two chains are packed in a herringbone type. However, of course, the orientation of CH2 unit cannot be determined automatically. The correct orientation of CH2 unit is determined by changing the position and direction of CH2 unit so that the observed X-ray diffraction intensities are reproduced well by the model calculation of the intensities in addition to the exact determination of the unit cell parameters.

1.3 Symmetry and Crystal Structure

1.3.2.5

35

2D Space Group Symmetry

So far we have learned the 1D and 3D space groups. Here, let us see the twodimensional cases [6]. Simply speaking, the 2D symmetry groups are the projections of the 3D space group symmetries along the particular direction. For example, Fig. 1.26 is the case of the 3D space group P1 21 /c 1 studied already in the previous section. The b-axis is a unique axis. The two points related by a 21 screw axis are different in the b-axial height, but, when the structure is projected along the b-axis, these two units take the same height of y = 0, meaning that the translation operation along this axis is lost. That is to say, the 21 screw axis changes to the twofold rotation symmetry 2. The symbols “+” and “−“ and “1/2, −1/2” are lost. The glide plane parallel to the b-axis is lost since the mirror becomes the paper sheet itself. As a result, the 2D space group originating from P21 /c viewed along the b-axis is expressed by the symbol p2. The operation matrix is useful for understanding such a situation. P 1 21/c 1

p2 [010]

a3

projection

a2

,

#2

#2

#4

#4

y

y+1/2 #1

y 1/4

b3

#3

y+1/2

#3

#1

c2

c3

p2 [010]

projection a2 #2

#2

#1 #1

c’2 = c2/2

Fig. 1.26 The transformation from the 3D space group P1 21 /c 1 to the 2D space group p2 along the b-axis

36

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

a −1 C2s along the b − axis = ⎝ 0 0 ⎛

b c ⎞ 0 0 a → ⎠ 1 0  b 0 −1 c

 C2 =

−1 0 0 −1



For the b-glide plane with the translation along the c-axis ⎛

⎞ ⎛ ⎞ 1 0 0 0 σg (ac|c) = ⎝ 0  -1 0 ⎠ ⎝  0 ⎠ 0 0 1 1/2

 →

E=

10 01

 

0 1/2



In this way, the twofold screw axis in the 3D lattice changes to the twofold rotation axis perpendicular to the sheet plane. The glide plane changes to the identity (E) symmetry. Therefore, the unit cell becomes half the original cell and contains the two units. These changes are illustrated in Fig. 1.26. The next is for the projection along the a-axis. The operation matrix is changed to

The 21 screw axis changes to the glide plane in the 2D lattice. For the b-glide plane with the translation along the c-axis, it changes to the glide plane. ⎛

1 0  ⎝ σg (ac|c) = 0 −1 0 0

⎞⎛ ⎞    0 0  −1 0 0 ⎠ ⎝ ⎠ → σg = 0 0 0 1 1/2 1 1/2

Similarly, the point of symmetry changes to the twofold rotation axis perpendicular to the bc-plane. The final result is to generate the 2D space group symmetry p2gg as shown in Fig. 1.27. ⎛

⎞ -1 0 0  i = ⎝ 0 −1 0 ⎠ 0 0 −1

 →

C2 =

−1 0 0 −1



The symmetry along the particular projection axis is given in the International Table for Crystallography [6]. For example, the abovementioned case is shown in Fig. 1.28. It must be noted that the positions of the symmetry elements shown above are different from those given in the International Tables.

1.3 Symmetry and Crystal Structure

37

Fig. 1.27 The transformation from the 3D space group P1 21 /c 1 to the 2D space group p2gg along the a-axis

along the b axis

along the a axis

P21/c

2D symmetry

p2

p2gg

Fig. 1.28 The relation between the three-dimensional space group P21 /c and the two-dimensional space group. The International Tables for Crystallography gives this relation for the structures projected along the a- and b-axial direction (Reproduced with permission from International Tables for Crystallography, Volume A (Theo Hahn edt), p. 83, 89, 174, IUCr (1983))

Hexagonal lattice and trigonal lattice Before closing this section, we need to note the difference between the hexagonal and trigonal systems clearly. The hexagonal lattice has a sixfold rotation (or screw) axis. The trigonal lattice is a little complicated, which has a threefold rotation (or

38

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.29 The unit cell of a trigonal lattice with the space group R3. Depending on the hexagonal and rhombohedral types, the fractional coordinates are different as mentioned in the text (Reproduced with permission from International Tables for Crystallography, Volume A (Theo Hahn edt), p. 486, IUCr (1983))

screw) axis but can be classified into the ordinary primitive lattice (hexagonal type, ¯ and so on). Both of them P3, P31 , and so on) and the rhombohedral lattice (R3, R3, have only the 3 or 31 symmetry elements. They do not have the higher 6 or 61 (62 , 63 ,…) symmetry. For example, the space group R3 can be a hexagonal type with a = b = c, and α = β = 90° and γ = 120° and also a rhombohedral type with a = b = c and α = β = γ = 90° as illustrated in Fig. 1.29. The angle α (= β = γ) can take arbitrary values. In the latter case or the rhombohedral cell, the positions generated by the symmetry operations are as follows: (x, y, z), (z, x, y) and (y, z, x). They are connected by C 3 axis. When compared with the hexagonal lattice, we notice that the two other sets of these 3 coordinates are placed (spontaneously) at the corners of heights +1/3 and +2/3 (of the hexagonal lattice), as seen in the right figure of Fig. 1.29. On the other hand, the hexagonal lattice case gives the positions at (x, y, z), (−y, x – y, z), (y, −x, −x, z), (x + 2/3, y + 1/3, z + 1/3), (−y + 2/3, x − y + 1/3, z + 1/3), (y − x + 2/3, −x + 1/3, z + 1/3), (x + 1/3, y + 2/3, z + 2/3), (−y + 1/3, x − y + 2/3, z + 2/3) and (y − x + 1/3, − x + 2/3, z + 2/3). Let us see the two examples (Fig. 1.30). One monomeric unit is located at (x, y, z) position. Then, in the R3c case, the right-handed (3/1) helices and the lefthanded (3/1) helices are generated and packed side by side. All of these helices direct into the same direction, upward (or downward): Ru and Lu. On the other hand, the ¯ the one monomeric unit at (x, y, z) position generates the (3/1) helices case of R3c, of the R and L handedness. One lattice site is occupied by Ru chain and Rd chain at 50% probability. Similarly, the Lu and Ld chains are existent at one site at 50% probability. The arrangement of R and L chains is disordered with respect to the upward and downward directions, being different from the case of R3c.

1.3 Symmetry and Crystal Structure

39

These discussions of the space group symmetries are quite important when we analyze the diffraction data and vibrational data from the symmetric point of view. The concrete examples will be shown in a later section.

Fig. 1.30 a The space group R3c of trigonal lattice. The blue and orange chains are generated spontaneously once when one monomeric unit (circle) is given at a certain position. The lefthanded upward (Lu) chain is of blue color and the right-handed upward (Ru) chain is of orange color. (Reproduced with permission from International Tables for Crystallography, Volume A (Theo ¯ of trigonal lattice. At one Hahn edt), p. 524, IUCr (1983)), b (see page 40) The space group R3c lattice site, the upward and downward chains of the same handedness are located at 50% probability. The left-handed upward (Lu) chain is of pale blue color and the left-handed downward (Ld) chain is of dark blue color. The similar situation is seen also for the Ru and Rd chains (orange and brown colors). (Reproduced with permission from International Tables for Crystallography, Volume A (Theo Hahn edt), p. 538, IUCr (1983))

40

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.30 (b continued)

1.4 Principle of Diffraction Phenomenon 1.4.1 Scattering and Diffraction of X-ray Beam 1.4.1.1

Diffraction by Electrons

X-ray beam is incident on a free electron. The electric field of the X-ray electromagnetic wave interacts with the electron and then the electron is forced to vibrate at the same frequency as that of the X-ray wave. The electron has a negative charge. According to the classic electromagnetic theory, the vibrating particle emits the electromagnetic wave into all the directions. As a result, the new X-ray wave is emitted from the electron at the same frequency as that of the incident X-ray beam. This secondary X-ray scattering is called the Thomson elastic scattering, which is utilized, in general, for the X-ray structure analysis. On the other hand, small amount of an initial X-ray beam may behave also as a photon particle and collides with the electron particle. Then, some energy of the X-ray photon transfers to the electron, and the X-ray with the lower energy or with a longer wavelength is scattered in a different

1.4 Principle of Diffraction Phenomenon

41

Fig. 1.31 The interference between the two X-ray waves scattered by electrons. The difference in path length of these waves r = r · cos(φ) + r · cos(ψ) = r · (s − so ), where so and s are the unit vectors along incident and scattered directions, respectively. The scattering vector k is defined as shown in the right side

direction from the incident beam. This can be detected as a diffuse background scattering in addition to the main Thomson scattering, and is called the Compton (inelastic) scattering. In the following sections, we assume that the X-ray wave is scattered elastically and the wavelength of the scattered X-ray beams is the same as that of the incident X-ray beam. The scattered X-ray beams are assumed not to attack the other electrons to cause the further scatterings from these electrons. When an X-ray beam is incident on the two electrons, then the scattered beams from each electron may interfere with each other as illustrated in Fig. 1.31. Since the positions of these electrons are different, the scattered X-ray waves may interfere with each other as shown below. The relative position of the two electrons is r. The total difference in the optical path length between the X-ray beams emitted from these two electrons is Δr. Then we have the following equation: Etotal = E(electron 1) + E(electron 2) = Eo exp[iω(t − ro /c)] + Eo exp[iω(t − ro /c + r/c)] = Eo exp[iω(t − ro /c)][1 + exp(iωr/c)]

(1.30)

where ω is an angular frequency of the X-ray beam and c is a velocity of light. The unit vector of an incident X-ray beam is expressed by so and the unit vector along the observed direction is s, and the Δr is given as r = r · cos(ψ) + r · cos(φ) = s · r − so · r = r · (s − so ) = (λ/2π )r · k (1.31) Here the vector k is a so-called scattering vector and defined as k = (2π/λ)(s − so )

(1.32)

42

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Therefore, using a relation of ω = 2π c/λ, we have Etotal = Eo exp[iω(t − ro /c)][1 + exp(iωr/c)] = Eo exp[iω(t − ro /c)][1 + exp(ir · k)]

(1.33)

In this way, the phase difference between the scattered two X-ray waves is given as r·k. This is important for the further consideration of the scatterings by the system of many electrons.

1.4.1.2

Diffraction from an Atom

Now let us consider the scattering from one atom containing many electrons. As shown in Fig. 1.32, these electrons scatter the incident X-ray individually and these scattered waves are interfered with each other to give a synthesized wave. The summation of the waves with the phase difference r·k is given as  Etotal = Eo exp[iω(t − ro /c)]Σj exp irj · k

(1.34)

An atomic scattering factor f (k) is defined as  f (k) = Σj exp irj · k = Σj

 ρj (r) exp(ir · k)dr

(1.35)

The second equation includes the distribution of electron clouds around a nucleus. The ρj (r) is a density distribution of the j-th electron. The summation  is made over all the electrons included in the atom. The f (sin(θ )/λ) of the several kinds of atoms are given in the next page.

Fig. 1.32 The interference between the X-ray waves scattered by electrons in the atom. The total sum is the atomic scattering factor

1.4 Principle of Diffraction Phenomenon

43

H

C

N

O

Atomic number

1

6

7

8

method

HF

RHF

RHF

RHF

sin(θ) / λ (1/Å)

fH

fC

fN

fO

0

1.000

6.000

7.000

8.000

0.01

0.998

5.990

6.991

7.992

0.02

0.991

5.958

6.963

7.967

0.03

0.980

5.907

6.918

7.926

0.04

0.01

–

–

–

The approximated function of f (k) is expressed as follows [18]: 

f (sin(θ )/λ) = ai exp −bi (sin(θ )/λ)2 + c For H atom (Hartree-Fock method), a1 = 0.48992 b1 = 20.6593

a2 = 0.262003 b2 = 7.74039 a3 = 0.196767

b3 = 49.5519 a4 = 0.049879 b4 = 2.2016

c = 0.001305

For C atom (RHF method) a1 = 2.31

b1 = 20.8439 a2 = 1.02

b3 = 0.5687 a4 = 0.865

b2 = 10.2075 a3 = 1.5886

b4 = 51.6512 c = 0.2156

Figure 1.33 shows the atomic scattering factors for C and H atoms. The x-axis is sin(θ )/λ for the diffraction angle 2θ and the wavelength of an incident X-ray beam λ. This can be related to the wave vector k as below. The angle between the unit vectors so and s is 2θ . Using a definition of k, we have the k value:

 |k|2 = (2π/λ)2 |(s − so )|2 = (2π/λ)2 |s|2 + |so |2 − 2|s||so | cos(2θ ) = (2π/λ)2 [2 − 2 cos(2θ )] = (2π/λ)2 4 sin2 (θ )

(1.36)

Therefore |k| = k = (4π/λ) sin(θ )

(1.37)

In Fig. 1.33, it must be noticed that the value f (k) at k = 0 is equal to the total number of electrons included in the atom. lim f (k) = Z

k→0

(1.38)

44

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.33 Atomic scattering factors for H and C atoms. The approximate equation shown in page 43 is convenient for the calculation of scattering intensity

where Z is the total number of electrons in the atom. As the number of electrons is increased, the scattering amplitude f (k) becomes larger. The X-ray structure analysis is difficult to extract the hydrogen atomic positions in the unit cell because of too small scattering amplitude of H atom. If the unit cell contains many iodine atoms, for example, the extraction of even the carbon atoms becomes difficult since the scattering factor of iodine atom is overwhelmingly large compared with that of carbon atom.

1.4.1.3

Diffraction from Molecule

The similar discussion can be made for the molecular scattering factor f M (k). The positions of the atoms are given by rj . The f M (k) can be formulated as below by taking the phase angle between the atoms into account:  fM (k) = Σj fj (k) exp irj · k

(1.39)

The X-ray diffraction pattern given by I = |f M (k)|2 is named a molecular transform.

1.4.1.4

Diffraction from Crystal

The X-ray diffraction by a single crystal is derived in the similar way as that seen for the molecular transform. The X-ray waves scattered by all the atoms in the crystal are interfered with each other and the total amplitude of the vibrating waves is expressed using a structure factor F(k). F(k) =

 j

 fj (k) exp irj · k

(1.40)

1.4 Principle of Diffraction Phenomenon

45

s

so

O

fj rj

b r oj

t

j r oj

a Fig. 1.34 X-ray scattering from the crystal lattice. All the atoms scatter the X-ray beam and the scattered X-ray waves interfere with each other to give the structure factor of the crystal lattice. The atomic position rj of the j-th atom in the unit cell located at the vector t is expressed as rj = t + roj . The roj is common to all the unit cells

where the summation is made for all the atoms contained in all the unit cells in the single crystal. As shown in Fig. 1.34, the position of the j-th atom in a particular unit cell rj measured from a common origin of the crystal may be expressed in the following way: rj = t + roj = La + M b + N c + roj

(1.41)

Here L, M, and N are the numbers of unit cells in the a-, b-, and c-directions, respectively. The roj is the coordinate vector of the j-th atom starting from the origin (corner) of the individual unit cell; rj o = (x j , yj , zj ). The summation in Eq. (1.40) may be expressed as follows:   F(k) = Σj fj (k) exp irj · k = ΣL ΣM ΣN Σj fj (k) exp irj · k

(1.42)

The term rj ·k may be expressed using the unit cell vectors a, b, and c as follows:

 rj · k = La + M b + N c + axj + byj + czj · k = La · k + M b · k + N c · k + a · k xj + b · k yj + c · k zj Then,

46

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

 F(k) = Σj fj (k) exp irj · k = L exp(iL a · k)M exp(iM b · k)N exp(iN c · k) 

 (1.43) fj (k) exp i a · k xj + b · k yj + c · k zi ∗ j

By the way, DL = L exp(iLa · k) = {exp(a · ki) − exp[(L + 1)a · ki]}/[1 − exp(a · ki)] = exp[ik · a(L + 1)/2] sin(La · k/2)/ sin(a · k/2) (1.44) which is called a Laue term. Therefore

 F(k) = DL DM DN j fj (k) exp i a · k xj + b · k yj + c · k zj

(1.45)

If the crystal size is infinitely large, the DL and so on become unity for the condition a · k = 2π h (h : integer) and so forth. Then, we have the so-called structure factor for the unit cell in the infinitely large crystal. The structure factor

  F(hkl) = j fj (k) exp 2π i hxj + kyj + lzj .

(1.46)

In the analysis of polymer crystallite of a small size, we need to consider the contribution of Laue terms. The Laue term is important and an origin of the Lorentz factor, the estimation of crystallite size, and so on. The details will be mentioned in a later section.

1.4.2 Diffraction Intensity and Symmetry The scattering intensity of X-ray wave by a crystal is expressed using the structure factor F(hkl) given by Eq. (1.46) as follows: I (k) ∝ F(k)F ∗ (k) = |F(k)|2 ∝ |F(hkl)|2 where the effects of X-ray absorption, Lorentz factor, polarization factor, multiplicity, and so on are neglected for simplicity. The detailed description of these effects will be made in the section of the actual quantitative treatment of the X-ray diffraction data. In the present section, we focus on the structure factor in relation with the symmetry property of the crystal. The structure factor F(hkl) becomes zero when the crystal lattice has a particular symmetry. Let us see some examples here.

1.4 Principle of Diffraction Phenomenon

1.4.2.1

47

Screw Axis

If the asymmetric units are repeated by the symmetry of 21 screw axis along the c-direction, the coordinates are given as X(1) = (x, y, z) X(2) = (−x, −y, z + 1/2) The structure factor is calculated as F(hkl) = f {exp[2π i(hx + ky + lz)] + exp[2π i(−hx − ky + lz + l/2)]} (1.47) where f is a scattering factor of a repeating unit. When the reflection 00l is focused here, F(00l) = f {exp[2π i(lz)] + exp[2π i(lz + l/2)]} = f exp[2π i(lz)][1 + exp[2π i(l/2)]]

∝ 1 + exp[π il] = 1 + cos(π l) + i sin(π l) = 1 + cos(π l) When l is odd, cos(π l) is −1, and so F(00l) is zero. That is to say, the X-ray diffraction can be observed only for a series of 00l reflections with the even l values, i.e., 002, 004, and so on. This is called an extinction rule for the 21 screw symmetry. Similarly, for the 31 helical symmetry, the extinction rule is as follows: only the 00l diffraction spots with l = 3, 6, 9,… can be detected.

1.4.2.2

C-centered Lattice

As shown in Fig. 1.8, in the C-centered lattice, the crystallographically asymmetric units are positioned at (0, 0, 0) and (1/2, 1/2, 0). The structure factor is given as follows. F(hkl) = f {exp[2π i(h · 0 + k · 0 + l · 0)] + exp[2π i(h/2 + k/2 + l · 0)]} = f {1 + cos[π(h + k)] + i sin[π(h + k)]} ∝ 1 + cos[π(h + k)]

(1.48)

Therefore, the diffraction intensity becomes zero if the sum of h and k is odd. The extinction rule is h + k = odd. Table 1.4 lists up the extinction rules for the various symmetry elements. In the actual X-ray structure analysis, we have to investigate which extinction rules are possible by analyzing the observed diffraction spots. For single crystals of small molecules, the several hundreds to thousands diffraction spots can be collected in the X-ray measurement, and so the various possible extinction rules can be extracted systematically. The thus-deduced rules allow us to estimate the combination of

48

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Table 1.4 Lattice symmetries and extinction rules [6] Reflection condition

Lattice

Reflection condition Symmetry element

hkl no condition

Primitive lattice (P)

h00, h = 2n

21 , 42 helix (// a)

h+k = 2n

C-face centered (C)

0k0, k = 2n

21 , 42 helix (// b)

k+l = 2n

A-face centered (A)

00l, l = 2n

21 , 42 helix (// c)

h+l = 2n

B-face centered (B)

00l, l = 4n

41 , 43 helix (// c)

h+k+l = 2n

Body centered (I)

00l, l = 3n

31 , 32 helix (// c)

All of h, k and l are odd All-face centered (F) or even

000l, l = 6n

61 , 65 helix (// c)

−h+k+l = 3n

Rhombohedrally centered obverse setting (hexagonal)

h0l, h = 2n

(010) Glide plane with translation along the a-axis

h−k+l = 3n

Rhombohedrally centered reverse setting (hexagonal)

h0l, l = 2n

(010) Glide plane with translation along the c-axis

h−k=3n

Hexagonally centered

h0l, h+l = 2n

(010) Glide plane with translation along the diagonal direction

symmetry elements or the candidates of the plausible space groups, which are indispensable for building up the molecular packing structures of the unit cell. Table 1.5 shows one example of the actually observed reflection data which were indexed after the search of the several thousand reflections with the various plausible unit cell parameters. A series of 00l reflections shows an alternate change of integrated intensity between the odd and even indices, indicating the systematic rule of 00l with l = 2n or suggesting the existence of 21 screw axis along the c-axis. Unfortunately, it is quite difficult to apply this method to the diffraction data of synthetic polymers, in general, because the total number of the observed diffraction spots is in the order of several tens at most. Besides, the relatively easy occurrence of the packing disorder in the small crystallite of polymers makes the estimation of the space groups more difficult. Rather it might be better to speculate as many candidates of the space group symmetries as possible and to extract the suitable models by comparing the observed diffraction intensity data with those predicted from the models.

1.4 Principle of Diffraction Phenomenon

49

Table 1.5 Example of actual indexing process Index

Intensity 0.625809

Error

0

0

3

0

0

4

0

0

5

0

0

6

0

0

7

0

0

8

0

0

9

0

0

10

0

0

11

0

0

12

0

0

0

0

0

0

15

0

0

16

0

0

17

0

0

18

0

0

19

0.0329677

0

0

20

9.8087

2.84205

0

0

21

−0.316258

3.25844

212.748 1.22547 7944.68 2.8202 3381.34 −0.364189 136.263 0.359078

0.68435 2.80755 0.5734 114.127 0.879176 45.3521 1.36067 3.49532 1.04435

173.709

5.19379

13

−0.63082

1.98423

14

300.814

7.79261

1.19759 1522.03 1.67239 374.328

4.00122 31.7411 2.39052 13.2632 1.80594

1.4.3 Fourier Transform and Phase Problem 1.4.3.1

Structure Factor and Phase Angle

As for the structure factor, it must be noted that the observed diffraction intensity is given by the square of the absolute value of F(hkl). The F(hkl) is a complex consisting of the real and imaginary terms.

  F(hkl) = j fj (k) exp 2π i hxj + kyj + lzj = |F(hkl)| exp[iα(hkl)]

(1.49)

I (hkl) ∝ F(hkl)F(hkl)∗ = |F(hkl)| exp[iα(hkI )] · |F(hkl)| exp[−iα(hkl)] = |F(hkl)|2

(1.50)

In this way, the information of a phase angle α(hkl) disappears and cannot be estimated from the observed intensity I(hkl) itself. Since the electron density distribution ρ(x, y, z) in the crystal lattice is related to the structure factor in the following way:

50

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

ρ(x, y, z) = Σh Σk Σl F(hkl) exp[−2π i(hx + ky + lz)] = h k l |F(hkl)| exp[iα(hkl)] exp[−2π i(hx + ky + lz)]

(1.51)

the information of phase angle α is indispensable to obtain the electron density distribution or the molecular packing structure in the lattice. How can we estimate the phase angle α(hkl) ? In other words, how do we know the packing structure models of atoms? There are several ways to get the initial structure models. (i) One is to build up directly a model of a proper structure as we speculate in our brain, and the structure factors are calculated using this model. This process is equivalent to the process of speculating the phase angles. But we need a good intuition. (ii) The second is to utilize the energy calculation technique so that the energetically stable structure is obtained. The details will be explained in a later section. (iii) The third one is to calculate the Patterson function. The Patterson function is useful and has been utilized for a long time. (iv) The fourth idea is to utilize the direct method.

1.4.3.2

Patterson Function

Let us see the Patterson function. The function is defined as follows: P(x, y, z) = ρ(r)∧ ρ(−r) = a convolution between ρ(r) and ρ(−r)  = ρ(r)ρ(x − r)dr (1.52) which corresponds to the correlation function of density ρ(r). Since ρ(r) is a Fourier transform of structure factor, then P(x, y, z) = a convolution between the Fourier transforms of F = F[F(k)]∧ F[F(−k)] = Fourier transform of the product of F(k) and F(−k)  = |F(k)|2 exp[i(rk)]dk  ∝ Fourier transform of diffraction intensity I (k) exp[i(rk)]dk (1.53) (Note) The convolution between the Fourier transforms of f and g is equal to the Fourier transform of the product f ×g. F(f )∧ F(g) = F(f × g)

1.4 Principle of Diffraction Phenomenon

51

(b)

(a)

x1-x3

x2-x3

x2

x1

x2-x1 x1-x2

x3

x3-x2

x3-x1

Fig. 1.35 a A model to show the atomic positions in the unit cell. The electron density of the atom 2 is higher twice than those of the atoms 1 and 3. b The corresponding Patterson map

The calculated Patterson function gives the vectors connecting the electron density peaks. Let us see one example. Figure 1.35 shows the two-dimensional map of three atoms. The positions of these atoms are given as x1 , x2, and x3 . The density distribution is expressed as a sum of delta functions in the following way. Here the difference in density is also taken into account. ρ(r) = δ(r − x1 ) + 2δ(r − x2 ) + δ(r − x3 )

(1.54)

Since the Fourier transform of the delta function is given as F[δ(r − x)] = exp(−ixk)

(1.55)

we have F[ρ(r)] = exp(−ix1 k) + 2 exp(−ix2 k) + exp(−ix3 k) F[ρ(−r)] = exp(ix1 k) + 2 exp(ix2 k) + exp(ix3 k)

(1.56)

Therefore, the intensity I(k) corresponds to F[ρ(r)]∗ F[ρ(−r)] 

= exp(−ix1 k) + 2 exp(−ix2 k) + exp(−ix3 k) × 

exp(ix1 k) + 2 exp(ix2 k) + exp(ix3 k) = 6 + 2 exp[−ik(x1 − x2 )] + 2 exp[ik(x1 − x2 )] + exp[−ik(x1 − x3 )] + exp[ik(x1 − x3 )] + 2 exp[−ik(x2 − x3 )] + 2 exp[ik(x2 − x3 )]

(1.57)

We need to notice here that exp(iy) + exp(−iy) = 2cos(y), which is a real term.

52

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

The Fourier transform of this equation gives the Patterson function as follows: P(x) = 6 + 2δ(x1 − x2 ) + 2δ(x2 − x1 ) + δ(x1 − x3 ) + δ(x3 − x1 ) + 2δ(x2 − x3 ) + 2δ(x3 − x2 )

(1.58)

As seen in Fig. 1.35b, the peak positions corresponding to the vectors connecting the atoms 1, 2, and 3 can be detected in the map. In this way, the Patterson function gives us a set of the interatomic vectors, from which some packing models may be speculated although the process might be quite difficult if we do not know the molecular shape at all. The peak heights of the Patterson function map are related to the heights of the electron density (atom 2 in the above case). Therefore, the introduction of heavy atoms in the unit cell might be effective for the detection of the heavy atoms. Then the positions of the lighter atoms related to these heavy atoms may be identified subsequently. This method is made often for the structure analysis of the protein single crystal into which such heavy atoms as mercury, bromine, etc. may be doped. The actual application to the synthetic polymer was made about the example of poly(ethylene oxide)-mercury chloride complex [19].

1.4.3.3

Direct Method

The direct method needs many number of the observed diffraction spots. Therefore, the application of the direct method to the polymer substances is quite difficult. But, some successful examples can be seen in the literatures. Let us see the outline of the direct method briefly [20]. Instead of the structure factor F(hkl), the normalized structure factor E(hkl) is used which is defined as follows: 1/2

= |E(hkl)| exp[iα(hkl)] E(hkl) = F(hkl)/ εfj 2

(1.59)

where f j is the scattering factor of the atom j. The coefficient ε is a normalization factor. The above equation avoids the effect due to the difference in the atomic number or the difference in the number of electrons. The thermal parameter is also ignored for simplicity. For the reflections of large |E(hkl)| values, the following relation may be assumed for the phase angles at a high probability:   α(h) = α h + α h − h

(1.60)

Here h and h are the reflection indices. For example, if α (h ) = α (213) =100° and α (h-h ) = α (1 -2 0) = 20°, then α (h) = α (3 -1 3) = α (213) + α (1 -2 0) = 120°. Furthermore, by assuming α (101) = 30o , the phase angle of another reflection can be predicted as α (4 -1 4) = α (101) + α (3 -1 3) = 30° + 120° =150°. In this way, by assuming an initial set of the phase angles of the several reflections with

1.4 Principle of Diffraction Phenomenon

53

the relatively strong intensity, the phase angles of the other indices can be generated. The probability of these sets is calculated and the electron density distribution ρ(xyz) is calculated for the set of the phase angles with the highest probability. In this process, the various methods were constructed (σ2 -method, symbolic addition method, etc.). The peak positions of the atoms are picked up from ρ(xyz) map and then the molecular structure can be speculated, which should be the same as that predicted from the chemical formula. The reliability factor R is calculated, which is a measure of the agreement of the structure factor |F(hkl)|2 between the observed and calculated reflections. The structure is refined so that the R factor becomes lower. The explanation of the R factor was already made (Sect. 1.1). In general, the number of the observed reflections must be overwhelmingly large compared with the total number of the adjustable parameters. In many cases of synthetic polymers, the observed reflections are only several tens to several hundreds in a lucky case. The application of the direct method was tried for the several polymers. Figure 1.36 shows the example of orthorhombic polyethylene to show the positions of the carbon atoms, where the total number of observed reflections was 32 using a Mo-Kα line of a shorter wavelength (0.71 Å) [21]. The similar application was made for trigonal polyoxymethylene. The polymer chain was deformed from the regular helix when derived by the direct method. The more number of observed Fig. 1.36 The crystal structure model of the orthorhombic polyethylene obtained by the direct method using 32 observed reflection data (b), and that after refinement (a) [21]

(a) refined structure

o

b

43.0

a

(b) structure by direct method

b

50.9o

a

54

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

reflections gave the more reasonable structure. The application of a high-energy Xray beam from the synchrotron radiation makes it possible to collect one-to-two-order larger number of the observed reflections. The atomic positions can be extracted more explicitly (see Sect. 1.13.1) [12].

1.4.4 Introduction of Thermal Factor The atoms are vibrating always, the amplitude of which depends on the temperature. The structure factor of the whole crystal is, as already described in Eq. (1.40),

  F(hkl) = j fj (k) exp 2π i hxj + kyj + lzj .

(1.46)

The atomic position is rj = La + M b + N c + roj = Rn + roj The Rn is the vector to specify the n-th unit cell and is given as Rn = La + M b + N c The structure factor is now expressed as    F(k) = j fj (k) exp(irj · k) = j fj (k) exp i Rn + roj · k = n Fn (k) exp(iRn · k)

(1.61)

  The summation j is over the whole crystal. The summation n is now made for all the unit cells included in the whole crystal. The structure factor of the n-th unit cell is  Fn (k) = m fm (k) exp irom · k

(1.62)

 The summation m is made for all the atoms (m) in the n-th unit cell. Why do we need to use such a modified expression for the same structure factor discussed in the previous section? This is needed if the structure of individual unit cell is different from each other. If all the unit cell structures are perfectly the same, then we can use the so-called (normal) structure factor and the Laue functions given in the previous section. However, the unit cell structures are not necessarily the same perfectly when the structural heterogeneity or the atomic vibrations are taken into consideration for the scattering intensity. In such a case, the averaging is needed to do. The structure factor averaged for all the unit cells included in the crystal is , then F n is given as Fn = F + Fn

(1.63)

1.4 Principle of Diffraction Phenomenon

55

where the Fn is the deviation of the structure factor from the averaged value [22]. The diffraction intensity can be expressed as 



I (k) ∝ F(k)F ∗ (k) = n Fn exp(iRn k) n Fn∗ exp(−iRn k) = n n Fn Fn∗ exp[i(Rn − Rn )k] Since the position vector of the n’-th unit cell is expressed as Rn’ = Rn + Rm and n’ = n + m, 

∗ exp(−iRm k) I (k) ∝ n n Fn Fn∗ exp(−iRm k) = m n Fn Fn+m 

∗ = m (Vm /v) < Fn Fn+m > exp(−iRm k) Here V m is a volume of the overlapped part between the original crystal and the crystal generated by a translational shift by Rm . The v is the volume of a unit cell. Therefore

 ∗ I (k) ∝ m (Vm /v) < Fn Fn+m > exp(−i Rm k)   ∗ > exp(−iR k) = m (Vm /v) < (< F > +Fn ) < F >∗ +Fn+m m ∗ = | < F > |2 m (Vm /v) exp(−iRm k) + m (Vm /v) < Fn Fn+m > exp(−iRm k)

(1.64)

The first term corresponds to the scattering from the perfect crystal with the averaged structure factor , and the second term is the so-called diffuse scattering due to the heterogeneous structure between the unit cells. If the correlation between the structural heterogeneity of the neighboring unit cells is quite low, the second term becomes zero. Now, let us consider the atomic thermal vibrations which affect the internal structure of the individual unit cell in a different way. The structure factor F n of the n-th unit cell is given as  Fn = m fmn (k) exp ironm k

(1.65)

The atomic scattering factor fmn (k) is almost equal to the f m (k). The position ronm of the m-th atom in the n-th unit cell is expressed as  e ronm = ronm + unm

(1.66)

Here we assume that the atom is vibrating with the displacement unm around the equilibrium position of the atom (ronm )e . Then Eq. (1.65) can be rewritten as   e

Fn = m fmn (k) exp i ronm k exp(iunm k)

(1.67)

56

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

The < Fn Fn∗ > is expressed in the following way: < Fn Fn∗ >  e    e    =< Σm fmn (k) exp i ronm k Σm fmn (k)∗ exp −i ron m k > × < exp(iunm k) exp(−i un m k) >  e   =< Σm Σm fmn (k)fmn (k)∗ exp i ronm − ron m k >< exp(i(unm − un m )k) > (1.68) The term < exp(i(unm −un m )k)> comes from the thermal vibrations of the atoms in the unit cells. If the atomic vibrations can be assumed as the harmonic vibrations, then the above equation is expressed in the following way (the concrete derivation is skipped. See the literature [22]): 

nn < exp(i(unm − un m ) · k) >= exp −Mm − Mm + Pm,m

(1.69)

Mm = < (unm · k)2 > /2

(1.70a)

Mm = < (unm · k)2 > /2

(1.70b)

where



nn Pm,m = < (unm · k)(un m k) >

(1.70c)

Finally, we have the scattering intensity as I = Io + I1 + · · · ¯ 2 |DL |2 |DM |2 |DN |2 |F|

(1.71)

  F¯ = j fj exp −Mj exp irj · k

(1.72)

Io (k) ∝

 nn I1 (k) ∝< [m m fmn (k)fmn (k)∗ exp(i(ronm − ron m )e k)] > exp Pm,m

(1.73)

The I o comes from the averaged structure factor F¯ of the unit cell, corresponding to the Laue-Bragg reflection. The I 1 corresponds to the primary thermal diffuse scattering due to the correlation of the atomic vibrational modes between the neighboring unit cells.

1.4 Principle of Diffraction Phenomenon

57

When the atomic vibration occurs isotropically and its amplitude is um , then the M m of the m-th atom is calculated as follows:  Since k = (2π/λ)(s − so ) and |k|2 = 4π2 /λ2 s2 + so2 − 2sso cos(2θ ) = 16 π2 sin2 (θ )/λ2 , 2 > sin2 (θ )/λ2 = Bm sin2 (θ )/λ2 Mm =< (unm · k)2 > /2 = 8π2 < um

(1.74)

where um is the amplitude component along the k-direction and the so-called isotropic 2 >. The term exp(-2M m ) is called the Debye temperature factor Bm = 8π2 < um factor. If we assume the same B value for all the atomic species in the unit cell, the B is called the global isotropic temperature factor. For the anisotropic vibrations, the atomic vibration amplitude vector u is expressed as a fraction of the unit cell parameters a, b, and c: u = aux + buy + cuz then, for the infinitely large crystal, 2M =< (unm · k)2 > =
= < hux + kuy + luz >

= h2 < ux >2 +k 2 < uy >2 +l 2 < uz >2 +2hk < ux uy > +2kl < uy uz > + 2hl < ux uz >  

= 1/8π2 h2 Bxx + k 2 Byy + l 2 Bzz + 2hkBxy + 2klByz + 2hlBxz (1.75) Bij is called the anisotropic temperature factor. In more general, the tensor component u11 is used for the 2 and so forth. According to Eq. (1.72–1.74), the atomic scattering factor f is affected by the thermal motion in a form of = f exp[-B sin2 (θ )/λ2 ] where the isotropic vibration is assumed. As the temperature increases, the B increases, resulting in the reduction of the scattering intensity with an increase of scattering angle [23]. Figure 1.37 shows a case of C atom with the isotropic temperature factor taken into account. The effect of the temperature factor on the diffraction profile is illustrated in Fig. 1.38. In this calculation, the crystallite size was assumed to be large, about 500 Å in all the directions. The temperature factor was of the global isotropic value (the H and C atoms take the same values). As the temperature factor becomes larger, the higher order reflections become weaker. In fact, the diffraction profile becomes weaker in the higher angle region at a higher temperature. This figure shows also the effect of crystallite size. As the size becomes smaller, the peak width becomes broader. It must be noted that even for the larger temperature factor, the profile is sharp as long as the crystallite size is large.

58

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.37 Scattering factor of C atom with the various temperature factors B

Fig. 1.38 The equatorial diffraction profile calculated for the orthorhombic polyethylene crystal with the various isotropic temperature factors (Biso ). The crystallite size along the a- and b-axes was assumed to be 500Å. The effect of the crystallite size is also shown in the lower pictures

1.4 Principle of Diffraction Phenomenon

59

1.4.5 Anomalous Scattering As mentioned, an atom consists of the positive nuclei and the negative electrons, which is assumed as a dipole moment. When an X-ray beam (photon) is incident to the atom, the X-ray electric field induces the forced oscillation of dipoles, and then the Thomson scattering occurs as a result. The scattering amplitude A by the forced oscillation is given as   A = e/mc2 v2 Eo / vB2 − v2 + ikv

(1.76)

where v is the frequency of the forced vibration and vB is the natural frequency of free electron (Bohr’s frequency). If v ≈ vB (resonance), the scattering amplitude becomes extremely large. The atomic scattering factor of an atom f is related to the ratio of the scattering amplitudes between the forced electron and the free electron: f = (the scattering amplitude by the forced electron)/(the scattering amplitude by the free electron) The resonance condition causes the anomalous f value. The f is expressed, in general, as f = fo + f + if

(1.77)

where f o is the normal atomic scattering factor and f + if is a correction term, which consists of the real part (f ) and the imaginary part (f ). The frequency dependence is shown in Fig. 1.39 [24]. In the actual case, the anomalous phenomenon is observed near the X-ray absorption edge region. The phase relation among these three terms is as follows: f o and f are in the π phase relation and f o and f are in the π/2 phase relation. The structure factors F(hkl) and F(h¯ k¯ ¯l) are given as follows:

Fig. 1.39 The frequency dependence of atomic scattering factor f o , f ’ and f ”. The v B is the Bohr’s frequency

60

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

F(hkl) = j fj e2π i(hxj +kyj +lzj )  = j fo + f + if e2π i(hxj +kyj +lzj ) = |F(hkl)|eiφ(hkl)  F(h¯ k¯ ¯l) = j fj e2π i(−hxj −kyj −lzj ) = j f0 + f + if e−2π i(hxj +kyj +lzj ) ¯ ¯¯ = |F(h¯ k¯ ¯l)|eiφ(hk l)

These terms including the phase angle φo = 2π (hx j + kyj + lzj ) can be expressed in the polar coordinate system as shown in Fig. 1.40, which is called the Argand diagram [25]. Focus on a pair of the reflections hkl and h¯ k¯ ¯l (a Bijvoet pair). As shown in Fig. 1.40, the starting fo vector (red arrow) is set on the R-axis if no phase angle is taken into account. Once the phase angle φo is introduced, the vector (red arrow) is rotated by φo . But, the rotation direction is opposite between fo (hkl) and

Fig. 1.40 The Argand diagram. a and b show the effect of an anomalous scattering term for the structure factor F(hkl) and F(h¯ k¯ ¯l). The initial F vector is positioned along the R axis. The phase angle φ(hkl) = φ(h¯ k¯ ¯l) but the intensity I(hkl) = I(h¯ k¯ ¯l). c and d show the combination of two atoms with (B) and without (A) such anomalous scattering effects to result in the relation of φ (hkl) = φ(h¯ k¯ ¯l) and the intensity I(hkl) = I(h¯ k¯ ¯l)

1.4 Principle of Diffraction Phenomenon

61

fo (h¯ k¯ ¯l) vectors since their phase angles are different in sign, φo and −φo , respectively (compare (a) and (b) in the figure). That is to say, these two vectors are symmetric with respect to the R-axis. When the anomalous scattering terms (f and f ) are taken into account, the directions of these two vectors change furthermore as indicated by blue arrows in Fig. 1.40 (a) and (b). The phase angle is much different between these two vectors, but the amplitude is the same, meaning that the scattering intensity is not different between them. However, as illustrated in Fig. 1.40 (c) and (d), if an atom having a large anomalous scattering term (B, blue vector) is connected to the atom without an anomalous scattering term (A, red vector), both of phase angle and scattering intensity are different between the Bijvoet pair (yellow vectors). The anomalous scattering effect becomes more serious for the atom with larger number of electrons. Table 1.6 shows some examples. The situation of Fig. 1.40 can be understood more easily by a concrete calculation. For example, in the case of Fig. 1.41 The crystal structure of ZnS

Zn S

Fig. 1.42 The diffraction of X-ray by the 111 and 1¯ 1¯ 1¯ planes

62

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Table 1.6 Anomalous scattering factors of atoms

Cu-Kα

fo

f

f

C

6

0.017

0.009

N

7

0.029

0.018

O

8

0.047

0.032

S

16

0.319

0.557

Cl

17

0.348

0.702

Br

35

−0.767

1.083

ZnS crystal, Zn atom has much larger anomalous scattering effect than S atom. The crystal is of cubic system with the unit cell parameters a = b = c = 5.4093 Å with ¯ the space group F 43m. The atomic positions are Zn (0, 0, 0); (1/2, 1/2, 0); (0, 1/2, 1/2); and (1/2, 0, 1/2) S (1/4, 1/4, 1/4); (3/4, 3/4, 1/4); (1/4, 3/4, 3/4); and (3/4, 1/4, 3/4) It must be noted here that this crystal does not have any points of symmetry. Let us calculate the structure factors of 111 and 1¯ 1¯ 1¯ reflections as examples [26]. The contribution of anomalous scattering effect of S atom is ignored compared with that of Zn. 



F(111) = fZn e0 + 3e2π i + foS e3π i/2 + 3e7π i/2    = 4 foZn + fZn − 4foS i = 4 foZn + fZn + 4i fZn + ifZn − foS 



¯ = fZn e0 + 3e−2π i + foS e−3π i/2 + 3e−7π i/2 F(1¯ 1¯ 1)    = 4 foZn + fZn + foS i = 4 foZn + fZn + 4i fZn + ifZn + foS Therefore, the phase angle tan[φ (111)]  = (f Zn −f oS )/(f oZn + fZn ) is not the same ¯ ¯ ¯ as that of tan[φ(111)] = fZn + foS / foZn + fZn . Similarly, the scattering intensity ¯ = F(1¯ 1¯ 1)F ¯ * (1¯ 1¯ 1). ¯ is also different between I(111) = F(111)F * (111) and I(1¯ 1¯ 1) ¯ ¯ ¯ In this case, the intensity I(111) is predicted to be lower than I(111). By comparing the intensity relation observed for the front and back surfaces of the ZnS crystal, we can distinguish these two surfaces or the polar direction of this crystal. This idea can be applied to determine the absolute structure of an optically-active single crystal. At first, the crystal structure is determined by the usual X-ray structure analysis. Then, the intensities of |F(hkl)| and |F(h¯ k¯ ¯l)| are calculated by this structure model. If the actually-observed intensity relation is opposite to the thus-predicted relation between these two diffraction peaks, the atomic coordinates of the initial model must be changed from (x, y, z) to (-x, -y, -z). For example, if the model is assumed to be D enantiomer, the correct structure should be L enantiomer.

1.5 Generation and Detection of X-ray

63

1.5 Generation and Detection of X-ray 1.5.1 X-ray Generators X-ray beam is created from the various types of generators. Historically a vacuum tube was utilized at first. Electrons are emitted from a heated tungsten filament in vacuum by applying a high voltage of about 40–60 kvolt and a current of 20–50 mA. Thus-generated thermal electron beam is incident on a metal block. The inner electrons at the various energy levels in the metal atom are emitted by the attack of externally incident thermal electrons. The electrons originally located at the higher energy levels transit to the thus-created empty energy level and the X-ray beams are generated with the corresponding energies. As illustrated in Fig. 1.43, when the transition occurs from the L and M levels to the lower K level, the emitted X-ray beams are called Kα and Kβ , respectively. If the transition occurs to the second lower energy level L from the higher levels, the emitted X-ray beams are called Lα , Lβ , and so on. These X-ray beams generated by the transitions between the particular energy levels are called the characteristic X-ray beams and they have the characteristic constant wavelengths as shown in Table 1.7. In the actual case, the continuous X-ray components overlap these characteristic lines. It must be noted that the total energy of Fig. 1.43 Emission of X-ray beams. In the case of Cu target, the wavelength of Kα1 = 1.5406 Å, Kα2 = 1.5444 Å, and Kβ = 1.3923 Å

Table 1.7 Characteristic X-ray beams Metal

X-ray lines α1

α2

(~(2α1 +α2 )/3)

β

Cu

1.540598Å

1.544426Å

1.5418Å

1.392250Å

Mo

0.709319Å

0.713609Å

0.7107Å

0.632305Å

Cr

2.289760Å

2.293663Å

2.2909Å

2.084920Å

64

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

the electrons incident on the metal block changes to the thermal energy at about 98% probability. The effective usage of the incident electron beam energy for the X-ray beam generation is only a few percentage. The overwhelmingly large thermal energy heats the metal block remarkably, and so the block must be cooled by circulating the water (or air) constantly. After the repeated generation of the X-ray beam by the thermal electron attack, the metal block surface is damaged by the incident electron beam and a small hole is dug finally. In order to depress such a damage, the metal block is rotated at a high speed during the incidence of the high-energy electron beams. This idea makes it possible to apply the higher voltage of 50–60 kvolt and higher current of about 300 mA. This system, as illustrated in Fig. 1.44, is called the rotating anode or the rotating target. The rotation of the metal cap is made by a high-speed motor under the high vacuum. At the same time, the water must be circulated to cool the metal cap. For separating the water and vacuum systems, the magnetic seal is utilized. The life time of the rotating anode system is about 3000 hrs in general. In the modern X-ray scattering system, the relatively weak X-ray beam is reflected on the flat or bent mirror surface and focused on the sample position (or the detector surface) as illustrated in Fig. 1.45. The focused X-ray beam is highly brilliant and has the small size of about several 10 micrometer even for the applied power of 50 kV and 30 mA.

(a)

(b)

(c)

Fig. 1.44 X-ray generator a Vacuum tube (Toshiba), b rotating anode (MAC Science), and c tungsten filament. In b the copper metal cap is rotated at a high speed to avoid the damage by high energy thermal electron beam. In this picture a trace on the copper surface is due to this damage

1.5 Generation and Detection of X-ray

65

X-ray

focal point

X-ray focal point Fig. 1.45 X-ray mirrors

The shape of the cross section of the incident X-ray beam is important. The usage of pin hole slits or a collimator of a round hole of several hundreds μ m diameter gives the X-ray beam of a round shape. The scattering from the sample occurs homogeneously as long as the sample does not have the deformed shape. The slit of a rectangular shape with the size of several hundreds μ m width and several millimeter height gives the beam of the thin and long square shape. The scattering occurs heterogeneously from the sample and the scattering signal must be corrected for this effect. By taking into consideration the diffraction phenomena occurring at the various positions of the slit, the detected scattering pattern is a convolution between the round-shape beam and the rectangular slit. This effect is particularly significant for the small-angle scattering case (see Chap. 4).

1.5.2 Synchrotron Radiation The usage of synchrotron radiation gives an X-ray beam of the remarkably high brilliance. A charged particle is emitted and speeded-up by a linear accelerator (LINAC) up to the high speed near the light velocity (99.9999…%). These particles are stored in the storage ring under a high vacuum of 10−7 Pa order. When the thus-speededup particle passes through a pair of magnet, the particle path is bent (the direction change means the acceleration). The bending of the particle path occurs following the Fleming’s left hand rule, as illustrated in Fig. 1.46. (Note that the direction of an electron with a minus charge is opposite to the direction of current. If the particle is a positively charged positron, the path direction of the particle is the same as the current path direction.). The bending of the high-speed charge or the acceleration of the charge causes the emission of the electromagnetic radiations with the various energies along the direction of particle path according to the electromagnetic theory. The thus-emitted radiation is called the synchrotron beam. The history of the

66

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

(a)

(b)

(c)

Fig. 1.46 a Fleming’s left hand law and synchrotron radiation, b undulator, and c storage ring

synchrotron facility is long. In the synchrotron system of the first generation, the beam is borrowed from the original accelerator facility. In the second generation, the synchrotron system was built up positively for the specialized purpose to supply the synchrotron beam to the users and the bending magnet was inserted. In the third generation, the undulators (or wigglers) are inserted to get the synchrotron beam of 5 GeV or higher energy. The undulator consists of many magnet pairs arrayed in the opposite directions and so the electrons are bent alternately many times and as a result the very strong radiation is generated along the undulator axis. After passing through the undulator, the coherency of the randomly generated X-ray beams is increased. In an extreme case, after the passage of many undulators, the X-ray beam of laser character can be generated, which is called an X-ray free electron laser beam (XFEL). There are several facilities in the world which can emit the XFEL beam (for example, SACLA, Japan). The energy of the synchrotron radiation distributes in a wide range of wavelength as shown in Fig. 1.47, making it possible to perform different types of experiment using the IR, visible, UV, vacuum-UV, and soft and hard X-ray beams. The synchrotron beam is almost parallel and polarized. As known from the circular motion

1.5 Generation and Detection of X-ray

67

Fig. 1.47 Synchrotron radiation spectrum in SPring-8, Japan. Reproduced from Ref. [28] with permission of Riken (Institute of Physical and Chemical Research, Japan) and the Japan Synchrotron Radiation Research Institute

of the charged particle in the storage ring, the synchrotron beam is a repetition of pulses of picosecond period, which might be applied to the research of short-time phenomena. We need to notice the difference between the brightness and brilliance of the light. The brightness is a total power of the photons. The brilliance B is a number of photons N produced per second (dt), per cross section (dxdy), per angular divergence (the degree of dispersion, dx’dy’), and per bandwidth (BW) of the 0.1% of the central wavelength (dw/w).  B = d6 N / dxdydx dy dtdw/w  = number of photons/ sec · mm2 · mrad2 · 0.1%BW

(1.78)

68

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

dx’dy’

Δω/ω

N dxdy

dt Fig. 1.48 Imaginary illustration of a brilliance [27]

An interesting comparison is made in Ref. [27] between a white lamp and a laser beam, as shown in Fig. 1.48. A 100 watt white light of 10 × 10 mm2 size with the wavelength range of 250–750 nm around a central wavelength 500 nm has the following total number of photons. The energy of one photon at λ = 500 nm is  ε = hv = hc/λ = 6.626 × 10−34 m2 kg/sec × 3 × 108 (m/sec)/500nm/photon = 4.0 × 10−19 m2 kg/sec2 /photon W = 100 watt = 100 × 0.102 kgf · m/sec = 10.2 × 9.8 N · m/sec = 10.2 × 9.8 kg m/sec2 · m/sec = 100 kg · m2 /sec3 Therefore, the number of photons emitted per sec is 

 N = 100 kg · m2 /sec3 / 4.0 × 10−19 m2 kg/sec2 /photon = 2.5 × 1020 photons/sec

For example, a white lamp emits the photons in all the directions (solid angle 4π). The wavelength width is 100% since the wavelength range λ = 750−250 = 500 nm and λ/λ = 500 nm/500nm = 100%. Therefore, the brilliance B is calculated as   B = 2.5 × 1020 photons/ sec / 100 mm2 / 4π rad2 /100%BW  = 2.0 × 108 photons/ sec ·mm2 · mrad2 · 0.1 %BW On the other hand, a 1 mwatt laser pointer of green color (λ = 500 nm wavelength) with the 0.01% BW, 1 mm2 area, and 1 mrad2 angular dispersion has the following values of N and B:

1.5 Generation and Detection of X-ray

69

W = 1 mwatt = 1 × 10−3 × 0.102 kgf · m/sec = 1 × 10−4 × 9.8 N · m/sec = 1 × 10−4 × 9.8 kg · m/sec2 · m/sec = 10−3 kg · m2 /sec3   N = 10−3 kg · m2 /sec3 / 4.0 × 10−19 m2 · kg/sec2 /photon = 2.5 × 1015 photons/sec.

 B = 2.5 × 1016 photons/ sec ·mm2 · mrad2 · 0.1%BW The laser beam of 1 mwatt power has a brilliance B higher by 108 than the 100 watt white lamp. Here we have the various measures to show the quality of photons. The emittance is a product of area and dispersion angle, which is a measure of an excellent beam: the lower emittance is better. The flux F, flux density d2 F/dx’dy’, and brilliance B are related to each other as follows:  ˜ 2 dy dx dy = Bdx dy dx dy Flux F= d F/dx ˜ 2 Flux density d F/dx dy = Bdx dy The electron is accelerated to the 99.9999998% of light velocity by the application of 8 GeV energy (at SPring-8, Japan). When it is inserted into a pair of bending magnet, the radiation of a power of 1 kwatt order is emitted, which is quite low, about the power of 10 lamps. However, the brilliance B is in the order of 1015 . When an undulator is inserted (32 mm pitch, 140 cycles), the brilliance becomes 1021 . The brilliance of the X-ray beam from the laboratory-level X-ray generator is about 108 . The Sun has a brilliance of 1010 . In the SPring-8, the beam size is about 300 μm x 6 μm in the horizontal and vertical directions, respectively. The dispersion angle is 12 × 1 μ rad, respectively, in the horizontal and vertical directions. That is, the beam is not circle but a thin and long square. By cutting this beam the size of the beam incident to the sample is changed. Here, the cutting of light by a slit causes the parasitic scattering from the slit edge, which must be cut by another slit (the production of microbeam is described in Sect. 4.4.1).

1.5.3 Detectors The detectors for the scattered X-ray signals have been developed remarkably for these years.

70

1.5.3.1

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

0D and 1D Detectors

The initial stage was a usage of a Geiger-Müller counter. After that, a scintillation counter has been used. As illustrated in Fig. 1.49, it is a one-point detector and can get the one-dimensional profile of the diffraction data. The detector and sample oscillate around the sample stage by 2θ and θ, respectively, so that the Bragg reflection law is satisfied. A one-dimensional PSPC (position-sensitive proportional counter) is useful to get the one-dimensional profile at once. As shown in Fig. 1.50, a tungsten wire is set horizontally at the center of the detector, and methane/Argon mixed gas is filled in there. The X-ray radiation is incident and ionizes the methane gas. The charged gas reaches the wire by applying the high voltage between the wire and the earth. Once when the gas reaches the wire, the electron jumps into the wire and the current is generated. The current reaches the time analyzers set at the left and right ends of the wire. If the ion reaches the center of the wire, the difference of the arrival time between these two time analyzers is zero. If the arrival position is deviated from the wire center, the time difference is detected between the two analyzers. The number of the electrons arrived at a particular position of the wire is proportional to the intensity of the incident X-ray signal. In this way, the position and intensity of the incident X-ray signals can be detected simultaneously. There are two types of the PSPC detector, one is a flat type and the other is a curved or cylindrical type. In the former case, the profile must be corrected depending on the incident position because the X-ray beam enters the wire at the different angle. The latter case is more convenient in such a sense, but the camera-to-sample distance is fixed.

Fig. 1.49 a 0D counter, b 1D PSPC detector, c 2D camera film (or imaging plate). (from Ref. [29] Copyright ©1997 by Wiley, Inc. Reprinted by permission of Wiley, Inc.)

1.5 Generation and Detection of X-ray Fig. 1.50 a 1D PSPC, b 2D PSPC

71

(a)

t1

t2

+ TAC

CH4

TAC

X-ray

(b) y1 x1 x2 x3 x4 … x1 x2 x3 x4 …

y2 y3 y4

1.5.3.2

2D Detectors

The two-dimensional detectors are more useful for the polymer samples. This is because the diffraction profile of polymer is not very sharp but has an anisotropic shape, from which the various information can be derived. The classic type of the 2D detector is a sheet of photographic film. The mechanically flexible film can be set into a flat camera or a cylindrical camera. As already explained, the flat camera gives the curved layer lines. The cylindrical camera giving the horizontal layer lines is easier for the analysis of the diffraction profiles along the layer lines. The silver chloride (or silver bromide) pasted on the film surface is electronically reduced by an irradiation of light and decomposed into silver and chlorine (bromine) (Ag+ X- → Ag + X). However, the X-ray diffraction image cannot be seen directly at this stage, and it is called a latent image. By immersing this film into a hydroquinone solution, silver chloride containing the Ag particles is started to be reduced quickly. The Ag area becomes larger and the image is “developed”. The non-irradiated AgCl region is not yet reduced. If you leave the film in the developing solution for a long time, both of the irradiated and non-irradiated film parts are reduced. In order to stop the further development, the film is immersed into an acetic acid aqueous solution. This process is called the fixing. The thus-chemically reduced film is transferred immediately to the aqueous solution of Na2 SO3 (hypo), which dissolves the notyet-reduced silver chloride particles on the film totally. Then the diffraction pattern

72

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

of the latent image can be seen visually since only the electronically reduced silver particles of black color remain on the film surface. The resolution of the image is excellently high because of the small size of silver chloride particles on the film surface, ca. 0.1–1 μm in diameter. In spite of quite high resolution, this method is not very popular now since the diffraction image must be chemically treated and a long time (about 10–15 min) is needed to visualize the image. The newer generation of the two-dimensional detector is a socalled imaging plate (IP) developed by a Japanese company (Fuji film) [30]. An imaging plate (IP), or a photostimulable phosphor (PSP) plate in more general, is a polyester plate into which BaFBr powder is mixed and Eu2+ ions are doped in there. When the IP is exposed to X-ray beam, Eu2+ is oxidized to Eu3+ and the emitted electron is trapped at a color center (or the bromine ion empty lattice part). The image obtained cannot be seen directly (a latent image). The irradiation of a red laser beam induces the transition of Eu3+ to Eu2+ and the visible blue-violet 400 nm luminescence is emitted. The intensity of the emitted light is proportional to the number of the trapped electrons or to the incident X-ray beam intensity. This visible light is read using two photomultipliers depending on the intensity of the signals. Then the originally obtained latent image can be converted into the visible image. Once the image is read correctly, the digital data is saved into the computer. The image data on the IP is erased by irradiating the white light on the plate for a few minutes. The excellent point of IP is a linear and wide dynamic range of the detected intensity: the X-ray signal of 1 to 106 photons can be read linearly. The resolution is high, about 30–100 μm per pixel. However, the weak point of IP is a long reading time of the latent image, and a few minutes are needed for the full scan of the red laser over the whole area, ca. 10 × 10 cm2 −30 × 30 cm2 . The alternate and repeated usage of the plural IPs may be useful to speed up the measurement time of the rapidly changing X-ray diffraction data: the first IP is used for the exposure, the second IP is for reading of the image, and the third IP is for erasing the already-read image [31]. The latent image is not stable but it fades away gradually. Therefore, it is better to read out the latent image as soon as possible after the end of the X-ray exposure. Another useful point of IP is its flexibility, just like a normal film. For example, the IP film can be set into a flat camera or a cylindrical camera. Opening a small hole at the center of the IP allows the passage of a scattered X-ray beam in a small-angle region (see Fig. 1.51). The SAXS component can be detected using another camera set at a long distance from the sample. In this way, a so-called simultaneous measurement of WAXD and SAXS signals can be made easily, though the rapidly time-resolved measurement is hard to do. The above mentioned 1D PSPC can be developed to the 2D detector. As shown in Fig. 1.50b, many horizontally taken-up wires are arrayed vertically at a constant spacing [32]. They are set inside the case filled with methane-Argon mixed gas. The (x, y) coordinates of the detected X-ray signals are defined by the number of a wire (y) and the position along the wire (x). The intensity is estimated as the

1.5 Generation and Detection of X-ray Fig. 1.51 Simultaneous measurement system of the 2D wide- and small-angle scatterings

73

(a) IP

WAXD

SAXS

X-ray

IP beam stopper

sample

(b)

WAXD

IP

SAXS

X-ray

beam stopper sample

(c) WAXD

1D-PSPC

1D-PSPC

X-ray sample

SAXS

beam stopper

integration of the signals incident on the position (x, y). Since the interval of the adjacent wires is quite narrow, the electric short occurs easily by moving the detector during the supply of the power. Recently, these wires are fixed safely using a pair of plastic films and this type of trouble is avoided [33]. A CCD (charge coupled device) is now used popularly because of the rapid data collection ability [34]. The photons incident on the electrode surface are converted to the electron charges, the number of which is proportional to the number of incident X-ray photons or the incident X-ray signal intensity. These electrons are trapped in the positively charged capacitors of the photo-active region (the interface between SiO2 and p-doped Si semiconductor). These charges are transferred step by step to the neighboring capacitors of the positive voltage. Once when the charges reach the last capacitor or a place of the charge amplifier, they are converted to the amplified voltage signals. These signals are digitized and stored to give the image. Depending on the 1D and 2D arrays of the photo-active regions, the visualized signals are 1D and 2D images, respectively. The ratio between the incident photon and the converted electron is (1 electron)/(1 photon), that is, the amount of charge is proportional to the number of the incident photons. Therefore, the reduction of dark current and noise is significant. Different from it, a single photon counting module (SPCM) counts the incident photons using an avalanche photodiode, which can convert the signal of one photon into more number of electrons (for example, 1000 electrons/1 photon) just like an avalanche and the obtained signal corresponds to one incident photon. The dark current and noise are ignored actually and the number of photons can be counted directly. The counting rate is quite high, about 50 nanoseconds. The 2D X-ray detectors are now being changed gradually from PSPC and CCD to SPCM (the currently commercially available SPCM is, for example, Pilatus (Dectris [35]) and HyPix (Rigaku [36])). Figure 1.52 shows the snapshots of

74

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

(a) 2D PSPC (Hi-star ® )

(b) CCD

(c) Hypix ®

Pilatus ®

Sophias ®

Fig. 1.52 Various X-ray 2D detectors. a Hi-Star® (Bruker), b CCD (Hamaphotonics), c Hypix® (Rigaku), Pilatus® (Dectris, 100k, 300k, 1M, etc), Sophias (JASRI, SPring-8)

1.5 Generation and Detection of X-ray

75

the several currently available detectors: a scintillation counter, 2D PSPC detector, CCD camera, and SPCM detector.

1.6 X-ray Diffraction Data Collection 1.6.1 Setting of Detector The detector is set on the X-ray generator so that the scattered X-ray signals can be collected effectively and correctly. Figure 1.53 shows the measurement system of a powder diffraction pattern. A scintillation counter is set on the rotation stage (ω scan stage) around the sample position. The sample and detector are rotated cooperatively in a so-called θ − 2θ scan mode so that the Bragg’s reflection law is satisfied. The 1D PSPC detector may be also set along a fixed direction and the diffraction data in a certain range of the diffraction angle is measured at once (Fig. 1.49). If the PSPC detector is of a flat type, the scattered X-ray signal enters at the different angle depending on the diffraction angle and the so-called tilting angle effect must be corrected, since it causes the deformation of the diffraction profile. The cylindrical PSPC detector can avoid this tilting angle effect, but the sample-to-camera distance must be fixed. Figure 1.54 shows the measurement systems of the 2D diffraction patterns. The detectors are a classic camera film, an imaging plate, a CCD, a 2D

Fig. 1.53 Powder diffractometer [Rigaku TTR-III® ]. In this picture, the sample is set horizontally and the tube (or rotator) and detector are scanned around the sample so that the θ − 2θ relation is satisfied always. The liquid sample can be measured

76

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.54 Setting of a flat camera and b cylindrical camera

PSPC, or an SPCM. These detectors have mostly the flat surface and need to correct the deformation of image. The flexible imaging plate can be used as a cylindrical detector, although the plate must be set inside the metal frame with a black sheet to protect the detector from the outside light. One important point in the usage of these detectors is to protect the detector from the incidence of the strong direct beam. The size of the beam stopper must be large enough to stop the beam without any leakage of the beam. But, at the same time, the stopper size must be as small as possible so that the scattered signals can be collected as many as possible up to the lowest scattering angle region. For the measurement of the powder diffraction pattern, as shown in Fig. 1.53, two types of measurement are possible. One is a focusing beam method. The scattered signal is collected by a detector set at the focal point. The signal intensity is relatively high. However, if the sample is not homogeneous in thickness or lifted more or less from the stage, the diffraction angle might be shifted from the correct position. The parallel beam method is better for such a case, although the signal is relatively weaker.

1.6.2 Sample Setting and Corrections The sample must be set carefully. The wrong setting results in the error of the correct scattering angle evaluation. The best way to set the sample is to use a goniometer head. The sample must be set perfectly at the center position without any tilting and precession motion. Figure 1.55 shows the correct and incorrect setting modes of the uniaxially oriented sample. By adjusting the four screws of the goniometer head,

1.6 X-ray Diffraction Data Collection

(a)

77

(b)

(c)

Fig. 1.55 Setting of a uniaxially oriented sample on the goniometer head. As shown in (a), the sample axis must be set perfectly so that the rotation axis and the sample axis are coincident perfectly without any tilt in all the directions. As seen in (b), the goniometer head (eucentric type) consists of the four movable parts: the bottom two parts are for the translation and the upper two parts are for the change of tilting angle of the sample. The technique is (i) the adjustment of the sample axis in parallel to the vertical direction using the upper two parts at first and (ii) the adjustment of the sample center to the rotation axis by using the lower two parts. c The telescope and the sample image

this centering can be made nicely. These operations must be checked by viewing a sample image with a telescope or video camera with cross lines. The best way of checking the tilt of the sample is to measure the 2D diffraction pattern.

1.6.2.1

Correction of Camera Distance

The accurate evaluation of the distance between the sample and the detector is made by measuring the reflection angles of a standard powder substance pasted on the sample surface. Si or Al fine powder is popularly used as a standard powder. As shown in Fig. 1.56, in the case of a flat camera, the roughly measured sample-to-camera distance Lapparent and the scattering angle 2θobsd are related by  xobsd /Lapparent = tan 2θobsd

(1.79)

78

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method camera camera sample

x

sample

x

2θ

2θ

L

R

Fig. 1.56 Camera distances

where xobsd is the distance of the observed reflection from the direct beam position on the flat detector. When the correct camera distance is Ltrue , we have the following relation for the correct scattering angle 2θtrue , because xobsd is common to these two cases.   xobsd = Lapparent tan 2θobsd = Ltrue tan 2θtrue

(1.80)

Ltrue can be obtained by knowing the standard diffraction angle 2θtrue . Similarly, in the case of a cylindrical camera of the raidus Rapparent , we have xobsd = Rapparent · 2θobsd = Rtrue · 2θtrue

(1.81)

The 2θ and lattice spacings are listed for Si, Al, and CeO2 in Table 1.8 (Cu-Kα line). Figure 1.57 shows the X-ray diffraction profiles calculated for Si, Al, and BeAg powders. When you want to calibrate the camera distance by reading the peak position, you need to notice that the peak consists of the two components, which come from the X-ray beams of the two different wave lengths, Kα1 and Kα2 . Sometimes Table 1.8 Structure information of standard samples (the diffraction angles are for Cu-Kα line) Sample

Si (a = 5.4307Å, ¯ cubic, Fd 3m)

Al (a = 4.0495 Å, ¯ cubic, Fm3m)

CeO2 (a = 5.4110 Å, ¯ cubic, Fm3m)

Ag Behenate (c = 58.380 Å triclinic, P1)

hkl, 2θ (relative intensity)

111 3.135 Å 28.47° (100%)

111 2.338 Å 38.50° (100%)

111 3.124 Å 28.57° (100%)

001 1.513°

220 1.920 Å 47.34° (49%)

200 2.025 Å 44.76° (42%)

200 2.706 Å 33.11° (27%)

002 3.027°

311 1.637 Å 56.17° (24%)

220 1.432 Å 65.16° (14%)

220 1.913 Å 47.53° (54%)

003 4.537°

400 1.358 Å 69.19° (4%)

311 1.221 Å 78.30° (11%)

311 1.632 Å 56.40° (44%)

004 6.051°

331 1.246 Å 76.45° (5%)

222 1.169 Å 82.52°

005 7.565°

1.6 X-ray Diffraction Data Collection

79

Fig. 1.57 X-ray diffraction profiles of standard samples

the intensity-weighted average is made for the wavelength [λ = (2λα1 + λα2 )/3 for Cu-Kα], as shown in Table 1.7.

1.6.2.2

Correction of Background

The observed diffraction data contains the unnecessary signals originating from the various sources. For example, the air scattering, the shadow of the sample holder, and so on may overlap the sample signals. The correction of the so-called background must be made. The correction is to subtract this contribution as given below: Iback-correct (2θ) = Iobsd (2θ) − k · Ibackeround (2θ)

(1.82)

The coefficient k is simply unity. But, the sample absorbs the X-ray beam more or less, then k may be a little smaller than unity (~0.8). Figure 1.58 shows one example of the oriented atactic poly(vinyl acetate). This is an amorphous polymer but oriented by stretching. The X-ray diffraction is quite diffuse but it is found to have an oriented pattern after the correction of air scattering. The coefficient k is quite important for getting the reasonably corrected signal.

1.6.2.3

Correction of Absorption Effect

If the absorption of the X-ray beam by the sample is not neglected actually, we have to correct the absorption effect by measuring the intensity of the direct beam. In particular, if the sample contains the metal elements the absorption effect becomes o and that after passing through serious. When the intensity of the direct beam is Idirect s the sample is Idirect , then we have  o s Iback−correct (2θ) /Idirect Icorrect (2θ) = Idirect

(1.83)

o s and Idirect must be measured by reducing the incident strong X-ray Of course, Idirect beam using an attenuator which is an aluminum plate or a copper plate, for example.

80

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method (a) PVAc

(b) air

(c) background-correction

Fig. 1.58 Exposure time is different between the sample (oriented atactic PVAc) and air scattering measurement. By choosing the coefficient k properly, we can erase the background nicely

The absorption, which is caused by the interaction of X-ray with the atoms in the sample, can be estimated theoretically using absorption coefficients. According to the Lambert-Beer law, the X-ray beam intensity is reduced as given below: I = Io exp(−μρx) = Io exp(−μ1 x)

(1.84)

where I o is the intensity without any absorption by the material of the thickness x, μ is the mass absorption coefficient or mass attenuation coefficient (cm2 /g), and ρ is the density (g/cm3 ). The linear attenuation coefficient μ1 = μρ. The μ value or μ1 /ρ is dependent on the wavelength of the X-ray beam and is given in the database (NIST, National Institute of Standards and Technology, [37]), which is assumed to be common to any state of the atom or solid, liquid, or gas. The mass absorption (or attenuation) coefficient is calculated as a weight average of the elements contained in the sample. μsample = i μi ωi

(1.85)

where ωi is the weight fraction of the i-th element. For example, let us consider the case of poly(vinyl chloride) [-(CH2 CHCl)n -] irradiated by Cu-Kα beam. The

1.6 X-ray Diffraction Data Collection

81

chemical formula of the basic monomeric unit is CH2 CHCl. The molecular weight of the unit is 12 × 2 + 1 × 3 + 35.5 × 1 = 62.5 g/mol. The weight fraction of each element is calculated as ωC = 12 × 2/62.5 = 0.38, ωH = 1 × 3/62.5 = 0.05, and ωCl = 35.5 × 1/62.5 = 0.57 According to the table given by the NIST, the mass attenuation coefficients at the X-ray energy of Cu-Kα line (the wavelength 1.54Å and the energy 8050.9 eV) are given as μ(C) = 4.576 cm2 /g, μ(H) = 0.391 cm2 /g, μ(Cl) = 107.5 cm2 /g The total mass attenuation coefficient is given as μsample = 0.38 × 4.576 + 0.05 × 0.391 + 0.57 × 107.5 = 63.03 cm2 /g The density of PVC is 1.4 g/cm3 , and then the incident Cu-Kα X-ray beam can transmit the sample of the 1 mm thickness by the ratio of I/I o = exp(−63.03x1.4x0.1) = exp(−8.82) = 0.015 %. If polyethylene of 0.9 g/cm3 density and thickness 1mm is used instead of PVC, the transmission of the incident X-ray beam is 69.9%. This small absorption of Xray is highly contrast to those of metal and inorganic substances. For example, the Cu-Kα X-ray beam is reduced to only 1.70 % after passing through an aluminum plate of 0.3mm thickness (μ = 50.33 g/cm2 , ρ = 2.70 g/cm3 , and I/I o = exp(−50.33 × 2.70 × 0.03) = 0.017). In an opposite way, we can calculate the thickness of PVC and aluminum  necessarily for reducing the incident X-ray beam by 50% as I /Io = 0.5 = exp −μsample ρl , and so l = ln 2/ μsample ρ = 79 μm and 51 μm, respectively. It must be noted that the sample does not necessarily have the same thickness in all the directions or it is anisotropic. In such a case, we need to correct the absorption effect by considering the difference in the path length in each direction or the sample geometry.

1.7 Sample Preparations 1.7.1 Unoriented Samples 1.7.1.1

Cast from Melt

A slide glass is set on a hot plate at a temperature higher than the melting point of the polymer. Pellets of the polymer are put on the glass plate, as shown in Fig. 1.59. Another glass slide is also heated on the hot plate. After some time passages, the

82

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

(a)

pellets

melt slow cooling quenching

slide glass

(b) spacer sample

(c)

Kapton

Fig. 1.59 Melt-pressing method. a The sample is pressed with a pair of slide glass. b The sample is surrounded with a spacer and pressed up to the fixed thickness. c The sample is covered with a Kapton film (or aluminum, PTFE)

second glass plate is put over the sample. Then the sample sandwiched between a pair of glasses is melted gradually. The upper glass slide is pressed lightly using a tweezer so that the molten sample becomes equally flat with a proper thickness. The molten sample is quenched into an ice water bath to get an amorphous sample or a sample with low degree of crystallinity. Sometimes, Dry-Ice®-methanol or liquid nitrogen is used as a quenching medium. The quenching temperature is dependent on the glass transition point of the polymer. Polyethylene crystallizes too fast to get a pure amorphous sample. In such a case, the use of a slide glass might not be suitable for quenching because of the low thermal conductivity. An aluminum foil might be better in some case. Polyester and polyamide are strongly stuck on the glass surface. In such a case, the sample is sandwiched with polytetrafluoroethylene sheet or Kapton film. The melt-quenched sample is relatively easily peeled off from the sheet. In order to peel off the film from the slide glass, some care is needed. A thin blade is sometimes used, which is inserted between the polymer film and the glass slide. In the worst case, the blade is broken and might hurt your finger. If the sample is stuck strongly on the glass, it is left for a long time at room temperature and then it might become easier to peel off the sample. The aluminum foil can be taken away using a tweezer. The aluminum foil can be dissolved by immersing the sample in a NaOH aqueous solution of high concentration. This method is convenient as long as the polymer is not damaged by the NaOH solution.

The heating of the sample at a high temperature may cause the oxidization and degradation of the sample. If the polymer sample absorbs a small amount of water, the sample must be evacuated above 100 °C over night to purge the water from the sample. If not, the absorbed water is boiled and many bubbles are produced inside the sample.

1.7 Sample Preparations

83 heater

Al spacer

pellets heater

Fig. 1.60 Melt-pressing method with an aluminum spacer for the preparation of a sample plate with homogeneous thickness

For a preparation of a sample sheet of a larger size, a hot press is useful (Fig. 1.60). The sample is sandwiched between a pair of parallel metal stages. In order to make a sheet of a fixed thickness, the sample is put into an aluminum plate spacer with a square hole. The sample is heated above the melting point and pressed by a metal plate, just when the sample thickness is adjusted to that of the aluminum plate. After the sample is pressed for a while, the pressure is released so that the bubbles inside the molten sample are taken away from the sample (because of the higher pressure inside the sample). This melt-press-release process is repeated a few times, and then the sample plate is cooled slowly or quenched into the ice water bath. The sample sheet of a fixed thickness can be obtained.

If you want to prepare an unoriented but highly crystalline film, you can anneal the melt-quenched sample at a high temperature for a fixed time. The sample is sandwiched with a pair of slide glasses using a metal clip and put in a silicone oil. After the heat treatment, the sample is taken out of the bath and cooled at room temperature, and then the oil on the sample surface is washed away by soaking the sample into a petroleum ether. The silicone oil must be washed away completely by dipping the sample many times. The residual oil gives the strong IR bands, which might lead to a misleading of the IR spectral analysis of the sample. The slow cooling from the melt gives the crystalline sample. In this case, the size of the crystallites in the sample might have a wide distribution. The so-called isothermal crystallization can control the size distribution more effectively. As shown in Fig. 1.61, a polymer sample is melted for a while at a high temperature and moved quickly into the bath controlled at a fixed temperature (Tc), then left for a fixed time at Tc. Then the sample is taken out of the bath quickly and quenched into an ice water bath to stop the further crystallization. The crystallite size is determined by the crystallization temperature Tc. The isothermal crystallization method is quite useful for the study of crystallization behavior of polymer, the details of which will be described in other chapters. The powder is also used as the unoriented sample. But the sample must be ground well to get a fine powder. If not, the X-ray diffraction pattern is not perfectly smooth but shows a pattern of heterogeneous intensity distribution with some spots (see Fig. 1.62). We need to grind the powder for a long time. The grinding process causes sometimes the transition of crystalline form. For example, PVDF powder obtained

84

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

T = Tc T > Tm

Fig. 1.61 Isothermal crystallization at Tc. A pair of hot oil bath is prepared and the sample is put into a bath at a temperature higher than the melting point. After melted for a constant time, the sample is moved quickly into the other oil bath kept at Tc. The sample is kept there for a fixed time and quenched into an ice water bath or room temperature. The sample must be washed with nonsolvent

(a)

(b)

Fig. 1.62 X-ray powder diffraction patterns. a Roughly-ground NaCl powder. b Finely-ground NaCl powder

from the polymerization reaction takes the crystalline form II. The long grinding of this powder at room temperature causes the transformation of form II to form I.

1.7.1.2

Cast from Solution

A polymer is dissolved into solvent to prepare the solution. This solution is gently poured on a slide glass set horizontally on the table. Covering the solution by a glass petri dish with a slight space is good for controlling the evaporation rate of the

1.7 Sample Preparations

85

solvent. The thus-cast thin film is difficult to peel off from the glass. The forceful manual peeling of the film may cause the partial deformation of the unoriented film. Prepare a petri dish of about 1 cm depth and fill it with water. The film stuck on the slide glass is dipped into the water gently, and then the film starts to float slowly in the water or onto the water surface, depending on the sample density. The thus-floated film can be taken out of the water using a filter paper (see Fig. 1.63). Use another dry filter paper to take out the water on the film. The film must be dried up perfectly. Sometimes the organic solvent remains in the film, which can be checked by the IR spectral measurement. The solvent used for the film casting depends on the kind of polymer. Polystyrene can be dissolved easily into chloroform, carbon disulfide (recently forbidden to use ?), and so on. In some cases, the crystalline modification of a polymer is affected sensitively by the choice of the kind of solvent. Typical case is poly(vinylidene fluoride) (PVDF). As listed in Table 1.9, the polarity of the solvent is important to control the crystalline form [14]. Hexamethyl phosphoric triamide (HMPTA) can dissolve PVDF quite easily at room temperature, but the casting of the film needs quite long time. After waiting for a few weeks at room temperature, an unoriented film of highly

(a) casting from the solution

(b) peel off the cast film from the glass film

solution dish water

glass plate

glass plate

Fig. 1.63 Casting from the solution. In a, a slight gap is useful for the control of the solvent evaporation rate. b The cast film on the glass plate is difficult to peel off. The film is immersed in water to float the film onto the surface

Table 1.9 Casting solvent and crystalline form of PVDF Solvent

Hot acetone

Acetone (room temp)

Cold acetone

DMSO

DMA

Form

II (α)

II (α)

I (β) + Disordered Disordered Disordered I (β) Disordered III (γ‘) III (γ‘) III (γ‘) III (γ‘)

Polarity--------------------------------------------------------->higher evaporation rate higher indicates the average depending on the various experimental conditions such as the powder diffraction measurement, the Laue method (the usage of white X-ray beam with the continuous distribution of wavelength), the full rotation method around a particular axis, etc. The integration over the region A is rewritten using the area S or the crossing region with the Ewald sphere surface. The relation between A and S is given as follows: for the solid angle Ω, A = r 2 Ω and S = (1/λ)2 Ω

1.8 Diffraction Data Analysis

117

Therefore A = λ2 r 2 S

(1.113)

Equation (1.111) is rewritten as given below: 

 < I (hkl) >=
=
(1.114)

The observed intensity comes from the total integration of the detector area A, which is now converted to the expression using the integration over the area S in the reciprocal lattice. Their relation is connected through the Lorentz factor. Since the integration over A is dependent on the experimental method, the Lorentz factor is also varied depending on the experiment. The factors K, p, |F(hkl)|, and λ are assumed to be almost constant in the small region of the Ewald sphere. Then, we have  < I (hkl) >= λ Kp|F(hkl)| < 2

2

S

G dS >

(1.115)

By defining the parameters ρ, χ, δ as shown in Fig. 1.90, the cross-sectional area dS is given as  dS ∼ ab = (1/λ)dχ(1/λ) sin(χ)dδ = sin(χ) 1/λ2 dχdδ  = 2 sin(χ/2) cos(χ/2) 1/λ2 dχdδ

(1.116)

Since the parameter ρ shown in Fig. 1.90 is given as ρ = (1/λ) sin(χ/2) × 2, then dρ = (1/λ) cos(χ/2)dχ or dχ = λdρ/ cos(χ/2)

(1.117)

As a result, we have dS = (2/λ) sin(χ/2)dρdδ and, in Eq. (1.115),  


=
∼ (2/λ) sin(θ)
(1.118)

118

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

where χ = 2θ. The total scattering intensity is given as   < I (hkl) >= (2λ) sin(θ)
Kp|F(hkl)|2

(1.119)

  The integration ( ρ δ ) and average < > of the Laue function G are different depending on the experimental method. (A)

Powder diffraction method

A powder sample is used in the X-ray diffraction experiment to give the DebyeScherrer rings. Among all the crystallites directing to all the directions, only the crystallites satisfying the Bragg   condition can scatter the incident X-ray beam. In the calculation of a term < ρ δ G dρ dδ> in Eq. (1.119), the average < > is made for the crystallites which satisfy the Bragg condition. The total surface area S of the Ewald sphere is 4π/λ2 and the surface dS corresponding to the dΩ is dΩ/λ2 , so the probability dS/S = dΩ/(4π). By the way, the volume V * of the reciprocal lattice with the vector length ρ in Fig. 1.91 is V ∗ = (4π/3)ρ3 For the small volume, we have dV ∗ = 4πρ2 dρ This part corresponds to the surface dS which satisfies the Bragg condition at the diffraction angle χ (= 2θ) in Fig. 1.91, and so a tiny volume part dω is given as 

dω = (the probability of dS) · dV ∗ = [dΩ/(4π)] × 4πρ2 dρ = dΩρ2 dρ or  dΩ = dω/ ρ2 dρ Fig. 1.91 Small cross section dS on the Ewald sphere and the small volume dω of the reciprocal lattice point with the vector ρ. The X-ray is scattered by a small reciprocal volume dV * for the powder sample but at the probability dΩ/4π to satisfy the Bragg condition

(1.120)

δ

1.8 Diffraction Data Analysis

119

The total solid angle (or the total surface of a sphere of radius 1) is 4π, and so the probability of dΩ is dΩ/4π, and the average of G for the variously oriented crystallites is made as follows:   

 
= (1/(4π))

G dρdδdΩ 

  = (1/(4π))

ω

ρ

δ

δ

   G 1/ρ2 dωdδ ∼ 1/ 4πρ2



 Gdω

ω

dδ δ

(1.121) As already shown, the Laue function is G = |DL |2 |DM |2 |DN |2 



= sin2 (Lπ t1 ) sin2 (M π t2 ) sin2 (N π t3 ) / sin2 (π t1 ) sin2 (π t2 ) sin2 (π t3 ) These terms can be approximated as Gaussian functions, as shown in Eqs. 1.102  and 1.104, and dω = a∗ d t1 · b∗ d t2 · c∗ d t3 = d t1 d t2 d t3 /v, and so the integration ω Gdω is finally given as  ω

Gdω = (aL/v)(bM /v)(cN /v) = LMN /v2 = V /v2

(1.122)

where the crystallite size is aL, bM, and cN along the a-, b-, and c-axes of the crystallite, respectively, and v is the unit cell volume (v = abc). As a result, we have  
= 1/ 4πρ dδV /v = (1/4π) λ / 4 sin (θ) dδV /v2 δ

δ

(1.123) The conclusion about the total intensity is   Gdρdδ > Kp|F(hkl)|2   

= (2λ) sin(θ)(1/4π) λ2 / 4 sin2 (θ) dδV /v2 Kp|F(hkl)|2 δ  3 = λ /[8π sin(θ)] dδV /v2 Kp|F(hkl)|2 (1.124)

< I (hkl) > = (2λ) sin(θ)
= λ3 /[4 sin(θ)]Kp|F(hkl)|2 V /v2 = LKp|F(hkl)|2 V /v2 (1.125)

120

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

The Lorentz factor for the powder diffraction ring is L = λ3 /[4 sin(θ)] If you measure only the finite arc part in the range δ = δ1 ∼ δ2 , then δ2 − δ1 = δ. The Lorentz factor is L = λ3 δ/[8π sin(θ)] (B)

(1.126)  δ

dδ =

(1.127)

The uniaxial rotation method

As shown in Fig. 1.92, the sample is rotated at the angular velocity γ around the y-axis. The average in Eq. 1.119 is made for the rotation (or measurement) time t.

Fig. 1.92 The case of uniaxially oriented sample around the y-axis. The angles α and φ and the vector ρ are used for showing the geometry of a small volume (dω), which can be converted using the angles δ and φ (and ρ). (upper) a flat camera and (lower) a cylindrical camera. The equatorial diffraction can be treated as shown in the lower left side. The spherical triangular angles are shown in the right upper parts. The point C is the projection of the point B along the x-axis. [modified from Fig. 5.2 of X-ray Crystallography (Ref. [74], 1961, Japanese) with permission from Maruzen Publisher.]

1.8 Diffraction Data Analysis

121

Since the rotation angle dφ = γdt,   

 
= t

ρ

   δ

Gdρdδdt = (1/γ)

Gdρdδdφ φ

ρ

(1.128)

δ

As likely the abovementioned (A) case, we need to calculate a small volume of the reciprocal lattice dω. An infinitesimally small volume is expressed as follows, when the spherical polar coordinates are defined (ρ, α, φ): dω = ρ2 sin αdρdαdφ

(1.129)

According to the property of the spherical triangle (note), the following equations are obtained: cos α = cos δ cos(χ/2) or dα = sin δ cos(χ/2)dδ/ sin α

(1.130)

Then we have dω = ρ2 sin δ cos(χ/2)dδdρdφ

(1.131)

Therefore, in Eq. (1.118),  


=
∼ (2/λ) sin(θ) < ρ    = (2/λ) sin(θ)(1/γ) Gdρdδdφ φ ρ δ   2 = (2/λ) sin(θ)/ γρ sin δ cos(χ/2) Gdω ω   = (2/λ) sin(θ)/ γ((2/λ) sin(θ))2 sin δ cos(θ) V /v2  = λ/(sin(2θ) sin δ) V /γ v2 ρ

δ

δ

Gdρdδ >

(1.132)

Then  < I (hkl) >= λ2 Kp|F(hkl)|2
= λ3 /(sin(2θ) sin δ) V /γ v 2 Kp|F(hkl)|2

(1.133)

The Lorentz factor for the flat camera is L = λ3 /[γ sin(2θ) sin δ]

(1.134)

The Lorentz factor for the cylindrical camera is L = λ3 /[γ cos(σ) sin τ]

(1.135)

122

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

where the relation sin(2θ)sinδ = cos(σ)sinτ is used. In the case of the equatorial line reflection, L = λ3 /[γ sin(2θ)]

(1.136)

since δ = 90°, σ = 0°, and τ = 2θ. *************************************** [Note] Spherical triangle Let’s assume that the three points A, B, and C are on one sphere of a radius R and the vectors are defined from the original point O. The inter-axial angles are defined as α, β, and γ, respectively. The external products of the vectors are AxB = S1 and AxC = S2 . The angle between these two vectors S1 and S2 is assumed to be θ. The following equations can be derived. Here |A| = A = R, |B| = B = R and |C| = C = R. S1 ·S2 = S1 ·S2 cos(θ) = [AB sin(γ)][AC sin(β)] cos(θ) = R4 sin(γ) sin(β) cos(θ). On the other hand, S1 ·S2 = [AxB]·[AxC] = Bx[AxC]·A = [(B·C)A – (A·B)C]·A = (B·C)(A·A) – (A·B)(C·A) = BCA2 cos(α) − A2 BC cos(γ) cos(β) = R4 [cos(α) − cos(γ) cos(β)]. From these two equations, we have cos(θ) = [cos(α) − cos(γ) cos(β)]/[sin(γ) sin(β)] In the case of point A of Fig.1.92, γ → α, β → δ, α → χ/2 and θ → φ, then we have the following relation: cos(φ) = [cos(χ/2) − cos(α) cos(δ)]/[sin(α) sin(δ)]

(1.137)

Similarly, at the point C, α → α, β → δ, γ → χ/2 and θ → 90◦ , then cos(90◦ ) = [cos(α) − cos(χ/2) cos(δ)]/[sin(χ/2) sin(δ)] = 0 Then, cos(α) = cos(χ/2) cos(δ)

(1.138)

*************************************** Equations 1.134 and 1.135 can be rewritten using the cylindrical coordinates (ξ, ζ ) [see the Sect. 1.9.1]. In Eq. 1.134 (and 1.135), sin(δ) is now modified as follows:

1.8 Diffraction Data Analysis

123

Fig. 1.93 Reciprocal cylindrical coordinates (ξ, ζ) and the angles α and ψ (refer to Fig. 1.92)

y

α

ψ

ζ

z

ξ -x

From Eq. 1.138, cos(α) = cos(χ/2) cos(δ) = cos(θ) cos(δ) By changing the angle α to the angle ψ defined in Fig. 1.93, ψ = 90° – α, we have sin(ψ) = cos(θ) cos(δ) or, cos(δ) = sin(ψ)/ cos(θ)

1/2 

1/2  = 1 − sin2 (ψ)/ cos2 (θ) sin(δ) = 1 − cos2 (δ)

(1.139) (1.140)

Using the ξ and ζ defined in Fig. 1.93, 1/2  sin(ψ) = ζ / ξ 2 + ζ 2

(1.141)

Besides, 2d sin(θ) = λ is rewritten as (in this section, the ξ and ζ are dimensionless)   sin2 (θ) = λ2 / 4d 2 = ξ 2 + ζ 2 /4

(1.142)

Therefore, using Eqs. (1.140)–(1.142), Eq. (1.134) changes to L = λ3 /[γ sin(2θ) sin(δ)]   

1/2  = λ3 /2γ / sin(θ) cos2 (θ) − sin2 (ψ)   2 −1/2  = λ3 /γ ξ 2 − ξ 2 + ζ 2 /4

(1.143)

[Note: Eq. (4) of L reported in Ref. [21] should be corrected by adding the “1/2” as shown above.]

124

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

The Laue method (Continuous λ Distribution, Time-of-Flight)

(C)

As well known, Dr. Max von Laue predicted “the diffraction of X-ray beam of short wavelength by the crystal consisting of the narrow atomic distances” in 1912 [Interferenz-Erscheinungen bei Röntgenstrahlen. Sitzungsberichte der Kgl. Bayer. Akad. der Wiss. 303–322 (1912)], and W. Friedlich, P. Knipping, and M. Laue discovered the X-ray diffraction phenomenon using the white X-ray beam irradiated to a copper sulfate crystal [Annalen der Physik, 346, 971–988 (1913)]. The white X-ray beam consists of the continuous distribution of X-ray beams of the various wavelengths. This method is called the Laue method. Since the synchrotron beam with the wide wavelength distribution fits this point, it becomes popular to use the Laue method to the X-ray structure analysis based on the synchrotron system. In the case of time-of-flight method, which is now utilized in the neutron scattering experiment with the pulse neutron beam (consisting of neutron waves in the wide wavelength range), the measurement corresponds exactly to the Laue method. Let us consider the Lorentz correction for the Laue method. The wavelength of the incident X-ray beam changes continuously. Different from the abovementioned (A) and (B) cases, a single crystal is assumed to be used as a sample. (Of course, the powder sample and the uniaxially oriented sample are also used with this method, but the average for the factor (G) becomes complicated.) When the single crystal is set with a certain orientation, no reflection might occur in the worst case as long as a monochromatized X-ray beam is used. However, in the Laue method, a chance to satisfy the Bragg condition may be existent for a certain X-ray beam, the wavelength of which is λ. However, the λ is not unique, but the X-ray beam components having the λ in some range (λ) are scattered since the crystallite size is finite. We need to average the G by taking such a situation into account. In Eq. 1.115,  


=
∼ (2/λ) sin(θ)
(1.144)

  

 
=

λ

ρ

 G dρdδ dλ

(1.145)

δ

Because ρ = (2/λ) cos ψ, λ = (2/ρ) cos ψ. Then, dλ = −(2/ρ) sin ψdψ  
=

ψ

ρ

δ

1.8 Diffraction Data Analysis

125

Fig. 1.94 A small portion on the Ewald sphere. [modified from Fig. 5.1 of X-ray Crystallography (Ref. [74], 1961, Japanese) with permission from Maruzen Publisher.]

(by exchanging the integral range between the upper and lower limits) As shown in Fig. 1.94, a small volume dω = ρ 2 sin ψdρdψdδ or dρdψdδ = dω/(ρ2 sin ψ), and so Eq. (1.146) is  
=

ψ ρ δ

 G(2/ρ) sin ψdψdρdδ =

ω

  G 2/ρ3 dω ∼ 2/ρ3 GV /v 2

Here ρ = (2/λ) sin(χ/2) = (2/λ) sin(θ),  
= 2GV /v2 /[(2/λ) sin(θ)]3

(1.147)

As a result,  < I (hkl) > = λ2 Kp|F(hkl)|2


  = λ2 Kp|F(hkl)|2 (2/λ) sin(θ) < Gdρdδ > ρ δ 

 = λ4 / 2 sin2 (θ) V /v2 Kp|F(hkl)|2

(1.148)

The Lorentz factor L is given as 

L = λ4 / 2 sin2 (θ)

(1.149)

In this way, the Laue function plays an important role to determine the integrated intensity of the observed diffraction, which changes depending on the crystallite size and their distribution (powder, uniaxially oriented sample, or single crystal) as seen in Scherrer’s equation or the shape of the Lorentz  factor. In particular, it must be noticed that the average of the Laue function < S GdS > is different depending

126

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

on the way of the average of the crystallites (wavelength dλ, rotation dφ, or powder dδ). Furthermore, it must be checked that the dependence on the wavelength λ is different for the various measurements.

1.8.6 Estimation of Integrated Intensity and Structure Factor The integrated intensity I(hkl) is given as I (hkl) = K



mhkl |F(hkl)|2 LpA

so the absolute value of the structure factor can be obtained by knowing the factors L, p, and A. For the usual polymer substance composed of carbon, oxygen, nitrogen, etc., the A factor can be ignored. The scale factor K is to adjust the total sum of the observed and calculated intensities to the equal value. K = Iobsd (hkl)/Icalc (hkl)

(1.150)

The usage of the thus-derived structure factor will be described in the following sections. As seen in the 2D X-ray diffraction pattern of a polymer sample, the reflections are broad as a whole and overlap sometimes. The estimation of the intensity of a diffraction spot is more or less hard. Sometimes the peak intensity is read and used for the structure analysis, but it is not necessarily recommended. (A)

Naked eyes method

In the old times using X-ray films, the intensity estimation was made with naked eyes. By comparing the darkness of an observed spot with that of a standard darkness, the observed darkness is estimated as a numerical value. Concretely, one reflection spot of an arbitrarily chosen sample (a sharp reflection from a single crystal is better) is measured on an X-ray film for a constant exposure time and the film position is shifted at a constant spacing. Repeating this process by changing the exposure time from 0.1 sec, 0.5 sec, 1sec, 5 sec, and so on, we can get a strip of film with a series of spots of different intensity, which is used as a standard intensity scale (Fig. 1.95). The spots are scored depending on the exposure time (#1, #5, #10, #50, etc. in the above example). This standard scale is put near the actually observed diffraction spot, and the standard spot with a darkness similar to the actually observed diffraction is found out. For example, if the darkness of the standard spot # 10 is comparable to the observed peak darkness, the intensity 10 is assigned to the peak. As a result, we have a table of scores for all the observed spots, which is scaled by the total sum of all the observed scores. This naked-eyes method looks quite rough but the careful estimation gives the intensity within an error of about 10%. Some people say this naked-eyes method is actually an evaluation of the “integrated” intensity. Of course,

1.8 Diffraction Data Analysis

127

Fig. 1.95 a Intensity estimation by naked eyes using a film strip with the standard darkness. b The diffraction spots are separated into the profiles along the horizontal lines. After the curve separation, the integrated intensity A is estimated and summed up to get the integrated intensity of a spot #1

this is only for the case of small and sharp diffraction spots. For the polymer samples, it is a little difficult to say so. Since the diffraction spots have the different intensities in a wide range, some of the strong peaks might be saturated in intensity. Usually, the X-ray diffraction patterns are measured by using the superimposed films. The top-surface film is exposed most strongly, and the second film shows a little weaker intensity for the same spot, and so on. The abovementioned naked-eyes method is applied for all the films of the different darkness and scaled among the same spots with different intensities. This method is called the multifilm method. (B)

Diffraction profile method

In these days, the digitized diffraction data can be used and the integration of the intensity becomes comparatively easier. There are the various ways. As will be shown concretely in a later section, the author developed a software to analyze the X-ray diffraction data of polymer substance. All the data points (pixels) on the image are saved as the digital data. The summation of the intensity values over a focused reflection spot can be made. For example, an observed spot is encircled using a mouse (a blue curve in Fig. 1.95b) and the numerical intensity values of the pixels existing inside this area are summed up to get an integrated intensity. Of course, the background correction cannot be forgotten. This method is rather rough since the overlapped diffraction spots are difficult to treat. We need to separate the overlapped spots. However, the separation of the overlapped spots is hard. Another way is a combination of 1D profile deconvolution method and intensity summation of the separated peak components. As shown in Fig. 1.95b, a series of the 1D profiles is obtained along the horizontal line of each pixel width (i = 1−m). The overlapped

128

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

peaks of an 1D profile are separated into the components including the background. This process is repeated for all the 1D profiles over a whole diffraction spot. Since the separated component belonging to the same diffraction spot is known (for example, reflection component # 1), the total sum of the integrated value A(1)[= i A(1)i ] is equal to the integrated intensity after the curve separation and background correction. The concrete demonstration will be shown in a later section.

1.9 Crystal Structure Analysis Once we know the methods how to estimate the various parameters (K, A, L, p, and m) for the calculation of the integrated intensity I and for the evaluation of the structure factor |F|, we can now proceed further to the structure analysis of a polymer sample.

1.9.1 Determination of Unit Cell 1.9.1.1

Polanyi’s Equation

One of the characteristic points of the two-dimensional X-ray diffraction pattern of a uniaxially oriented polymer sample is the existence of the many horizontal layer lines. This comes from the fully rotated pattern of the one-dimensional crystal lattice with the regular repeating period I (Fig. 1.85). As illustrated in Fig. 1.96a, the incident X-ray beam is diffracted by all the repeating points. These diffracted waves are interfered with each other and go to the directions determined by the following equation. The difference of the optical path between the neighboring diffracted waves is I sin(φ), which should be equal to the integral multiple of wavelength λ.

(a)

(b)



rising-up angle

 X-ray

sample

Fig. 1.96 Polanyi’s equation

R camera distance

ym

1.9 Crystal Structure Analysis

129

Fig. 1.97 2D X-ray diffraction pattern of the uniaxially oriented polyethylene sample measured with Mo-Kα X-ray beam

Y1 y1 Y0

I sin(φ) = mλ

(1.151)

The rising-up angle φ is given as tan(φ) = y/R

(1.152)

where y is the distance between the equatorial line and the layer line, and R is the camera radius, as illustrated n Fig. 1.96b. Therefore, it is easy to estimate the repeating period I using these two equations. Equation (1.151) is called Polanyi’s equation [77], and quite useful for the estimation of the repeating period along the chain axis. This information is the starting point of the X-ray structure analysis of a polymer. For example, Fig. 1.97 is the two-dimensional X-ray diffraction pattern of the uniaxially oriented polyethylene sample. The X-ray wavelength is 0.71073 Å (Mo-Kα). The distances between the equatorial and layer lines are given below: Pixel position along the y-axis equatorial Y0 = 1505.95 pixel 1 st layer Y1 = 1876.04 pixel Y2 = 2366.26 pixel 2nd layer

(pixel size 0.1 × 0.1 mm2 ) → 150.595 mm → 187.604 mm → 236.626 mm

The spacing between l =0 and l =1 or 2 is as follows: y1 =Y 1 – Y o = 37.01 mm and y2 = Y 2 – Y o = 86.03 mm. The camera distance R = 127.4 mm. Then tan φ1 = y1 /R = 37.01/127.4 = 0.291 φ1 = 16.20◦ tan φ2 = y2 /R = 86.03/127.4 = 0.675 φ1 = 34.03◦ Using Polanyi’s equation, we can get the repeating period as I1 = 2.54 Å (from y1 ) and I2 = 1.27x2 = 2.54 Å(from y2 )

130

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

The averaged value is I = 2.54 Å. If we assume the planar-zigzag chain conformation for PE chain, then the model gives the calculated repeating period I I = 2r(CC) sin(∠CCC/2) = 2 × 1.54 Å · sin(109.5◦ /2) = 2.52 Å This value is almost equal to the observed value, indicating the conformation of PE chain is essentially of the planar-zigzag type.

1.9.1.2

Cylindrical Coordinates

The two-dimensional X-ray diffraction pattern measured for the uniaxially oriented sample is used for analysis by reading the peak positions and evaluating the integrated intensities of all the observed diffraction spots. The (x, y) coordinates of the diffraction spots are read as accurately as possible. The coordinates have the unit of mm. It must be careful to notice that the X-ray diffraction pattern is displayed usually using the matrix of pixels. One pixel size is 0.05–0.1 mm depending on the detector. The positions expressed by the pixel unit must be transformed to the coordinates of the mm unit. At first, the (x, y) coordinates are transformed to the (ξ, ζ ) coordinates of the cylindrical coordinate system of the reciprocal lattice on the basis of the following equations (Fig. 1.98). For the cylindrical camera used in the measurement,   1/2  2 2 ξ = 1 + 1/ 1 + (y/R) − 2 cos(x/R)/ 1 + (y/R) 

 1/2 ζ = y/ 1 + (y/R)2

(1.153) (1.154)

where R is the camera radius and λ is the wavelength of the incident X-ray beam. For the flat camera, 

   2 1/2 2 2 2 2 2 ξ = 1 + R + x / R + x + y − 2R/ R + x + y 

2

2

 ζ = y/ (R2 + x2 + y2 )

(1.155) (1.156)

As already mentioned, Polanyi’s equation treats the repeating period along the chain axis by assuming the one-dimensional lattice. The ζ value is defined for the axis parallel to the drawing direction, and it is equal to the spacing between the equatorial and layer planes in the reciprocal lattice. If the chain axis is parallel to the drawing direction of the sample, these two values correspond to each other although the dimension is inversed. For the ζ value of the first layer line, I = c = λ/ζ

(1.157)

1.9 Crystal Structure Analysis

131

Fig. 1.98 Relation between the cylindrical coordinates (ξ, ζ ) and the (x, y) coordinates fixed on the flat plate (upper) and cylindrical camera (lower)

1.9.2 Estimation of Unit Cell and Space Group (Orthorhombic Cell) The thus-calculated ξ and ζ values of orthorhombic polyethylene are listed in Table 1.10, where the data were measured using a cylindrical camera and the calculation was made using Eqs. 1.153 and 1.154. At first, let us focus the equatorial line with ζ = 0. The circles of the radii ξ are drawn for the equatorial spots (#1 - #10) on a sheet of paper. A set of any plausible reciprocal lattices (a*, b*, and γ∗ ) is drawn also on this paper. You can find the crossing points of the circles with the reciprocal lattice points. One circle must cross at least one reciprocal lattice point. This must be satisfied for all the observed diffraction spots. In other words, the reciprocal unit cell parameters can be determined so that all of the observed circles cross any one lattice point at least. The lengths of the a*- and b*-axes and the biaxial angle between them (γ∗ ) are varied in a trial-and-error way and finally some candidates of the reciprocal lattice are obtained. The thus-obtained equatorial a*b*-plane must fit to the circles drawn for the 1st layer-line diffractions with the corresponding ζ value. The similar situation must be satisfied for all the layer lines.

132

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Figure 1.99 shows one example: the equatorial reflections observed for the uniaxially oriented PE sample. Usually we start by assuming that the reflection of the lowest angle (or the smallest x value, #1 in Table 1.10) should have the simplest index, for example, 100. The second one (#2) should have the similar one, 010. If this assumption cannot give the reasonable answer for the other reflections, then we have to abandon this candidate of the unit cell, and assign #1 and #2 to the 110 and 200, for example. In the PE case, we found that the 1st circle (reflection #1) crosses the 110 position. The 2nd one fits the 200 point. Then, all the other reflections were found to cross the reciprocal lattice points as seen in Fig. 1.99. The thus-determined parameters are

a∗ · λ = 0.1891/2 = 0.0946, b∗ · λ = 0.2864/2 = 0.1432 and γ∗ = 90.0◦ The hk0 indices were obtained as the crossing points of the reciprocal lattice, which are listed in Table 1.10. The unit cell parameters of the real lattice system are Table 1.10 The positions of the diffraction spots observed for PE sample No.

x/mm

y/mm

ξ

ζ

hkl

d obs /Å

d calc / Å

1 (equatorial)

21.89

0

0.1714

0

110

4.147

4.142

2

24.16

0

0.1891

0

200

3.759

3.759

3

30.31

0

0.2375

0

210

2.993

2.996

4

36.59

0

0.2864

0

020

2.482

2.482

5

40.71

0

0.3182

0

120

2.338

2.357

6

43.99

0

0.3432

0

220

2.071

2.071

7

48.77

0

0.3795

0

400

1.873

1.879

8

51.81

0

0.4049

0

410

1.755

1.758

9

56.41

0

0.4394

0

130

1.618

1.616

10

60.37

0

0.4696

0

420

1.514

1.498

11 (first layer)

17.77

37.01

0.1422

0.280

011

2.263

2.261

12

21.81

37.01

0.1722

0.280

111

2.162

2.165

13

23.93

37.01

0.1880

0.280

201

2.107

2.105

14

30.56

37.01

0.2378

0.280

211

1.935

1.938

15

39.11

37.01

0.3022

0.280

121

1.725

1.728

16

41.07

37.01

0.3170

0.280

311

1.680

1.679

17

44.83

37.01

0.3453

0.280

221

1.599

1.605

18

49.58

37.01

0.3810

0.280

401

1.503

1.511

19

52.95

37.01

0.4063

0.280

411

1.440

1.445

20

56.20

37.01

0.4306

0.280

031

1.384

1.386

1.9 Crystal Structure Analysis

133

(b)

(a)

b*

b*

020 120

010

220

110 210 110

O* 100

200

a*

O*

310

200

a*

Fig. 1.99 Indexing of the equatorial line reflections of the uniaxially oriented PE sample using the ξ values. (a) Not good indexing (b) correct answer

a = 1/a∗ = 7.517 Å, b = 1/b∗ = 4.963 Å, and γ = 180◦ − γ∗ = 90.0◦ The first layer-line data satisfied also these unit cell parameters. By knowing all the indices and the rough values of a-, b-, and c-axial lengths, we can refine the parameters by the least squares calculation. The indexing of all the observed data is shown in Table 1.10. The lattice spacings (d) are calculated as 1/2  dobs = λ/ ξ 2 + ζ 2

(1.158)

The d values are calculated using the above-determined lattice constants a, b, and c. (The c value was already known as mentioned in Sect. 1.9.1.1.)

−1/2  dcalc = (h/a)2 + (k/b)2 + (l/c)2

(1.159)

The comparison between the observed and calculated d values is made in Table 1.10 and in Fig. 1.100. From Table 1.10, we may extract some extinction rules about the observed reflections. For example, the spots of 200 and 400 are observed and those of 100 and 300 are not detected, suggesting the extinction rule of h00 with even number. Similarly, the spot of 020 was detected but 010 could not be found, suggesting the rule of 0k0 reflections with the even number. If these rules are correct, the unit cell must have the 21 screw axes along the a- and b-directions. The hkl spots did not show any systematic rule, indicating that the Bravais lattice should be primitive (P). The 0kl

134

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.100 Comparison between the observed and calculated d values (Table 1.10)

3

d

calc

/A

4

2

1 1

3

2 d

obs

4

/A

with k+l = even might be satisfied also. The h0l with h = even seems also possible. These rules suggest the existence of the glide planes: the normal vector // a with the translation along the (b + c)/2 axis (diagonal direction) and the normal vector // b with the translational vector // a-axis, respectively. By combining all of these extinction rules, the space group of Pna* might be possible. More concretely, the candidates are Pnam and Pna21 . The packing of CH2 units into the unit cell must be considered now. Since the observed density of the PE sample was ρ = 0.97 g/cm3 , the total number of CH2 units Z is given as  Z = ρabcNA /M (CH2 ) = 0.97 g/cm3 × 7.517 × 4.963 × 2.54  × 6.02 × 1023 Å3 /mol /14(g/mol)   = 39.5 g/cm3 × 1023 × 10−24 cm3 /mol /(g/mol) = 3.95 Since the Z value should be an integer and the sample contains the amorphous region with the lower density (in general), then the Z might be 4. As already known, PE takes the zigzag conformation. The zigzag chain contains the 2 CH2 units in a repeating period along the chain axis, and so the unit cell contains the two zigzag chains as a result. Now we need to know how these two chains are packed in the unit cell. As mentioned above, the candidates of the plausible space group are Pnam and Pna21 . The CH2 units are packed in such a way as mentioned in Sect. 1.3.2. At first, one CH2 unit (rather C atom) is put into an arbitrary position and the other three units are generated automatically by applying the various symmetry operations included

1.9 Crystal Structure Analysis

135

in these space group symmetries. Figure 1.25 shows the possible packing modes for the space group Pnam. The structure factors are calculated for these models and the coordinates of the C atoms are modified so that the observed data are reproduced as well as possible. The structural refinement will be described in the next section. Even after many trials of the structure modification, the reproduction of the observed diffraction intensity data might not be good in some cases. In such a case, we need to doubt the space group symmetry and/or the indexing of the observed diffraction spots. The indexing cannot be necessarily perfectly made for all the observed reflections. Some reflections might not be fitted well to the estimated lattice network. One possibility originates from the wrong unit cell parameters. Another possibility is the mistake in the estimation of the peak positions of these observed reflections. The other possibility is the mixing of the other crystalline form as a contamination. In such a case, we need to prepare the sample of the pure crystalline form again.

1.9.3 Estimation of Unit Cell and Space Group (Monoclinic Cell) The second example is about the monoclinic unit cell. In this case, the indexing of the non-equatorial layer lines needs some care. In the abovementioned PE crystal, the rectangular unit cell is a correct answer. However, if the unit cell is not rectangular and α ∗ or/and β ∗ angle is different from 90°, the c*-axis is not parallel to the c-axis as shown in Fig. 1.101. For example, in the case of monoclinic unit cell with β ∗ =

Real Lattice

c

Reciprocal Lattice

draw-axis

c*

Ewald sphere

c c*

O*1

molecular chain

hk0

b* a*

b a

hk1

O*0

b*

a*

Fig. 1.101 Chain tilting and crossing of reciprocal lattice points with Ewald sphere. Reprinted from Ref. [110]. Copyright 2014, with permission from Elsevier

136

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

90°, the c*-axis will be tilted from the c-axis in the ac-plane. In the triclinic unit cell, the tilt of the c*-axis may occur in more complicated way since both of α ∗ and β ∗ angles are different from 90°. In the monoclinic case of β ∗ = 90°, the origin of the reciprocal lattice network on the first layer must be shifted to the a*-axial direction by the amount c*cos(β ∗ ).

At first, as similarly to the case of PE data, the observed diffraction spots on the equatorial line are treated. The circles with the radii of ξ values are drawn around the center of the reciprocal lattice of a*- and b*-axes. Then, for the m-th layer line, the circles are drawn using the ξ values of the m-th layer line as the radii around the center which is the same as that of the equatorial line. But, the origin of the reciprocal lattice plane must be shifted by the amount of m·c*cos(α ∗ ) or m·c* cos(β ∗ ). This can be seen in the case of the α crystal form of trans-1,4-polyisoprene or Gutta Percha [78]. Figure 1.102 shows the 2D X-ray diffraction pattern measured for the uniaxially oriented α form, where Mo-Kα line was used as an incident X-ray beam and the cylindrical camera for the detector. The 104 reflections were detected in total from this X-ray diffraction pattern. Table 1.11 shows the (x, y) positions of the observed diffraction spots along the equatorial and 1st layer lines as examples. The (ξ, ζ ) values were calculated using Eqs. 1.153 and 1.154. The indexing was made at first for the equatorial line data of ζ = 0. The basic reciprocal unit cell plane Fig. 1.102 X-ray diffraction pattern measured for the uniaxially oriented Gutta Percha α form. Reprinted from Ref. [78]. Copyright 2012, with permission from Elsevier

1.9 Crystal Structure Analysis

137

Table 1.11 Experimental data (part) of 2D X-ray diffraction diagram of uniaxially oriented Gutta Percha (The ξ and ζ have the dimension of λ−1 (Å−1 ) in this table) No

x(pixel)

y(pixel)

ξ (Å−1 )

ζ (Å−1 )

d obs /Å

h

k

l

d calc /Å

1

1538

1270

0.126

0

7.84

1

0

0

7.75

2

1606.4

1270

0.201

0

4.94

1

1

0

4.91

3

1654.1

1270

0.254

0

3.92

2

0

0

3.88

4

1695.4

1270

0.299

0

3.34

2

1

0

3.31

5

1708.3

1270

0.313

0

3.18

0

2

0

3.17

6

1731.5

1270

0.339

0

2.94

1

2

0

2.93

7

1770.2

1270

0.381

0

2.62

3

0

0

2.58

8

1797.1

1270

0.411

0

2.43

2

2

0

2.45

9

1873.8

1270

0.494

0

2.02

3

2

0

2.00

10

1909.2

1270

0.533

0

1.87

2

3

0

1.86

1

1567.8

1373

0.159

0.114

5.11

0

1

1

5.10

2

1592.8

1373

0.186

0.114

4.58

−1

1

1

4.54

3

1620.4

1373

0.216

0.114

4.09

1

1

1

4.02

4

1675.6

1373

0.277

0.114

3.34

−2

1

1

3.30

5

1712.5

1373

0.317

0.114

2.97

2

1

1

2.91

0

2

1

2.97

6

1741.4

1373

0.349

0.114

2.72

1

2

1

2.71

[Reproduced from Ref. [78] with permission of Elsevier, 2012]

satisfying all the observed equatorial line spots was a* = 0.126 Å−1 and b* = 0.313 Å−1 and γ ∗ = 90.0° as a candidate. This reciprocal unit cell plane should be applied effectively to the 1st layer-line data, too. The circles were drawn similarly and the a*b* network was put on these observed circles. But, this network cannot satisfy the observed circles well as long as the origin of the reciprocal lattice of the 1st layer line was put at the same position as the origin of the equatorial lattice plane. It is found the data can be satisfied better by shifting the reciprocal network by 0.0243 Å−1 along the a*-direction. This shift value is equal to c*cos(β ∗ ). Then we have the following two relations: c*cos(β ∗ ) = 0.0243 Å−1 and also c*sin(β ∗ ) = ζ = 0.114 Å−1 = 1/c, which comes from the interlayer spacings or the inverse of the repeating period along the chain axis. Therefore, we get the following values after refinement: a∗ = 0.126 Å−1 , b∗ = 0.313 Å−1 , c∗ = 0.117 Å−1 , α ∗ = 90◦ , β ∗ = 78.0◦ , and γ ∗ = 90◦ a = 7.948 Å, b = 6.339 Å, c(chain axis) = 8.787 Å, and β = 102.716◦

138

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Table 1.11 lists up the indices of the observed diffraction spots. When the observed indices hkl are investigated totally, we find the following systematic rules: 0k0(k = 2n) and h0l(l = 2n) These extinction rules are some of the rules expected for the space group of the 5 . monoclinic system P12l /c1 − C2h

The lattice spacing d is given as d ∗2 = (1/d )2 = (ξ 2 + ζ 2 )/λ2 The unit cell parameters obtained by the indexing procedure are refined so that all the observed d values could be reproduced as exactly as possible. Usually, the least squares refinement is made so that the standard errors between the observed and calculated d values become the smallest. The d* value is given as

|p∗i |2 = (1/d )2 = |hi a∗ + ki b∗ + li c∗ |2 = h2i a∗2 + k12 b∗2 + li2 c∗2 + 2hi ki a∗ b∗ cos γ ∗ + 2hi li a∗ c∗ cos β ∗ + 2ki li b∗ c∗ cos α ∗

(1.160)

δi = (1/di )2obs − (1/di )2calc = (1/di )2obs  − h21 a∗2 + ki2 b∗2 + li2 c∗2 + 2hi ki a∗ b∗ cos γ ∗ + 2hi li a∗ c∗ cos β ∗ + 2ki li b∗ c∗ cos α ∗

The total error  = i δ2i is minimized by adjusting the parameters a*, b*, c*, α *, β *, and γ *. The process is to derive the first derivatives ∂/∂a∗ = 0, ∂/∂b∗ = 0, ∂/∂c∗ = 0, ∂/∂α ∗ = 0, ∂/∂β ∗ = 0, and ∂/∂γ ∗ = 0. The details of the derivation of the nonlinear equation for getting the refined parameters are referred to in the literature. (Note) The unit cell determination of cubic NaCl crystal (reference) For reference, let us see the other method to estimate the unit cell parameters and the indices of the observed reflections by using a simple example of NaCl powder sample. The following method may be used under such an assumption that the crystal system is already known. In the case of NaCl, the crystal system belongs to the cubic system. Table 1.12 lists up the diffraction angles of the observed peaks where the CuKα line was used as an incident X-ray beam. At first, the lattice spacing (d) is calculated from the Bragg angle [d = λ/(2 sin(θ))]. The results are shown in Table 1.12. The reflection peaks in the lowest angle region should have the simplest indices such as

1.9 Crystal Structure Analysis

139

Table 1.12 Indexing of the X-ray diffraction peaks observed for NaCl powder sample 2θ /degree

(a/d)2

d/Å

(a/d)2 × 2

(a/d)2 × 3

h2 + k 2 + l 2

hkl

m

Relative intensity

a = 3.238Å assumed

(Mo-Kα) 27.55

3.238

1.000

2.000

3.000

3

111

8

8.0

31.89

2.806

1.332

2.664

3.994

4

200

6

100.0

45.65

1.987

2.656

5.312

7.968

8

220

12

37.0

54.04

1.697

3.641

7.282

10.923

11

311

24

2.0

56.66

1.624

3.975

15.801

11.926

12

222

8

9.0

66.39

1.408

5.289

10.577

15.866

16

400

6

5.0

100, 200, 110, etc. Let us assume that the first peak is assigned to the index 100, and ainitial = 3.238 Å (= b = c). The other observed reflections must satisfy the following equation as long as the assumed ainitial value is correct: (ainitial /d )2 = h2 + k 2 + l 2 = integer The actually calculated values of (a/d)2 are shown in Table 1.12. They did not satisfy the above equation. Then the thus-calculated (a/d)2 values were multiplied by 2, 3, and so on, until all the observed reflections satisfy the above “integer” condition. The 3(ainitial /d)2 was found to give the integers for all the observed reflections. Therefore, the real a value is a2 = 3(ainitial )2 and a = 5.608 Å. The index hkl can be estimated easily as long as the cubic lattice is assumed. For example, for the peak ¯ ¯ 111, ¯ 1¯ 11, ¯ … 111, with h2 + k 2 + l 2 = 3, h, k, and l must be +1 or −1; 111, 111, (multiplicity m = 8). The results are shown in Table 1.12. Now, we need to find the Bravais lattice from the extinction rule. The indices given for all the observed reflections satisfy such a rule that the h, k, and l of an hkl index are even or odd number together; in other words, they satisfy the conditions of h + k = even, k + l = even, and l + h = even. This extinction rule corresponds to that of the face-centered lattice. Since the density of NaCl crystal is 2.18 g/cm3 , the number of NaCl pairs (mass = 58.44 g/mol) in the unit cell is calculated as below:  2.18 g/cm3 = 58.44Z/ a3 NA and Z = 3.95 ∼ 4 The number of the representative points contained in the face-centered lattice is 4, and one pair of Na and Cl atoms corresponds to one representative point. The next job is to determine the coordinates of the Na and Cl atoms in the cell by calculating the structure factors of all the reflections and compare them with the observed values. In this calculation, the space group symmetry is needed to be used. However, the total number of the observed reflections is too small as seen in Table 1.12 and it is

140

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

actually difficult to derive one unique space group. The already-established space ¯ the extinction rules of which are as follows: group of NaCl cubic crystal is Fm3m, hkl : h + k, h + l, k + l = 2n 0kl : k, l = 2n hhl : h + l = 2n h00 : h = 2n The indices shown in Table 1.12 look to satisfy all of these rules well.

1.9.4 Modeling of Crystal Structure and Calculation of Diffraction Intensity Now let us return to the analysis of polymer crystal. We have already determined the unit cells of orthorhombic polyethylene and Gutta Percha. The polymer chains are packed in these cells. At first, we need to estimate the shape of molecular chain, or molecular chain conformation. The construction of a molecular chain conformation is not easy. By using the various structural parameters, i.e., the bond lengths, bond angles, and torsional angles, we can estimate the stereo structure using the so-called equation of helical parameters (Sect. 1.12.4). For this purpose, the computer program was written. In the actual calculation of the chain conformation of Gutta Percha, we must have the condition that the monomeric units are repeated by the glide plane symmetry (see Figs. 1.103 and 1.104). Because of such a geometrical constraint, the torsional angle between the adjacent monomeric units is not independent but is given as a subjective condition. In

(a)

(b) -179.1o T -15.5o C -177.1o T -

-116.9o S 179.1o T 15.5o C 177.1o T 116.9o S

Fig. 1.103 Chain conformation of Gutta Percha α form. Reprinted from Ref. [78]. Copyright 2012, with permission from Elsevier

1.9 Crystal Structure Analysis

141

these days, we can utilize the convenient commercial software, by which a polymer chain with the infinitely repeated monomeric units can be generated. For example, one monomeric unit is drawn on the computer display screen and the crystal structure with a particular symmetry operation (21 , 31 ,..., glide plane,…) is created. The energy calculation is performed and the model of the lowest energy is produced. In this calculation, the standard geometrical parameters are used and the suitable potential function parameters are adapted. Depending on the parameters, the minimized chain conformation is more or less different. The details of the energy calculation will be described in a later section. Figure 1.103 shows the thus-constructed conformation of the Gutta Percha α form with a glide plane σg (ac|c).

In the present case of Gutta Percha α form, we have an information about the space group (P21 /c). Once if we can get a rough shape of one crystallographically asymmetric unit (which does not necessarily correspond to one monomeric unit), the symmetry operations included in this space group can generate the other units and then the crystal structure model is built up. Since the observed density of the sample is about 0.91 g/cm3 , we can estimate the total number of the monomeric units included in the unit cell Z using the information of the density ρsample ≤ ρcrystal = ZM /(Vcell NA ); Z = 3.8–4. The four monomeric units are included in the unit. The 2 monomeric units are used for the molecular chain with a glide symmetry σg (ac|c). So the two chains are contained in the cell, which are related by a point of symmetry, requiring that these two chains should be directed into the upward and downward directions along the chain axis. The definition of the upward and downward chains can be made by focusing on some atomic groups, for example, a methyl group. The methyl group is attached to the skeletal carbon atom and the vector of C-CH3 bond can be directed toward the positive (upward) or negative (downward) direction along the chain axis. The thus-obtained structure must be refined so that the observed X-ray diffraction data are reproduced as well as possible. The total number of adjustable parameters of one crystallographically asymmetric unit is 21. (One monomeric unit contains the five carbon atoms, each of which has the x-, y-, z-coordinates and the isotropic temperature factor Biso . Therefore, the 5 x 4 = 20 parameters. Including the scale factor, the total parameters are 21. The hydrogen atoms are not taken into account.)

Using the thus-built up model, the structure factors F(hkl)s are calculated. As an example, the calculation of F(210) is performed as given below. The structure factor F(hkl) with a global isotropic temperature factor B is given as

   F(hkl) = j fj (k) exp 2π i hxj + kyj + lzj exp −B sin2 θ/λ2  

 F(210) = j fj (k) exp 2π i 2 · xj + 1 · yj exp −B sin2 θ/λ2

(1.161)

142

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

The summation is made for all the atoms in the cell. The atomic scattering factor f j (k) at the scattering angle 2θ is needed to be known from the curve of the atomic scattering factor plotted against the 2θ. The f j (2θ) is expressed using an equation  fj (2θ/λ) = aj exp −bj (sin θ/λ)2 + c (j = 1 ∼ 4) where the coefficients are given in the table shown in Sect. 1.4.1.2. For example, the following values are assigned to a carbon atom: a1 = 2.310, a2 = 1.020, a3 = 1.589, a4 = 0.865 b1 = 20.844, b2 = 10.208, b3 = 0.569, b4 = 51.651 c = 0.216 Since the diffraction angle 2θ for the 210 spot is 12.2°, sin(θ)/λ = 0.149 and f (C) is given as 4.32, where λ = 0.71 Å. The f value of the H atom is small and can be ignored here for simplicity. Therefore, we can calculate F(210) as 



F(210) = 4.32 exp(2πi(2x1 + y1 )) + exp(2πi(2x2 + y2 )) + . . . exp −B × 0.1492

The fractional coordinates (x 1 , y1 ,), (x 2 , y2 ),… of the initial model are input into this equation.

It must be careful to use the fractional coordinates in the monoclinic unit cell. The initial B value used was 10 Å2 , as an example. The observed intensity I(210) is obtained from the 2D X-ray diffraction data. In the refinement, the |F|2 value is used. The observed |F(210)|2obs value can be calculated as

1.9 Crystal Structure Analysis

143

|F(210)|2obs = Iobs /(KmLpA) As already mentioned, L, p, A, and m are known.

 p = 1 + cos2 (2θ) /2 = 0.978 L = λ3 /[γ sin(2θ) sin(δ)] ∝ 1/ sin(2θ) = 4.732 A=1 m=2 In the present example, K is not known, which is estimated by comparing the total sum of the observed structure factors with that of the calculated values. The actually calculated |F|2 values are changed by modifying the atomic positions and the temperature factor so that the calculated structure factors are in good agreement with the corresponding observed values. In the original paper [78], the anisotropic temperature factors were used. The thus-refined structure gave the R factor 18.4%. Comparison of the observed and calculated d spacings and the structure factors of the hk0 reflections are listed in Table 1.13. The fractional coordinates Table 1.13 Comparison of d and |F| of several equatorial peaks between observed and calculated vales (Gutta Percha α). Reprinted from Ref. [78]. Copyright 2012, with permission from Elsevier h

k

l

d obs [Å]

d calc [Å]

|F|obs

|F|calc

1

0

0

7.84

7.75

7.40

9.71

1

1

0

4.94

4.91

48.56

34.95

2

0

0

3.92

3.88

45.81

51.15

2

1

0

3.34

3.31

35.28

32.81

0

2

0

3.18

3.17

6.75

7.19

1

2

0

2.94

2.93

10.88

10.84

3

0

0

2.62

2.58

7.09

7.35

2

2

0

2.43

2.45

2.28

4.09

3

1

0

2.39

2.42

5.57

3

2

0

2.02

2.00

5.93

5.93

2

3

0

1.87

1.86

3.95

4.18

4

1

0

1.85

5.45

7.66

4

2

0

1.65

5.89

6.09

3

3

0

1.64

8.23

9.17

5

0

0

1.55

4.23

2.96

1

4

0

1.55

3.59

3.79

1.64 1.55

144

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Table 1.14 Fractional coordinates of Gutta Percha α form (Reprinted from Ref. [78]. Copyright 2012, with permission from Elsevier.) Atom

x

y

z

U eq [Å2 ]

C1

0.8356 (20)

0.2960 (32)

0.0597 (14)

0.1182 (74)

C2

0.8358 (24)

0.2866 (28)

0.2366 (17)

0.1543 (84)

C3

0.7448 (30)

0.4228 (30)

0.3081 (18)

0.1889 (119)

C4

0.7546 (21)

0.4153 (31)

0.4854 (16)

0.1146 (68)

C5

0.6587 (20)

0.6184 (28)

0.2266 (18)

0.1286 (72)

The parentheses show the standard errors. For example, 0.8356(20) = 0.8356 ± 0.0020 U eq = equivalent isotropic mean-square atomic displacement parameter = (1/24π2 )i j Bij ai aj , where ai is the unit cell vector (i = 1, 2, or 3)

Fig. 1.104 Crystal structure of Gutta Percha α form. Reprinted from Ref. [78]. Copyright 2012, with permission from Elsevier

and temperature factors are listed in Table 1.14. The crystal structure is shown in Fig. 1.104. The chain conformation is basically trans − CTS − trans − CT S with the glide-plane symmetry. The positions of H atoms were not determined in this analysis. Usually the H atoms are added to the C atoms using the standard geometries (CH = 1.02 Å, ∠HCH = ∠HCC = 109.5°). The R factor was calculated after the addition of the H atoms.

1.10 Structure Refinement and Least Squares Method

145

1.10 Structure Refinement and Least Squares Method The refinement of the crystal structure is made by changing the fractional coordinates and temperature factors of all the atoms (full-matrix least squares method). In many cases, however, we get the geometrically curious structures. This comes from such a situation of polymer crystal that the total number of the observed reflections is much smaller than the total number of the adjustable parameters. For example, n atoms are contained in one asymmetric unit in the unit cell. If the coordinates (x, y, z) of these atoms and one global (common) isotropic thermal factor B are assumed as the adjustable parameters, 3n + 2 parameters are needed by adding a scale factor K for the full structural refinement. Ideally, the total number of the observed diffraction spots should be three times larger than that of the parameters: 3(3n+2) = 9n + 6. If n = 5, then the observed diffraction spots must be about 50. Even in the simplest orthorhombic PE case (one CH2 unit), 30 diffraction data are necessary for the determination of C and H atomic positions. More number of data is needed for the determination of anisotropic temperature factors of the individual atoms. In order to keep the stereochemically reasonable model even after the least squares refinement, therefore, we have to reduce the total number of the adjustable parameters. It is made usually by introducing the constraining conditions. The easiest way is to fix the bond distances and/or bond angles of the targeted atomic groups. Only the torsional angles of the skeletal atoms are assumed to be the parameters. The temperature factor is also fixed. The least squares calculation is made under such constraining conditions, being called the constrained least squares method [79–81]. Before the description of this method, let us see the normal least squares method.

1.10.1 Least Squares Method (General) The least squares method is one method to determine the most suitable parameters that can reproduce the observed data as well as possible. The errors between the

2 observed and calculated values (f ) are calculated,  = j fj obs − fj calc , which must be minimized by changing fj calc , which is a function of the parameters x i . Do

 not use the error in the form  = j fj obs − fjcalc because the + and – errors of the individual f values may cancel each other and the  value may become quite small apparently. In the actual calculation in the structure analysis, the error between the observed and calculated structure factors |F(hkl)obs | and |F(hkl)cal | is minimized by varying the adjustable parameters (sometimes the |F|2 or intensity I is used). The total error  is expressed as follows: =

 m=1∼M



2 wm |Fobs |m − |Fcalc |m

(1.162)

146

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

where wm is the weight for the m-th structure factor and it is usually unity or 1/|Fobs |. M is the total number of the observed diffraction data. The structure factor is given as  

 F(hkl) = Fm = Kfj (sin θ/λ) exp −Bj sin2 θ/λ2 exp 2πi hxj + kyj + lzj (1.163) F m is a function of the parameters K, Bj , x j , yj , and zj . The temperature factor used is anisotropic, in general, but it is assumed here to be an isotropic value common to all the atoms. All of these parameters are now expressed simply as uj with j = 1 ~ N. N is the total number of the parameters. The minimization condition of Φ is expressed as ∂/∂uj = 0 (j = 1, . . . , N )

(1.164)

That is,  

2  ∂/∂uj = ∂ m wm |Fobs |m − |Fcalc |m /∂uj 

= −2m wm |Fobs |m − |Fcalc |m · ∂|Fcalc |m /∂uj

(1.165)

Here,  

   ∂|Fcalc |m /∂uj = ∂ Kfj (sin θ/λ) exp −Bj sin2 θ/λ2 exp 2πi hxj + kyj + lzj /∂uj

(1.166) The derivatives are different depending on the type of parameters uj , which is K, Bj , x j , yj , or zj (j = 1 ~ N). The structure factor is developed with respect to the parameters uj and the terms of linear order are used in the approximation.  ◦ 

   |Fcalc |m = F  + n [∂|F|m /∂un ]un + (1/2)p q ∂ 2 |F|m /∂up ∂uq up uq + . . .  ◦ m  ◦     ≈ F  + n [∂|F|m /∂un ]un = F  + n Fmn un (1.167) m

m

Here F mn = ∂|F|m /∂un Therefore, Eq. 1.165 becomes   

un · Fmj ∂/∂uj ≈ −2m wm |Fobs |m − F o m − n Fmn 

= −2m wm Fm − n Fmn un · Fmj

(1.168)

  Fm = |Fobs |m − F o m

(1.169)

where

1.10 Structure Refinement and Least Squares Method

147

The minimization condition is ∂/∂uj = 0 m wm [Fm − n Fmn un ] · Fmj = 0 (j = 1, 2, . . . , N )

(1.170) (1.171)

Using the matrix expressions, we have 

u1 − F12 u2 − F13 u3 − . . . − F1N uN F11 j = 1 w1 F1 − F11 

+ w2 F2 − F21 u1 − F22 u2 − F23 u3 − . . . − F2N uN F21 

+ w3 F3 − F31 u1 − F32 u2 − F33 u3 − . . . − F3N uN F31 + ... 

+ wM FM − FM1 u1 − FM2 u2 − FM3 u3 − . . . − FMN uN FM1 =0 

j = 2 w1 F1 − F11 u1 − F12 u2 − F13 u3 − . . . − F1N uN F12 

+ w2 F2 − F21 u1 − F22 u2 − F23 u3 − . . . − F2N uN F22 

+ w3 F3 − F31 u1 − F32 u2 − F33 u3 − . . . − F3N uN F32 + ... 

+ wM FM − FM1 u1 − FM2 u2 − FM3 u3 − . . . − FMN uN FM2 =0 ………….. These equations can be rearranged in the following way by moving the Fm terms to the right side: j=1 F u + w F F u + w F F u + . . . + w F F u w1 F11 1 1 12 11 2 1 13 11 3 1 1 N 11 N 11

F u + w F F u + w F F u + . . . + w F F u + w2 F21 1 2 22 21 2 2 23 21 3 2 2N 21 N 21

F u + w F F u + w F F u + . . . + w F F u + w3 F31 1 3 32 31 2 3 33 31 3 3 3 N 31 N 31

+ ... F u + w F F u + w F F u + . . . + w F F u + wM FM1 1 M M2 M1 2 M M3 M1 3 M MN M1 N M1 + w F F + w F F + . . . + w F F = w1 F1 F11 2 2 21 3 3 31 M M M1

j=2 F u + w F F u + w F F u + . . . + w F F u w1 F11 1 1 12 12 2 1 13 12 3 1 1 N 12 N 12

F u + w F F u + w F F u + . . . + w F F u + w2 F21 1 2 22 22 2 2 23 22 3 2 2N 22 N 22

F u + w F F u + w F F u + . . . + w F F u + w3 F31 1 3 32 32 2 3 33 32 3 3 3 N 32 N 32

+ ...

148

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

F u + w F F u + w F F u + . . . + w F F u + wM FM1 1 M M2 M2 2 M M3 M2 3 M MN M2 N M2 + w F F + w F F + . . . + w F F = w1 F1 F12 2 2 22 3 3 32 M M M2

….. j=N F u + w F F u + w F F u + . . . + w F F u w1 F11 1 1 12 1N 2 1 13 1N 3 1 1 N 1N N 1N

F u + w F F u + w F F u + . . . + w F F u + w2 F21 1 2 22 2N 2 2 23 2N 3 2 2N 2N N 2N

F u + w F F u + w F F u + . . . + w F F u + w3 F31 1 3 32 3N 2 3 33 3N 3 3 3 N 3N N 3N

+ ... F u + w F F u + w F F u + . . . + w F F u + wM FM1 1 M M2 MN 2 M M3 MN 3 M MN MN N MN + w F F + w F F + . . . + w F F = w1 F1 F1N 2 2 2N 3 3 3N M M MN

These equations can be summed together in the following matrix form: j=1 ⎛

⎞ F + w F F w1 F11 2 21 21+···+ wM FM1 FM1 11 ⎜ w F F + w F F ⎟ ⎜ 2 22 21+···+ wM FM2 FM1 ⎟ [u1 , u2 , · · · , uN ]⎜ 1 12 11 ⎟ ⎝ ⎠ .......... F + w F F F w F w1 F1N 2 M 11 2N 21+···+ MN M1 + w F F + · · · + w F F = w1 F1 F11 2 2 21 M M M1

j=2 ⎛

⎞ F + w F F w1 F11 2 21 22+···+ wM FM1 FM2 12 ⎜ w F F + w F F ⎟ ⎜ 2 22 22+···+ wM FM2 FM2 ⎟ [u1 , u2 , · · · , uN ]⎜ 1 12 12 ⎟ ⎝ ⎠ .......... F + w F F F w F w1 F1N 2 M 12 2N 22+···+ MN M2 + w F F + · · · + w F F = w1 F1 F12 2 2 22 M M M2

….. j=N ⎛

⎞ F + w F F w1 F11 2 21 2N+···+ wM FM1 FMN 1N ⎜ w F F + w F F ⎟ ⎜ 2 22 2N+···+ wM FM2 FMN ⎟ [u1 , u2 , · · · , uN ]⎜ 1 12 1N ⎟ ⎝ ⎠ .......... w1 F1N F1N + w2 F2N F2N+···+ wM FMN FMN + w F F + · · · + w F F = w1 F1 F1N 2 2 2N M M MN

1.10 Structure Refinement and Least Squares Method

149

Furthermore, ⎞⎛ √ ⎞ ⎛ √ √w F . . . √w F √w F . . . √w F w1 F11 w1 F11 2 21 M M1 1 12 1 1N √ √ √ √ √ ⎟ ⎜ ⎟ ⎜ √ . . . w F . . . w F w2 F22 w2 F21 w2 F22 ⎜ w1 F12 M M2 ⎟⎜ 2 2N ⎟ [u1 u2 , . . . , uN ]⎜ ⎟⎜ ⎟ ⎠⎝ ⎠ ⎝ .......... .......... √ √ √ √ √ √ w1 F1N w2 F2N . . . wM FMN wM FM1 wM FM2 . . . wM FMN ⎞ ⎛ √ √w F . . . √w F w1 F11 1 12 1 1N √ √ √ ⎟ . . . w F √

⎜ √ √ w2 F21 w2 F22 ⎟ ⎜ 2 2N = w1 F1 , w2 F2 , . . . , wM FM ⎜ ⎟ ⎠ ⎝ .......... √ √ √ . . . w F wM FM1 wM FM2 M MN

Therefore, U MtM = F M

(1.172)

M t is the transpose of the matrix M. The matrix components F mn = ∂|F|m /∂un and Fm = |Fobs |m − |F ◦ |m can be calculated concretely, and we can get the answer U or uj by solving Eq. 1.172.

1.10.2 Constrained Least Squares Method The abovementioned least squares method can refine the atomic coordinates in the unit cell so that the observed X-ray diffraction data are reproduced as well as possible. However, the thus-determined molecular geometry is not necessarily reasonable from the stereo-chemical point of view. In particular, in the case of polymers, the total number of the observed diffraction spots is too small to obtain the reliable structure, leading to the unacceptable structure parameters; too long bond length, too wide or narrow bond angle, etc. One method to escape this matter is to fix the bond lengths and/or bond angles during the least squares process, which are not very much affected by the change in the environment surrounding the atoms. We call this method the least squares method under the constraining conditions or the constrained least squares method [79–81]. At first, we need to know the principle of the constrained least squares method. Figure 1.105 shows a triangle with the angles a, b, and c. These angles were actually Fig. 1.105 A triangle model

b

a c

150

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

measured with a protractor and the following result was obtained. Measurement

ai

bi

ci

i=1

59.2°

80.2°

40.4°

i=2

60.4

79.5

39.5

i=3

58.7

81.0

42.8

i=4

61.5

78.8

38.9

If the correct angles are a, b, and c, the total error f(a, b, c) is expressed as 4 4 4 (bi − b)2 + i=1 (ci − c)2 f(a, b, c) = i=1 (ai − a)2 + i=1

(1.173)

4 Since the simple summation of errors i=1 (ai − a), for example, gives the apparently small error, the squared values must be used as above.

(A)

No constrained condition

The most probable values of a, b, and c are obtained by minimizing the total error f(a,b,c) using the following equations: ∂f(a, b, c)/∂a = 0, ∂f(a, b, c)/∂b = 0, and ∂f(a, b, c)/∂c = 0

(1.174)

Then, we have the following equations: 4 4 4 (ai − a) = 0, i=1 (bi − b) = 0, and i=1 (ci − c) = 0 i=1

(1.175)

Thus, we have the answer as follows: 4 4 4 a = i=1 (ai )/4 = 60.0◦ , b = i=1 (bi )/4 = 79.9◦ , c = i=1 (ci )/4 = 40.4◦ (1.176)

Equation 1.176 indicates that the answers are simple averages of the experimental data ai , bi , and ci . (B)

Introduction of constraining condition

The total sum of the three angles a, b, and c must be 180°. The above-solved three values give the total amount of 180.3°, which must be corrected by taking this matter into consideration in the least squares minimization process. The constraining condition in the present case is a + b + c = 180.0◦

(1.177)

1.10 Structure Refinement and Least Squares Method

151

The condition is expressed as g(a, b, c) = a + b + c − 180(= 0)

(1.178)

This constraining condition is a kind of error, and it is introduced into the error f(a,b,c) as follows: F(a, b, c) = f(a, b, c) + λg(a, b, c)

(1.179)

The second term indicates the contribution of g(a,b,c) to the error F with the weight λ. The parameter λ should be suitably determined so that the condition g(a,b,c) = 0 is satisfied as much as possible. The total sum of error F must be the smallest. Then we have the minimal condition as follows: ∂F(a, b, c)/∂a = 0, ∂F(a, b, c)/∂b = 0, ∂F(a, b, c)/∂c = 0 and ∂F(a, b, c)/∂λ = 0

(1.180) 4 − 2i=1 (ai − a) + λ = 0 4 − 2i=1 (bi − b) + λ = 0 4 − 2i=1 (ci − c) + λ = 0 a + b + c − 180 = 0

(1.181)

Therefore, 4 8a − 2i=1 (ai ) + λ = 0 4 8b − 2i=1 (bi ) + λ = 0 4 (ci ) + λ = 0 8c − 2i=1 a + b + c − 180 = 0

(1.182)

Using the matrix expression, ⎛

8 ⎜0 ⎜ ⎝0 1

00 80 08 11 A

⎞ ⎛ ⎞ ⎞⎛ ⎞ ⎛ 4 479.6 1 a 2i=1 (ai ) ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 4 1⎟ ⎟⎜ b ⎟ = ⎜ 2i=1 (bi ) ⎟ = ⎜ 639.0 ⎟ 1 ⎠⎝ c ⎠ ⎝ 2 4 (ci ) ⎠ ⎝ 323.2 ⎠ 0

λ U

i=1

180

AU = B and so U = A−1 B

(1.183)

180 B (1.184)

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1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.106 Bond angle φijk and distances

The inverse matrix A−1 is given as ⎛

A−1

⎞ 1/12 −1/24 −1/24 2/3 ⎜ −1/24 1/12 −1/24 1/3 ⎟ ⎟ =⎜ ⎝ −1/24 −1/24 1/12 1/3 ⎠ 1/3 1/3 1/3 −8/3

(1.185)

We have the final answer, ⎛ ⎞ ⎛ ⎞ a 59.9◦ ⎜ b ⎟ ⎜ 79.8◦ ⎟ ⎟ ⎜ ⎟ U=⎜ ⎝ c ⎠ = ⎝ 40.3◦ ⎠ λ 0.6

(1.186)

In fact, the total sum a + b + c = 180.0° satisfies the condition (1.177).

1.10.2.1

Introduction of Constraining Conditions

When the atomic coordinates are defined as the structural parameters, the simplest constraining condition may be the fixing of the interatomic distances between the two atoms i and j. 2  2  2  d2ij = xi − xj + yi − yj + zi − zj

(1.187)

or 2  2  2  g(x, y, z) = d2ij − xi − xj − yi − yj − zi − zj = 0

(1.188)

For keeping the bond angle φijk shown in Fig. 1.106, all the bond distances are fixed.  d2ik = d2ij + d2jk − 2dij djk cos φijk

(1.189)

1.10 Structure Refinement and Least Squares Method

153

This situation can be expressed using the following three constraining conditions. 2  2  2  g1 = d2ij − xi − xj − yi − yj − zi − zj = 0 2  2  2  g2 = d2jk − xj − xk − yj − yk − zj − zk = 0 2  g3 = d2ik − (xi − xk )2 − yi − yk − (zi − zk )2 = 0

(1.190)

In the preceding section, we know that the minimized quantity  is given by the following equation: =





2 ωm |Fobs |m − |Fcalc |m

(1.191)

m=1∼M = ∂|F|m /∂un , Since ∂/∂uj = 0 (j = 1, . . . , N) and Fmn then we have

 ◦ 

 |Fcalc |m = F m + n [∂|F|m /∂un ]un + (1/2)p q ∂ 2 |F|m /∂up ∂uq up uq + . . .  ◦ (1.192) ≈ F  + n [∂|F|m /∂un ]un = |Fobs |m − Fm + n F un mn

m

and  

2  ∂/∂uj = ∂ m wm |Fobs |m − |Fcalc |m /∂uj 

= −2m wm |Fobs |m − |Fcalc |m · ∂|Fcalc |m /∂uj 

≈ −2 m wm Fm − n Fmn un · Fmj

(1.193)

Now, by introducing the constraining conditions, we have the following error :  

2 wm |Fobs |m − |Fcalc |m + λh Gh



=

m=1∼M

(1.194)

h=1∼H

Since N (∂Gh /∂uj ) uj (1.195) Gh (u1 + u1 , . . . , uN + uN ) ≈ Gh (u1 , . . . , uN ) + j=1

then we have ∂/∂uj ≈ −2



 

wm Fm − n Fmn un · Fmj + λh (∂Gh /∂uj ) = 0

m=1∼M

h=1∼H

(j = 1 ∼ N) ∂/∂λh ≈ Gh +

(1.196)  j=1∼N

 uj ∂Gh /∂uj = 0 (h = 1 ∼ H)

(1.197)

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1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Using the matrix expression, 

M t M N/2 [U, L] N t /2 0

 = [FM, G/2]

(1.198)

where L = [λ1 , λ2 , . . . , λH ] ⎛

⎞ ∂G1 /∂u1 . . . ∂GH /∂u1 ⎠ N=⎝ ... ∂G1 /∂uN . . . ∂GH /∂uN

(1.199)

G = [−G1 , −G2 , . . . , −GH ] By inputting the initial values of each matrix components, we can get the answers U or uj under the constraining conditions.

1.10.2.2

Constraining by Internal Coordinates

In the above equations concerning the structure factor and its derivatives, we notice that the parameters uj to be determined are the fractional coordinates (x j , yj , zj ), scale factor K, and temperature parameters Bj . The constraining condition is the fixing of the interatomic distances. However, this type of constraining is difficult to set. Rather the constraining of the geometry using the internal coordinates such as bond length, bond angle, and torsional angle is much easier and clearer in the physical meaning [82, 83]. Here we see the corresponding equations. For this purpose, the least squares method must be rewritten based on the various internal coordinates such as bond distances, bond angles, torsional angles, out-of-plane angles, and so on. The above-derived minimal conditions   

wm Fm − n Fmn un · Fmj + λh (∂Gh /∂uj ) = 0 (j = 1 ∼ N) −2 m=1∼M

h=1∼H

(1.200) Gh +

 j=1∼N

uj (∂Gh /∂uj ) = 0 (h = 1 ∼ H)

(1.201)

1.10 Structure Refinement and Least Squares Method

155

are expressed using the internal coordinates, which are now represented as uj . The uj are the bond distances, bond angles, torsional angles, and so on. Since the structure factor F is a function of fractional coordinates x j , yj , and zj , the derivatives ∂|F calc |m /∂x j , etc. must be changed on the basis of the internal coordinates uj . For example, for the bond distance r j , which is expressed using the atomic coordinates (x i , yi , zi ) (i = 1 and 2, for example), ∂|F|m /∂uj = ∂|F|m /∂rj 2 (∂|F| /∂x )∂x /∂r + (∂|F| /∂y )∂y /∂r + (∂|F| /∂z )∂z /∂r

= n=1 m n n m n n m n n j j j

(1.202)

The derivatives (∂x n /∂r j ) are calculated using the so-called B matrix components, which are used in the normal modes calculation (Chap. 5). We know the relation between the change in the Cartesian coordinate x and the change in the internal coordinate R(= u). R = Bx

(1.203)

x = B−1 R

(1.204)

Here the matrix B is calculated easily. Brief explanation is made here with some examples. The bond length r ij is given as 2    rij2 = xi − xj = (xi · xi ) + xj · xj − 2 xi · xj

(1.205)

Therefore, the small change in r ij is given as     rij = xi − xj · xi − xj /rij = xi − xj /rij · xi + xj − xi /rij · xj Similarly, the change in the bond angle φijk is given below. Since  rji · rjk = rji rjk cos φijk

(1.206)

or    xi − xj · xk − xj = rji rjk cos φijk then we have    φijk = −pji /rji + pjk /rjk xj + pji /rji xi − pjk /rjk xk

       pjk = −ejk cos φijk + eji / sin φijk , pji  = pjk  = 1

(1.207)

156

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

The e is a unit vector. Refer to Fig. 1.106.  From these equations, the terms (∂x n /∂r ij ) and ∂xn /∂φijk are expressed, respectively, as a component of B−1 matrix. For the simple case of r ji = r jk = r, B is given as x yi zi xj yj zj xk yk zk ⎛ i ⎞ rji s 0 c −s 0 −c 0 0 0 B= rjk ⎝ −s 0 c 0 0 0 s 0 −c ⎠ φijk 0 0 −2s/r0 −c/r0 0 s/r0 c/r0 0 s/r0

(1.208)

Here s = sin(φ/2) and c = cos(φ/2). The atomic coordinates (x i , yi , zi ) can be calculated from the internal coordinates using Eyring’s relation as explained in the Sect. 1.12.3.4. In that case the x-, y-, and z-coordinates are defined on the basis of the bond length and bond angle of the first three atoms. However, these coordinates must be transferred to the coordinates defined by the unit cell system. The details will be referred to in a later section.

1.10.2.3

Concrete Example

Figure 1.116 shows the X-ray diffraction pattern of oriented polyoxymethylene (POM) prepared by the γ-ray-irradiated solid-state polymerization reaction of trioxane single crystal [10, 12]. The diffraction pattern was measured using a highenergy X-ray beam from the synchrotron radiation with the wavelength 0.328 Å as an incident beam. The diffraction spots observed in the first quadrant zone are about 500 in number. The reflection indexing was performed successfully using the trigonal (hexagonal-type) unit cell of a = b = 4.46 Å and c (fiber axis) = 17.35 Å (γ = 120°). The observed diffraction spots are strong along the 9th, 5th, … layer lines, consistent with the equation requested to a regular (9/5) helical chain. One (9/5) chain is contained in the unit cell [10]. If we assume the uniform helical chain with the same torsional angles around the C-O and O-C bonds, the so-called equation of helical parameters can be applied. For the helical pitch d, helical rotation angle per repeating unit θ, and the radius of helix ρ, we have the following relations: cos(θ/2) = cos(τ/2) sin(φ/2) d sin(θ/2) = r sin(τ/2) sin(φ/2) 2ρ2m (1 − cos(θ)) + d 2 = r 2

(1.209)

where r, φ, and τ are bond length (C−O), bond angle (C−O−C, O−C−O), and torsional angle, respectively. Since the number of skeletal atoms in the repeating period is 9 × 2 = 18, the helical pitch d = 17.35Å/18 = 0.964 Å and θ = 5×360°/18

1.10 Structure Refinement and Least Squares Method

157

= 100°. By substituting the concrete values for r = 1.43 Å, φ = 110°53’ we can calculate the τ value as τ = 77.47◦ (More strictly speaking the bond angles COC and OCO are different slightly and the torsional angles around the COCO and OCOC are also different from each other.) In this way, the uniform POM chain with the simple repetition of CH2 and O atomic units is built by setting the torsional angles of gauche form. This torsional angle 77° is deviated slightly but significantly from the exact gauche torsional angle 60°. The latter generates the (2/1) helical form with the rectangular square shape projected along the chain axis (which corresponds to the chain conformation of the orthorhombic modification, as will be explained in later [44]). The Cartesian coordinates of the C and O atoms are calculated using Eyring’s transformation equation. The space group of POM trigonal form is P32 (C 3 ). So, one asymmetric unit contains three CH2 O units. For the calculation of structure factors, the concrete method of which will be explained in a later section, we need the fractional coordinates of the atoms. The fractional coordinates of the first asymmetric unit must be transformed to the second unit using a transformation matrix. It must be noted that the transformation of the coordinates must be made on the basis of the fractional coordinates of the trigonal unit cell, not the Cartesian coordinate system, making the transformation a little complicated. *******************************

As shown in Fig. 1.107, the 3 points (#1, #2 and #3) are projected along the z axis (or the center axis of a (3/1) helix) as an example. The fractional coordinates are (x 1 , y1 ), (x 2 , y2 ), and (x 3 , y3 ), respectively. The point x 1 (#1) along the x-axis is rotated by −120° to the point #2 as shown in orange-color arrow, and the y coordinate of #2, y2 = −x 1 . Similarly, the point #3 is obtained by rotating #1 by 120°, and y3 = |y2 | – |y1 | = x 1 – y1 . In addition, x 3 = −y1 . x 2 = −(|x 1 | − |x 3 |) = −x 1 + y1 . The geometrical determination of the coordinates is not very easy for the general helix. If this graphical method is difficult to understand, the transformation via the Cartesian coordinate is used. The C 3 rotation operation is considered here. In the xy-plane (see Fig. 1.108), the Cartesian coordinate of the #1 atom is  X1 =

 X1 Y1

 =

       x1 − y1 cos 60◦ 1 −1/2 x1 − y1 /2 x1  = A x1 = = √ √ y1 y1 sin 60◦ y1 3/2 0 3/2

The Cartesian coordinate of the #2 atom is      √ −1/2 3/2 X1 cos(240◦ ) − sin(240◦ ) √ = X2 = X 1 = BAx1 Y1 sin(240◦ ) cos(240◦ ) − 3/2 −1/2

158

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.107 The rotational transformation of the points on a helical chain

Fig. 1.108 The rotational transformation of the points on a helical chain based on the Cartesian coordinate system

On the other hand, X 2 = Ax2 , and then  x2 =  =



 x2 y2

= A−1 X 2 = A−1 BAx1 =

 −1 1 x1 −1 0

  √  √ 1 1/ 3 −1/2 3/2 1 −1/2 x1 √ √ √ 0 2/ 3 − 3/2 −1/2 0 3/2

1.10 Structure Refinement and Least Squares Method

159

The thus-obtained relation between x 1 and x 2 can be summarized as follows: x2 = −x1 + y1 and y2 = −x1 These relations are already shown in Fig. 1.107. Similarly, you can get the fractional coordinate transformation matrix from x1 to x3 .     √ −1/2 − 3/2 cos(120◦ ) − sin(120◦ ) X1 = X 1 = B Ax1 , X 3 = Ax3 √ 3/2 −1/2 Y1 sin(120◦ ) cos(120◦ )      √  √ x3 1 1/ 3 −1/2 − 3/2 1 −1/2 −1 −1 x3 = = A X 3 = A B Ax1 = x1 √ √ √ y3 0 2/ 3 3/2 −1/2 0 3/2   0 −1 = x1 1 −1 

X3 =

As a result, we can obtain the rotation matrices in the fractional coordinate system as given below: ⎛

⎞ ⎛ ⎞ ⎛ ⎞ 100 −1 1 0 0 −1 0 M(1 − 1) = ⎝ 0 1 0 ⎠ M(1 − 2) = ⎝ −1 0 0 ⎠ M(1 − 3) = ⎝ 1 −1 0 ⎠ 001 0 01 0 0 1 The translation vector ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 0 0 T(1 − 1) = ⎝ 0 ⎠ T(1 − 2) = ⎝ 0 ⎠ T(1 − 3) = ⎝ 0 ⎠ 0 2/3 1/3 x(1) = M(1 − 1)x(1) + T(1 − 1) x(2) = M(1 − 3)x(1) + T(1 − 2) x(3) = M(1 − 3)x(1) + T(1 − 3)

(1.210)

************************ The generation of all the atoms belonging to the three asymmetric units can be made in this way by starting from the atomic fractional coordinates of the first unit. The structural refinement without any constraint gives the R factor of 13.8 %, where the isotropic temperature factors of C and O atoms were also refined. However, one problem is about the connection of the chain segments between the neighboring unit cells along the chain axis, which should be stereochemically reasonable. The constraining conditions were introduced now.

160

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

O(5, unit 3, 000) − C(0, unit 1, 001) = 1.43 Å O(5, unit 3, 000) − O(1, unit 1, 001) = 2.36 Å C(4, unit 3, 000) − C(0, unit 1, 001) = 2.36 Å

C4

C0 O1

O3

O5

C2 unit 3 in the cell (0, 0, 0)

(1.211)

C0

O1

C2

unit 1 in the cell (0, 0, 1)

The refinement under these constraining conditions gave the R factor of 14.6% [12]. This is only one demonstration of the least squares refinement of POM crystal under the constraining conditions. The actual refinement was made by introducing the anisotropic temperature factors as reported in the literature [12].

1.11 Calculation of 2D X-ray Diffraction Patterns 1.11.1 Using Atomic Coordinates As already learned, the X-ray diffraction intensity I is expressed as I = KALpm|F(k)|2 |DL |2 |DM |2 |DN |2

(1.212)

where the structure factor is given as (under the assumption of isotropic temperature factor) 

 F(k) = j fj (k) exp −B sin2 (θ)/λ2 exp i a · k xj + b · k yj + c · k zj where the scattering vector k is given as (ξ , η, ζ ). By assuming the fractional coordinates (x j , yj , zj ) of all the atoms of one asymmetric unit and the space group symmetry (or the fractional-coordinate-based transformation matrices consisting of both the rotation matrices and translational vectors) and by assuming the crystallite size along the a-, b-, and c-directions (which give the Laue terms |DL |, |DM |, and |DN |) and also the Lp factors, we can calculate the diffraction intensity distribution in the three-dimensional reciprocal lattice (ξ , η, ζ ). If we assume the uniaxially oriented sample and the rotation axis or the fiber axis is parallel to the ζ axis, then many reciprocal lattice points of (ξ , η) cross the Ewald sphere except the points near the meridional direction. The 2D diffraction pattern is

1.11 Calculation of 2D X-ray Diffraction Patterns

161

calculated using Eq. 1.212. A commercial software Cerius2 (Accelrys), for example, supplies the program of this type of calculation to give the two-dimensional and one-dimensional diffraction patterns for the uniaxially oriented crystal. Since (1/d)2 = ξ 2 + η2 + ζ 2 = 4sin2 (θ)/λ2 , the diffraction profile of a particular layer line ζ can be calculated as a function of 2θ.

1.11.2 Finding Cross-points with Ewald Sphere If the crystallites have the three-dimensionally anisotropic orientation, we need to calculate the crossing plane of the 3D reciprocal lattice with the Ewald sphere. For this purpose, we developed a software to calculate the 3D diffraction intensity distribution in the reciprocal lattice by performing the Fourier transform of the real crystal structure model on the basis of the abovementioned equation. The XYZcoordinate system is fixed on the center of the reciprocal lattice. The X-ray is incident along the X-axis. Depending on the orientation direction of the crystallite, the YZplane crossing the Ewald sphere is different. For the radius of Ewald sphere R (= 1/λ), the reciprocal lattice points included in the region of R ~ R + R are searched and the 2D image projected onto the YZ-plane is obtained. As shown in Fig. 1.109, each reciprocal lattice point has the weight of |F|2 . This method can be applied to the system consisting of many crystallites with the individual orientations on the basis of the Monte Carlo simulation technique [84]. Depending on the spatial orientation of each crystallite, the plane of the reciprocal lattice cutting the Ewald sphere is different, meaning the different diffraction patterns correspondingly. The total sum of the diffraction patterns calculated for all the crystallites gives the diffraction pattern of the whole system. By comparing the totally summed diffraction pattern with the observed 2D pattern, the orientation of the first-chosen crystallite is changed. Then the agreement between the observed and calculated patterns is checked, and then the second crystallite orientation is changed into the direction of the better agreement. This process is repeated until the agreement between the observed and calculated patterns becomes the best. (More elegant way is to apply the Monte Carlo Metropolis method, where the temperature is controlled for the calculation of the probability exp(-F/T), where F is the difference between the observed and calculated structure factor. F is the difference of F between the i-th and i+1-th steps. If F is negative, the i+1-step structure is better. Even when the F is positive, the i+1-th model is employed if the probability exp(-F/T) is larger than a random number generated at this step. This method accelerates the conversion to the correct answer) (Figs. 1.109, 1.110, 1.111, 1.112).

162

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

(a)

(b) c

a’

c*

b* a*

b’ b

Fig. 1.109 a Crystal structure of PET [85] and b the corresponding reciprocal lattice points with the weights of |F|2 [84]

Fig. 1.110 For an orientation of the crystallite, the crossing plane with the Ewald sphere is calculated by finding the points of the reciprocal lattice included in the region of R ~ R + R

Ewald sphere

R+ R R

E 2θ

X-ray



D

Fig. 1.111 Depending on the orientation angle  of the crystallite from the incident X-ray beam, the cross plane of the reciprocal lattice is different [84]

1.12 Characteristic Diffraction Pattern of Helical Chains

163

(a)

(b)

obsd

calcd

 r Through

r

r

r Edge

Fig. 1.112 a Monte Carlo method with many crystallites. The orientation of the crystallites is changed step by step by generating a random number so that the total diffraction pattern is in good agreement with the observed pattern. b The comparison between the observed and calculated diffraction patterns for the various settings of the doubly oriented PET sample, where ψ is the angle between the X-ray beam and the rolled plane [84]

1.12 Characteristic Diffraction Pattern of Helical Chains 1.12.1 X-ray Diffraction Pattern of Helical Chain One characteristic molecular shape of a polymer chain is a helical conformation. The X-ray diffraction pattern of the helix is characteristic correspondingly.

164

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

(a)

(b)

Fig. 1.113 a a continuous helix, b a discontinuous helix

The diffraction pattern of a helical chain can be predicted theoretically using the so-called helical equation derived by Cochran, Crick, and Vand [86, 87]. As shown in Fig. 1.113, one point at the coordinate (ρ, 0, 0) is rotated by the angle φ and translated by z along the helical axis. The thus-obtained 2nd point is expressed by the coordinate (x, y, z). A pitch P is defined as the period of the helix after the 360o rotation around the helical axis. Since P : 2π = z : φ, we have φ = 2πz/P. The (x, y, z) is given as follows: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x ρ · cos(φ) ρ · cos(2π z/P) ⎝ y ⎠ = ⎝ ρ · sin(φ) ⎠ = ⎝ ρ · sin(2π z/P) ⎠ z z z

(1.213)

These points on the continuous helical curve are expressed using the delta function as follows: H (x, y, z) = δ(x − ρ · cos(2π z/P))(y − ρ · sin(2π z/P))

(1.214)

1.12 Characteristic Diffraction Pattern of Helical Chains

165

The structure factor F(x*, y*, z*) for the function H(x, y, z) is given by  F x∗ , y∗ , z ∗ =

˚

 

H (x, y, z) exp 2πi x · x∗ + y · y∗ + z · z ∗ dxdydz

(1.215)

where (x*, y*, z*) is the reciprocal lattice point, which is expressed as follows using the cylindrical coordinate (ρ*, φ*, z*): ⎛

⎞ ⎛ ∗ ⎞ x∗ ρ · cos(φ ∗ ) ⎝ y∗ ⎠ = ⎝ ρ ∗ · sin(φ ∗ ) ⎠ z∗ z∗

(1.216)

By combining Eqs. 1.214–1.216, we have ˚   

F ρ ∗ , φ ∗ , z∗ = δ(x − ρ · cos(2π z/P))(y − ρ · sin(2π z/P)) exp 2πi x · x∗ + y · y∗ + z · z ∗ dxdydz    = exp 2πi(ρ · cos(2π z/P) · x∗ + ρ · sin(2π z/P) · y∗ + z · z ∗ dz 

 exp[2πi{ρ · cos(2π z/P) · ρ∗ cos(φ ∗ ) + ρ · sin(2π z/P) · ρ ∗ sin(φ ∗ ) + z · z ∗ } dz



  exp[2πi ρ · ρ ∗ cos(2π z/P − φ ∗ ) + z · z ∗ } dz

= =

(1.217)

The z-coordinate is expressed as z = jP + z (j = 0, ±1, ±2,… and 0  z  P) because the z-position of the unit is repeated along the helical axis. Then,  +∞ F ρ ∗ , φ ∗ , z ∗ = j=−∞



+∞ −∞

!

"

# $   2π i jP + z ∗ ∗ exp 2π iρρ cos − φ + 2π i jP + z z dz P ∗

(1.218) Modifying this equation in the following way,  F ρ ∗ , φ ∗ , z∗ = %   &  +∞

 2π z +∞ − φ ∗ + 2π iz z ∗ dz exp 2π ijPz ∗ exp 2π iρρ ∗ cos 2π j + j=−∞ P −∞  & %   +∞

 2π z +∞ ∗ ∗ − φ ∗ + 2π iz z ∗ dz = j=−∞ exp 2π ijPz exp 2π iρρ cos P −∞  & %   

+∞ 2π z +∞ ∗ ∗ − φ ∗ + 2π iz z ∗ dz = j=−∞ exp 2π ijPz exp 2π iρρ cos P −∞    & %   P 1 2π izn 2π z +∞ δ(z ∗ − n/P) − φ∗ + dz = n=−∞ exp 2π iρρ ∗ cos P P P 0

(1.219)

166

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

The Bessel function J n (m) is  Jn (m) =

1 2π



2π 0

 π  dϕ exp(im cos ϕ) exp in ϕ − 2

(1.220)

Then, compare the variables in the following way: m = 2πρρ ∗ and ϕ = 2πz/P − φ ∗ , and dz = (P/2π)d ϕ

(1.221)

The integral range is changed as below. z:0→P ϕ : −φ ∗ → 2π − φ ∗ (i.e., 0 → 2π ) Finally we have     2π 1  ∗ +∞ δ z − n/P F ρ ∗ , φ ∗ , z ∗ = n=−∞ exp[im cos(ϕ)] exp[in(ϕ + φ ∗ )](P/2π )dϕ P 0   n +∞ jn (2πρρ ∗ ) exp[in(π/2 + φ ∗ )] = n=−∞ δ z∗ − (1.222) p

where the atomic scattering factor is ignored. The diffraction intensity I from the helical chain is proportional to J n (2πρρ*)2 .  I ρ ∗ , φ ∗ , z ∗ = n/P

∝

 2 Jn 2πρρ ∗

(1.223)

Figure 1.114 shows the calculated results. As seen from the existence of delta function δ(z*-n/P) in Eq. 1.222, the layer lines are observed only at z* = n/P with a spacing of 1/P.

Fig. 1.114 The 2D diffraction pattern predicted for a simple helix, where the magnitudes of J n (2πρρ*)2 , as shown in (b), are arrayed at the different layer lines

1.12 Characteristic Diffraction Pattern of Helical Chains

167

In the above discussion, the helical pitch P or the period per one rotation of a helix is used as a repeating period. However, in the real case of polymer chain, the several repeating units are included in the repeating period of the chain axis c. The total number of repeating units is M. In the repeating period c, the helix may rotate by N times. This helical form is named the (M/N) helix. The total rotation angle in the repeating period c is 360° × N. The rotation angle per one repeating unit is φ = 360° × N/M. The helical pitch P is equal to c/N. The translational length per one repeating unit p = c/M = PN/M. In equation 1.213, the x- and y-coordinates are used for expressing the position of a repeating period. If the z-coordinate of the 1st repeating unit is 0, the z-coordinate of the other repeating unit is expressed as z = jp (j = 0, ±1, ±2, ….). Then the coordinate of the j-th unit is expressed as ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ x ρ · cos(jφ) ρ · cos(2 π jp/P) ⎝ y ⎠ = ⎝ ρ · sin(jφ) ⎠ = ⎝ ρ · sin(2 π jp/P) ⎠ z

jp

(1.224)

jp

Equations 1.214 and 1.215 are expressed as follows: H (x, y, z) = δ(x − ρ · cos(2π jp/P))(y − ρ · sin(2π jp/P))(z − jp) 

∗

∗

F x ,y ,z

∗



˚ =

(1.225)

 

H (x, y, z) exp 2π i x · x∗ + y · y∗ + z · z ∗ dxdydz (1.226)

For the reciprocal coordinates (ρ*, φ*, z*), the structure factor is given in Eq. 1.222.    +∞ +∞ n=−∞ δ z ∗ − n/P − m/p Jn 2πρρ ∗ exp(in(ϕ + π/2)) F ρ ∗ , φ ∗ , z ∗ = m=−∞ (1.222) If the repeating period consists of various kinds of atoms, the atomic positions are expressed as  

xi = ρj cos φj + 2π zj /P ,

 

yj = ρj cos φj + 2π zj /P ,

zj = jp + zj (1.227)

and Eq. 1.222 changes to  F ρ ∗ , φ ∗ , z∗

      2π ilzj n π  m +∞ +∞ n=−∞ fj δ z ∗ − − = j m=−∞ Jn 2πρj ρ ∗ exp in φ ∗ − ϕj + exp P p 2 c     2π ilzj π  1 ∗ ∗ (1.228) = ( )j n fj Jn 2πρj ρ exp in φ − ϕj + exp P 2 c

168

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

From this equation, we have the following important condition. z ∗ = l/c = nN /c + mM /c or l = nN + mM

(1.229)

The f j is an atomic scattering factor of the j-th atom. The X-ray diffraction intensity I is proportional to the square of F as given below:   2    I ∼ |F|2 ∼ j n fj 2 Jn 2πρj ρ ∗ + 2j k fj fk n Jn 2πρj ρ ∗ Jn 2πρk ρ ∗ (1.230) The constrained condition l = nN + mM is used as follows. The X-ray diffraction patterns calculated for the helical chains are shown in Fig. 1.115, where M is the number of repeating units in the period c and N is the rotation time around the chain axis. For example, in the case of (3/1) helix, M = 3 and N = 1. Therefore, the helical condition of l = n + 3m is obtained (Fig. 1.115a). The n and m values satisfying this condition are given as follows: l l l l

=0 =1 =2 =3

(n, m) = (0, 0) (n, m) = (1, 0) (n, m) = (2, 0) (n, m) = (0, 1)

(−3, 1) (3, −1) (6, −2), . . . . . . . (−2, 1) (−5, 2) , . . . . . . . (−1, 1) (−4, 2) , . . . . . . . (3, 0) (−3, 2) , . . . . . . .

The helical radius is not very large, in general, and so we can focus the Bessel function in the narrow region of 2πρρ*(= x in Fig. 1.115a). The Bessel function of the low-order n takes a large value in the low region of 2πρρ*. For example, the maximum is detected at x = 0 (A) and x = 8 for n = 0, x = 2 (B) and x = 9 for n = 1 and x = 3.5 (C) for n = 2. As shown above, the 1st layer line (l = 1), the smallest n is 1 and so the peak should be detected at around the position B. Similarly, on the second layer line (l = 2), the smallest n = 2 and so the maximal intensity is detected at around the C point. The intensity maxima predicted from these relations correspond well to the actual tendency. In the case of (4/1) helix, M = 4 and N = 1 and the helical condition of l = n + 4m is obtained as shown below. As shown in Fig. 1.115b, the maximal intensity is detected at the x positions predicted for the smallest n values (n = 0, 1, 2,... for l = 0, 1, 2,... layer lines). l l l l l

=0 =1 =2 =3 =4

(n, m) = (0, 0) (n, m) = (1, 0) (n, m) = (2, 0) (n, m) = (3, 0) (n, m) = (0, 1)

(−4, 1) (4, −1) (8, −2), . . . . . . . (−3, 1) (−6, 2) , . . . . . . . (−2, 1) (−6, 2) , . . . . . . . (−1, 1) (7, −1) , . . . . . . . (4, 0) (−1, 2) , . . . . . . .

1.12 Characteristic Diffraction Pattern of Helical Chains

169

Fig. 1.115 a 2D diffraction pattern and Bessel functions predicted for a simple (3/1) helix. In the actual calculations, the non-normalized Bessel functions are used: J 0 (x) = sin(x)/x, J 1 (x) = sin(x)/x 2 – cos(x)/x, J 2 (x) = (3/x 2 – 1)sin(x)/x – 3cos(x)/x 2 , and so on. The scattering intensity is related to the square of J, b The 2D diffraction pattern predicted for a (4/1) helix

Characteristic Points of X-ray Diffraction Pattern Predicted for Helical Chain Conformation In general, the radius of helical chain ρ is in an order of 1 Å. If we treat the X-ray diffraction data up to the Bragg scattering angle 2θ~50° for the wavelength 1.5 Å, d = 1.5/(2sin(25°)) ~ 1.8 Å, and 1/d = R ~ 0.5 Å−1 and so 2πρR ~ 2 × 3.14 × 1 × 0.5~3, meaning that the discussion can be made in the range of lower order of Bessel function which gives the relatively strong scattering intensity (refer to Fig. 1.115). As summarized in Ref. [88], some characteristic points are speculated to appear in the X-ray diffraction pattern of a helical chain.

170

➀

➁

➂

➃

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

The M monomeric units (more strictly speaking, the number of the repeating units) are included in the repeating period c. If the homogeneous helix is assumed, the translational height per one monomeric unit p is given as p = c/M. When Polanyi’s equation is applied to the spacing between the equatorial and first layer lines, the fiber identity period I or the c-axial length is estimated directly. About the condition l = nN + mM, n = 0 and m = 1 for l = M, and so the Bessel function with n = 0 becomes strong near the meridional directional area on the M-th layer line. This means that the M-th layer line with the strong near-meridional diffraction gives the information about the number M of the monomer units included in the fiber period. By inputting m = 0 into the equation of l = nN + mM, l = nN. Therefore, the layer lines (l) with the order n close to 1~2 may give the strong diffraction intensity. Polanyi’s equation applied to the spacing between the equatorial and the strong layer line should allow us to estimate the helical pitch P. The rotation number along the helical axis N is given as N = c/P. The observation of the relatively strong layer line may make it possible to estimate the radius (ρj ) of the helical chain from the peak position ρ* of the maximal intensity. The equation 2πρj ρ* = Rmax , where Rmax is the peak position of the n-th Bessel function.

1.12.2 Several Examples of X-ray Analysis of Helical Chains 1.12.2.1

Polyoxymethylene

Figure 1.116 shows the 2D X-ray diffraction pattern measured for the γ-raypolymerized POM sample, which was collected using the high-energy synchrotron X-ray beam (λ = 0.328 Å) and the sample was rotated fully by 360° around the chain axis to avoid the preferential orientation of the crystallites. The 5th layer line shows the strongest intensity [10]. The 9, 18, and 27 layer lines show the relatively strong peaks near the meridional line. By putting l = 9 to the abovementioned equation, we have 9 = nN + mM and n = 0 and m =1, giving M = 9. The helical chain consists of nine monomeric units (repeating units) in a fiber period. The interlayer spacing gives the fiber period c = 17.35 Å. The interspacing between the equatorial and 9th layer lines gives p = 1.93 Å. Since the 5th layer line is quite strong, N may be equal to 5. That is to say, POM chain takes a (9/5) helical conformation as a possibility. The equation for the 5th layer line is l = 5 = 5n + 9m, and so we know n = 1 and m = 0. This means that the peak position of the first-order Bessel function should correspond to the strongest peak position along the 5th layer line. The strongest position is around ρ* = 0.3 Å−1 . This gives the helical radius ρ of about 1.1 Å using the equation of 2πρρ* = Rmax = 2.0, which is the peak position of the J 1 function. The strongest intensity position along the layer line corresponds relatively well to the peak position of the n-th Bessel function. The plot of l versus n (l = Nn +

1.12 Characteristic Diffraction Pattern of Helical Chains

171

(b)

(a)

POM (9/5) helix

12 l = 30

l =9

l = 25 l = 20

l =5

l = 15

8

n =2 n =0 n = −2

4

n =1 n = −1

0

n =2 n =0

fiber axis

l = 10 l= 5

l =0

l= 0

-4 -8 -12 -4

-2

0 n

2

4

Fig. 1.116 a X-ray diffraction pattern of POM and b Bessel functions. The solid straight lines of gray color are the equations l = 5n + 9m with m = 0, 1, 2, …. The broken lines are overlapped by considering the symmetric pattern of the observed data due to the full-rotation measurement. The numerical values shown along the right edge indicate the order n showing the maximal intensity position along the layer lines

Fig. 1.117 Uniform helix of POM

Mm) may allow us to predict the diffraction pattern of the (M/N) helix if we know the M an N values. For example, in the case of POM chain, the diffraction pattern predicted for the (9/5) helix is shown in Fig. 1.116b, which is comparatively close to the observed pattern. For example, the strongest peak positions on the l = 4, 5, and 9 lines correspond to the strong peaks of J n curves with the order n = −1, 1, and 0, respectively (refer to Fig. 1.115). However, careful observation rises some question: the relative intensity of the 4th and 5th layer lines is almost equally strong in the predicted pattern, which is different from the actually observed relation (the

172

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

5th layer line >> the 4th layer line). POM chain must be assumed to consist of the helices of C and O atoms. The interference between the diffraction waves from these two helical chains should modify the relative intensity of these layer lines. In fact, according to the calculation by Tadokoro et al. [89], the structure factor of these layer lines is given as follows by inputting the relative z heights of the C and O atoms; zC = 0, 2/18, 4/18,…, 16/18 for the C atoms and zO = zC + 1/18 (for (9/5) helix). The rotation angle φC = 0.0° and φO = 2π × 5 × 1/18, where the C and O atoms are assumed to distribute homogeneously around the helical axis (5 turns for 18 atoms). For the one monomeric unit, the structure factors are given as follows (Eq. 1.228). Here, “r j ” is used for “ρj ” in Eq. 1.228. 

  F(l = 5; n = 1) ∝ j n fj Jn 2π rj R exp in π/2 − φj exp(2π i × l × zj /c) = fC J1 (2π rC R) exp[i(π/2 − φC )] exp(2π i × 5 × zC /c) + fO J1 (2π rO R) exp[i(π/2 − φO )] exp(2π i × 5 × zO /c) = fC J1 (2π rC R) exp(iπ/2) exp(2π i × 5 × 0) + fO J1 (2π rO R) exp(−π i/18) exp(2π i × 5/18) 

= fC J1 (2π rC R) + fO J1 (2πrO R) exp(i π/2) 

= i fC J1 (2π rC R) + fO J1 (2πrO R)

(1.231)

F(l = 4; n = −1) ∝ fC J−1 (2π rC R) exp[−i(π/2 − φC )] exp(2π i × 4 × zC /c) + fO J−1 (2π rO R) exp[−i(π/2 − φO )] exp(2π i × 4 × zO /c) = fC J−1 (2π rC R) exp(−iπ/2) exp(2π i × 4 × 0) + fO J−1 (2π rO R) exp(πi/18) exp(2π i × 4 × 18) = fC J−1 (2π rC R) exp(−iπ/2) + fO J−1 (2π rO R) exp(iπ/2) 

= i fC J1 (2π rC R) − fO J1 (2π rO R) (1.232) Here, J−n (x) = (−1)n Jn (x). The F(l = 5) is given as the positive summation of C and O diffraction waves, while these two diffraction waves are cancelled for the F(l = 4). This relation (l = 5 reflection intensity >> l = 4 reflection intensity) is consistent with the observed data.

1.12.2.2

Isotactic Polypropylene

In this case, the 1st layer line is overwhelmingly strong and the meridional diffraction is observed on the 3rd layer line (Fig. 1.118). That is, the helical chain may take the (3/1) conformation. According to the equation l = n + 3m, we have the sets of n and m with the actually observed relative intensity as shown in Table 1.15. The fiber period obtained from the interspacing between the 1st and equatorial lines is about 6.50 Å. The spacing between the 3rd and equatorial lines gives the

1.12 Characteristic Diffraction Pattern of Helical Chains

173

Fig. 1.118 a X-ray diffraction pattern of it-PP fiber, measured in SPring-8 using a high-energy X-ray beam and b Bessel functions. The order n giving the maximal intensities is shown at the right side

Table 1.15 X-ray diffraction and Bessel functions for it-PP sample

Layer line

Bessel function order n

Scattering intensity

l=1

1 (m = 0), 4 (m = −1)

Very strong

2

2 (m = 0), 5 (m = −1)

Weak

3

0 (m = 1), 3 (m = 0)

Near meridional

4

1 (m = 1), −2 (m = 2)

Strong

5

2 (m = 1), 5 (m = 0)

Weak

6

0 (m = 2), 3 (m = 1)

Near meridional

7

1 (m = 2)

Weak

8

2 (m = 2)

Weak

9

0 (m = 3), 3 (m = 2)

Near meridional

period 2.17 Å, which should be the c-axial-translation length p of one monomeric unit. The three rotations and the translation of p satisfy the fiber period c.

1.12.2.3

Polytetrafluoroethylene

The interlayer spacings are quite narrow, suggesting the chain conformation of a long repeating period. Overwhelmingly strong diffractions are observed on the 6th and 7th layer lines (Fig. 1.119). The near-meridional diffractions are observed on the 13th layer line. Therefore, the helix is assumed to take a (13/6) or (13/7) conformation [90–92]. The reason why only l = 6, 7 and 13 layer lines are strong can be understood from the equation of l = 6n + 13m or l = 7n + 13m. The translational height per one monomeric unit, estimated from the spacing between the 13th layer line and

174

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.119 X-ray diffraction pattern of highly oriented PTFE sample measured at 173K with a synchrotron high-energy X-ray beam (λ = 0.328 Å)

l=6 7 l = 13

the equatorial line, is 1.30 Å, and the pitch of one helical rotation is 2.81 Å or 2.41 Å as estimated from the spacing between the equatorial line and the 6th or 7th layer lines. The averaged value between these two values is 2.61 Å, which is near the repeating period of the planar-zigzag conformation 2.55 Å. This suggests that the PTFE chain may take the conformation close to the planar-zigzag form, but actually the chain takes the helical form through the CC torsional angles. One of the most significant reasons for this helical conformation is the nonbonded interatomic interactions between the F and F atoms of the neighboring CF2 groups.

1.12.2.4

Doubly Stranded Helix (Polyethylene Imine)

The structure factor of one helical chain was already given above. If these two helical chains are coupled together to form a doubly stranded helix, the structure factor of the latter is expressed by taking into account the phase relation between these two components. As an example, let us see the case of poly(ethylene imine) (PEI) [93]. PEI forms the various crystalline phases depending on the humidity. Different from the planar-zigzag chain conformation in the hydrate states, the PEI chains in the dry state form the doubly stranded helix as shown in Fig. 1.120. The observed 2D X-ray diffraction pattern of the dry PEI sample is composed of the two characteristic points: (i) the strong layer lines give the repeating period 4.79 Å and (ii) the diffuse layer lines are observed between the main layer lines. We have to interpret these data consistently. The basic unit cell is a hexagonal type with a = 8.60 Å (the finally determined unit cell is much larger, a = 29.8 Å and b = 17.2 Å, with the space group Fddd). The two chains of five monomeric units along the chain axis are assumed as the candidate. The geometrically reasonable single chain is that of the (5/1) conformation with the repeating period 9.58 Å. By referring to the observed period, the doubly stranded helix is assumed as the next candidate. Two possible models are built (Fig. 1.121). In the case A, one chain is shifted by a half of the repeating period along the chain axis. In the case B, one chain is rotated by 180°

1.12 Characteristic Diffraction Pattern of Helical Chains

175

H2O

Double helix (5/1 helix)

Planar-zigzag

Fig. 1.120 Double helix of PEI (dry state) and its X-ray diffraction pattern. Reprinted from Ref. [93]. Copyright 1982, with permission from the American Chemical Society

Fig. 1.121 Two types of double helix model of PEI (dry state). Reprinted from Ref. [93]. Copyright 1982, with permission from the American Chemical Society

around the chain axis to generate another chain. As we know already, the structure factor of a helical chain is expressed as follows:  F ρ ∗ , φ ∗ , z ∗ = Fm (1)

  

 = (1/P)j n fj Jn 2π ρj ρ ∗ exp in φ ∗ − ϕj + π/2 exp 2π ilzj /c (1.233)

with the constraining l = nN + mM = n + 5m. In the case A, the structure factor of another chain is given as   

 

Fm (2) = (1/P)j n fj Jn 2π ρj ρ ∗ exp in φ ∗ − ϕj + π/2 exp 2π il zj + c/2 /c = Fm (1) exp(π il)

(1.234)

176

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

The structure factor of a double helix of the model A is FA = Fm (1) + Fm (2) = Fm (1)[1 + exp(π il)]

(1.235)

Similarly, F m (2) in the model B is given as   



Fm (2) = (1/P)j n fj Jn 2π ρj ρ ∗ exp in φ ∗ − ϕj + π/2 + π exp 2π ilzj /c = Fm (1) exp(π in)

(1.236)

Then, the structure factor of a double helix of the model B is FB = Fm (1) + Fm (2) = Fm (1)[1 + exp(π in)] = 2Fm (1) (n = even)

(1.237)

In this case, all the layer lines can appear though the order n should be even. The repeating period is 9.58 Å. In the model A, FA = 2Fm (1) for even l, and FA = 0 for odd l In this case, the interlayer spacing becomes twice wider apparently, indicating the fiber period is reduced to c/2 = 4.79 Å. The main part of the observed diffraction pattern is consistent with this model. But, different from the prediction of the perfectly zero intensity of the odd layer lines, the actually observed layer lines are not zero, although they are diffuse. The model B should be employed since the equatorial line profile can be reproduced quite well by this model B. By introducing the statistically disordered structure for the model B, the diffuseness of the odd layer lines was reproduced well (the diffuse scattering of the layer lines 1 and 3 and the strong 2 and 4 layer lines). Besides the model B can form the strong NH…N hydrogen bonds of 3.10 Å (Fig. 1.120). The finally derived packing structure of the double helices B is shown in Fig. 1.122. Fig. 1.122 Crystal structure of double helical chains of PEI (dry state). Reprinted from Ref. [93]. Copyright 1982, with permission from the American Chemical Society

1.12 Characteristic Diffraction Pattern of Helical Chains

177

1.12.3 Prediction of Chain Conformation by Energy Calculation As already mentioned, the energy calculation is useful for the prediction of a conformation of a polymer chain. The details of the energy calculation method will be described in a later section. Here we will learn about the setting of a helical chain to predict the stable conformation and also will experience some training about the conformational analysis.

1.12.3.1

Energy Terms Needed for the Conformational Prediction

The potential energy of an isolated chain is assumed to be a summation of the various possible energy terms as listed below. E(total) = E(bond length) + E(bond angle) + E(torsional angle) + E(electrostatic) + E(van der Waals) + E(intramolecular hydrogen bond) + . . . . The detailed explanation of the individual energy terms will be made in Chap. 6.

1.12.3.2

Calculation of Conformational Energy

The conformational energy of an isolated chain is calculated by summing up all the possible interactions mentioned above. In the actual calculation, the limitation of the range of interactions must be assumed for each term. The electrostatic interaction is one of the so-called long-range interactions, whereas the van der Waals interaction is a short-range interaction. For the prediction of the conformation of a chain in the unit cell, we introduce the cyclic boundary condition, that is, the arrangement of the atoms included in a repeating period along the chain axis is repeated cyclically. More simple approximation is that the monomeric units are repeated regularly with the same local structure along the chain axis. For example, as shown in Fig. 1.123, isotactic polypropylene is assumed to have the following torsional angles τ1 , τ2 , and τ3 for one monomeric unit and this set of geometry is repeated for the other monomeric units. In the rough calculation, the contribution of only the torsional angles is taken into account and such other almost constant interactions as bond length, bond angle are ignored for simplicity. The CH2 and CH3 units are sometimes assumed as the spherical balls without any hydrogen atoms, which are called the united atoms (the potential function coefficients for these united atoms are different from the original C and H atoms). In this case, the usage of torsional angle τ3 is meaningless. For the discussion of the anisotropic interactions, the assumption of united atom is not recommended to use.

178

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.123 Torsional angles defined for it-PP chain. a hydrogen atoms are positively taken into account, b united atom type

(a)

τ3

τ2 τ1 (b)

τ2 τ1

τ3

τ2 τ1 1.12.3.3

τ3

τ3

τ2 τ 1

A Simple Helical Chain (Model 1)

Let us try to calculate the conformational energy for one simple model shown below. We can get the potential energy as a function of two torsional angles τ1 and τ2 and find the conformation of the minimal energy. The model consists of the five atoms which are connected by the bond length r and the bond angle φ. The bond angle φ is 90°, as an exercise, which is, of course, different from the normal value of CCC bond angle 109.5o . The bond length r = 1.54 Å. The distances r14 and r25 change depending on the torsional angles τ1 and τ2 . The other distances such as r 13 , r 24 , and r 35 are constant independently of the torsional angles. Therefore, the following energy terms are taken into account in the estimation of the conformational energy: E 1 = the nonbonded interatomic interaction between the atoms 1 and 4, E 2 = the nonbonded interatomic interaction between the atoms 2 and 5, and E 3 = torsional energies of τ1 and τ2.

1.12 Characteristic Diffraction Pattern of Helical Chains

179

The coordinate of the fourth atom is (r, r(1-ct), rst) on the coordinate system of the first atom, where ct and st represent cos(τ) and sin(τ), respectively. The distance r 14 is given as r14 = r[3 − 2 cos(τ1 )]1/2 The distance r 25 is expressed in the same way, r25 = r[3 − 2 cos(τ2 )]1/2 As will be explained in Chap. 6, the total energy E(total) is expressed as E(total) = Vo [1 + cos(3 τ1 )]/2 + Vo [1 + cos(3 τ2 )]/2 6 6 +A exp(−Br14 ) − Cr14 + A exp(−Br25 ) − Cr25

(1.238)

The actual calculation is performed using the following parameters: V o = 8.4 kJ/mol A = 11000 kJ/mol B = 3.7 Å−1 C = 115.0 kJÅ6 /mol The calculated energy E(total) is a function of the two torsional angles τ1 and τ2 . The contour map is shown in Fig. 1.124. Point A or the conformation of τ1 = 0° and τ2 = 0° is the most unstable because of the collision of the atoms. The conformation at point B with τ1 = 180° and τ2 = 180° is of the lowest energy. The molecule takes a fully extended zigzag shape. The points C and D with the TG and GG conformations, respectively, are also stable but with relatively high energies. The lower figure of Fig. 1.124 is the case of the torsional potential with the shape of E(torsion) = (V o /2)[1 + cos(τ)]. The points C and D do not appear, different from the case of the equation with cos(3τ) form.

180

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.124 2D contour maps of potential energies calculated for a simple n-pentane model. Torsional potential function used in the calculation is (upper) cos(3τ) type and (lower) cos(τ) type

1.12.3.4

Eyring’s Transformation Matrix (Model 2)

Figure 1.125 is another example of the normal chain form. According to the equation given by Eyring [94], the atomic coordinates are calculated in the following way: For example, let us focus the 4th atom, the coordinate of which is expressed as ⎛

⎞ r3 x3 (4) = ⎝ 0 ⎠ = Br (3) on the coordinate system based on the 3rd atom (red solid line). 0

1.12 Characteristic Diffraction Pattern of Helical Chains

181

Fig. 1.125 Definition of coordinates for C–C torsional rotation

In order to express this coordinate on the basis of the x3’ coordinate system (blue dotted line), the 3rd atomic coordinate system is rotated around the z3 -axis by the angle 180° – φ 3 . ⎞ ⎛   ⎞ cos 180◦ − φ3 − sin 180◦ − φ3 0 − cos φ3 − sin φ3 0  ⎟ ⎟ ⎜  ⎜ x3 (4) = ⎝ sin 180◦ − φ3 cos 180◦ − φ3 0 ⎠ x3 (4) = ⎝ sin φ3 − cos φ3 0 ⎠x3 (4) 0 0 1 0 0 1 ⎛

(1.239) or

x3 (4) = Aφ (3)x3 (4) The coordinate system of x3’ atom is now rotated by the torsional angle τ23 around the bond r 2 or around the x 2 -axis of the coordinate system of the 2nd atom (black solid line). Since the origin of this 2nd atomic coordinate system is shifted from that of the 3’ atomic coordinate system, the final coordinate of the 4th atom is expressed as follows: ⎞ ⎛ ⎛ ⎞ 1 0 0 r2 2 3 ⎠ ⎝ ⎝ x (4) = 0 cos τ23 − sin τ23 x (4) + 0 ⎠ = Aτ (23)x3 (4) + Br (2) 0 sin τ23 cos τ23 0 ⎞⎛ ⎛ ⎞ ⎛ ⎞ − cos φ3 − sin φ3 0 r2 1 0 0 = ⎝ 0 cos τ23 − sin τ23 ⎠⎝ sin φ3 − cos φ3 0 ⎠x3 (4) + ⎝ 0 ⎠ ⎛

0 sin τ23 cos τ23

0

0

⎞

1

⎛

0 ⎞

− cos φ3 − sin φ3 0 r2 = ⎝ cos τ23 sin φ3 − cos τ23 cos φ3 − sin τ23 ⎠x3 (4) + ⎝ 0 ⎠ sin τ23 sin φ3 − sin τ23 cos φ3 cos τ23 0 = Aτ (23)Aφ (3)x3 (4) + Br (2) = Aτφ (23)x3 (4) + Br (2) = Aτ (23)Aφ (3)Br (3) + Br (2)

(1.240)

182

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

By repeating this process for the 5th atom, the x2 (5) is expressed as follows: x4 (5)

→



x4 (5)

→

x3 (5)

→



x3 (5)

→

x2 (5)



x4 (5) = Aφ (4)x4 (5)

x3 (5) = Aτ (34)x4 (5) + Br (3) = Aτ (34)Aφ (4)x4 (5) + Br (3) = Aτφ (34)x4 (5) + Br (3)

x3 (5) = Aφ (3)x3 (5)   = Aφ (3) Aτ (34)Aφ (4)x4 (5) + Br (3) = Aφ (3)Aτ (34)Aφ (4)x4 (5) + Aφ (3)Br (3)

x2 (5) = Aτ (23)x3 (5) + Br (2) = Aτ (23)Aφ (3)Aτ (34)Aφ (4)x4 (5) + Aτ (23)Aφ (3)Br (3) + Br (2) = Aτφ (23)Aτφ (34)x4 (5) + Aτφ (23)Br (3) + Br (2) = Aτφ (23)Aτφ (34)Br (4) + Aτφ (23)Br (3) + Br (2)

(1.241)

In the above-illustrated simple model case (p.179), the coordinates of atoms 4 and 5 are expressed below. Since the x3 (4) is expressed simply as x3 (4) = Br (3) = (r 3 , 0, 0)t , the x2 (4), x2 (5), and x2 (1) are given as ⎛

⎞ r(1 − cos φ) x2 (4) = Aτφ (23)Br (3) + Br (2) = ⎝ r cos τ23 sin φ ⎠ r sin τ23 sin φ

(1.242)

x2 (5) = Aτφ (23)Aτφ (34)Br (4) + Aτφ (23)Br (3) + Br (2)  ⎞ ⎛ r 1 − cos φ + cos2 φ − r sin2 φ cos τ34 ⎟ ⎜ = ⎝ r sin φ cos φ cos τ23 (cos τ34 − 1) − r sin φ sin τ23 sin τ34 + r sin φ cos τ23 ⎠ r sin φ cos φ sin τ23 (cos τ34 − 1) + r sin φ cos τ23 sin τ34 + r sin φ sin τ23

(1.243) ⎛

⎞ r cos(φ) x2 (1) = ⎝ r sin(φ) ⎠ 0

(1.244)

The distances r 14 and r 25 are calculated as follows:   

1/2 r14 = x2 (1) − x2 (4) = r 3 − 4 cos(φ) + 2 cos2 (φ) − 2 sin2 (φ) cos(τ23 ) (1.245)

  %  2   r25 = x2 (2) − x2 (5) = r 1 − cos φ + cos2 φ − r sin2 φ cos τ34

+ [r sin φ cos φ cos τ23 (cos τ34 − 1) − r sin φ sin τ23 sin τ34 + r sin φ cos τ23 ]2 1/2 +[r sin φ cos φ sin τ23 (cos τ34 − 1) + r sin φ cos τ23 sin τ34 + r sin φ sin τ23 ]2

(1.246)

1.12 Characteristic Diffraction Pattern of Helical Chains

183

Fig. 1.126 2D contour map of potential energy calculated for the model (n-pentane) used in Fig. 1.124. The CCC bond angle is now 109.5°

If the bond angle φ = 90°, the distances are more simply expressed as follows: r14 = r[3 − 2 cos(τ23 )]1/2

(1.247)

r25 = r[3 − 2 cos(τ34 )]1/2

(1.248)

These equations are the same as those derived in the previous section. Figure 1.126 shows the energy contour map calculated for the same molecule with the bond angle φ = 109.5o and bond length r =1.54 Å. The points A (τ23 = τ34 = 180.0° etc.) are energetically stable. The point B is an unstable saddle point through which the conformation transfers between the points A.

1.12.4 Construction of Helical Chain Models For the crystal structure analysis of the helical chains, we need to construct the helical conformation starting from the internal parameters such as torsional angles. The general way of construction of a helical chain is to repeat the abovementioned process for the skeletal chain atoms [95, 96]. (The side group coordinates can be expressed in a similar way.) The coordinate X 1m (i, m) of the i-th atom of the m-th repeating unit along the helical axis is given as X 1 m (i, m) = Aτφ (12)Aτφ (23) . . . Aτφ (i − 2, i − 1)Br (i − 1) + Aτφ (12)Aτφ (23) . . . Aτφ (i − 3, i − 2)Br (i − 2) + . . . + Aτφ (12)Aτφ (23)Br (3) + Aτφ (12)Br (2) + Br (1) i−2 τφ A (01)Aτφ (12)Aτφ (23) . . . Aτφ (n − 1, n)Aτφ (n, n + 1)Br (n + 1) (i > 2) = n=0

(1.249)

184

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

where Aτφ (01) = E (identity matrix) and the coordinate system is based on the 1st atom of the m-th unit. The coordinate X 1m (i, m+1) of the i-th atom of the m+1st unit is written as follows, because this atom is numbered as p + i counted from the 1st atom of the m-th unit using the total number of the repeating unit included in a repeating period p. X 1 m (i, m + 1) = X 1m (p + i, m) =

p+i−2 

Aτφ (01)Aτφ (12)Aτφ (23) . . . Aτφ (n − 1, n)Aτφ (n, n + 1)Br (n + 1)

n=0 p−1 = n=0 Aτφ (01)Aτφ (12)Aτφ (23) . . . Aτφ (n − 1, n)Aτφ (n, n + 1)Br (n + 1) p+i−2 + n=p Aτφ (01)Aτφ (12)Aτφ (23) . . . Aτφ (n − 1, n)Aτφ (n, n + 1)Br (n) = X1 m (1, m + 1) + Aτφ (01)Aτφ (12)Aτφ (23) . . . Aτφ (p − 1, p)Aτφ (p, p + 1)Br (p + 1) + Aτφ (01)Aτφ (12)Aτφ (23) . . . Aτφ (p, p + 1)Aτφ (p + 1, p + 2)Br (p + 2) + Aτφ (01)Aτφ (12)Aτφ (23) . . . Aτφ (p + 1, p + 2)Aτφ (p + 2, p + 3)Br (p + 3) ............................ + Aτφ (01)Aτφ (12)Aτφ (23) . . . Aτφ (p, p + 1) . . . Aτφ (p + i − 3, p + i − 2)Aτφ (p + i − 2, p + i − 1)Br (p + i − 1) = X 1 m (1, m + 1)

 + Aτφ (01)Aτφ (12)Aτφ (23) . . . Aτφ (p − 1, p)Aτφ (p, p + 1) Br (p + 1) + Aτφ (p + 1, p + 2)Br (p + 2)+ + Aτφ (p + 1, p + 2)Aτφ (p + 2, p + 3)Br (p + 3) + . . .

+Aτφ (p + 1, p + 2)Aτφ (p + 2, p + 3) . . . Aτφ (p + i − 3, p + i − 2)Aτφ (p + i − 2, p + i − 1)Br (p + i − 1)



= X 1 m (1, m + 1) + Aτφ (01)Aτφ (12)Aτφ (23) . . . Aτφ (p − 1, p)Aτφ (p, p + 1)  x Br (p + 1) + Aτφ (p + 1, p + 2)Br (p + 2) + Aτφ (p + 1, p + 2)Aτφ (p + 2, p + 3)Br (p + 3) + . . . + Aτφ (p + 1, p + 2)Aτφ (p + 2, p + 3) . . . Aτφ (p + i − 3, p + i − 2)Aτφ (p + i − 2, p + i − 1)Br (p + i − 1)



= X 1 m (1, m + 1) + Aτφ (01)Aτφ (12)Aτφ (23) . . . Aτφ (p − 1, p)Aτφ (p, p + 1)X 1m (1, m) = X 1 m (1, m + 1) + A · X l m (i, m)

That is to say, X 1 m (i, m + 1) = B + A · X 1 m (i, m) A = Aτφ (01)Aτφ (12)Aτφ (23) . . . Aτφ (p − 1, p)Aτφ (p, p + 1)

(1.250) (1.251)

B = n=0 Aτφ (01)Aτφ (12)Aτφ (23) . . . Aτφ (n − 1, n)Aτφ (n, n + 1)Br (n + 1) p−1

= X 1 m (1, m + 1)

(1.252)

Inversely, 

X 1m (i, m) = A˜ · X 1m (i, m + 1) − B where A˜ is a transpose of the matrix A.

(1.253)

1.12 Characteristic Diffraction Pattern of Helical Chains

185

Fig. 1.127 Bond vectors and helical parameters (1)

Now, let us consider the relation between the helical parameters and the internal coordinates [95, 96]. As shown in Fig. 1.127, the vectors connecting the two neighboring units are created, which are named X(m-1, m) and X(m, m+1). By focusing on the 1st units, we have the following relations. Since X 1m (1,m) = 0, X(m − 1, m) = X 1m (1, m) − X 1m (1, m − 1) 

= X 1m (1, m) − ' A · X 1m (1, m) − B = ' A · B ≡ B(m − 1, m) X(m, m + 1) = X 1m (1, m + 1) − X 1m (1, m) = B + A · X1m (1, m) − X 1m (1, m) = B ≡ B(m, m + 1) X(m + 1, m + 2) = X 1m (1, m + 2) − X 1m (1, m + 1) 

= B + A · X1m (1, m + 1) − B + A · X 1m (1, m) 

= A · X 1m (1, m + 1) = A · B + A · X 1m (1, m) = A · B ≡ B(m + 1, m + 2) The two vectors C and C’, defined below, direct to the radial direction from the helical axis. C = B(m − 1, m) − B(m, m + 1) = ' A · B − B = (' A − E) · B

(1.254)

C = B(m, m + 1) − B(m + 1, m + 2) = B − A · B = (E − A) · B

(1.255)

The angle between the vectors C and C is equal to the helical rotation angle θ. C · C = C 2 cos(θ)

(1.256)

In a similar way, we have the following equations: d 2 + 2ρ2 (1 − cos θ) = B2

(1.257)

186

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.128 Bond vectors and helical parameter (2)

 d sin θ = B · C × C / C 2 sin θ

(1.258)

Let us consider about these equations for the simple helical chain (Fig. 1.128). The C 123 , C 234 , B12 , B23 , and B34 vectors defined in the above part are given as follows for the helical chain: ⎛ ⎞ −2rc2 (φ/2)  1



C 123 = B12 − B23 = X (2) − X 1 (1) − X 1 (3) − X 1 (2) = ⎝ −rs(φ) ⎠ 0 (1.259) ⎞ 2rc2 (φ/2)  1



= X (3) − X 1 (2) − X 1 (4) − X 1 (3) = ⎝ −rc(τ)s(φ) ⎠ ⎛

C 234 = B23 − B34

−rs(τ)c(φ) (1.260) At first, we have the following equation (Eq. 1.256): C 123 · C 234 = |C 123 ||C 234 | cos(θ) Since |C 123 | = |C 234 | = 2rc(φ/2) and C 123 ·C 234 = 4r 2 c2 (φ/2)[2c2 (τ/2)s2 (φ/2)-1], we can get the following equation: cos(θ/2) = cos(τ/2) sin(φ/2)

(1.261)

1.12 Characteristic Diffraction Pattern of Helical Chains

187

Next is to derive the equation about the helical pitch d. Since the C vectors are perpendicular to the helical axis, we have C 123 × C 234 = |C 123 ||C 234 | sin(θ) ed

(1.262)

Here, ed is a unit vector parallel to the helical axis. The vector d is expressed as d = ded . The spacing between the units 1 and 2 (or 2 and 3, and so on) along the helical axis is d. Therefore, we have the following relation: d = B12 · ed = B12 · (C 123 × C 234 )/|C 123 ||C 234 | sin(θ) The simple calculation gives d · sin(θ) = r · sin(τ) sin2 (φ/2)

(1.263)

Using the above-derived equation [Eq. 1.261, cos(θ/2)], we get d · sin(θ/2) = r · sin(τ/2) sin(φ/2)

(1.264)

The third equation is about the radius ρ. The simple consideration leads to the following equation of the vectors: ρ3 + d = ρ2 + B23 2  (B23 )2 = r 2 = ρ3 + d − ρ2 The final result is given as follows: 2ρ2 (1 − cos θ) + d 2 = r 2

(1.265)

By combining these equations, the ρ is given as ρ · sin2 θ = 2r · sin2 (φ/2) · cos(φ/2) · cos2 (τ/2)

(1.266)

ρ · sin2 (θ/2) = r · cos(φ/2)

(1.267)

The similar calculation can be made for the more complicated helical chains such as -(A1 -A2 )n -, -(A1 -A2 -A3 )n -, etc. The details are referred to in the literature [96].

188

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

In the actual process of building up the helical chain, we need to know the atomic coordinates based on the helical axis coordinate system. The coordinates derived in the above section are based on the internal coordinate system. The transformation from the internal coordinate system to the helical coordinate system can be made by using the B and C vectors derived above. As already shown, the d vector parallel to the helical axis is given in Eq. 1.262, from which the unit vector ed or the zo -coordinate axis is obtained.

 ezo = ed = (C 123 × C 234 )/ |C|2 sin(θ)

(1.268)

The xo -axis perpendicular to the helical axis is defined as a unit vector parallel to C 123 . exo = C 123 /|C|

(1.268)

The yo -axis is perpendicular to both of the unit vectors ezo and exo . eyo = ezo × ex0

(1.269)

Once we know the Cartesian coordinate system fixed on the helical axis, the transformation from the coordinate system defined by the internal coordinates to the Cartesian coordinates is made as follows. The coordinate of the i-th atom fixed on the helical chain system is X i = [x i , yi , zi ]. When the unit vectors are defined as ex , ey , and ez , the coordinate X i can be expressed as follows: ⎛ ⎞

xi  X i = xi ex + yi ey + zi ez = ex , ey , ez ⎝ yi ⎠ zi On the other hand, the atomic coordinate based on the helical chain axis X oi is expressed as follows: ⎛ o⎞  xi  X oi = xio eox + yio eoy + zio eoz = eox , eoy , eoz ⎝ yio ⎠ zio As shown in Fig. 1.129, the transformation of the two kinds of the coordinate systems must be made by shifting the origin. This translation is made by shifting the coordinate origin by ρ along the x-axis. The translation vector is L. Then we have the following relation: X oi = X i + L, X i → (rotational transformation) → X i

1.12 Characteristic Diffraction Pattern of Helical Chains

189

4 3 ez d

e yo L

1

3

4

e yo

o

e xo

ey

e z’ e x e y’

e zo

ex e xo L

e x’

2

1

ez

ey

e y’ 2 e z’

e x’

C 123

Fig. 1.129 Transformation of two coordinate systems

What we have to do is to know the concrete expression about the transformation matrix T.     

X i = TX i or eox , eoy , eoz = e x , e y , e z = ex , ey , ez T t ⎛

⎞ ⎛ ⎞ eox ex ⎝ eoy ⎠ = T ⎝ ey ⎠ ez eoz where T t is the transpose of the matrix T. The rotational transformation can be made by setting the coordinate axes of the X o system as shown in Fig. 1.129. The concrete definition was already given in the previous paragraph. That is,

 eoz = ed = (C 123 × C 234 )/ |C|2 sin(θ) eox = C 123 /|C| eoy = ezo × exo The concrete calculation can lead to the following relations:

 eoz = ed = (C 123 × C 234 )/ |C|2 sin(θ) = t31 ex + t32 ey + t33 ez eox = C 123 /|C| = t11 ex + t12 ey + t13 ez eoy = ezo × exo = t21 ex + t22 ey + t23 ez

190

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

⎛

⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ eox t11 t12 t13 ex ex ⎝ eoy ⎠ = ⎝ t21 t22 t23 ⎠⎝ ey ⎠ = T ⎝ ey ⎠ t31 t32 t33 ez ez eoz The actual calculation for the simple helical chain system gives the following transformation matrix: ⎛

−(sφ/2) ⎜ (sφ/2)(sφ)(cτ/2)2 ⎜ T=⎜ sin θ ⎝ (sφ/2)2 (sτ) sin θ

−(sφ/2) −(cφ/2)(sφ)(cτ/2)2 sin θ −(sφ)(sτ) 2 sin θ

0 −(sφ)(sτ) sin θ (sφ)(cτ/2)2 sin θ

⎞ ⎟ ⎟ ⎟ ⎠

(1.270)

where the abbreviated expressions are made for various terms. sφ = sin(φ), sφ/2 = sin(φ/2), cτ = cos(τ), sτ = sin(τ ), cτ/2 = cos(τ/2) Using this T matrix, the coordinates can be calculated as X oi = TX + L

(1.271)

Let us see the concrete calculation for the atoms 2, 3, and 4 shown in Fig. 1.128. The coordinates of these atoms based on the coordinate system fixed on the atom 2 are as follows: (atom 2) x2 = 0, y2 = 0 and z2 = 0 x2o = ρ, y2o = 0 and z2o = 0 (atom 3) ⎛ ⎞ ⎞ x3◦ = ρ − r cos(φ/2) = ρ cos(θ) x3 = r ⎝ y3 = 0 ⎠ use equation 1.271⎝ y◦ = r(sφ/2)(sφ)(cτ/2)2 / sin(θ) = ρ sin(θ) ⎠ 3 → z3 = 0 z3◦ = r(sφ/2)2 (sτ)/ sin(θ) = d (1.272) ⎛

1.12 Characteristic Diffraction Pattern of Helical Chains

191

The second terms are perfectly equal to the results given from the helical transformation equation. X o3 = RX o2 + D ⎛

⎛ ⎞ ⎞ ⎛ ⎞ cos(θ) − sin(θ) 0 ρ 0 R = ⎝ sin(θ) cos(θ) 0 ⎠, D = ⎝ 0 ⎠, X o2 = ⎝ 0 ⎠ 0 0 1 0 d (atom 4) ⎞ x4 = 2r(sφ/2)2 ⎟ ⎜ ⎝ y4 = r(cτ/sφ) ⎠ use equation 1.271 → z4 = r(sτ/sφ) ⎛ ◦ x = ρ − 2r(cφ/2)(sφ/2)2 − r(sφ/2)(sφ)(cτ) = 

⎜ 4◦ ⎝ y4 = r 2(sφ/2)3 (sφ)(cτ/2)2 − (cφ/2)(sφ)2 (cτ)(cτ/2)2 − (sφ/2)(sφ)(sτ)2 / sin(θ) = 

◦ z4 = r 2(sφ/2)4 (sτ) − (sφ)2 (sτ)(cτ)/2 + (sφ)2 (sτ)(cτ/2)2 / sin(θ) = ⎛

⎞ ρ cos(2θ) ⎟ ρ sin(2θ) ⎠ 2d

(1.273)

The second terms are perfectly equal to the results given from the helical transformation, X o4 = RX o3 + D = R2 X o2 + 2D ⎛

⎛ ⎞ ⎞ ρ cos(θ) ρ X o3 = ⎝ ρ sin(θ) ⎠ , Xo2 = ⎝ 0 ⎠ d 0 The more complicated helical chains of -(A1 -A2 )n -, etc. can be treated in a similar way to that mentioned for the simplest helical chain of -(A)n - [96]. The side chain atoms can be generated and calculated in a similar way to the abovementioned treatment. These equations can be applied to the building up of the helical chains, for example, the energy calculation for the purpose to find the energetically stable chain conformation. The details of the conformation analysis will be explained in Chap. 6.

1.13 Examples of Actual Crystal Structure Analysis Some examples are presented here for showing the concrete processes of the crystal structure analysis and for solving the logical problems deduced from the analyzed structure models.

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1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

1.13.1 Orthorhombic Polyethylene In the previous section, we demonstrated the brief process of the structure analysis of orthorhombic PE crystal [21]. Some points not mentioned in the previous section will be shown here.

1.13.1.1

The X-ray Data Analysis

The 2D X-ray diffraction pattern of the uniaxially oriented and annealed HDPE sample was analyzed using the software produced by ourselves. The detector was a flat imaging plate set at 8 cm from the sample position. Figure 1.130 shows the transformation of the diffraction pattern from the flat camera [(a)] to the cylindrical reciprocal lattice [(b)]. The (x, y) positions of the observed reflections were converted to the cylindrical coordinates (ξ, ζ ) in the reciprocal lattice. Figure 1.130c is the further conversion of (b) by performing the Lp corrections for all the pixel points. (d) (a) 2D X-ray diffraction pattern by a flat IP camera

(b) transformation to (ξ,ζ ) coordinate system

(c) Lp corrected pattern

(d) Indexing of equatorial line reflections

Fig. 1.130 X-ray diffraction pattern of uniaxially oriented PE measured with a flat imaging plate camera, b the pattern transformed to the cylindrical reciprocal lattice, c the pattern b with the Lp correction, d indexing of the equatorial line reflections

1.13 Examples of Actual Crystal Structure Analysis

193

shows a set of the circles with the radii of the ξ values of the equatorial diffraction spots. The detection of the crossing points between the circles and the reciprocal lattices was made automatically, from which the reciprocal lattice parameters (a*, b*, c*, and so on) were determined. By investigating the indices of the observed peaks, the extinction rules (actually, the appearance rules of reflections) were derived as follows: hkl : h00 :

no rule h = 2n

0kl : k + l = 2n 0k0 : k = 2n

h0l : h = 2n 00l : l = 2n

These rules suggest us the possibilities of the space group Pnam or Pna21 . As already mentioned in the previous sections, the two planar-zigzag chains were packed in the cell but the orientation and position of the CH2 units in the unit cell were not known at this stage. The next job was to estimate the observed structure factors from the observed reflection intensities. The reflections were overlapped in some places and so the separation of the overlapped reflections was needed. This was performed using the previously mentioned curve separation method (see Sect. 1.8.6). In the actual program, the 1D profile was separated into the several peak components and the integrated intensities were evaluated (see Fig. 1.131). One important measure for the accuracy of the intensity evaluation is to calculate the so-called Rmerge , which is a measure of the intensity equivalence of the separately observed same reflection spots

Fig. 1.131 Integration of the observed reflections by combining with the curve separation method (Reprinted from Ref. [97] with permission of Elsevier, 1999)

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1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

observed in the different plates (or different positions of one plate). For example, the 110 reflections are detected in the several frames (or positions). The integrated intensity should be the same for these equivalent reflections. The Rmerge is defined in the following equation:   2 Rmerge = j j (Iobs,j − < Iobs,j >)2 /i Iobs,i /N

(1.274)

where I obs,j is the integrated intensity of the j-th reflection. The ’ covers the same reflections. < > is their average. The first j is made for all the observed reflections j. The Rmerge should be as low as possible, ca. 0.05–0.03 for the ideal low-molecularweight compound single crystal with sharp and separated reflection spots. For the polymer materials, Rmerge is difficult to estimate but it is in the range of 0.05–0.1. In the present example, Rmerge was 5%, indicating the reasonableness of the intensity evaluation method. The structure factors (absolute values) were evaluated from the equation I obs = KLpAm|F obs |2 .

Since the abovementioned convenient software for the data analysis of 2D X-ray diffraction pattern of a polymer substance is not very popular, we have to obtain the information of the (x, y) position and integrated intensity of all the observed diffraction spots by hands. The method is not very difficult as long as we have the several programs for the treatment of the 2D-digitized pattern and the 1D curve separation. (i)

(ii)

(iii)

(iv)

The integration of the 2D diffraction data is made along each layer line to obtain the 1D diffraction profiles. If the orientation is high, a rectangular box is used as illustrated in Fig. 1.132. Since the diffraction arcs in a higher angle region are longer, in general, the integration using a fan shape is better to get the whole area of the diffraction spot. The thus-obtained 1D profiles are separated into the peak components using a curve separation software. The peak position and integrated intensity can be evaluated as a result. The position coordinates (x, y) are transformed to the (ξ, ζ ) coordinates as already mentioned in Sect. 1.9.1, from which the indexing of the observed peaks and the estimation of the unit cell parameters can be made. The integrated intensities are converted to the structure factors after the correction of the various factors as mentioned above.

1.13 Examples of Actual Crystal Structure Analysis

(a)

(c)

195

(b)

(d)

Fig. 1.132 a Integration of the peaks using a rectangular box along the equatorial line of the 2D X-ray diffraction pattern of an oriented PE sample to obtain the 1D diffraction profile shown in (c). b the integration using a fan shape. d The curve separation to estimate the peak position and the integrated intensity using the 1D diffraction profile (c)

1.13.1.2

Structure Determination

The direct method was applied to the thus-collected |F obs | data. In the present example of HDPE, the total number of the observed reflections is 32, much larger than the total number of the adjustable parameters. If only C atom is taken into account in the analysis, the variables are four: the fractional coordinates (x, y), the isotropic thermal parameter B, and the scale factor K. A result of the direct method was already shown in Fig. 1.36. The position of C atoms can be detected clearly. Starting from this structure model, the refinement was made so that the R factor became minimal. The final result is R = 12% for the 32 observed reflections. In this example, since the total number of the observed diffraction spots was many, even the hydrogen atomic positions were detected by performing the difference Fourier calculation (F obs – F calc ), where F calc is the structure factors calculated for only the C atoms. The thermal parameters used were anisotropic ones. The finally obtained fractional coordinates are shown in Table 1.16.

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1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Table 1.16 Atomic fractional coordinates of orthorhombic PE crystal (Reproduced from Ref. [21] with permission of Wiley, 1997) x/a

y/b

z/c

U eq /Å2 *

C

0.4604 (13)

0.5640 (17)

0.2500

0.060

H(A)

0.31412

0.53706

0.25000

0.101

H(B)

0.48892

0.78196

0.25000

0.101

Atom

* Equivalent thermal parameters U eq = (4/3)ij U ij (ai· aj ) where ai and aj are the unit cell vectors (i, j = 1, 2, and 3). U ij are the components of the anisotropic thermal parameters (U ij = Bij /8π2 ), not given in this table

1.13.1.3

Temperature Dependence of the Structure

The 2D X-ray diffraction measurement of the oriented HDPE sample was performed also using the synchrotron high-energy X-ray beam of the wavelength 0.328 Å as shown in Fig. 1.133, where the data measured at the different temperatures are compared [97]. Comparing with the Cu-Kα and Mo-Kα cases, the layer-line reflections were detected up to the fifth layer line. The structure analysis was carried out using these data. The direct method gave quite clearly the positions of C and H atoms. The thus-clarified structure parameters are plotted against temperature as shown in Fig. 1.134. The interesting point is that the unit cell parameters, in particular, the a-axial length, the thermal parameters, and the setting angle of the zigzag chain measured from the b-axis, show the deflection point at around 10 °C. The molecular dynamics calculation gave the similar structural change (see Sect. 6.3.4.4). -150oC

25oC

100oC

Fig. 1.133 The 2D X-ray diffraction patterns of ultra-drawn HDPE sample measured at the different temperatures at SPring-8 BL04B2 with an incident X-ray beam of 0.328 Å wavelength

1.13 Examples of Actual Crystal Structure Analysis

197

φ: Setting angle

(a) b

φ

a

(b)

(c)

Fig. 1.134 Temperature dependence of a the unit cell parameters a and b, b the thermal parameters of C and H atoms, and c the setting angle of the planar-zigzag chain plane obtained by the data analysis. (Reprinted from Ref. [97] with permission of Elsevier 1999)

1.13.2 Trigonal Polyoxymethylene 1.13.2.1

Mo-Kα X-ray Beam

POM sample was prepared by the γ-ray-irradiated solid-state polymerization reaction of trioxane single crystal of needle shape [21]. Figure 1.135 shows the 2D diffraction pattern taken with a flat camera using Mo-Kα X-ray beam. The total number of the observed reflections is 72. The Rmerge was about 5%. The indexing of the observed reflections gave the following unit cell parameters: a = b = 4.464 Å, c (fiber period) = 17.389 Å, and γ = 120.0°. The hexagonal-type trigonal crystal system with the space group P32 was considered as a candidate. The application of the direct method was tried using the observed reflections. A deformed helix was obtained when the small number of the observed reflections was used. By increasing the number of the reflections up to 40, the relatively regular helix with (9/5) form was extracted. The chains are packed in the hexagonal-type cell and the refinement was performed by changing the positions of 3 C and 3 O atoms. The full-matrix least squares method was applied with the anisotropic temperature factors taken into account. The R factor is 5.0%. Fig. 1.136 shows the thus-obtained crystal structure of POM trigonal phase.

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1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.135 2D X-ray diffraction pattern of trigonal POM crystal using an incident X-ray beam of Mo-Kα line and a flat imaging plate camera

Fig. 1.136 Crystal structure of trigonal POM derived by analyzing the 2D X-ray diffraction pattern shown in Fig. 1.135. (Reprinted from Ref. [21] with permission of Wiley 1997)

1.13.2.2

Usage of High-Energy Synchrotron X-ray Beam

The high-energy synchrotron X-ray beam of 0.328 Å was used for the same sample [12]. The 2D diffraction pattern was measured with a cylindrical imaging plate camera of 12 cm radius. The γ-ray-polymerized POM sample is in a multi-crystalline state composed of many single crystals with perfect orientation along the chain axis. Therefore, in order to collect the 2D X-ray diffraction pattern of the homogeneously uniaxially oriented sample, the multi-crystalline sample was rotated around the fiber axis during the measurement. Figure 1.137 shows the thus-measured 2D X-ray diffraction pattern at room temperature. As already mentioned (Sect. 1.8.2.5), the short-wavelength X-ray makes the Ewald sphere larger, giving higher chance to detect more number of diffraction spots in a wider diffraction angle region. In fact, the layer lines up to the 28th layer line were detected, and the 428 reflections were observed. The number of unique reflections is 107, where the unique reflection

1.13 Examples of Actual Crystal Structure Analysis

199

(a)

(b)

Fig. 1.137 2D X-ray diffraction pattern of trigonal POM crystal using a high-energy synchrotron X-ray beam and a cylindrical imaging plate detector. (a) The electron density distribution derived from the Fourier-transform of the observed structure factors. (b) The difference Fourier (F o – F c ) calculation to reveal the H atomic positions, where F o and F c are the observed and calculated structure factors, respectively. (Reprinted from Ref. [12] with permission of Springer Nature, 2007)

means the reflection with only one hkl index. (For example, 111 and 1¯ 1¯ 1¯ reflections are not the same reflections. They are different, but symmetrically equivalent “unique” reflections. Sometimes the reflections with the same lattice spacing are accidentally overlapped, the intensity of which is difficult to separate into the components reasonably. The diffraction spots without such an accidental overlapping are named the independent reflections). The trigonal unit cell parameters were estimated in the similar way mentioned above: a = b = 4.464 ± 0.004 Å, c (chain axis) = 17.389 ± 0.002 Å at room temperature The initial model deduced by the direct method was refined using a full-matrix least squares method with anisotropic thermal parameters. The final R factor was 6.9 % (room temperature). The F o – F c gave the positions of the H atoms clearly (Fig. 1.137). So far the (9/5) helical chain conformation was assumed. In the literatures [11, 98], the more complicated conformation was proposed; the (29/16) or even (324/179) helix. These conformations are almost equal to (9/5), but slightly deviated as known from the number of monomeric units per one turn; (1.800/1), (1.813/1), and (1.810/1), respectively. If the (9/5) helix is perfectly homogeneous, the structure factor F(00l) is given as F(00l) ∝ exp(0) + exp(2πil/9) + exp(4πil/9) + exp(6πil/9) + . . . + exp(16πil/9) ∝ 1/[1 − cos(2π l/9)]

(1.275)

200

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.138 a The Weissenberg pattern measured for a series of 00l reflections using a Norman method and b the low-angle region of 2D X-ray diffraction pattern of POM crystal. The horizontal line is along the chain axis. (Reprinted from Ref. [12] with permission of Springer Nature, 2007)

Therefore, the structure factor can be quite large for l = 9, 18,… As seen in Fig. 1.138, the 00l reflections were measured using a Norman’s method. Not only 009, 0018,… reflections expected for the (9/5) helix, the additional reflections were detected. The observed reflections are indexed as 0058 (corresponding to 0018), 0029 (corresponding to 005), and so on. ***************

Figure 1.139 illustrates the Norman’s method. An oriented sample is set perpendicularly to a glass rod, which is set to the goniometer head. The sample is oscillated repeatedly in the range of +170o ~−170o around the glass rod, during which the cylindrical camera (or a Weissenberg camera) is translated along the rotation axis direction at a constant speed. A narrow slit is set between the sample and the camera so that only the 00l reflection spots can pass through it toward the camera. Since the camera is moved during this phenomenon, a series of reflection spots is detected in a line with a tilting angle. The rotation angle 180° corresponds to the translation of 90 mm for a commercial Weissenberg camera. When the radius of the cylindrical camera R is 57.3 mm, the tilting angle of the reflection array line measured from the horizontal axis (or along the translational direction) of the film is 63.4o [= tan−1 (π R/90mm)]. In this way, the reflections of the various 00l lattice planes can be detected at the different positions of the camera. ****************** Another measurement was made by focusing the lower reflections by increasing the camera distance (see Fig. 1.138b). The reflections detected near the beam stopper can be indexed as 001 and 002, which correspond to the lattice spacings 56.0 Å and 28.0 Å, respectively. It is reasonable to conclude that all the 00l reflections can be assigned using a (29/16) helical conformation with the repeating period 56.0 Å (at room temperature). In addition, it must be noted that not only 29, 58,… reflections, which can be predicted for the regularly repeated homogenous helical conformation, but also the reflections of 0016, 0026, 0052, etc. are detected, indicating that the helix

1.13 Examples of Actual Crystal Structure Analysis

201

Fig. 1.139 a The Weissenberg camera and the Norman method. b The 00l reflections measured with the method (a)

does not take a regular form but the monomeric units are more or less irregularly repeated. The chain does not have a symmetry except C 1 . The structure analysis was tried for the (29/16) helical model: the R factor was about 9%, where the 184 unique reflections were used for the refinement. The number of the observed reflections is smaller than the total number of the adjustable parameters 224 [29 monomeric units x 2 atoms (C and O) x 4 variables (x, y, z and isotropic temperature factor) + scale factor)], making the analysis ambiguous. In fact, the refined structure is appreciably deformed from the circle when viewed along the chain axis. In such a sense, the approximate (9/5) helical model may be a limitation of the analysis.

1.13.3 atactic Poly(vinyl Alcohol) atactic Poly(vinyl alcohol) (PVA) is a crystalline polymer in spite of the random arrangement of the side OH groups. This is because the small OH groups do not disturb the good packing of the chains in the unit cells, and the OH…O hydrogen bonds stabilize the packing structure. The crystal structure was proposed based on the X-ray structure analysis [99–102]. The fiber diffraction pattern measured for the uniaxially oriented PVA sample is shown in Fig. 1.140. The rolled sample shows the double orientation, the X-ray diffraction patterns of which were taken along the three mutually perpendicular directions as shown in Fig. 1.141. The indexing of the

202

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

(b) calculated

(a) observed

Fig. 1.140 X-ray diffraction pattern measured for the uniaxially oriented at-PVA sample. A cylindrical IP camera was used. (right) computer simulated. (In the calculation, the isotropic temperature factor 8.0 Å2 , the crystallite size is 90 Å (// a), 90 Å (// b), and 90 Å (// c), and the lattice strain is 0% // a x 0% // b x 0% // c. Reprinted from Ref. [103] with permission of the American Chemical Society, 2020

end

a*

_

210 _

110

a* 200

100

b* 110

010

b*

rolled plane

through

edge

Fig. 1.141 X-ray diffraction patterns measured for the doubly oriented at-PVA sample (water cast film, 150°C hot rolled, annealed). The flat IP camera was used. Reprinted from Ref. [103] with permission of the American Chemical Society, 2020

1.13 Examples of Actual Crystal Structure Analysis

203

observed reflections is easier by analyzing these anisotropic diffraction patterns. The resultant unit cell parameters are as follows: a = 7.81 Å, b = 5.51 Å, c(fiber axis) = 2.55 Å and γ = 93.0◦ As shown in Fig. 1.141, the reflection positions calculated using these lattice parameters agree well with those of the end diffraction pattern measured by irradiating the X-ray beam along the chain axis of the doubly oriented sample, supporting the reasonableness of the unit cell parameters. The observed fiber period indicates that the molecular chain takes an extended planar-zigzag conformation similar to that of polyethylene. The density of PVA samples is about 1.3 g/cm3 , suggesting the existence of the two zigzag chains in the cell. The measurement of the 00l reflections indicated that only 002 reflection was detected, suggesting the existence of 21 screw axis along the c-axis. Here the space group P21 or P21 /m is a candidate. Since a zigzag chain of PVA itself cannot have such a 21 screw axis, different from the PE case, the two chains are considered to be connected by a 21 screw axis. Therefore, the structure determination is to know the orientation and relative position of these two chains in the unit cell once when the rigid zigzag form is assumed. The OH groups are considered to be set to the skeletal chain randomly. The alternately right- and left-set OH groups along the chain axis are an ideal syndiotactic chain, which gives the fiber period of 5.1 Å. The observed repeating period (2.55 Å) indicates the statistically random arrangement of the OH groups, i.e., the OH groups can be assumed to be located at the right and left sides of the zigzag plane at 50% probability. In such a case, we say that the occupancy of the OH group is 0.5. As a possibility, we may assume the space group P21 /m, since the 21 screw axis along the c-axis, and the zigzag chain takes a mirror symmetry in the plane perpendicular to the chain axis. This means only a half of one monomeric unit is enough as an asymmetric unit. But the analysis is made by using the zigzag chain of one monomeric unit taken into account as an asymmetric unit here. The H atoms are ignored in the calculation of the diffraction intensities since the scattering power of H atom is small, compared with those of C and O atoms. Some of the chain setting modes are illustrated in Fig. 1.142, which were reported as the possible models in the literature [99–102]. The diffraction profiles are calculated for these models and compared with the observed data as shown in Fig. 1.143 [103], where the intensity is calculated using the equation of I(hkl) = KALpm|F(hkl)|2 as already explained before. As noticed in these calculated profiles, the equatorial line profile in the lower diffraction angle region is not very sensitive to the change of chain orientation and the profile in the higher angle region may distinguish these structure candidates. The 1st and 2nd layer-line profiles are also sensitive to these structures. The conclusion is that the face-to-face packing mode of the two chains along the diagonal direction gives the best agreement between the observed and calculated diffraction profiles for all the 0th, 1st, and 2nd layer lines. As will be discussed in a later section (neutron), the thus X-ray-analyzed crystal structure model was found, unfortunately, not to reproduce the wide-angle neutron diffraction data very well. The reason comes from the several important factors: (i) the X-ray diffraction data is not enough to determine the structure uniquely and

204

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

(a) a

a γ

b b a γ

b

c

c

b

a’

(b)

(c) a

a

b

b

Fig. 1.142 The chain packing modes in the at-PVA crystal lattice. In the case (a), the direction of the zigzag planes in the diagonal axis must be noted (o and x). Reprinted from Ref [103] with permission of the American Chemical Society, 2020

1.13 Examples of Actual Crystal Structure Analysis

205

Fig. 1.143 X-ray diffraction profiles observed for a uniaxially oriented PVA-h sample (solid lines) in comparison with the profiles calculated for the structure models shown in Fig. 1.142 (a), (b), and (c), respectively. However, the observed and calculated neutron equatorial line profiles shown in the right bottom are not in good agreement with each other even for the model of Fig. 1.142 (a). Reprinted from Ref [103] with permission of the American Chemical Society, 2020

206

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

accurately at the final stage because of the small number of the relatively broad diffraction peaks, (ii) the neutron diffraction watches the crystal structure from the different viewpoint because the atomic scattering factors are different between the Xray and neutron scattering cases. In such a lucky case, as poly(L-lactic acid) α form [73], the synchrotron high-energy X-ray beam gave about 700 diffraction spots, making it possible to perform a unique structure determination. The thus-derived crystal structure did not have any ambiguity and could be transferred reasonably to the neutron data analysis. Therefore, for a polymer substance, in particular, it is more ideal to use the hybridized data of the different types of diffraction data (X-ray, electron, and neutron). The concrete analysis of at-PVA by the combination of X-ray and neutron diffraction data will be explained in a later Sect. 2.3.3.

1.13.4 Giant Single Crystal of Polymer Quite limited but important case is the crystal structure analysis of a giant single crystal of polymer. The most typical case is seen for the photo-induced solid-state polymerization reaction of monomer single crystal. As shown in Fig. 1.144, a single crystal of diacetylene monomer, grown as a needle crystal from the toluene solution, is subjected to an irradiation of γ -ray for several tens hours to obtain the giant single crystal of polydiacetylene [65, 104, 105]. The shape and size of the single crystal are almost kept unchanged before and after the polymerization reaction. The side group R is changeable. In the present example, R is a carbazolyl group. It is a good chance

C

R

H

C C

R

R

R

R

R

R

C C

R

R

C C C

0

1

2

3 cm

C

R

R

C C C

R= CH2 N

C

R

R

CH

Fig. 1.144 The photo-induced solid-state polymerization reaction of DCHD monomer single crystal. The size of the thus-produced polymer single crystal is almost the same as that of the initial monomer crystal

1.13 Examples of Actual Crystal Structure Analysis

(a)

207

(b)

goniometer head

Fig. 1.145 a Setting of a single crystal on a glass capillary (see Fig. 1.146), b the sample shape, which should be covered totally by the irradiated X-ray beam in an ideal case. If not, the effective volume is different depending on the sample direction

now for us to know the concrete process how to collect the data and how to analyze the crystal structure using a single crystal.

1.13.4.1

How to Set a Single Crystal

Since the X-ray beam is absorbed by a sample, we have to avoid a usage of too large crystal. Ideally, the crystal size is smaller than the X-ray beam width so that the whole volume of the sample is exposed to the X-ray beam. The usage of a crystal of the anisotropic sizes should be avoided since the effective scattering volume is different depending on the direction of the sample (Fig. 1.145). The best sample is of a spherical shape, which can be prepared by rolling a crystal on a sheet of sand paper with a fine surface gently and for a long time. In the present case, the sample is rod and cannot be cut into a small piece, because it is easily split to fibrils. We should prepare a small monomer single crystal at first before the γ-ray irradiation. If we have to use the anisotropic sample, the absorption correction must be made necessarily in the evaluation of the observed structure factors. A single crystal is mounted on a thin glass rod using an adhesive, as illustrated in Fig. 1.145. This glass rod is set to a metal rod, which is fixed on a goniometer head. If the crystal is unstable by some reason, it can be input into a glass capillary filled with an inert solvent or any other medium. The type of adhesive is important. We have to choose an amorphous adhesive after being solidified. Some adhesives give the X-ray scatterings from the crystalline parts. Different from the oriented polymer sample, the orientation of the sample is arbitrary. The important point is a centering of the crystal: the center of the sample must not be shifted during the rotation of the goniometer head (refer to Fig. 1.55).

208

1.13.4.2

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

How to Collect the Diffraction Data

The collection of the diffraction data is made with an oscillation method. As already explained, the observation of a diffraction spot is equivalent to find the crossing point of the corresponding reciprocal lattice point with the Ewald sphere. However, the probability for the reciprocal lattice point to cross the Ewald sphere is not very high. The oscillation method is used for the single crystal to increase the crossing chances of the reciprocal lattice points with the Ewald sphere, as illustrated in Fig. 1.146a. Instead of oscillation of the reciprocal lattice, the rotation of the Ewald sphere around the center of the reciprocal lattice is easier to understand the situation [(b)]. The suitable range of oscillation angle φ (or ω) is 1–5°. The wider oscillation width causes the overlap of many reflections, making the indexing difficult. The wavelength of an incident X-ray beam was Mo-Kα line with 0.7107 Å, which is better than Cu-Kα with 1.5418 Å wavelength because more number of reflections can be collected. The 2D detector is now popular, compared with the conventional four-circle diffractometer. One shot of the thus-detected 2D diffraction pattern is shown in Fig. 1.147. The orientation of the crystal is changed by setting the χ and φ angles. At the first stage, the 2–3 shots are collected for the different sample orientation of φ = 0, 45, and 90°. The search of the spot positions can be made automatically during (or after) the continuous measurement of the diffraction data (in the present case, the software AutoRun (Rigaku) was used, where the spots with the intensity higher than 5σ ~ 10σ are picked up (σ is the standard error of intensity)). The thus-collected data give us the several tens reflections in total. The (x, y) coordinates of the reflections and the orientation angles ω, χ, and φ are recorded, from which the orientation matrix (UB), the possible candidates of the unit cells, and the indices (h, k, l) are listed up. The diffractometer coordinate system fixed on the crystal is related to the reciprocal lattice coordinate using UB matrix. p∗ = ha∗ + kb∗ + lc∗ = xex + yey + zez 

(1.276)

⎛ ⎞ ⎛ ⎞ x  ∗ ∗ ∗ h ex , ey , ez ⎝ y ⎠ = a , b , c ⎝ k ⎠ z l

or ⎛ ⎞ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ x h h h a∗ · ex b∗ · ex c∗ · ex   −1 ∗ ∗ ∗ ⎜ ⎟ ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎜ a , b , c ⎝ k ⎠ = ⎝ a∗ · ey b∗ · ey c∗ · ey ⎠⎝ k ⎠ = UB⎝ k ⎠ ⎝ y ⎠ = ex , ey , ez a∗ · ex b∗ · ex c∗ · ex z l l l

(1.277) This UB matrix consists of the two matrices, U and B. The matrix B relates the orthogonal coordinate system of the crystal to the oblique coordinate system defined by a reciprocal unit cell parameters.

1.13 Examples of Actual Crystal Structure Analysis

(a)

209

(b)

Δφ

(c)

Δφ

χ = 0, 45o ω = 0 – 180o Δω = 2 ~ 5o

ω

Shots 50 ~ 300

φ χ

ω Fig. 1.146 a Oscillation method of X-ray diffraction spots, b the oscillation of Ewald sphere, equivalent to (a), and c the setting of the sample on a goniometer head

Fig. 1.147 X-ray diffraction pattern of a PDCHD single crystal. The blue circles are the results of the peak search

210

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

⎛

⎞ c* cos β ∗ a* b* cos γ ∗ B = ⎝ 0 b* sin γ ∗ −c* sin β ∗ cos α ⎠ 0 0 1/c

(1.278)

where z // c and x ⊥ bc-plane. The a*, b*, β *, and γ * are of the reciprocal lattice and c and α are of the real lattice. The matrix U relates the orthogonal diffractometer coordinate system to the orthogonal crystal coordinate system through the rotation matrices R. U = Rx Ry Rz

(1.279)

The plausible candidates of the unit cell parameters and the Bravais lattices are given as follows: a

b

c

α

β

γ

volume probability

The unit cell parameters with the highest probability are chosen. In the present case, the most possible unit cell is as follows: a = 12.873 Å, b = 4.893 Å, c = 17.364 Å, β = 108.390◦ (probability 97.6 %) Once the unit cell parameters and UB matrices are estimated, then the measurement of all the possible reflections is performed by changing the angles ω, χ , and φ. If we want to measure the diffraction data with an oscillation angle ω = 5o in the range of ω = 0 – 180°, then we may collect 36 shots (= 180°/5°) of the diffraction images in total. On the basis of the UB matrix and unit cell parameters, the positions of all the possible reflections are predicted (see Fig. 1.148). We can judge whether the selected unit cell parameters and UB matrix are reasonable or not by comparing the predicted (green spots) and actually observed reflection spots. The integrated intensity is evaluated for each reflection enclosed by a green circle. The intensity is corrected for the L, p, A, and K and then the structure factors |F(hkl)obs | are estimated. In this process, the unit cell parameters are refined at the same time. The Rmerge was 0.026. Table 1.17 shows the table of |F(hkl)obs |2 , from which the extinction (or appearance) rules of the reflections are derived and the possible space groups are picked up. The space group of PDCHD is P21 /c. The refined monoclinic unit cell parameters are as follows:

1.13 Examples of Actual Crystal Structure Analysis

211

Fig. 1.148 Integration of diffraction spots. The green circles indicate the positions of the spots predicted from the UB matrix. Yellow circles are the spots partially crossing the Ewald sphere

Table 1.17 Indices h, k, l, structure factors |F(hkl)obs |2 and σ (F(hkl)) of PDCHD single crystal 2

0

-1

6.42

3.11

2

0

0

164382.80

3437.43

1

0

2

87160.14

1575.05

2

0

-2

24323.39

349.07

1

0

-3

2.73

3.06

0

0

3

-0.07

2.74

2

0

1

4.54

3.92

2

0

-3

2.36

4.40

0

1

0

2.41

1.97

0

1

1

574.30

11.93

1

1

0

29370.50

425.78

1

1

-1

24403.68

345.72

1

0

3

3.31

4.54

1

0

-4

150036.00

3142.73

2

0

2

359840.56

9373.08

3

0

-1

1.21

7.31

1

1

1

332704.62

8347.95

3

0

-2

176851.14

3871.05

……………

……………

a = 12.8542(4) Å, b = 4.88730(10) Å, c = 17.3408(6) Å, β = 108.3673(13)o where the numerical figure in parenthesis indicates the error, for example, 12.8542(4) = 12.8542 ± 0.0004.

1.13.4.3

How to Determine the Crystal Structure

The initial structure model necessary for the structure determination is obtained using the direct method (SHELX program), where the information about the number of constituent atoms is input as well as the data shown in Table 1.17. One of the results is shown in Fig. 1.149. C and N atoms are assigned reasonably. In this example, even the H atoms are also picked up already.

212

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.149 Initial structure model of PDCHD chain obtained by the direct method. One asymmetric unit is shown here

Fig. 1.150 Crystal structure of PDCHD after the refinement

This initial model is refined so that the R factor becomes the lowest. Using the software SHELX, the refinement was performed with a full-matrix least squares method and the anisotropic temperature factors were introduced for C, N, and H atoms. The finally obtained crystal structure is shown in Fig. 1.150. The R factor was 0.030 for the 1764 reflections used for the structure refinement. The atomic coordinates and anisotropic thermal parameters as well as the geometries of the polymer chain are recorded as a CIF (crystallographic information framework) file. (The program SHELX is a great software for the crystal structure analysis of a single crystal, which was developed by Dr. G. M. Sheldrick about a half century ago, since

1.13 Examples of Actual Crystal Structure Analysis

213

Fig. 1.151 The result of Fo – Fc calculation (X–X method). The peaks detected in the contour map should be the maximal positions of the bonded electron density. The peaks around the CC triple bond is abnormal. The correct result is shown in Fig. 2.20 as obtained by the X-N method [104]

then the contents and functions have been improved step by step. This software can be used by academic researchers with no charge after getting a license.) The Fourier transform of the structure factors calculated using the atomic coordinates obtained by the crystal structure analysis gives the electron distribution map around the atoms. Similarly the Fourier transform of the observed structure factors can be transformed to the observed electron density. The subtraction between these two electron density distributions gives us the information about the electron density distribution between the atomic peaks. More exactly speaking, the electron density information between the neighboring bonded atoms or the bonded electron density distribution can be deduced. But, sometimes the thus-obtained electron density subtraction gives the curious result. As shown in Fig. 1.151, the electron density does not show the maxima at around the center positions along the CC bonds but some small peaks are detected out of the bond axis. This comes from the atomic coordinates obtained by the X-ray analysis which are not the positions of the atomic nuclei but the electron density maxima. The details will be mentioned in the neutron section.

1.13.4.4

Comparison of X-ray Diffraction Pattern Between the Single-Crystal and the Semicrystalline Sample

It is a good timing to notice that the structure analysis of a giant polymer single crystal is somewhat different from that of the general synthetic polymers. In the latter case, the total number of the observed reflections is quite low compared with

214

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.152 The fully rotated X-ray diffraction patterns of PEMU: a the single crystal, b the meltquenched, stretched, and annealed sample, c the equatorial diffraction profiles obtained from the 2D patterns. (Reprinted from Ref. [108] with permission of Wiley, 2003)

that of the single crystal. One good example to show this difference is the case of poly(cis,cis-diethylmuconate) (PEMU). This polymer is prepared by the photoinduced solid-state polymerization reaction of the corresponding monomer single crystal [48, 106, 107]. The structure analysis of this giant single crystal was performed with the same process as that mentioned above for PDCHD. On the other hand, different from PDCHD, a giant single-crystal PEMU can be melted and cooled to the room temperature to give the semicrystalline sample. As shown in Fig. 1.152, the X-ray diffraction profile is quite different in the reflection width between the single-crystal and the melt-cooled sample of PEMU, although the profile itself (and the 2D diffraction pattern) is more or less similar to each other [108]. The difficulty in the crystal structure analysis of the semicrystalline polymer sample comes from such a quite poor diffraction data compared with the sharp diffraction data of the single crystal covering a wide diffraction angle.

1.14 Characteristic Structural Features of Polymer Crystals 1.14.1 Tilting Phenomenon 1.14.1.1

X-Ray Diffraction Pattern of Tilted-Chain Sample

Poly(ethylene terephthalate) (PET) is well known to show the so-called tilting phenomenon where the sample is uniaxially oriented and annealed at a high temperature [85, 109]. The X-ray diffraction pattern shows the displacement of the reflections from the normal layer lines, as shown in Fig. 1.153 [110]. This position shift of the

1.14 Characteristic Structural Features of Polymer Crystals

(a)

215

(b) c

Real Lattice

draw-axis

c*

tilt angle Ф

molecular chain

a*

c*

tilt plane

nhkl

b* b

a

draw-axis

c

a*

b* b

a

Reciprocal Lattice Ewald sphere

draw-axis

c

c c*

Ф

hk1

c*

O*1

O*1

hk0

hk0

O*0

hk1

O*0

b*

b*

a*

a*

2-D WAXD hk1

hk1

hk0

hk0

Fig. 1.153 a Tilting phenomenon, b the rotation of the reciprocal lattice points, and c the Xray diffraction pattern with and without a tilting phenomenon. (Reprinted from Ref. [110] with permission of Elsevier, 2014)

reflection spots can be interpreted by assuming the tilt of the c-axis from the draw axis at a constant angle φ in a particular lattice plane (ht k t l t ). Therefore, the a- and b-axes and so the a*- and b*-axes are shifted from the equatorial plane. All the reciprocal lattice planes are rotated by an angle φ around the tilt axis, which is the

216

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.154 Various parameters to derive the equation showing the shift of reciprocal lattice point due to the tilting phenomenon

straight line connecting the origin and ht k t l t point or the normal to the ht k t l t plane in the real space. As a result, some of the hkl reciprocal lattice points cross the Ewald sphere surface at the positions higher than the original equatorial line and are located upward in the X-ray diffraction pattern. The other hkl points may shift downward. The ht k t l t reflection does not shift at all. If the reciprocal unit cell (a*, b*, c*, α*, β*, and γ *) is known already, the tilt plane and tilt angle may be estimated because the shift of the diffraction spots from the horizontal equatorial and layer lines can be predicted arithmetically. In the actual case, however, the reciprocal lattice cannot be known at the initial stage of structure analysis. The reciprocal unit cell, the tilt angle, and the tilt plane must be speculated and determined by a trial-and-error method so that the observed reflection positions are reproduced as well as possible. The reciprocal lattice system is more convenient for the derivation of equations about the reflection position shifts or the ξ and ζ coordinates of the reciprocal lattice point p*. As shown in Fig. 1.154, the reciprocal lattice vector po * shifts to p* by rotating around the tilting line parallel to the reciprocal lattice vector t*. What we need to get are ξ and ζ after the tilting occurs. The following equations are obtained: ⎞ a∗ p∗o = (h, k, l)⎝ b∗ ⎠ c∗ ⎛

⎞ a∗ t ∗ = (ht , k t , l t )⎝ b∗ ⎠ c∗ ⎛

(1.280)

The vector p∗o is rotated around the tilting axis to give the vector p*. The line p3o , the distance of p∗o measured from the tilting axis at a point A, moves to p3 by keeping the value itself (p3o = p3 ). The angle β shown in Fig. 1.154 is given by

1.14 Characteristic Structural Features of Polymer Crystals

217

 cos(β) = p∗0 · t ∗ / p∗ t ∗

p∗o · t ∗

= hht a∗2 + kk t b∗2 + ll t c∗2    + hk t + kht a∗ · b∗ + kl t + lk t b∗ · c∗ + lht + hl t c∗ · a∗ 1/2  p0∗ = h2 a∗2 + k 2 b∗2 + l 2 c∗2 + 2hka∗ · b∗ + 2klb∗ · c∗ + 2hla∗ · c∗ 1/2  t ∗ = ht2 a∗2 + k t2 b∗2 + l t2 c∗2 + 2ht k t a∗ · b∗ + 2k t l t b∗ · c∗ + 2ht l t a∗ · c∗

(1.281)

p3o = po∗ sin(β)

(1.282)

p1o = p3o cos(φo ) = po∗ sin(β) cos(φo )

(1.283)

The movement of p3o to p3 is equal to the transformation of the coordinate (p1o , ζo ) to the coordinate (p1 , ζ ) by a rotation of φ (the tilting angle). 

p1 ζ



 =

cos(φ) − sin(φ) sin(φ) cos(φ)



p1o ζo

 (1.284)

Using Eq. 1.283 and ζo = lc∗ , then p1 = p1o cos(φ) − ζo sin(φ) = po∗ sin(β) cos(φo ) cos(φ) − ζo sin(φ)

(1.285)

ζ = p1o sin(φ) + ζo cos(φ) = po∗ sin(β) cos(φo ) sin(φ) + ζo cos(φ)

(1.286)

p2 = po∗ cos(β)

(1.287)

ξ 2 = p22 + p12



2 = po∗2 cos2 (β) + po∗ sin(β) cos(φo ) cos(φ) − ζo sin(φ)

(1.288)

Therefore  

2 1/2 ξ = po∗2 cos2 (β) + po∗ sin(β) cos(φo ) cos(φ) − lc∗ sin(φ) ζ = po∗ sin(β) cos(φo ) sin(φ) + lc∗ cos(φ)

(1.289) (1.290)

218

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

If we know already the unit cell parameters and the indices of the observed reflections (for example, for the non-tilted sample), the angle β can be calculated by assuming the tilt axis (or the corresponding reflection), and then the coordinate (ξ, ζ ) can be estimated by giving a tilting angle φ . For the reflections positioned originally on the equatorial line, φo = 0 and l = 0. Then, the reflection positions can be predicted using Eqs. 1.289 and 1.290. For the layer-line reflections, the equation p3◦ sin(φo ) = ζo gives the angle φo as long as the angle β is known. The thus-predicted shift of the reflections must be compared with the actually observed pattern. The reflection positions on the camera screen can be predicted using the following equations. For the cylindrical camera, the (x, y) coordinates of the reflection are expressed as follows:   1/2  x = R cos−1 2 − ξ 2 − ζ 2 /2 1 − ζ 2

(1.291)

 1/2 y = Rζ / 1 − ζ 2

(1.292)

where R is a radius of the camera. If the flat camera is used at a distance D from the sample, the (x, y) coordinates are expressed as follows: 1/2    2 

1/2 = D 1/ cos2 (Φ) − 1 = D tan(Φ) x = D 4 1 − ξ2 / 2 − ξ2 − ζ2 − 1 (1.293) where

   1/2  cos(Φ) = 2 − ξ 2 − ζ 2 / 2 1 − ζ 2

 y = 2Dζ / 2 − ξ 2 − ζ 2

(1.294)

However, for the polymer crystal without any knowledge of the unit cell parameters and so on, what should we do? If the sample of low degree of chain orientation is used for the X-ray diffraction measurement, the reflections distribute along the arcs just likely the case of poorly oriented sample. If such a sample is not used, the sample can be set with the drawing axis tilted from the rotation axis and the diffraction pattern is measured to get the similarly poorly oriented sample. The centers of the observed reflections are read and the indexing is tried in a normal way to get the rough estimation of the candidates of the unit cell parameters. If the original 2D X-ray diffraction pattern is used for the analysis, a horizontal line passing through the averaged center of the observed upward- and downward-shifted reflection spots is estimated and the reflection positions on the central line are read. This process is made for all the layer lines, and the indexing is tried. Using the thus-obtained indexing information containing the unit cell parameters and assuming the tilting direction and tilting

1.14 Characteristic Structural Features of Polymer Crystals

219

angle, the reflection positions are predicted and compared with the observed result. The estimation of the tilting direction is made by carefully searching the reflections which do not deviate very much from the averaged central lines.

1.14.1.2

Example 1 (Vinylidene Fluoride Copolymers)

Vinylidene fluoride-trifluoroethylene (VDF-TrFE) copolymer with VDF molar content of 60–30 % shows the tilting diffraction pattern as seen in Fig. 1.155 [111– 113]. The uniaxially oriented sample of the low-temperature phase (or the ferroelectric phase) shows the non-tilted diffraction pattern when it is annealed under tension. If the sample with free tension is heated up to the temperature above the phase transition point from the low-temperature phase to the high-temperature paraelectric phase and then cooled to the room temperature, the so-called cooled phase is obtained, which shows the tilting pattern (see Fig. 1.155b). The comparison of the diffraction pattern between the low-temperature phase and the cooled phase shows the similarity of the pattern, making the indexing of the observed reflections relatively easy.

Fig. 1.155 X-ray diffraction diagrams measured for VDF-TrFE copolymer sample with VDF 55% content. a the low-temperature phase and b the cooled phase. (Reprinted from Ref. [111] with permission of Elsevier, 1984)

220

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

However, the reflections of the cooled phase are detected at the positions upward and downward from the horizontal equatorial line due to the tilting phenomenon. The apparently long arc of the 1st layer-line reflection is also due to the tilting phenomenon. The tentative unit cell parameters are obtained as follows by referring to those of the low-temperature phase: a = 9.16 Å, b = 5.43 Å, c(fiber axis) = 2.53 Å, α = β = γ = 90.0◦ a∗ = λ/a = 0.168, b∗ = 0.284, c∗ = 0.609, α∗ = β∗ = γ∗ = 90.0◦ where λ = 1.5418 Å was used as the wavelength of the incident X-ray beam. The indexing is made as follows: #1 110, 200 #4 130, 420, 510

#2 020, 310 #5 330, 600

#3 220, 400

One of the #4 reflections dose not shift from the equatorial line. For example, let us assume that the 130 reflection corresponds to a titling line. Then ht = 1, k t = 3, l t = 0,

p∗ (130) = 0.868

The φo = 0o for the equatorial reflections. The angle β is estimated using Eq. 1.281. For example, for the #1 reflections, the vectors are p∗o (200) = 2a∗ = [0.336, 0.0, 0.0], p∗o (110) = a∗ + b∗ = [0.168, 0.284, 0.0], ¯ = a∗ − b∗ = [0.168, −0.284, 0.0], p∗o (110) t ∗ (130) = a∗ + 3b∗ = [0.168, 0.852, 0.0]. And so, p∗o (200) · t ∗ (130) = 0.336 × 0.168 = p∗o (200) · t ∗ (130) cos(β) = 0.336 × 0.868 cos(β) cos(β) = 0.193, and β = 78.9◦ p∗o (110) · t ∗ (130) = 0.168 × 0.168 + 0.284 × 0.852 = p∗o (110) · t ∗ (130) cos(β) = 0.330 × 0.868 cos(β) cos(β) = 0.943, and β = 19.4◦ ¯ · t ∗ (130) = 0.168 × 0.168 − 0.284 × 0.852 = p∗o (110) ¯ · t ∗ (130) cos(β) p∗o (110) = 0.330 × 0.868 cos(β) cos(β) = −0.746, and β = 138.2◦

1.14 Characteristic Structural Features of Polymer Crystals

221

If the tilting angle φ = 18.0° is assumed, the ξ and ζ values are calculated as follows: 200 ξ = 0.320, ζ = 0.102 x = 32.2 mm, y = 10.2 mm 110 ξ = 0.328, ζ = 0.003 x = 33.0 mm, y = 3.40 mm ¯ ξ = 0.328, ζ = 0.003 x = 32.5 mm, y = 6.81mm 110 where the (x, y) is the position on the cylindrical camera with 100 mm radius. Fig. 1.156 compares the observed and calculated results in good correspondence about the positions of the reflections. Similar calculation is made for the meridional 00l reflections, which are split into two as seen in the Weissenberg patterns measured using a Norman method. The tilting of the chain axis occurs in the 130 plane. By rotating the chains by about 18°, the trans-zigzag segmental parts are tilted. Poly(vinylidene fluoride) (PVDF)

Fig. 1.156 a Comparison of the observed X-ray diffraction pattern of the cooled phase of VDF-TrFE copolymer sample with those calculated by assuming the tilting phenomenon: (left) the normal 2D diffraction pattern and (right) the Weissenberg pattern. b shows the actually measured Weissenberg patterns of LT, HT, and Cl phases. (Reprinted from Ref. [111] with permission of Elsevier, 1984)

222

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

¯ and its copolymers can take both of trans and gauche conformations. If the TGTG sequences are generated in the all-trans-zigzag chain, the zigzag segments are tilted and oriented to the 60°-rotated directions. This TG conformational disorder may be related to the 130-direction tilting phenomenon [111].

1.14.1.3

Example 2 (Arylate Polyesters)

PET and its group of polyesters are known to show the tilting phenomenon. The crystal structure analysis was succeeded by taking the upward and downward shifts of the observed reflections into account. Depending on the type of polyester, or even by changing the sample preparation condition, the tilting angle and the tilting direction are affected sensitively [114]. Fig. 1.157 shows the case of uniaxially oriented and

Fig. 1.157 X-ray diffraction pattern of 10GT compared with that calculated by taking the tilting phenomenon into account. b The crystal structure of 10GT. (Reprinted from Ref. [110] with permission of Elsevier, 2014)

1.14 Characteristic Structural Features of Polymer Crystals

223

annealed poly(decamethylene terephthalate) (10GT, -[OCO−−COO−(CH2 )10 ]n -) sample [110]. The shift of the observed reflections is quite small but can be detected clearly. The indexing of the observed reflections and the estimation of the unit cell parameters were at first performed by ignoring the tilting phenomenon and then refined so that the (x, y) positions of all the observed reflections were reproduced well by considering the tilting phenomenon. In this case, the crystal belongs to the triclinic system with the parameters a = 4.77 ± 0.01 Å, b = 5.66 ± 0.01 Å, c (f.a.) = 20.7 ± 0.1 Å, α = 108.2 ± 0.1o , β = 122.7 ± 0.2°, and γ = 97.1 ± 0.1°. The ¯ plane by about 1° from the draw direction. The comparison tilting occurs in the 210 of the 2D X-ray diffraction pattern is shown in Fig. 1.157. The chain conformation is almost fully extended. The tilting phenomenon gradually becomes ambiguous as the methylene segmental length becomes longer. For example, 20GT does not show the tilting phenomenon. The polyester with shorter methylene segment shows clearer tilting phenomenon, typically seen for PET sample, suggesting the important role of terephthalate group part. The as-drawn polyester takes the repeating structure of the crystalline and amorphous phases with some tensile strain along the draw direction. As illustrated in Fig. 1.158, the heat treatment causes the shrinkage of the structure and the crystal lattice is slightly tilted due to the generation of shear stress. In the case of VDF-TrFE copolymer, the tilting phenomenon is related to the TG conformational disorder. The polyester case may be also related to the change of methylene segmental conformation. (a)

as-drawn

(b)

annealed

Fig. 1.158 An illustration of the tilting phenomenon of crystallites caused by the thermal relaxation of stressed lamellar structure

224

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

1.15 Diffraction and Structure Disorder 1.15.1 Thermally Induced Disordering In the most ideal case, the molecules in the crystal lattice take the regular structure and the packing mode of these molecules are also regular. However, the actual crystal has many kinds of disorder. As already mentioned, the structure factor F is given as Fn =< F > +Fn

(1.295)

where Fn is the deviation of the structure factor from the averaged value . As shown in Sect. 1.4.4, the diffraction intensity can be expressed as ∗ I (k) ∝ | < F > |2 m (Vm /v) exp(−iRm · k) + m (Vm /v) < Fn Fn+m > exp(−iRm · k)

(1.296)

The first term corresponds to the scattering from the perfect crystal with the averaged structure factor , and the second term to the so-called diffuse scattering due to the heterogeneous structure between the unit cells. Atoms are vibrating always even at 0 K (zero point vibrational energy). Such thermal vibration causes the displacement from the equilibrated positions on the lattice. As long as the thermal vibration occurs harmonically, the average position is always at the center of the lattice point. Therefore, the so-called thermal expansion (or thermal contraction) does not occur. If the vibration occurs an-harmonically, as the vibrational amplitude becomes larger at the higher temperature, the average positions of atoms are deviated from the original centers and the thermal expansion occurs. The thermal vibrations are the origin of the thermal parameter u or temperature factor B, as already mentioned. If the crystal is quickly frozen from such a vibrationally disordered state, then the atomic positions are randomly deviated from the original lattice points. This kind of structure disorder is called the first kind of disorder (or distortion). The higher angle diffraction peaks become lower in intensity because of this kind of disorder (refer to Fig. 1.38). In the case of crystalline polymers, the thermal displacement may occur in various ways in the crystal lattice of polymer. The molecular chains are disordered from the regular conformation. Mostly the torsional angles are changed from the regular values. If the conformationally irregular (but not disordered) chains are packed in a regular manner, the crystal structure is not said to be disordered. The X-ray diffraction gives the clear spots. Some of the typical cases are seen for poly(ethylene oxide) (Fig. 1.159) [82] and poly(L-lactic acid) [115].

1.15 Diffraction and Structure Disorder

225

Fig. 1.159 X-ray diffraction diagram and the crystal structure of poly(ethylene oxide). The chain conformation is irregular, but the chain packing itself is ordered. (Reprinted from Reference [82] with permission of the American Chemical Society, 1973)

1.15.1.1

High-Temperature Conformational Disordering

The disordering should be statistically random. This can be realized by the thermal motion of polymer chains in the lattice. One case is the thermal motion of the conformationally flexible chain. Vinylidene fluoride-trifluoroethylene random copolymer shows the ferroelectric-to-paraelectric phase transition at a Curie temperature, where the originally almost planar-zigzag chains change to the conformationally disordered ¯ bonds [14, 111–113]. At the chains of statistical combinations of TT, TG, and TG same time, these gauche-type chains rotate around the chain axis, resulting in the nonpolar crystal lattice of the hexagonal packing system. The X-ray diffraction pattern changes as shown in Fig. 1.160. The repeating period of the low-temperature phase is 2.54 Å, corresponding to the planar-zigzag conformation. In the high-temperature phase, the new layer line is detected between the original equatorial and first layer lines. This shows the existence of the conformer of gauche form with the averaged repeating period of 4.60 Å. The equatorial line is sharp and the peak positions correspond to those of the hexagonal pacing structure. On the other hand, the layer-line diffractions become quite diffuse, showing the lower correlation in the relative height between the neighboring disordered chains.

226

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.160 Xray diffraction pattern of VDF-TrFE copolymer sample measured below and above the Curie transition temperatures [112]. (Reprinted from Refs. [111, 112] with permission of Elsevier, 1981, 1984, respectively)

1.15.1.2

Rotational Motion of n-alkanes

n-alkane molecules are packed in various ways in the crystal lattice. For example, the planar-zigzag chains are packed in the orthorhombic manner as shown in Fig. 1.161. By heating the crystal, the thermal motion of the chains becomes enhanced and the zigzag plane of the chain starts to liberate around the stable position at a small amplitude [116–119]. As the temperature rises furthermore, the amplitude of the librational motion becomes larger and starts to rotate apparently freely at each lattice point. This is called the rotator phase. Strictly speaking, the end parts of the chain are said to change their conformation between trans and gauche forms. But this local disorder of chains is here ignored approximately. The details of the thermal motion can be described from the different points of view. In one case, the planar chains as rigid bodies rotate perfectly freely around the chain axis (Fig. 1.161). Another case is the hopping motion of the rigid zigzag chains: the zigzag plane direction changes discontinuously from one direction to the other direction by every 60°. The motional correlation between the neighboring chains is needed to be considered also. One is the gear-type motion: the neighboring chains rotate in the opposite direction to each other. Another is the random motion. The neighboring chains rotate without any correlation.

1.15 Diffraction and Structure Disorder

227

Fig. 1.161 a Hexagonal packing structure of cylinder. b c Thermal motions of zigzag chains

All these different types of the thermal motion give the apparently hexagonal packing structure of the cylindrical chains in the static X-ray diffraction measurement. The hexagonal peaks observed at the diffraction angles corresponding to √ are √ the lattice spacing of 1, 1/ 3, 1/ 4, … The phase transition from the orthorhombicto-hexagonal lattices occurs in the thermodynamically first-order manner: these two

228

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

phases coexist together at a transition temperature, resulting in the coexistence of the two different diffraction patterns. When the 2D diffraction pattern is measured in the rotator phase of polyethylene, the fiber period might be shorter because of the conformation disordering from alltrans form to the trans-gauche disordered conformation. The n-alkane chains are considered to rotate with the zigzag conformation kept almost unchanged except the end parts.

1.15.1.3

Hexagonal Phase of Polyethylene

Polyethylene shows the orthorhombic-to-hexagonal transition in a high temperature near the melting point. This phenomenon can be seen more easily by constraining the oriented sample at a constant length, where the ultra-drawn high-molecular-weight polyethylene is better [120]. As shown in Fig. 1.162, the DSC thermogram shows the two peaks: the lower endothermic peak corresponds to the transition point and the higher endothermic peak is the melting of the hexagonal crystal. The X-ray peaks of the orthorhombic and hexagonal phases overlap in the transition region (Figs. 1.162 and 1.163). Polyethylene chains in the hexagonal phase change the conformation from the fully extended all-trans conformation to the irregular form consisting of the relatively long trans segments and the gauche-type bonds (TG, GG, etc.), as revealed clearly by the observation of vibrational spectral change (Fig. 1.164). The wagging band at around 1360 cm−1 is originated from the disordered gauche form and is not detected in the orthorhombic phase. In the transition region to the hexagonal phase, the gauche bands at 1304, 1352, 1366 cm−1 with the parallel polarization increased in intensity. The totally free rotational motion is difficult to imagine for polymer chains, but rather the locally twisted conformational disordering is easier to imagine. Therefore, the observation of T-G conformational exchange seems reasonable.

1.15.1.4

Helical Polymers

Another example is seen for polytetrafluoroethylene (PTFE), where the conformational change and rotation motion are coupled [8, 9, 90–92, 121–123]. As shown in Fig. 1.165, the X-ray diffraction pattern at low temperature consists of many clear spots, where these data were measured using an X-ray beam of short wavelength (0.328 Å) in SPring-8. The molecular chain takes the (13/6) conformation. The observation of strong diffractions is on the 6 and 7th layer lines and the strong near-meridional diffraction is on the 13th layer line (see Sect. 1.12.2.3). When the temperature reaches near the room temperature, the X-ray diffraction pattern becomes diffuse on the several layer lines. The strong intensity is observed on the 7 and 8th layer lines and the sharp reflection of the meridional 15th line, indicating the conformational change to (15/7) form. The molecular chains are considered to experience a small angular displacement around the chain axis to give the characteristic pattern. When the chain rotates by an angle φ, then the intensity becomes weaker by a factor

1.15 Diffraction and Structure Disorder

229

Fig. 1.162 a How to bind the oriented PE sample around a metal holder, b the DSC thermograms measured under tension: (upper) oriented gel film and (lower) ultra-drawn fiber (Dyneema), and c the temperature dependence of X-ray diffraction pattern [120]. (Reprinted from Ref. [120] with permission of the American Chemical Society, 1996)

exp[-n2 ] where n is an order of Bessel function and is the averaged rotation angle of the helix. The layer lines except of low n (= 0, 6, 7, 8 and 15) become weaker due to this situation. The diffraction pattern measured at a higher temperature shows the diffuse layer lines and the originally strong 7th and 8th layer lines are merged into apparently one spot. As proposed by the vibrational spectral data analysis, the local contribution of the planar-zigzag or (2/1) helical conformation becomes higher to give the diffraction pattern similar to that of the fully rotating zigzag chain conformation of hexagonal-type polyethylene. These trans-zigzag segments are said to exist as a boundary between the left-handed and right-handed helical chain segments and shuttle along the chain axis just like a soliton motion [124].

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1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.163 Typical X-ray diffraction peaks measured for the orthorhombic, triclinic, and hexagonal phases of polyethylene crystals

Fig. 1.164 Polarized infrared spectral change in the heating process measured for the oriented PE film. The oriented gauche bands appeared before the melting. (Reprinted from Ref. [120] with permission of the American Chemical Society, 1996)

1.15 Diffraction and Structure Disorder 173 K

231 293 K

423 K

Fig. 1.165 Temperature dependence of X-ray diffraction diagrams measured for a ultra-drawn PTFE sample, where the measurement was performed using a high-energy synchrotron X-ray beam of λ = 0.328 Å at SPring-8, Japan

The similar analysis was made for the thermal motion of POM [125]. The integrated intensity of the Bragg reflections such as 208, 215, and 213 started to decrease the value at around −110 °C. The logarithm of the intensity [ln(I/Io), Io the intensity at −140 °C] was plotted against the order n2 of the Bessel function where n was derived for the various layer lines. These plots gave the straight lines, the slopes of which corresponded to the squared amplitude . The started to increase at around −110 °C, indicating the start of the rotational motion of POM chains.

1.15.2 Disorder in Relative Height of Chains In the above section, the molecular rotation was considered as one factor of the structural disordering. But the rotation does not occur solely but it may be coupled with the translational motion along the chain direction. Such translational motion of chains causes the disorder in the relative height between the neighboring chains. This type of disorder gives the streak lines along the layer lines. The extreme case is seen for an isolated chain which does not have any relative height correlation with the neighboring chains. The X-ray diffraction pattern predicted for such an isolated chain consists of a series of continuous horizontal lines with the spacing corresponding to the repeating period as illustrated in Fig. 1.166, which is derived as the Fourier transform of the isolated chain with a repeating period I. If the molecular chains in the crystal lattice give such streak lines along the layer lines, we may say that these molecular chains are aggregated together to form the crystal lattice but they are more or less random in their relative height. Only the equatorial line shows the spots with the spaces corresponding to the distances between the chains. The layer lines are streaky. The appearance of these streak lines depends on the situation of the molecular chains in the crystal lattice. To know the systematic relation between the X-ray streak patterns and the structural disorder of polymer chains, the various types of

232

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

(b) random height packing of chains

(a) an isolated chain and

and the diffraction pattern

the diffraction pattern

l=2

l=2 I

1/I

l=1 l=0

1/I

l=1

I 1/a

l=0

a Fig. 1.166 X-ray diffraction pattern of a an isolated chain and b the chain packing with the disordering in the relative height along the c-axis

translational disorder model are constructed and the corresponding X-ray diffraction profiles are calculated, as shown in Fig. 1.167. For this purpose, one large unit cell is built up and the 25 chains of (4/1) helical conformation are packed into the cell, as an example [115]. The arbitrarily chosen 12 chains are shifted along the chain axis in the various ways. For example, if these selected chains are shifted randomly but with the constant shift by c/2, the reflections on the even-numbered layer lines are spot-like but the odd-numbered layer lines are streaky. When these randomly chosen 12 chains are shifted by c/4 along the chain axis, the 0, 4, and 8th layer lines are spot-like but the other layer lines consist of the overlap of spots and continuous distribution of intensity. For the shift of c/3, the 0, 3, 6, and 9th layer lines are spots and the others become streaky. In this way, if the translational shift is made for randomly chosen chains by the fixed value (c/m), the layer lines of m, 2m, 3m, … are spot-like but the other layer lines are streaky. If all the chains in the unit cell are randomly shifted along the chain axis [case (e)], it gives streaky patterns along all the layer lines. The whole diffraction pattern is similar to that of an isolated (4/1) helical chain [(g)]. If the several chains are packed together to form a finite domain and these domains are gathered together to form a crystallite with some statistically random translational shift along the c-axis, then the X-ray diffraction pattern of (f) is obtained. The X-ray diffraction pattern is similar to that of the regular packing structure as a whole, but some layer lines are strong and diffuse, corresponding to those of the isolated chain. (h) is the case of an isolated domain consisting of 16 chains only. The X-ray pattern is similar to that of the original regular pattern but with some diffuse lines. As the domain size becomes larger, the X-ray diffraction pattern becomes sharper. Another type of disorder is about the rotational orientation of the chains. If the (4/1) helical chains are oriented randomly around the chain axis with arbitrary angles, all the layer lines are streaky in addition to the spot-like Bragg reflections because of the slight difference of the relative height due to the chain rotation, but the degree of streak is not very significant compared with the above cases.

1.15 Diffraction and Structure Disorder

233

Fig. 1.167 X-ray diffraction patterns predicted for the packing structures of 4/1 helical chains. (Reprinted from Ref. [115] with permission of Elsevier, 2011)

We may summarize these simulation results as follows [115]: Type (I)

Type (II)

Type (III)

Higher degree of random translational shift along the chain axis gives the X-ray diffraction pattern similar to that of an isolated chain except the spot-like equatorial line. The streaks along the layer lines become remarkable. Different from the type (I), the upward/downward random shift of chains by a fixed value (c/m) results in the spots on the layer lines of m, 2m, 3m, etc. and the streaks on the other layer lines. The disorder in relative height between the neighboring domains gives also the streak layer lines, but the X-ray diffraction pattern of the basic unit cell structure is kept unaltered seriously.

234

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

+ _ 0.22c

Fig. 1.168 X-ray diffraction pattern and crystal structure of poly-p-phenylene benzobisoxazole (left: Reprinted from Ref. [130] with permission of the American Chemical Society, 1998, right: Reprinted from Ref. [131] with permission of Wiley, 2001)

These types are seen for the actual examples. The type (I) is seen for the uniaxially oriented poly(β-propiolactone) [126]. All the layer lines are streaky except for the spot-like equatorial line, just likely the case seen in Fig. 1.166b. The remarkable case is seen for poly(vinyl alcohol)-iodine complex, where the diffraction pattern is almost determined by the iodine (I3 ) species because of the overwhelmingly many number of electrons originating from I atoms. But the layer lines coming from the one-dimensional array of I-3 ions are strongly streaky because of the relative height disorder of these one-dimensional iodine columns [127]. Poly-p-phenylene benzobisoxazole shows the characteristic X-ray fiber pattern (Fig. 1.168), in which some of the layer lines include both of the spot-like reflections and the continuous intensity distribution, but the other layer lines are streak [128– 131]. This characteristic pattern was interpreted reasonably in such a way that the neighboring chains are shifted by a fixed value (0.22c) along the chain axis but the plus/minus direction of shift is random [131]. This type of disorder was called the registered disorder, and it corresponds to the abovementioned type (II). The relative height shift of domains is seen for poly(L-lactic acid) α and δ forms [73, 115] and the ethylene-tetrafluoroethylene alternate copolymer [132]. The X-ray crystal structure analysis was made successfully with the low R factor. However, the streak lines are observed for them and the relative intensity of the observed 00l reflections cannot be interpreted reasonably using the thus-analyzed crystal structures. The problem was speculated to originate from the relative height disorder being discussed here. By introducing the perfectly random shift into the crystal lattice, the X-ray diffraction profile must be changed totally. But, as mentioned above, the observed

1.15 Diffraction and Structure Disorder

235

Fig. 1.169 The X-ray 00l reflection profile observed for the uniaxially oriented ethylenetetrafluoroethylene alternating copolymer sample in comparison with those calculated for the various kinds of disordered structures. The domain height disorder structure reproduces the observed data well. Reprinted from Ref. [132] with permission of the American Chemical Society, 2011

X-ray diffraction data can be analyzed by the apparently regular chain packing structure. By keeping the diffraction pattern, we need to introduce the disorder to cause the streak lines. The 00l reflection profile is sensitive to the shift disorder of the domains. Very slight shift of domain height gives the large change of the 00l reflection profile and can reproduce the observed 00l pattern. The shift of the domains can keep the whole crystal structure of the unit cell and so the Bragg diffraction pattern. As shown in Fig. 1.169, the slight shift of the adjacent domains reproduces the observed 00l profile quite well.

1.15.3 Packing Disorder of Helical Chains Helical chains have two types of handedness: right-handed (R) and left-handed (L). We have already learned the definition of helical handedness. Figure 1.170 might be good for checking it as an exercise. In addition to the R and L helical definitions,

236

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.170 Helical chains (L and R: left- and right-handed, u and d: upward and downward) Reprinted from Ref. [157] with permission of the American Chemical Society, 2016

we have to know the directionality of the helices. If the side groups orient to the upward direction, we say this chain is an upward helix. The downward helix is defined oppositely. In some cases, we can focus the direction of functional groups of the skeletal chain. For example, in the polymer chain containing CO-O ester groups, we can define the directionality of this chain using the direction of CO→O vectors: the upward helix contains the vectors oriented in upward direction. These four types of the helical chains are packed in various ways into the crystal lattice depending on the space group symmetry. As already mentioned, some space group may give the chain packing mode in which the R and L chains are positioned at the same crystal lattice point at 50% probability (Fig. 1.171). We can say this crystal takes a statistically disordered structure with respect to the R and L helical senses. Similarly, for example, we say that the crystal takes a disordered structure about the upward and downward directions of the R (and/or L) helices. When these disordered structures are viewed locally, of course, the two chains cannot be displaced at the same point at the same time (as long as the doubly stranded helix is not considered). They may be aggregated by forming the domains composed of only the R chains and those of only the L chains at the same ratios. These domains are assumed the optically active domains. But the domains with the opposite activities are collected together in a statistically disordered mode, resulting in the apparently optically inactive crystal. In a case, the R and L chains may be packed in an alternate manner at the neighboring positions of the common crystal lattice (optically inactive domain or racemic domain), but the thus-created regular racemic domains are aggregated together in a different region from each other. In the cases (a) and (b), the X-ray diffractions come from the basic unit cells. In the case (b-1), the unit cell structure consists of only R or L chain stems and so the structures are in the mirror relation. In the case (b-2), the domains give the X-ray diffractions originating from the unit cells of the regularly packed structure but the coherency between the neighboring domains gives the modification of the diffraction

1.15 Diffraction and Structure Disorder

237

(b) Statistically-disordered packing of R and L chains

(a) Regular racemic lattice R

R L

R

R L

R

R

R L

L R L R

R L

R

R L

R

R

R

R

(b-1) Aggregation of optically-active domains of the same handedness

R R

L

R

R

L

R L

R R

L R

L

R

L R

L R

L

R

L

L

R

L

R

R

R

L

L

R

R L

R

R

L R

L

R

R

R L

R/L

L R

R

L R

L

R

R

R

L

R/L

L R

L R

R

R/L R/L

R/L

R

R

R

L L

R R

R L

R

R

R/L R/L

R/L

R/L R/L

(b-2) Aggregation of racemic domains of the opposite helical chain handedness with the phase angle π at the boundaries

R

L

R

R

L L

R

R

R

L

R R

R/L R/L

R/L R/L

R/L

R

R

R/L R/L

R/L

R/L

R/L

R/L

R/L

L

R/L

R/L R/L

R/L R/L

R

R

R/L R/L

R/L

L

L

R/L R/L

R/L

R/L R/L

R/L

R/L

L

L

R/L

R/L R/L

R

R

R

R/L

L

L

L

R

R L

L L

R

Fig. 1.171 Various packing modes of helical chains

pattern or the streak lines depending on the mode of disorder. The diffraction theory of the layer stacking disorder may be applied to the case of (b-1) and (b-2). The probabilities of the arrangement of the neighboring domains (or layers) are taken into account in the calculation of the diffraction patterns (refer to Sect. 1.15.6).

1.15.4 Disorder in Copolymers At first, we assume that the monomeric unit sequence in the crystal lattice is created by arranging the monomeric units A and B without any constraining condition about the arrangement. In other words, the probability of arranging the monomeric units is determined by only the molar content X. The averaged structure factor is given by the weighted average of the structure factors of various structure units. < F >= wi Fi

wi = 1

(1.296)

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1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

I l = | < F > |2 The diffuse scattering is given as I2 =< F 2 > − < F >2

(1.297)

< F 2 >=< ΣΣwi wj Fi Fj > One example is shown here. Figure 1.172 shows the X-ray fiber diagrams of a series of the uniaxially oriented ethylene-tetrafluoroethylene (E-TFE) copolymers with the various E/TFE contents [133]. For E/TFE 50/50 copolymer sample, the layer lines can be counted as the 0th (equatorial), 1st, 2nd, 3rd, and 4th lines. The interlayer spacings give the fiber period of 5.10 Å, indicating that the chain conformation is of the trans-zigzag form. PTFE itself does not take the perfect planar-zigzag conformation (16.88 Å for 13/6 helix and 19.50 Å for 15/7 helix). But the TFE segments included in the copolymer chains are considered to take essentially the planar-zigzag conformation as long as the TFE segmental length is not very long, different from PTFE homopolymer. As the E/TFE ratio changes from 50/50 to 61/39, 35/65, and 29/71, the 1st and 3rd layer lines become weaker and more diffuse. If these weak reflections of the 1st and 3rd layer lines are ignored, then the apparent fiber period becomes ca. 2.55 Å for E/TFE 35/65 and 29/71 samples, almost the same as that of the fiber period of pure PE. In this way, the layer-line reflections change systematically with the E/TFE content, but the 1st and 3rd layer-line reflections become extremely weaker when the E and TFE contents deviate from 0.5. The similar interpretation can be made for ethylene-vinyl alcohol (EVOH) copolymers [134]. As shown in Fig. 1.173, the X-ray fiber pattern changes systematically with the E/VA content. The PVA (VOH100%) sample shows the layer lines corresponding to the planar-zigzag conformation, as mentioned before. The ideal configuration of PVA is syndiotactic or isotactic chain. The syndiotactic segments take the repeating period of 2.55 × 2 = 5.10 Å and the isotactic one is 2.55 Å. However, the OH groups of the syndiotactic configuration jetty to the right and left sides at 50% probability (in a sense, an atactic PVA is assumed to be a kind of copolymer with the different tacticities), and so the PVA chain can be assumed to take the chain structure of -CH2 C(OH)2 - with the 50% occupancy of OH groups, meaning that the average repeating period is 2.55 Å, and the layer line corresponding to the 5.10 Å period does not appear. The random array of comonomer sequences can be observed for all the members of EVOH copolymers. However, the equatorial and first layer-line patterns change gradually depending on the E/VOH ratio. Another important feature of the X-ray diffraction patterns is the horizontal streaks along the layer lines. Since the E and VOH monomeric units are distributed randomly along the chain axis, the relative height of the neighboring chains becomes more or less random, causing the streak lines due to the reduction of the correlation between the neighboring chains.

1.15 Diffraction and Structure Disorder

239

As the VOH content is lower, the relative height disorder causes the isolation of some VOH units, and the intermolecular hydrogen bonds reduce in probability and the OH groups free of the hydrogen bonds increase in number. VDF-TrFE and VDF-TFE random copolymers are another examples to check the effect of random monomeric sequence on the X-ray diffraction pattern [14]. VDF 100% copolymer or PVDF form I is a polar and ferroelectric crystal. As the content of TrFE (and TFE) units increases, the unit cell volume becomes larger because of the coexistence of relatively bulky TrFE (and TFE) units. As a result, the ferroelectric phase transition temperature becomes lower, and the change of X-ray diffraction pattern with temperature can be detected clearly. In this section, however, the details PE

50/50

35/65

29/71

PTFE

H

2.55 Å H H

F+H

H

PE (100/0)

F +H

H

H H

H

E/TFE 61/39

5.10 Å H H+F F H ETFE 50/50

H+F

F

F

F H

F F F F

F +H

F

PTFE (0/100)

Fig. 1.172 X-ray diffraction patterns and repeating period of E-TFE copolymers. Reprinted from Ref. [133] with permission of Elsevier, 2008

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1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.173 X-ray diffraction patterns of EVOH copolymers (EVOH90: the copolymer with VOH 90 ml% content). Reprinted from Ref. [134] with permission of the American Chemical Society, 1999

of the transition behavior are not described. Let us focus on the X-ray diffraction pattern at room temperature, or the pattern of the polar low-temperature phase with the planar-zigzag chain conformation. As seen in Fig. 1.174, where the VDF 37% copolymer case is shown, the diffraction pattern of the highly oriented sample is similar to that of the planar-zigzag chain conformation. When the sample is heated and cooled to the room temperature, the X-ray diffraction pattern becomes diffuse and the equatorial reflections shift upward and downward directions. The tilting phenomenon can be detected also for PTrFE sample. The TrFE parts induce the structure defect which causes the generation of gauche bond easier because of the bulk TrFE units, resulting in the tilting phenomenon (see Sect. 1.14.1.2).

1.15 Diffraction and Structure Disorder

241

Fig. 1.174 X-ray diffraction patterns of VDF-TrFE copolymer (VDF37%) a under free tension, b under tension, and c of PTrFE under free tension. Reprinted from Ref. [113] with permission of Taylor & Francis, 1984

1.15.5 Kink and Streaks PVDF forms I and II show the streaks in the diffraction patterns [135–137]. They are originated from the insertion of conformational defect into the skeletal chains. Figure 1.175 shows the X-ray [135] and electron [137] diffraction patterns of PVDF form II sample with the streak lines, which can be compared with that of the normal sample without this type of streak. The streak is observed along the c*-axis from the 110 reflection. Another streak is between the 002 and hk0 reflections. The VDF monomeric units can take TT and TG conformations. The regular conformational ¯ ¯ sequence TTTT… is that of the form I crystal. The TGTGTGT G… is the conformation of form II. On the basis of the irregular stacking layer theory, the streaks detected for the form II sample are interpreted as the conformational defect of …. ¯ ¯ ¯ TGTGTGT GTTTTTGT GTG…. or the kink structure. Here the layer consists of the ¯ Similarly, the form I crystal shows the streak monomeric units of TT or TG or TG. along the line connecting 001 and 110, 200 reflections, although it is a little difficult to detect since the amorphous halo overlaps with them. This streak is interpreted ¯ similarly using the kink structure of the form … TTTTTTTTGTGTTTTTTTTT… (Fig. 1.176).

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1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.175 a X-ray diffraction pattern (Reprinted from Refs. [135] with permission of the American Chemical Society, 1976). and b electron diffraction pattern of PVDF form II sample with the streak lines along the c*-axis. Reprinted from Ref. [137] with permission of Elsevier, 1981

Fig. 1.176 Kink structures of PVDF a form II and b form I. The kink is generated by the insertion ¯ respectively [135, 137] of TTTT and TGTG,

1.15 Diffraction and Structure Disorder

243

1.15.6 Faults in Stacked Layer Structure In the above example, the theory of stacking layer disorder was used [138, 139]. Let us see more details. The equation to show the diffraction from the irregular structure is given as below (refer to Eq. 1.64). ∗ I (k) ∝ | < F > |2 m (Vm /v) exp(−iRm · k) + m (Vm /v) < Fn Fn+m > exp(−iRm · k)

(1.298)

If the structure in the unit cell is perfectly the same among all the cells, the average ||2 is equal to that of |F|2 and the F is 0. Only the Bragg reflections appear and no diffuse scattering occurs. If the structure in the ab-plane or the layer structure is regular but there are several different types of layer structures and these layers are stacked together at the corresponding probabilities along the c-axis, then we have the disorder of the layer stacking structure, which is called the (one-dimensional) irregular layer lattice (see Fig. 1.177). If the regularity occurs along the a-axis and the repetition of the bc-plane structure is disordered, we have the two-dimensional (a)

2D layer stacking disorder

l

X-ray

c +Δ z

h a

b +Δy (b)

l

-k

1D layer stacking disorder

h c+ Δz

X-ray

-k a b Fig. 1.177 Stacking disorder and the corresponding diffraction pattern. a 2D and b 1D disorder

244

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

irregular layer lattice. The three-dimensional disorder is defined in a similar way. Equation 1.295 may be changed to One dimension I (k) ∝ La Lb {| < F > |2 Lc + Nc [< F 2 > − < F >2 ] Two dimension I (k) ∝ La {| < F > |2 Lb Lc + Nb Nc [< F 2 > − < F >2 ] Three dimension I (k) ∝ | < F > |2 La Lb Lc + Na Nb Nc [< F 2 > − < F >2 ] (1.299) Here L a , L b , and L c are the Laue functions (La = sin2 πNa ξ/ sin2 π ξ, etc. Refer to Eq. 1.101). In the case of three-dimensionally disordered structure, the whole reciprocal lattice shows the continuous intensity distribution. The cross section with the Ewald sphere may give the totally diffuse diffraction pattern although the degree of diffuseness may depend on the degree of disorder. In the case of the two-dimensional disorder, the term L a confines the distribution only on the plane of the index h. The diffuse scattering is observed along the lines generated by the crossing of Ewald sphere with the kl-planes at the h points. The averages and are calculated by knowing the probability of each structure. If the stacking of a layer is affected by the already existing layers, the probability becomes complicated. If there is no influence of the last layer on the new layer, then the probability is ωi , which is the probability of the i-th layer structure. The average can be obtained as = i ωi Fi and < F 2 >= i i ωi ωi Fi Fj . If the stacking of a new layer is affected by the one layer which is already existing, then the probability ωi = j ωj Pji where j Pji = 1. In the abovementioned PVDF cases, the one layer corresponds to one CH2 CF2 monomeric unit of TG or TT conformation. ¯ ¯ The stacking layer structure is (TG)(TG)(TG)(T G)(TT)(TT)(TG)… The streaks can be reproduced well by this kink structure [135].

1.15.7 Domain Structure Domain consists of the crystal lattices with a finite size. These domains are gathered together to form one crystallite. The domains may be coherent or incoherent with each other in the X-ray diffraction. If they are coherent, the diffraction pattern may be sensitively affected by the spatial change in the domains, as seen in the case of PLLA α and δ forms [73, 115]. In the VDF-TrFE copolymers, the X-ray equatorial diffraction profile is broad at room temperature or in the polar crystal phase [140]. By heating, the phase transition to the hexagonal chain packing occurs and the reflection becomes quite sharp (see Fig. 1.178). The molecular chains are rotating in the hightemperature (HT) phase and the difference between the domains disappears to result in the large single domain. By cooling again, the polar crystal domains appear and the multiple domains are recovered. From the half-width of the 110/200 reflection, the X-ray coherent domain size is reasonably estimated on the basis of Scherrer’s equation.

1.15 Diffraction and Structure Disorder

(a)

245

(b)

Fig. 1.178 Domain structure of VDF-TrFE copolymer. a temperature dependence of integrated width of X-ray diffraction spots measured for the LT and HT phases. b The illustration of domain structures dependent on temperature. Reprinted from Ref. [140] with permission of Taylor & Francis, 1995

1.15.8 Disorder in Polymer Blends The disorder caused by the atomic substitution can be seen in many metal alloys and the cocrystallizing organic substances where the different types of atoms or molecules coexist in the same crystal lattice in a perfectly random way or in some statistically disordered way. The copolymers may be assumed to be one of the typical compounds. The different types of the monomeric units are coexistent together to form the crystalline region, as already seen in many examples. The similar situation can be seen for the blended polymer substances, which is called sometimes a polymer alloy. The polymers in the polymer alloy may be a mixture of the semicrystalline and amorphous polymers, a mixture of the amorphous polymers, or a mixture of crystalline polymers. The degree of coexistence is dependent on the miscibility of these polymer components. For a pair of immiscible polymers, the macro-phase separation occurs between the crystalline and/or amorphous regions. In a diblock (triblock, …) copolymer, the micro-phase separation might occur in which the two polymer species are aggregated in their individually different regions to form the large-scale morphology. The details can be analyzed by measuring the transmission electron microscopic image or the small-angle X-ray scattering.

246

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

The cocrystallization of the crystalline polymers is seen in relatively many examples. They may be a kind of crystalline complex. The two types of polymer chains are coexisting in the common lattice in a regular manner or in a random manner depending on the polymer species. If the two polymers are stereoisomers, for example, poly(L-lactic acid) and poly(D-lactic acid), the crystalline complex is the stereocomplex [141, 142]. The effect on the diffraction pattern is dependent on the aggregation state of the two chains.

1.15.9 Disorder of the Second Kind (Paracrystal) 1.15.9.1

Paracrystal Theory

If we call the disorder caused by the thermal motions of the atoms around the equilibrated positions the disorder of the first kind, the lattice spacing or the unit cell parameters are constant in time average. The crystallite size of the disordered crystal can be estimated using Scherrer’s (or Bragg’s) equation as already mentioned. On the other hand, the spatial fluctuation of the unit cell length must be treated in a different way. This type of disorder is called the disorder of the second kind [143]. The case of the one-dimensional lattice will be explained in a brief manner [5, 143, 144]. Let us see the one-dimensional lattice with the averaged unit cell parameter a. If all the unit cells are repeated at the same spacing a, then we can derive the structure factor of a finite crystal by the method mentioned before. Since the scattering factor of the atoms is expressed as f (q), the total structure factor F t (q) is given as   F t (q) = j f(q) exp 2π iqxj = f(q)Σj exp 2πiqxj = f(q)Lx (q)

(1.300)

where L x (q) is a Laue function or the so-called form factor dependent on the size of the crystallite [q = k/2π = (2/λ) sin(θ) in the previous chapters. k = (4π/λ) sin(θ)]. In this way, the structure factor of the total system is expressed using the multiplication of structure factor and form factor. The shape or the half-width of the Laue function (Fig. 1.179) is dependent on the crystallite size, then Scherrer’s equation can be derived (Sect. 1.8.5.1). L2x (q) = sin2 (La · k/2)/ sin2 (a · k/2) = sin2 (La π q)/ sin2 (a π q)

(1.301)

In the above case, the unit cell spacing is constant. As a result, the form factor is repeated constantly without any change in the peak position and peak width, although the half-width changes depending on the crystal size L. If the unit cell spacing fluctuates depending on the position, we need to know the correlation between the distances of the neighboring cells, resulting in the change of the form factor. As illustrated in Fig. 1.180, the spacing between the neighboring unit cells is calculated as d j,j+1 = x j+1 – x j , where the x j is the atomic position in the j-th unit

1.15 Diffraction and Structure Disorder

247

Fig. 1.179 Laue function

d j,j+1

xj

p(x)

p(x)^p(x)

x j+1

x j+2

Fig. 1.180 Position probability along the x axis

cell (only one atom is assumed to exit in the cell for simplicity). The probability of positioning the atom between x j and x j +xj is given by p(x j )xj . For example, the position j-1 is fixed and the probability of the j-th atom to locate at the distance d j-1,j is p(x j ). The probability of the j+1-th atomic position from the j-th atom is p(x j+1 ). The correlation between d j-1,j and d j,j+1 or γ(j-1, j, j+1) is expressed as γ(j − 1, j, j + 1) =< dj−1,j dj,j+1 > / < a >2   ∝< p xj p xj+1 >=< p(x)p(x + x) >= p(x)∧ p(x) ≡ p2 (x)

(1.302)

where the distance is now expressed as x and the symbol ˆ indicates the selfconvolution of the two functions p(x). If all the distances are , then γ should be 1 independent of the positions. As the two distances are more random, the γ value becomes smaller and diminishes. The similar treatment can be made for the pairs between the three distances, which is related to the probability p2 (x)∧ p(x) = p(x)∧ p(x)∧ p(x) ≡ p3 (x) . The total correlation is expressed as γ(x) ∝ δ(x) + pj (x) + pj (−x) and pj (x) = p(x)∧ p(x)∧ . . .∧ p(x)

(1.303)

248

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Here the plus and minus distances are taken into account, which should be symmetric. The Fourier transform of the correlation function gives the structure factor. As well known the Fourier transform of the convoluted functions gives the multiplication of the Fourier transforms F of the individual functions. Then, we have the paracrystalline lattice factor as follows:





 Z(q) = F δ(x) + pj (x) + pj (−x) = F[δ(x)] + F pi (x) + F pj (−x) = 1 + i {F[p(x)]}j + j {F[p(−x)]}j = 1 + j z(q)j + j z ∗ (q)j 

= 1 + z(q)/[1 − z(q)] + z ∗ (q)/ 1 − z ∗ (q) 



= 1 − z(q)z(q)∗ / 1 + z(q)z(q)∗ − z(q) − z(q)∗

(1.304)

The z(q) = F[p(x)] = Fourier transform of p(x). The asterisk * is the conjugated component. Since z(q) is a complex, z(q) = m + in, z(q)* = m – in, z(q)z*(q) = m2 + n2 , and z(q) + z(q)* = 2m = 2Re(z). Then,   Z(q) = 1 − m2 − n2 / 1 + m2 + n2 − 2m

(1.305)

By the way, we know the following equation:   (1 + m + in)/(1 − m − in) = 1 − m2 − n2 / 1 + m2 + n2 − 2m  + i2n/ 1 + m2 + n2 − 2m so, Eq. 1.305 is equal to the first term of this equation, i.e., the real term of the complex (1+m+in)/(1-m-in). That is to say, Z(q) = Re[(1 + m + in)/(1 − m − in)] = Re[(1 + z(q))/(1 − z(q))]



 (1.306) = 1 − |z|2 / 1 − 2|z| cos(χ) + |z|2 where z(q) = |z(q)| exp(iχ). If we assume the Gaussian distribution of the atomic position, we have such a relation  

  p(x) = 1/ (2π)1/2 σ exp −(x− < a >)2 / 2σ2

(1.307)

 z(q) =

p(x) exp[2πiqx]dx

(1.308)

where is an averaged unit cell constant and σ is a standard deviation. The full width at the peak height maximum (FWHM) is equal to 2σ(2 ln 2)1/2 = 2.35σ.

1.15 Diffraction and Structure Disorder

249

By putting the following relations: (x− < a >)/σ = r, dx = σdr, x = rσ+ < a> z(q) = F [p(x)]  



 = p(x) exp[2π iqx]dx = 1/(2π)1/2 exp −r 2 /2 exp[2π iq(r σ+ < a >)]dr 







 = 1/(2π)1/2 exp[2π iq < a >] exp −r 2 /2 exp[2π iqrσ ] exp (2π qσ)2 /2 exp −(2π qσ)2 /2 dr 





 1/(2π)1/2 exp −(r − 2π iqσ)2 /2 dr = exp[2π iq < a >] exp −(2π qσ)2 /2 

= exp[2π iq < a >] exp −(2π qσ)2 /2 [the integration of Gaussian function is 1]

Then, the Z(q) is given as (refer to Eq. 1.304)   Z(q) = 1 − zz ∗ / 1 + zz ∗ − z − z ∗  1 − exp −4π2 q2 σ2 =  2  1 − exp −2π2 q2 σ2 + 4 sin2 (π q < a >) exp −2π2 q2 σ2

(1.309)

If we define the vectors k and q as before (q = k/2π= (2/λ)sin(θ), and the exponential phase term is 2πiqx = ikx), Z(q) should become  1 − exp −k 2 σ2

Z(k) =   2   1 − exp −k 2 σ2 /2 + 4 sin2 (k < a > /2) exp −k 2 σ2 /2

(1.310)

Figure 1.181 shows the plot of paracrystal lattice factor Z(q) against q (= 1/a order). The highly-regular peaks become weaker and broader with the increase of q or scattering angle. This behavior is similar to the effect of the temperature factor, which comes from the change in coherency between the unit cells by the atomic vibrational displacements with keeping the averaged atomic positions unchanged. But, as already pointed out, the temperature factor itself does not induce the increase of the half-width dependent on the diffraction angle. In the treatment by Scherrer, the peak width becomes larger when the crystallite size becomes smaller, which is due to the contribution of Laue function. The paracrystal lattice gives the effect on both of peak height and peak width, in contrast to the case of Laue function, even when the crystallite size is not taken into account in the derivation of the form factor Z(q). The FWHM (δ) of each curve with the different order h is estimated by calculating the area A of the curve (= 1/ for Gaussian function) and the peak height Z max measured from the background (Z min ). As shown in Fig. 1.181b, δ = A/(Zmax − Zmin ) ∼ 1/(< a > Zmax )

250

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.181 a plot of paracrystalline lattice factor Z(q) against q. b Estimation of width by assuming a Gaussian curve

The following result is obtained.   δ = 1/(< a > Zmax ) ∼ 1/ < a >) π2 h2 σ2 / < a >2 = π2 h2 g 2 / < a > (1.311) where g = σ/ is a lattice strain. The half-width δ is proportional to h2 . In fact, when the numerically calculated curves shown in Fig. 1.181 are investigated carefully, we notice that the FWHM (δ) is proportional to h2 , where h is an index along the a-axial plane (d h00 = /h), becoming a measure of the existence of the paracrystalline disorder in the crystal lattice. How can we combine the effects of the crystallite size and the disorder of the second kind? The diffraction intensity with these two types of disorder taken into account is expressed as

 I (q) = N < |FN |2 > − |< FN >|2 e−2M + (1/v)|< FN >|2 e−2M Z(q)∧ |L(q)|2 (1.312)

1.15 Diffraction and Structure Disorder

251

Fig. 1.182 Plot of the square of the FWHM (full width at half maximum) estimated from Fig. 1.181 against h4

Here the AˆB indicates the convolution between A and B, N is the number of the unit cells, F is the structure factor of the unit cell, L(q) is the shape factor, and M is the temperature factor. The half-width originates from both of the half-width of Z(q) and L(q). Roughly, Gaussian function Lorentzian function

δ2 ∼ δ(L)2 + δ(h)2 ∼ (1/D)2 + h4 g 4 π4 / < a >2 (1.313a) δ ∼ δ(L) + δ(h) ∼ (1/D) + h2 g 2 π2 / < a >

(1.313b)

D is an averaged crystallite size along the a-axis. The second term is from Eq. 1.311. It must be noted that the equation changes depending on the usage of Gaussian function or Lorentzian function. For example, in the case of Eq. 1.313a, the square of half-width is plotted against h4 , the intercept and slope of the straight line may give the D and g information, respectively. In the actual treatment, the half-width of the reflection should be an integrated width (= the integrated area/the peak height) with the correction of the broadening effect by the measurement system. The correction is made in the following two ways: q(correct) = q(obsd) − q(system)

1/2  q(correct) = q(obsd)2 − q(system )2 where the system width is given by measuring the width of a sharp reflection of a standard metal powder.

252

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

The one-dimensional paracrystal theory can be extended to the three-dimensional paracrystal. γ(x) ∝ δ(x) + pj (x) + pj (−x) and pj (x) = p(x)∧ p(x)∧ . . .∧ p(x), and so the lattice factor Z(q) is given as Z(q) = Z1 Z2 Z3 

Z1 (q) = Re (1 + z1 (q))/(1 − z1 (q)) , etc. By assuming the Gaussian functions, Z 1 (q), for example, is given as   2 2

2 2 2 X + σ12 Y 2 + σ13 Z Zl (q) = exp −2π2 σ11

1.15.9.2

(1.314)

Application of Paracrystal Theory

The application of the paracrystal theory to the SAXS data analysis is made for interpreting the lamellar stacking disorder. The concrete example will be given in a later section. Here we will see one example in which the paracrystal concept was applied to interpret the disorder in the crystal lattice of the orthorhombic polyethylene [145]. A series of PE samples were prepared under various conditions; cooled from the melt, quenched from the melt, drawn by 11 times the original length followed by heat treatment, crystallized under high pressure, and so on. The unit cell parameters and the setting angle of the planar-zigzag chain measured from the b-axis (Fig. 1.134) are evaluated by performing the crystal structure analysis based on the intensity data for all the observed diffractions. The lattice strain (g) was estimated using the abovementioned equation. The result is shown in Fig. 1.183. The unit cell parameters are almost in a linear relation with the lattice strain, in particular, the change of the a-axial length is remarkable. The setting angle is also affected by the lattice strain sensitively. Besides, the solution-grown sample and the high-pressure crystallized sample show the higher setting angle (~47°) of the chain but with the smaller lattice strain (~0.2%), the origin of which might come from the difference in the degree of chain entanglement.

1.16 Crystal Structure Analysis Using Powder Diffraction Data

253

Fig. 1.183 Effect of paracrystalline lattice distortion on a unit cell parameters and b setting angle of zigzag chain of orthorhombic polyethylene crystal. ◯● melt-crystallized,  single crystal mat, ▲ soluton-cast,  pressure-crystallized. Reprinted from Ref. [145] with permission of the American Chemical Society, 1980

1.16 Crystal Structure Analysis Using Powder Diffraction Data The powder sample is seldom used for the detailed crystal structure analysis. The reason is simple: the diffractions are overlapped in a complicated manner to make the separation of these diffractions difficult. If there are many observed peaks in one diffraction profile, the possibility of successful analysis might be higher. Usually the structure analysis of the powder diffraction data is carried out for the low-molecular-weight organic compounds and inorganic substances. The method is called, in general, the Rietveld method [146]. It is difficult to utilize this method for the polymer substance. The process of the Rietveld method is as follows: (i)

(ii) (iii) (iv)

The diffraction data of a perfectly unoriented powder sample is collected in as wide range of diffraction angle as possible. The “perfectly unoriented” is actually impossible, and so the sample is rotated around the ω−, χ−, and/or φ− axes during the measurement for averaging the anisotropic intensity. The thus-obtained powder data is used as it is. Sometimes the Compton scattering contribution is erased using an equation shown in Eq. (1.345). The peak positions are read out for all the observed diffractions. Using the data (iii), the candidates of the unit cell parameters and the indexing of the observed peaks are listed up. The indices of the several main peaks are

254

(v)

(vi)

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

assumed and the diffraction angles of the other peaks are predicted. The list of the unit cell parameters is shown with the comparison of the positions between the observed and calculated peaks. The thus-obtained unit cell parameters are refined. The atoms are positioned in the unit cell by using the random numbers, by referring to the energy calculation or by referring to the geometry of the model compound. The space group is assumed. The diffraction profile is calculated. The atomic positions are varied so that the observed diffraction profile is reproduced as well as possible. It must be noted that the thus-derived structure model is one of the candidates derived from the observed diffraction profile. Sometimes people try to use the Rietveld method to the low-molecular-weight organic compounds even when the single crystals can be obtained in a normal way. We have to make an effort to prepare the single crystals to get the diffraction data set for the complete structure analysis. The usage of the powder sample must be a last means.

This method may be applied also to the polymer substances, but the possibility to extract the most plausible candidate is much lower than the low-molecular-weight compounds. Let us see one example. The sample is an unoriented polyethylene film prepared by cooling the melt. The analysis was made by using a commercial software “Reflex” of Materials Studio (Biovia). The concrete process is as follows. (i)

(ii) (iii)

Install the experimental data shown in Fig. 1.184. The data format is explained in the HELP of the software. In the present case, the powder diffraction data was collected using a Rigaku TTR-III X-ray diffractometer. The data was transferred to the ASCII data. Set up the various conditions including the wavelength of an incident X-ray beam (Cu-Kα, 1.54178 Å). Peak search was performed (method: ITO15), where the crystal system was set to the orthorhombic system. The list of the candidates of the unit cell parameters was shown on the display screen (see Table 1.18). The top one was similar to those of the correct values, but the b-axial length, 9.87536921

Fig. 1.184 X-ray powder pattern measured for the melt-cooled HDPE sample. The red lines are the results of the peak search

1.16 Crystal Structure Analysis Using Powder Diffraction Data

255

Fig. 1.185 Refined crystal structure of PE

(iv)

(v) (vi) (vii) (viii)

Å, and the c-axial length, 5.71111638 Å, are twice larger than the real ones. The second candidate is almost the same as that of the reported values: a = 7.42 Å, b = 4.94 Å, c = 2.54 Å, α = β = γ =90° (#2 in Table 1.18). These values were refined furthermore using the Pawley method. The indexing was performed. The candidates of the possible space groups were given with the probabilities. The Pna21 was selected here (since we know already the answer). It must be noted that the CH2 unit was used as one asymmetric unit (not (1/2)CH2 unit). The unit cell with this space group symmetry was created on the screen. Separately the model of an asymmetric unit, one CH2 unit, was prepared on the screen by ourselves. The thus-drawn CH2 unit was input into the unit cell of (v). The structure was refined using the various methods. The Pawley method gave the best result as shown in Fig. 1.185. The R factor was 7.1%. However, some of the main peaks did not appear in the calculated profile, which might come from the reflections of the aluminum base.

The various parameters are varied so that the observed diffraction pattern is reproduced as well as possible. However, it must be noted that the thus-derived parameters are not necessarily physically meaningful. For example, the crystallite sizes along the a-, b-, and c-axes became the tremendously curious values in some cases. In the present calculation, these values were 219 Å (// the a-axis), 308 Å (// the b-axis), and 79 Å (// the c-axis). The lattice strains were 1.2%, 2.5%, and 4.0 % along the a-, b-, and c-axes, respectively. The degree of the agreement of the diffraction profile is dependent also on the method used in the simulation. For example, the Pawley method and the Rietveld method were tried in the refinement, but the latter did not give very good result (R = 10.7 %). The details of these methods are referred to the home page of Materials Studio. The powder diffraction data are, in general, difficult to give the unique answer of the unit cell parameters, etc. The R factor of 2–3 % is required for the reliable structure analysis. The total number of peaks should be as many as possible in the wide range of diffraction angle up to 90° or more. Even when the answer of structure is obtained,

35.60000000

33.10000000

33.09999000

12.40000000

10.50000000

1

2

3

4

5

20 of 20

20 of 20

22 of 27

19 of 20

18 of 20

Peaks Found

FOM: figure of merits

FOM

#

Orthorhombic

Orthorhombic

Hexagonal

Orthorhombic

Orthorhombic

System

9.89614068

9.90487473

11.39999962

7.41799766

7.42290057

a/Å

11.29817316

11.29961539

11.39999962

4.93500000

9.87536921

b/Å

4.94039556

4.94474227

7.42000008

2.54000000

5.71111638

c/Å

90.00000000

90.00000000

90.00000000

90.00000000

90.00000000

α/degree

Table 1.18 List of the plausible unit cell parameters after the indexing of the searched peaks

90.00000000

90.00000000

90.00000000

90.00000000

90.00000000

β/degree

90.00000000

90.00000000

120.00000000

90.00000000

90.00000000

γ/degree

552.37728311

553.42185938

831.07000732

417.83978212

418.64701067

Volume/Å3

256 1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

1.16 Crystal Structure Analysis Using Powder Diffraction Data 020 210

257

011 120

310 111

110

? ? 211

121

200 400

311

Fig. 1.186 Comparison of the X-ray diffraction profile of PE between the observed (red) and calculated (blue) ones

we have to check the reasonableness of the results from the various viewpoints. As already said, we have to make an effort to prepare a single crystal before we use the powder sample for the crystal structure analysis. This can be said also for the synthetic polymer samples. We have to prepare, not the unoriented samples, but the highly-oriented and highly-crystalline samples for finding the crystal structure with high reliability.

1.17 X-ray Analysis of Amorphous and Liquid Structures The amorphous region may be assumed simply to consist of the random arrays of the atoms, but the atoms in polymer chains are connected to each other and the chain segments may be arranged more or less in some systematic way. In order to estimate the atomic arrangement in the amorphous phase, it is needed to estimate it in a statistical way [147–150].

1.17.1 Randomly Oriented Gas Molecule The X-ray scattering from an isolated gas molecule occurs by the interference of the X-ray waves scattered from the electrons of the molecule. The contribution of I e or the scattering intensity by an electron is ignored here (I = I e |F|2 ).

258

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.187 Surface area of a small portion ds

k

l nmsin α dφ l nmd α

α ds

l nmsin α d φ

l nm

 F(k) = j fj (k) exp irj · k

(1.315)

I (k) = F(k)F ∗ (k) =  nm fn (k)fm∗ (k) exp[i(rn − rm ) · k] =  nm fn (k)fm∗ (k) cos[(rn − rm ) · k]

(1.316)

The sine terms are erased because the pair of rn – rm and rm – rn is necessarily existent, which are erased to zero. Equation 1.316 is for the molecule fixed in a space at a moment. We need to average them for all the possible orientation directions. As shown in Fig. 1.187, the angle between the vector k and rn – rm is α. The interatomic distance is expressed as lnm = |rn – rm |. Then Eq. 1.316 becomes I (k) =  nm fn (k)fm∗ (k) cos[lnm k cos α]

(1.317)

We have to obtain the averaged value . The interatomic distance  2 is given as lnm is constant. The small surface ds on the total sphere surface 4π lnm ds = lnm sin αdφlnm dα. Then,   2 cos[lnm k cos α]ds < cos[lnm k cos α] > = 1/4π lnm  2π  π  2 2 = 1/4π lnm cos[lnm k cos α]lnm sin αdαdφ 0 0  π = (1/4π)(2π) cos[lnm k cos α] sin αdα (1.318) 0

Inputting xnm = lnm k cos α, dxnm = −lnm k sin αdα and so 

π 0

 cos[lnm k cos α] sin αdα = −

−lnmk lnmk

cos[xnm ]dxnm /(lnm k) = 2 sin(lnm k)/(lnm k)

1.17 X-ray Analysis of Amorphous and Liquid Structures

259

Therefore < cos[lnm k cos α] >= sin(lnm k)/(lnm k) Finally, by putting unm = lnm k = (4π lnm /λ) sin(θ), I (k) =< |F|2 >=  nm fn (k)fm∗ (k) sin(unm )/unm

(1.319)

1.17.2 Aggregation of Randomly Oriented Gas Molecules The interference of X-ray waves scattered from the neighboring gas molecules is taken into account now. Let us consider the mono-atomic molecules as an example. The scattering intensity is given as I (k) = F(k)F ∗ (k) =  nm fn (k)fm∗ (k) exp[i(rn − rm ) · k] = |f |2  nm exp [i(rn − rm ) · k]

(1.320)

Here f n = f m = f . The average of I(k) is made for all the spatial arrangements of the molecules. The probability for the n-th atom to exist in the small volume dV n is dV n /V. Then,   <  nm exp[i(rn − rm ) · k] >=  nm

V

V

exp[i(rn − rm ) · k](dVn /V )(dVm /V )

(1.321) For the n = m, the contribution to the I(k) is calculated as N |f |2 since the number of the pair is N. For the pair of n = m, if we assume that the two neighboring molecules (atoms) cannot invade inside the molecule each other, the part of the integration in the range of interatomic distance |ri – rj | = 0~2a must be erased from the total integration shown in Eq. 1.321, where a is a radius of the atom. < exp[i(rn − rm ) · k] >   = exp[i(rn − rm ) · k](dVn /V )(dVm /V ) V V   2a exp[i(rn − rm ) · k](dVn /V )(dVm /V ) − V

(1.322)

0

The first term is almost zero since the whole volume V is overwhelmingly larger than the X-ray wavelength λ . The second term can be calculated in the following

260

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

way:   The second term =

Vn

Vm

  exp[i(rn − rm ) · k] dVn /V dVm /V

in which the integration (Vn ) is made for the spherical surface of the n-th atom and for all the orientation direction of the m-th atom. That is to say, 

The second term B = 1/V 2

 Vn

Vm

exp[i(rn − rm ) · k]dVn dVm

= (the integration for the orientation angles φ and α for the m-th atom)×(the integration for the spherical volume of the radius lnm = 0 ∼ 2a) 

= 1/V 2 {integration for φ and α of the m-th sphere}dVn

= 4π/V 2

V n



Vn



2a 0

 2 {sin(unm )/unm } lnm dlnm dVn [unm = (4π llm /λ) sin(θ)]

Here 2θ is a scattering angle. In the integration of dVm , it must be noted that the 2 sin αdαdφdlnm and φ = 0 ∼ 2π and small volume for the integration, dVm = lnm α = 0 ∼ π and lnm = 0 ∼ 2a.    B = 4π/V 2 Vn [sin(h) − h cos(h)]/(h/2a)3 dVn [h = (8aπ/λ) sin(θ)]     



= 4π/V 2 Vn (2a)3 (h)/3 dVn (h) = 3/h3 (sin(h) − h cos(h)) 

= (1/V ) (4π/3)(2a)3 (h) In Eq. 1.321, the summation  nm must be taken into account. The total number of the term B is N(N-1), and finally we have

 I (k) = |f |2 N − |f |2 (N (N − 1)/V ) (4π/3)(2a)3 (h) 

 ≈ |f |2 N − |f |2 N 2 /V (4π/3)(2a)3 (h)

(1.323)

By putting the term N(4π/3)(2a)3 for , which is a total sum of the effective volume of the molecule, we have   I (k) = |f |2 N − |f |2 (N (N − 1)/V ) (4π/3)(2a)3 (h) ≈ |f |2 N [1 − (/V )(h)]

(1.324) By introducing the electron diffraction term, we have   2 

I (k) = Io 1/r 2 (μo /4π )2 e2 /m 1 + cos2 (2θ) |f |2 N [1 − (/V )(h)]/2 (1.325)

1.17 X-ray Analysis of Amorphous and Liquid Structures

261

Let us calculate the scattering intensity of Ne gas. The atomic radius a = 1.05 Å. The X-ray scattering factor of Ne atom is numerically given as follows (RHF):  fj (2θ/λ) = aj exp −bj (sin θ/λ)2 + c (j = 1 ∼ 4) a1 = 3.9553, a2 = 3.1125, a3 = 1.4546, a4 = 1.1251 b1 = 8.4042, b2 = 3.4262, b3 = 0.2306, b4 = 21.7184 c = 0.3515 Figure 1.188 shows the thus-calculated X-ray scattering profile of Ne gas by the irradiation of X-ray beam of 0.7 Å wavelength. If the gas concentration is dilute or the small /V value, the scattering profile decreases following the shape of the scattering factor. As the pressure of gas is increased or /V is larger, the intermolecular interference becomes stronger and the maximal peak is observed at a certain scattering angle. In the case of multi-atomic molecule, the atomic scattering factor is changed as follows: 

 I (k) =< |f |2 > N − < |f | >2 N 2 /V (4π/3)(2a)3 (h) ≈ N [< |f |2 > − < |f | >2 (/V )(h)]

(1.326)

4

Ω/V=0 120 Intensity (arbitrary)

Fig. 1.188 X-ray scattering patterns of randomly aggregated Ne molecules.(h) = (3/h3 )(sin(h) − h cos(h)) and h = (8aπ/λ) sin(θ)

3

Ω/V=0.1

2

80

Ω/V=0.5 Φ(h)

1

40 0 0 0

20

40

60

80

2theta/degree

100

120

262

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

1.17.3 X-ray Diffraction from Liquid The similar treatment can be made for the liquid sample. However, the aggregation of the liquid molecules may vary at place to place. The probability W (l) is introduced for the calculation of the averaged diffraction intensity. The intensity I(k) is now changed to the following equation, where the mono-atomic molecules are treated: I (k)/|f |2 =<  nm exp[i(rn − rm ) · k] >   =  nm exp[i(rn − rm ) · k]W (lnm )(dVn /V )(dVm /V ) V

(1.327)

V

For the n = m case, the integration is 1. The total number of atoms is N. For the n = m case, the integration is modified in the following way:   exp[i(rn − rm ) · k]W (lnm )(dVn /V )(dVm /V ) = X1 − X2

X = V

V

  X1 =

exp[i(rn − rm ) · k](dVn /V )(dVm /V ) V

  X2 =

(1.329)

V

exp[i(rn − rm )k][1 − W (lnm )](dVn /V )(dVm /V ) V

(1.328)

(1.330)

V

(X 1 ) The integration is made for the volume V. The shape of  the liquid changes with time. Here the spherical shape is assumed. The integration V exp[ir·k](dV /V ) is expressed in the following equation: 



2π

exp[ir · k](dV /V ) = (1/V )



0

V



2π 0



R

cos(kr cos α)r 2 sin αdrdαdφ

0

R

[sin(kr)/(kr)]r 2 sin αdr 

= (4π/V ) sin(kR)/k 3 −R cos(kR)/k 2 = (4π/V )

0

(1.331)

Since V = (4π/3)R3 , we have 

 exp[ir · k](d V /V ) = 3/(kR)3 [sin(kR) − kR cos(kR)]

(1.332)

V

Using this equation, the X 1 is given as

2  X1 = 3/(kR)3 [sin(kR) − kR cos(kR)]2

(1.333)

As for the summation  nm , the total number of n-m pairs is N(N – 1) ~ N 2 . In general, the spherical size R >> λ, and so R/λ >> 1 or Rk >>1.

1.17 X-ray Analysis of Amorphous and Liquid Structures Fig. 1.189 Derivation of small volume dV

263

k

rsinα

dφ

r sin α dφ r d



r

dV dr

2 2 2    X1 = 3/(kR)3 [sin(kR) − kR cos(kR)]2 by using a molar fraction wj .

1.17.4 Amorphous Solid State For the amorphous sample including the amorphous polymers, the equation to be used is the same as that shown in Eqs. 1.341 and 1.342 [151]. However, we need to remember that these equations are under the assumption of spherical shape of liquid. In the actual liquid, the shape should change with time. In the solid amorphous sample, the disordered structure is frozen (by ignoring the micro-Brownian motion). The atomic scattering factors are needed to use the averaged values for the various atomic species by considering the molar fractions.

1.17.5 Actual Calculation of g(r) The method to derive the radial distribution function g(r) of polymer samples is described here [151]. (1) (2)

(3)

Measure the X-ray diffraction pattern I sample of a polymer sample. Collect the background I back which is obtained by measuring the X-ray diffraction pattern of the air under the perfectly same condition as (1) without sample. Subtract the background from the sample data. Since the X-ray is absorbed more or less by the polymer sample, the contribution of the air scattering might be larger than the value expected from the same measurement condition.

266

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Icorrect (1) = Isample − p Iback where p = 0.5 (in this example) (4)

The correction of Lorentz (L), polarization (p), and absorption effect (A) is made for I correct (1). Icorrect (2) = Icorrect (1)/(ALp)

(5)

(1.343)

(1.344)

In a case, the Compton scattering must be corrected from the I sample data. The Compton scattering I Compton is originated from the inelastic scattering of X-ray beam from the electrons.  a   a

ICompton = Kj cj Zj bj k j / 1 + bj k j

(1.345)

Here Z j is the atomic number of the j-th atom, cj is the composition ratio of the j-th atom, and aj = 1.2450 + 2.6917/Z j  2  3 bj = 0.1075 + 1.1870/Zj + 0.00436Zj − 0.01543Zj + 0.01422Zj  

K(Recoil factor) = 1 + (2h/mc) sin2 θ/λ with h/mc = 0.02426 Icorrect (3) = Icorrect (2)/N − ICompton

(1.346)

Here N is a normalization factor. (6)

Calculate i(k) = [I correct (3) – ]/2 where < |f |2 >= wj fj 2 and < |f | >= wj fj

(1.347)

f j and wj are the j-th atomic scattering factor and the corresponding molar ratio, respectively. (7)

Calculate the g(r) using i(k).  g(r) = 4π r 2 [ρ(r) − ρ0 ] = (2r/π)

∞

k · i(k) sin(kr)dk

(1.348)

0

In the actual calculation, the k range in the integration is limited, and the calculated curve g(r) gives many ripples. In such a case, the insertion of a so-called window function M(k) is useful to reduce this effect (see Fig. 1.191). However, the peaks may become broader than the true ones. Depending on the M(k) function chosen, the calculated curve is modified more or less. For example,  g(r) = (2r/π) 0

∞

ki(k)M (k) sin(kr)dk

1.17 X-ray Analysis of Amorphous and Liquid Structures M(q)

267

obsd

g(r)obsd

qmax

q

r g(r)corrected

modified

qmax

q

r

Fig. 1.191 Effect of cutoff on the distribution function g(r)

M (k) = sin(π k/kmax )/(π k/kmax ) (k < kmax ) M (k) = 0 (k >= kmax )

(1.349)

Figure 1.192 shows the X-ray diffraction patterns of the pure amorphous and pure crystalline polyethylene models and the corresponding g(r) curves. The sharp distribution of the crystalline phase changes to the broad curve for the amorphous sample.

1.18 Degree of Crystallinity The degree of crystallinity is important for the discussion of the physical property of the bulk polymer sample. There are many methods to estimate the degree of crystallinity. The density method is basically important. The two-phase model consisting of the amorphous and crystalline phases is often used. The densities of these phases are ρa and ρc , respectively. The bulk sample density ρb is given as 1/ρb = Xc /ρc + (1 − Xc )/ρa (Xc defined for the weight fraction)

(1.350)

ρb = Xc ρc + (1 − Xc )ρa (Xc defined for volume fraction)

(1.351)

The thermal estimation of X c is made as follows. The melting enthalpy Hm (obsd) ◦ is measured. If the Hm of 100% crystalline sample is known, the crystallinity is defined as ◦

Xc = Hm (obsd)/Hm

(1.352)

The X-ray method is also useful [2, 152]. The total diffraction profile is decomposed to the crystalline peaks and the amorphous peaks. The total integration of these two profiles is given as follows:

Intensity

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Intensity

268

2θ/degree (Mo-Kα)

g(r)

g(r)

2θ/degree (Mo-Kα)

distance/Å

distance/Å

Fig. 1.192 X-ray diffraction profile and the corresponding radial distribution function calculated for the amorphous and crystalline PE samples



∞

The total integration



∞

Itotal (v)dv =

0

 Itotal (k)k dk = 2

0

∞

< f 2 >k 2 dk

0

(1.353)  The crystalline part 0

∞

 Ic (v)dv = 0

∞

 Ic (k)k 2 dk = Xc

∞

< f 2 >Dk 2 dk

0

(1.354) where the integration is made for the whole reciprocal lattice space and the squared average of the structure factors of the constituent atoms is . The D is a parameter

1.18 Degree of Crystallinity

269

of the disorder of the crystalline region, containing the first and second kinds of disorder, the positional disorder, and so on: as a possibility, D = exp(−wk 2 ). The Xc is given as  Xc =

∞

 Ic (k)k 2 dk/

0

∞

 Itotal (k)k 2 dk

0

∞



∞

< f 2 > k 2 dk/

0

 < f 2 >Dk 2 dk = X P

0

(1.355) Here X =



∞



0



∞

P=

∞

Ic (k)k 2 dk/ 

0 ∞

< f 2 >k 2 dk/

0

Itotal (k)k 2 dk < f 2 >Dk 2 dk

0

k 2 . The k 1 and k 2 In the actual integration, the integration range is finite, k 1 to  ∞ values are determined so that the calculated total integration value 0 Itotal (k)k 2 dk becomes almost the same. The D value in P is determined also similarly so that the X c value is not changed. This Ruland method is difficult in the actual case and we need to erase the incoherent scattering. In simpler calculation, the ratio between the integrated values of the crystalline peaks and the total curve, that is, the X is assumed to be a measure of the crystallinity X c , although the separation between the crystalline and amorphous components is difficult. Xc ∼ X =

 0

∞



∞

Ic (k)dk/

Itotal (k)dk

(1.356)

0

Another convenient method is a so-called Hermann method under the assumption of the two-component system of the crystalline and amorphous phases. The weights of the sample, the crystalline, and amorphous phases are M b , M c , and M a , respectively. ⎫ Mb = Ma+ Mc ⎬ ∞ Mc = kc 0 Ic (k)dk = kc Ac ⎭ ∞ Ma = ka 0 Ia (k)dk = ka Aa

(1.357)

Ac = −(ka /kc )Aa + (Mb /kc )

(1.358)

Therefore,

The various samples with the different crystallinities are prepared for the same polymer and the integration of the crystalline and amorphous peaks gives the Ac and Aa values. As shown in Fig. 1.193, these sets of (Ac , Aa ) are plotted to give the straight line, the slope of which is equal to –(k a /k c ). The degree of crystallinity is calculated

270

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

Fig. 1.193 A plot of Ac against Aa

Ac

Sample 1

-(ka/kc)

2 3

4

Aa

for all the sample pairs using the following equation: Xc = Mc /Mb = kc Ac /(kc Ac + ka Aa ) = Ac /[Ac + (ka /kc )Aa ]

(1.359)

1.19 Degree of Crystallite Orientation In the present book, the estimation of the degree of orientation of the crystallites is not described in detail. Only the brief description is made here [2].

1.19.1 Simple Estimation The two-dimensional X-ray diffraction pattern of an oriented sample shows the diffraction spots of more or less long and broad arc, as illustrated in Fig. 1.194. The arc length of the reflection changes depending on the degree of the orientation of crystallites in the sample. The convenient and qualitative way to express the degree of chain orientation is to estimate the arc length of the strongest equatorial line reflection or an angle of circumference (β). The degree of orientation DOri ∝ β angle width (β) at the half-height of the arc profile/180º DOri ∝ β/180◦ or (180◦ − β)/180◦

(1.360)

In the former definition, the perfect unorientation gives DOri = 1, while the latter definition gives DOri = 0 for the unoriented sample. The latter may be easier to judge the degree of orientation intuitively (Fig. 1.194).

1.19 Degree of Crystallite Orientation

271

(a) Melt-cooled C

(b) Poorly-oriented

(c) Melt-drawn

(d) Highly-stretched

(a)

(b)

(c)

(d)

Fig. 1.194 X-ray diffraction patterns measured for the variously oriented PE samples. The βscanned profiles of the 200 reflection. The β angle is counted from the horizontal direction. Note that the melt-drawn sample shows the peak at 90°

272

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

1.19.2 Meridional Reflection As an example, let us consider the crystal of the orthorhombic type. The sample is set vertically and the X-ray is incident from the horizontal direction (refer to Fig. 1.54). The sample consisting of the perfectly oriented chains does not show the meridional diffractions (see Fig. 1.195). But, the sample of relatively high degree of stretching ratio gives the 00l reflection. However, it must be noted that the observed diffraction intensity does not necessarily reflect the total 00l planes but only some portions are related. For more exact measurement of the intensity and profile of the 00l diffraction, following the Norman’s method (Sect. 1.13.2.2), we need to oscillate the oriented sample by changing the titling angle from the X-ray incident direction in some range of the angle θ, where 2θ is the Bragg angle expected for the 00l plane. From the arc length of the observed 00l reflection, the degree of orientation of the c-axis can be estimated.

Fig. 1.195 The measurement of 00l reflection of the orthorhombic crystal. a The perfect orientation of the chain axis. The 00l reflection can be measured by rotating the sample by the angle θ . b The poorly oriented sample case, where the 00l reflection can be observed but only partially. c The exact measurement of the 00l reflection of the poorly oriented sample

1.19 Degree of Crystallite Orientation

273

1.19.3 Orientation Function Let us consider the case of the uniaxially oriented sample, where the c-axis distributes around the draw axis (Z) homogeneously. The orientation function is defined as follows for the orientation of the c-axis [153]:  fc =< P2 (cos ϕ) >= 3 < cos2 ϕ > −1 /2

(1.361)

where the angle ϕ is between the drawing axis and the c-axis and P2 (cos ϕ) is the second-order Legendre function of cos ϕ . If we have the sample of the perfectly oriented c-axis, f c = 1 since ϕ = 0o and = 1. For the perfectly unoriented sample, f c = 0 since = 1/3. By the way, cos ϕ = cos θ001 cos φ

(1.362)

where the angle φ is measured from the Z-axis to the observed 00l reflection recorded on the flat detector. The average is written as < cos2 ϕ >= cos2 θ00l < cos2 φ > By using the observed intensity I(00l) as the weight of the average, we have  90◦ < cos φ >= 2

0

I (00l) sin φ cos2 φdφ  90◦ I (00l) sin φdφ 0

(1.363)

The I(00l) is assumed to distribute homogeneously around the Z-axis or the draw axis. By measuring the intensity distribution I(00l) (see the preceding section), we can calculate the orientation function f c of the c-axis once when the Bragg angle 2θ00l is known. However, the measurement of I(00l) by changing the tilting angle φ of the sample stepwise is not very easy. Usually the fiber diagram is measured with the sample set vertically without any tilt. As pointed out by Hermans et al. [153], the direct usage of the 00l diffraction intensity from such a normal diagram does not give any good result. Rather, the usage of the intensity profiles of the equatorial diffractions along the arcs and the evaluation of the average values of and may give better estimation of by using the following equation. < cos2 ϕc > + < cos2 ϕa > + < cos2 ϕb >= 1

(1.364)

Here ϕa , ϕb , and ϕc are defined as the angles of the a-, b-, and c-axes to the Z-axis, respectively. These orientation functions can be related also to the estimation of infrared dichroism, etc., which will be described in a later section.

274

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

1.19.4 Higher Order Parameters of Crystal Orientation Detailed information is obtained by evaluating an orientation function ω(θ, φ, ϕ) where the orientation angles are defined as shown in Fig. 1.196 [2, 154, 155]. Actually, in stead of angle θ , we use ξ = cos θ : the orientation function is ω(ξ, φ, ϕ). The coordinate system of the sample is O-xs-ys-zs. The draw axis is parallel to the zs-axis. The sample plane is ys-zs-plane. The coordinate system of crystallites is O-Xc-Yc-Zc. The Euler angles θ, φ and ϕ are for the orientation of crystallites in the sample. The direction of a reciprocal lattice vector rj is defined using the angles Θj and Φj and also using the angles χj and ηj . These two coordinate systems are related using the following transformation matrix T(φ, θ, ϕ) . Here sx and cx indicate sine and cosine of the variable x, respectively. ⎛ ⎞ ⎞ sΘj c Φj sχj c ηj ⎝ sχj s ηj ⎠ = T −1 (φ, θ, ϕ)⎝ sΘj s Φj ⎠ c χj c Θj ⎛

⎛

cφcθ cϕ − sφsϕ T(φ, θ, ϕ) = ⎝ −cφcθ sϕ − sφcϕ cφsθ

sφcθ cϕ + cφsϕ −sφcθ sϕ + cφcϕ sφsθ

(1.365) ⎞ −sθ cϕ sθ sϕ ⎠ cθ

(1.366)

A function of these Eulerian angles is often expressed using the associated Legendre function Plm (x) and also the generalized associated Legendre function Plmn (x). The associated Legendre function Plm (x) is defined as follows: m/2 1+m  2   l x − 1 /dxl+m d Plm (x) = (−1)m / 2l l! 1 − x2

(1.367)

where l and m are the integers showing the degree and order of this function, respectively. For n = 0, Plm (x) = Plm0 (x) [154, 155]. The orientation function ω(ξ, φ, ϕ) or ω(cos θ, φ, ϕ) is expressed by ∞ l l m=−l n=−l Mlmn Plmn (ξ ) exp(−im φ) exp(−in ϕ) ω(ξ, φ, ϕ) = l=0

(1.368)

where M lmn is a coefficient. By estimating the coefficients M lmn , ω(ξ, φ, ϕ) can be obtained concretely. The coefficients are obtained  by using the observed intensity distribution of the X-ray diffraction spots Ij ζj , ηj , where  angles ηj and χj are drawn in Fig. 1.196 and ζj = cos(χj ) . A reduced intensity Ijred ζj , ηj is defined as follows: Ijred

  ζj , η j = I j ζj , η j /



1 −1



2π

  Ij ζj , ηj dηj dζj

0

 Using the Legendre function, Ijred ζj , ηj is expressed as

(1.369)

1.19 Degree of Crystallite Orientation

Sample coordinates

(a)

275

Crystallite coordinates

Zc

zs

xs ys

Yc Xc

Zc

(1) Xc

zs

(2) φ

xs

zs

Zc xs

Xc

(3)



Zc

Yc

ys

ys Yc zs

(4)

xs

zs Zc

xs ys

ys

Xc

Yc Yc

Xc

χ j rj

(b) ηj

 rj

(c)

j

j

Fig. 1.196 a Coordinate systems of sample and crystallite. In order to define the Eulerian angles (θ, φ, ϕ), the concrete processes are shown here. (1) adjust the coordinate axes, (2) rotate green axes around the z-axis by angle φ, (3) rotate the axes around the yc -axis by θ, and (4) rotate the axes around the zc -axis by ϕ . b and c: the polar coordinates to define the reciprocal lattice vector r on the sample and crystallite coordinate systems. Note that the lattice plane is normal to the vector r

276

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

  j ∞ ∞ Ijred (ζj , ηj ) = l=0 m=−l Qlm Plm ζj exp −im ηj

(1.370)

j

The coefficient Qlm can be calculated using the orthogonality of the associated Legendre functions. The result is  j Qlm

= (1/2π)

1 −1

 0

2π

Ijred

   ζj , ηj dηj Plm ζj d ζj

(1.371)

The Q values are related to the above-shown M lmn in the following equation.  j l Qlm = 2π(2/2l + 1)1/2 n=−l Mlmn Pln (Ξj ) exp in Φj

(1.372)

Ξj = cos(Θj ) The process to evaluate the orientation function ω(ξ, φ, ϕ) or M lmn is as follows: (i) (ii) (iii) (iv)

Intensity distributions Ij (ζj , ηj ) of the several  diffraction  spots (j = 1–N) are ζ χ = cos χj . measured by changing the angles ηj and j j  The reduced intensity distributions Ijred ζj , ηj are calculated using Eq. 1.369. j

For the diffraction spot j, the coefficients Qlm are calculated for the various pairs of l (= 0, 1, 2, …) and m (= -l ~ l) based on Eq. 1.371. The angles Θj (or Ξj = cos Θj and Φj are the angles of the reciprocal lattice point j and are calculated from cell parameters and the indices (h,k,l).  the unit Inputting aj,l,n = Pln (Ξj ) exp in Φj , we get the following simultaneous equations for n = -l~l (the total number of equations is 2l + 1) at the constant pair of l and m:

 1 Qlm / 2π(2/2l + 1)1/2 = a1,l,−l Mlm,−l + a1,l,−l+1 Mlm,−l+1 + . . . + a1,l,0 Mlm,0 + . . . + a1,l,l Mlm,l

 2 Qlm / 2π(2/2l + 1)1/2 = a2,l,−l Mlm,−l + a2,l,−l+1 Mlm,−l+1 + . . . + a2,l,0 Mlm,0 + . . . + a2,l,l Mlm,l ...

 N Qlm / 2π(2/2l + 1)1/2 = aN ,l,−l Mlm,−l + aN ,l,−l+1 Mlm,−l+1 + . . . + aN ,l,0 Mlm,0 + . . . + aN ,l,l Mlm,l

(1.373)

By solving these simultaneous equations, the coefficients M lmn (n = −l ~ l) are obtained. By changing the various pairs of (l, m), all the M lmn values are obtained. Since the total number of the solvable M lmn is 2l + 1, which is equal to the total number of the observed diffraction spots N: 2l + 1 = N or l = (N – 1)/2. In order to obtain the M lmn values of many l values, the number of the observed diffraction spots N must be increased as many as possible. (v)

By inputting these M lmn values into Eq. (1.368), the orientation function ω(ξ, φ, ϕ) is evaluated.

1.19 Degree of Crystallite Orientation

277

Fig. 1.197 Examples of the pole figures (the 3D distribution of the observed diffraction intensities)

278

1 Crystal Structure Analysis by Wide-Angle X-ray Diffraction Method

The case of a uniaxially oriented sample is easier in the treatment. This is because the crystallites are oriented homogeneously around the zs-axis and so the angles φ and ηj are not contained in the equations. The orientation function ω(ξ, ϕ) is expressed as ∞ l m=−l Mlm Plm (ξ ) exp(−im φ) ω(ξ, ϕ) = l=0 ∞ ∞ l = l=0 Ml0 Pl0 (ξ ) + 2l=1 m=1 Mlm Plm (ξ ) exp(−im φ)

Ijred

 ζj = Ij (χ )/



2π

Ij (χ ) sin χ dχ

(1.374)

(1.375)

0

 Using the Legendre function, Ijred ζj is expressed as   j ∞ Ijred ζj = l=0 Ql Pl0 ζj  j Ql

= (1/2π)

1 −1

  Ijred ζj Pl0 ζj dζj



 j l Ql = 2π(2/2l + 1)1/2 Ml0 Pl0 (Ξj ) + 2n=1 Mln Pln (Ξj ) exp in Φj

(1.376)

(1.377) (1.378)

The process to estimate the orientation function ω(ξ, ϕ) is the same as that mentioned above. Equation (1.375) → Eq. (1.377) → Eq. (1.378) → solving the simultaneous equations, M ln are obtained. → estimation of ω(ξ, ϕ). The plot of ω(ξ, φ, ϕ) or ω(ξ, ϕ) gives the so-called pole figure (see Fig. 1.197) [2, 154–156]. Usually the pole figure is obtained by measuring the scattering intensity of a particular reciprocal vector and by plotting the 2D figures of intensities with the machine direction (MD) and the transverse direction (TD). The normal direction (ND) is perpendicular to the paper.

References 1. H. Tadokoro, Structure of Crystalline Polymers (Wiley, New York, 1979) 2. L.L. Alexander, X-ray Diffraction Methods in Polymer Science (Wiley, New York, 1969) 3. B.K. Vainshtein, Diffraction of X-rays by Chain Molecules (Elsevier Publ. Co., Amsterdam, 1966) 4. Fiber Diffraction Methods, edt. A.D. French, K.H. Gardner, Am. Chem. Soc. Symposium, 141 (1980) 5. N. Kasai, M. Kakudo, X-ray Diffraction by Macromolecules (Springer, Berlin, 2005) 6. T. Hahn, Ed. International Tables for Crystallography, Volume A “Space-Group Symmetry”, D. Reidel Publ. Co., Dordrecht, Holland (1983) 7. F.A. Cotton, Chemical Applications of Group Theory. 3nd edn. (Wiley, New York, 1990) 8. E.S. Clark, The crystal structure of polytetrafluoroethylene, Forms I and IV. J. Macromolecu. Sci. Part B: Phys. 45, 201–213 (2006) 9. E.S. Clark, The crystal structure of polytetrafluoroethylene, Forms I and IV. J. Macromolecu. Sci. Part B: Phys. 45(2), 201–213 (2006)

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Chapter 2

Structure Analysis by Wide-Angle Neutron Diffraction Method

Abstract The crystal structure analyzed by X-ray diffraction method must be checked from the different point of view. The wide-angle neutron diffraction is one of the most powerful methods for this purpose. The principle of wide-angle neutron diffraction phenomenon is leaned at first in comparison with the X-ray diffraction. The experiment of neutron diffraction is made using a neutron beam from the atomic reactor with a nuclear fission of Uranium or using neutron pulses generated by spallation. The principles of these different experimental methods are learned. At the next stage, the process to determine the crystal structure using the neutron diffraction data is learned by focusing the difference from the X-ray diffraction analysis. The Fourier transformation of the X-ray and neutron diffraction data gives the electron density distribution and the nuclear atomic positions, respectively. The combination of these two different information (X-N method) allows us to know the distribution of electrons contributing to the covalent bonds along the polymer chains. Keywords Neutron diffraction · Scattering cross-sectional area · Crystal structure analysis · Time-of-flight method · X-N method

2.1 Neutron 2.1.1 Neutron as Wave and Particle Neutron possesses the property as a particle and as a wave [1–3]. As a particle, the mass mo is almost equal to that of proton, ca. 1 atomic unit and 2000 times heavier than one electron. The size of a neutron particle is 1.5 × 10–13 cm, which is tremendously smaller than the broadening size of electron cloud. A neutron has a spin ±h/4π and follows the Fermi–Dirac statistics. As the wave, a neutron has an energy E = hν = (1/2)mo v2 for the velocity v (Einstein relation). The momentum p = h/λ = mo v (de Broglie relation). From these equations we have λ(Å) = h/(m o v) = 3956/v(m/s) = 0.2860/(E(eV))1/2

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 K. Tashiro, Structural Science of Crystalline Polymers, https://doi.org/10.1007/978-981-15-9562-2_2

(2.1)

287

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2 Structure Analysis by Wide-Angle Neutron Diffraction Method

Table 2.1 Property of various radiation beams X-ray

UV

Visible

IR

Far IR

Microwave

λ (Å)

1~100

~1000

104

105

106

107

ν (cm−1 )

108 ~ 106

~105

104

103

102

10

E (eV)

12,400

12.4

1.24

0.124

0.0124

0.00124

v (cm/s)

1.55 ×

1.55 × 106

4.9 × 105

1.55 × 105

4.9 × 104

λ(Å)

0.0025

0.081

0.255

0.807

2.552

8.073

Intermediate neutron

Slow neutron

Neutron 108

4.9 ×

106

Thermal neutron

Cold neutron

v(1/sec) = 2.418 × 1014 E(eV) As shown in Table 2.1, the neutron of the wavelength 1 Å has a velocity v = 3.96 km/s, corresponding to E = 12,400 eV. In other words, this neutron takes 252 μs for passing the distance 1 m. This speed can be detected relatively easily, making it possible to perform the experiment of the time of flight (TOF). Table 2.1 compares the velocity v, wavelength λ, and energy E among X-ray, neutron and electromagnetic wave. Sometimes the thermal neutron energy is expressed using temperature (T ) since it exists in a thermally equilibrated state. For example, using the relation E = k B T, the neutron of 1 Å corresponds to the temperature 950 K. The cold neutron is called for the neutron with the energy smaller than 0.002 eV and the thermal neutron is for the energy smaller than 0.5 eV.

2.1.2 Cross-Sectional Area If a neutron beam with the total intensity  is incident to the plate, which has the thickness dx and consists of N atoms per unit volume, the neutron intensity is reduced by d. By taking into account the probability σT of the collision of neutron with atoms, we have d = −σT N dx

(2.2)

 = 0 exp(−N σT x)

(2.3)

2.1 Neutron

289

The unit of the probability σT is [cm3 ][cm−1 ] = [cm2 ]. Then, the σT is treated with the name of cross section. The following unit is used often: 1 barn = 10–24 cm2 . If the scattered neutron is detected in the direction of a solid angle d, the number of the scattered neutron is d = N (dσ/d)ddx

(2.4)

The (dσ/d) is named the differential cross section, which is a scattering proba bility into the d direction. The total scattering is expressed as σ = (dσ/d)d. If the energy of the scattered neutron is changed by dE (or the inelastic scattering), we have d = N (dσ/dE)dEdx

(2.5)

The (dσ/dE) is called the inelastic scattering cross section. Similarly, the inelastic scattering cross-sectional area corresponding to the energy change dE in the direction d is given as (d2 σ/dEd). Since the incident neutron is absorbed and scattered by the sample, the total cross section is the sum of the absorption cross section (σa ) and scattering cross section (σs ). σ T = σa + σs

(2.6)

The neutron scattering occurs by the interaction between neutron and nuclei and also between the magnetic spins of neutron and nuclei. The magnetic scattering is related to the magnetic moment of the nuclei, useful for the study of the spatial distribution of magnetic spins. The former case is treated here. The scattering intensity (or cross-sectional area) is expressed using the nuclear scattering amplitude a. The cross section (dσs /d) is divided into the coherent and incoherent scatterings. When the scattering amplitude of the m-th nucleus is am and its position is Rm , then (dσs /d) =

 

am an exp[i k(Rm − Rn )] m n  2    |δam |2 > exp(i k Rm ) + N< = m

= (dσs /d)coherent + (dσs /d)incoherent

m

(2.7)

where am = + δam . The former is the coherent scattering and the latter the incoherent scattering. The Bragg scattering by the nuclei in the crystal lattice is needed for the structure analysis. The corresponding coherent scattering amplitude is given as , which is not dependent on the scattering angle because the quite

290

2 Structure Analysis by Wide-Angle Neutron Diffraction Method

small nuclei contribute to this scattering. This point is largely different from the X-ray scattering by the electron clouds.  2   exp(ik Rm ) (dσs /d)coherent = 

(2.8)

m

Structure factor F(k) =

 m

exp(ik Rm )

(2.9)

The value is listed in Table 2.2 and Fig. 2.1. Table 2.2 Neutron scattering amplitude (b) and cross-sectional area (σ) of nuclei σ(coh) /10–24 cm2

σ(incoh) /10–24 cm2

Nucleus

b(coh) /10–13 cm

b(incoh) /10–13 cm

H

−3.739

25.274

1.7583

80.27

D

6.671

4.04

5.592

2.05

C

6.646

0

5.551

0.001

N

9.37

2.0

O

5.803

0

4.232

0

I

5.28

1.58

3.5

0.31

Fig. 2.1 Coherent scattering factors (amplitude) of the various atoms. Note that the neutron scattering factor (and the cross-sectional area) is constant for sin(θ)/λ, different from the case of X-ray scattering

11.03

0.5

2.2 Collection of Neutron Diffraction Data

291

2.2 Collection of Neutron Diffraction Data 2.2.1 Atomic Reactor Neutron is generated by the nuclear fission of 235 U. 235

U + n →95 Mo +139 La + 2n

The reaction occurs in a chain reaction mode and the number of neutrons is increased with time. The atomic reactor is built up for taking out neutrons from 235 U (Fig. 2.2). Since the thus-generated neutrons fly at a speed close to that of light, the neutron beams are cooled by passing through the D2 O moderator tank. The reactor is covered perfectly by thick concrete walls and the neutrons are taken out through Aluminum pipes. These neutrons enter the guide tube, which is a tube with Ni layer coated on the flat glass surface. The total reflections occur on the surface of the guide tube and the neutrons with the energies lower than some critical value are used for the experiments.

Fig. 2.2 a Atomic reactor, b rotation selector, c an outward aspect of Japan Atomic Energy Agency JRR-3, and d an inside map of installments of (c) (Reprinted from https://jrr3.jaea.go.jp/jrr3e/2/21. htm, copyright JAEA)

292

2 Structure Analysis by Wide-Angle Neutron Diffraction Method

The energies of the generated neutron beams distribute following the Maxwell distribution, and so the neutron component with a particular wavelength is taken out using a monochromator. One is to use the crystal plates (for example, Si, Ge, and graphite), on the surface of which the neutrons are reflected and only the component satisfying the Bragg equation is lead to the sample position using a collimator. Another type is to use the velocity selector. The two circular disks are rotated at the different timings and only the neutron beam components passing through the two holes of these disks can be taken out. There are various kinds of detectors of neutron signals. One of the most typical detectors is a gas-filled detector. The filled gas in an aluminum vessel is 3 He (3 He + n → 3 He ion) or BF3 (10 B + n → α + 7 Li). For the measurement of the 1D diffraction profile, these detectors are arrayed along one line. A ZnS fluorescent powder is used popularly as a scintillator, which is mixed with 6 LiF, etc. It emits the fluorescence light by an irradiation of neutron (6 Li + n → α, Zn + α → visible light), which is detected by such a sensitive detector as a photomultiplier. Many optical fibers with ZnS powder are arranged in 2D array, which is now used as the 2D detector. The neutron signals are incident the optical fibers and the emitted fluorescent lights are amplified and detected by the photomultipliers, as mentioned above. Another useful 2D detector is an imaging plate for neutron signals, as shown in Fig. 2.3 [4–6]. As mentioned in Chap. 1, the imaging plate is a thick polyester plate containing Gd2 O3 , Pb (γ-ray) IP(Gd2O3+ Ba2BF+Eu2+) neutron beam

slit

sample

beam stopper

B4C (external neutron) IP reader neutron beam

B4C (external neutron) shutter

Si(111)

beam stopper monitor

Pb (γ-ray) IP(Gd2O3+ Ba2BF+Eu2+)

Fig. 2.3 A schematic illustration of neutron imaging plate camera, BIX-3 system. Reprinted from Ref. [12] with permission of the American Chemical Society, 2004

2.2 Collection of Neutron Diffraction Data

293

Ba2 BF, and Eu2+ . The film was defended from γ-ray by covering with a lead plate and also defended from neutron scatterings originated from outer sources by covering with B4 C plate. The 2D scattering image is read by irradiating a rotating He–Ne laser beam on the imaging plate moved continuously along the vertical direction.

2.2.2 Accelerator As shown in Fig. 2.4, the highly accelerated electrons or protons with a high speed close to the light velocity are incident to a target (W, U, Hg etc.) [7–10]. The nuclear fission decay occurs inside the target and the neutrons are emitted. This phenomenon is called the spallation. The emitted neutrons are pulses, since the protons are circularly accelerated in a similar way to that of synchrotron system. The pulse consists of the neutron components with the different wavelengths (or different energies). The velocity of the neutron beams of the various wavelengths is different from each other and so the time to reach the detector is also different among them. By measuring the time needed to fly from the source to the detector, we can know the wavelength of this neutron beam component. The experimental method using these pulse neutrons is called the time-of-flight (TOF) method (see Sect. 2.4). Figure 2.5 is a snap shot of J-PARC (Japan Proton Accelerator Research Complex), Tokai, Japan. synchrotron

Experimental hatch

proton Linear accelerator Hg

Fig. 2.4 Neutron pulse generation by spallation

Guide tube

294

(a)

2 Structure Analysis by Wide-Angle Neutron Diffraction Method

(b)

(c)

Fig. 2.5 MLF, J-PARC a, b Hg target, c experimental hall (https://mlfinfo.jp/ja/blmap.html)

2.3 Crystal Structure Analysis by Neutron Diffraction Data The crystal structure analysis based on the neutron diffraction data is essentially the same as that of the X-ray structure analysis [11, 12]. The structure factor is given as    F(hkl) = j f j (2θ) exp 2πi hxj + kyj + lz j

(2.10)

I (hkl) = K L Am|F(hkl)|2

(2.11)

Here the definition of the parameters is the same as those in the X-ray diffraction case. However, the property of these parameters is appreciably different. The atomic scattering factor f j (= ) and the cross-sectional area of a nucleus are not dependent on the scattering angle of neutron as already illustrated in Fig. 2.1 and Table 2.2. The absorption of neutron by a nucleus is almost negligible (A = 1), though a sample with too large thickness needs some correction. The Lorentz factor (L) correction is needed to do, but the correction of polarization factor (p) in the X-ray scattering case is not needed to take into account. About the cross-sectional area for neutron scattering, we have already studied in the previous sections. In the structure analysis of a polymer crystal, the deuterated sample is the most ideal since the cross-sectional area of the D species is almost comparable to that of C atom, and the H atom gives

2.3 Crystal Structure Analysis by Neutron Diffraction Data

295

appreciably high incoherent scattering as a background and the coherent scattering is relatively weak with the negative value of the cross-sectional area.

2.3.1 High-Density Polyethylene Figure 2.6 shows the 2D neutron diffraction patterns measured for the uniaxially oriented samples of the normal or hydrogenous high-density polyethylene (H4HDPE) and the fully deuterated high-density polyethylene (D4-HDPE) samples [12, 13]. The samples were melted and quenched into ice-water bath, which were stretched by about 7 times the original length on a hot plate, followed by heat treatment at 120 °C for a few hours under tension. These samples were cut into several pieces of rods, and these oriented rods were bundled together in parallel to prepare the thick samples for the neutron diffraction measurements (Fig. 2.7). The size of the samples was 5 mm (l) × 2 × 2 mm2 (w). The usage of thin wire was easy to bundle these rod specimens, where the material should be Aluminum tape because neutron beam passes Al with very weak scattering. If Cu wire is used for bundling of rods, the wire part must be covered with Cd plate to avoid the radioactivation of wire by neutron. Fig. 2.6 2D wide-angle neutron diffraction patterns measured for the uniaxially oriented D4-HDPE and H4-HDPE samples. Reprinted from Ref. [12] with permission of the American Chemical Society, 2004

D4-HDPE

H4-HDPE

296

2 Structure Analysis by Wide-Angle Neutron Diffraction Method

monochromatized neutron beam

Al tape Cd plate goniometer head

Fig. 2.7 Setting of a bundle of rods on Cd plate, which is set on a goniometer head to adjust the c-axis to the vertical direction

The D4-HDPE shows the clear diffraction pattern with the weak background, while the H4-HDPE shows the relatively weak Bragg diffraction spots with the strong background. The positions and integrated intensity were estimated for all the D4-HDPE

H4-HDPE

Fig. 2.8 Nuclear density distribution obtained for D4-HDPE and H4-HDPE crystals. Reprinted from Ref. [12] with permission of the American Chemical Society, 2004

2.3 Crystal Structure Analysis by Neutron Diffraction Data

297

observed Bragg diffraction spots and the structure analysis was performed for these two samples. In the calculation of structure factors, the atomic scattering factors or the cross-sectional areas were constant as said above. The unit cell parameters were almost the same as those determined by the X-ray diffraction data (It is noted that the X-ray diffraction pattern is of course the same between the H4- and D4-HDPE samples since the number of electron distributing to the X-ray scattering is essentially the same for them). Figure 2.8 shows the nucleus density distribution maps obtained for D4- and H4-HDPE samples. In the case of D4-HDPE map, the positive peaks are detected at the positions of C and D atomic nuclei. The H4-HDPE shows the negative peaks for the H atomic positions due to the negative cross-sectional area of the H nucleus. The random copolymer samples of D and H ethylene monomeric units show systematic change of the WAND profiles as seen in Fig. 2.9a, where the uniaxially oriented samples were used for the measurement of the equatorial diffraction patterns. The 1D profiles are shown in Fig. 2.9b which were obtained by integration of each

Fig. 2.9 a Wide angle neutron diffraction patterns measured for a series of D/H random copolymers of PE with the different ratio. b The 1D profiles along the equatorial line. Reprinted from Ref. [13] with permission of the Springer Nature, 2014

298

2 Structure Analysis by Wide-Angle Neutron Diffraction Method

Fig. 2.10 Nuclear density distributions derived by the WAND data analysis of a series of D/H random copolymer of HDPE. Reprinted from Ref. [13] with permission of the Springer Nature, 2014

profile. The relative intensity of the various reflections is changed systematically with a change in D/H ratio, and the incoherent scattering becomes large with the higher content of H component. The structure analysis was made using the 1D diffraction profiles to obtain the structures projected along the chain axis. The H positions are detected clearly but the peak height changes from positive to negative value as the hydrogen atoms are changed from D to H (Fig. 2.10). The simulation of the diffraction pattern was made by assuming the statistically random distribution of D and H atoms on the polyethylene zigzag chains in the orthorhombic crystal. The calculated profiles are reproduced quite well (Fig. 2.11). These data show clearly the correctness of the analysis procedure and the perfectly random distribution of the CH2 CH2 and CD2 CD2 monomeric units along the chain axis.

2.3 Crystal Structure Analysis by Neutron Diffraction Data

299 PE_WANS (D70H30) 2.35 A

Intensity

Intensity

PE_WANS (D100) 2.35 A Calcd

Calcd

Obsd

20

30

40

50

60

70

80

90

100

20

30

40

50

40

50

70

90

100

Calcd

80

Obsd

*

Obsd

60

80

PE_WANS (D30H70) 2.35 A

Calcd

30

70

Intensity

Intensity

PE_WANS (D50H50) 2.35 A

20

60

2theta/degree

2theta/degree

*

Obsd

*

*

90

100

20

30

40

2theta/degree

50

60

70

80

90

100

2theta/degree

Intensity

PE_WANS (H100) 2.35 A

Calcd

Obsd

20

30

40

50

60

70

80

90

100

2theta/degree

Fig. 2.11 Comparison of the observed and calculated WAND equatorial line profiles for a series of D/H random copolymer of HDPE. Reprinted from Ref. [13] with permission of the Springer nature, 2014

2.3.2 Polyoxymethylene The similar analysis was made for POM-h2 and POM-d2 crystals [14]. For each sample, a bundle of γ-ray-polymerized multi-crystals, the diameter of which was about 2 mmφ , was rotated around the chain axis and the diffraction pattern was recorded on the 2D imaging plate. As shown in Fig. 2.12, the background of the POM-h2 sample is relatively high (although the contribution is not very remarkable) and the Bragg reflections are weak, different from the case of POM-d2 sample. The data analysis was made in a similar way as that mentioned above. The results are shown in Fig. 2.13. In the analysis of POM-h2 data, the H peaks are detected as negative peaks, and the D peaks as positive peaks in the analysis of POM-d2 data, the positions of which correspond well to the X-ray structure analysis based on the synchrotron X-ray diffraction data (see Sect. 1.13.2).

300

2 Structure Analysis by Wide-Angle Neutron Diffraction Method WAND

(a) POM-h 2

(b) POM-d 2

Fig. 2.12 2D WAND patterns measured for the oriented POM-h2 and POM-d2 . Reprinted from Ref. [14] with permission of the Springer nature, 2014 (a) POM-h2 Fo(+)

Fo(-)

(b) POM-d2 Fo(+)

Fig. 2.13 Chain conformation of POM showing the nuclear density distribution. Reprinted from Ref. [14] with permission of the Springer nature, 2014

2.3 Crystal Structure Analysis by Neutron Diffraction Data

301

2.3.3 Atactic-PVA and Its Iodine Complex One of the most effective usages of the neutron structural analysis technique is the study of hydrogen bonds. The X-ray crystal structure analysis of atactic PVA sample was already mentioned in Sect. 1.13.3. The neutron diffraction data measured for the uniaxially oriented deuterated PVA sample [-CD2 CD(OH)n -] were used for checking the reliability of the X-ray analyzed structure of PVA. The crystal structure shown in ball-stick model in Fig. 2.14, which was derived from the X-ray data analysis, gives the neutron diffraction profile of Fig. 2.15a, which is not in good agreement with the observed data, in particular, the relative intensity of the 110 and 110 peaks. In order to satisfy both of the X-ray and neutron diffraction data consistently, an introduction of packing disorder was needed. A pair of zigzag chains included in the unit cell is copied and pasted in the unit cell to create another pair shown in green color in Fig. 2.14. These two pairs (ball-stick model and green-color model) are existent at 50% statistical probability. This statistically disordered model satisfies both of the X-ray and neutron data consistently, as shown in Fig. 2.15b [15]. The coexistence

(a)

(b)

Fig. 2.14 a Crystal structure of at-PVA (ball-stick model) and disordered model (green color). b The structure disorder due to the slippage along (110) plane. Reprinted from Ref [15] with permission of the American Chemical Society, 2020

302

2 Structure Analysis by Wide-Angle Neutron Diffraction Method

Fig. 2.15 Comparison of the observed equatorial line profiles of (a-1) X-ray and (a-2) neutron data with those calculated for the regular model shown in Fig. 2.14a. (b-1) and (b-2) are for the structurally disordered model shown in Fig. 2.14a (green model). Reprinted from Ref [15] with permission of the American Chemical Society, 2020

of the ball-stick model and green-colored model in the unit cell indicates actually that PVA crystallites consist of the domains of regular packing structure with the 1/2 slippage along the 110 plane, as illustrated in Fig. 2.14b.

2.3 Crystal Structure Analysis by Neutron Diffraction Data

303

Fig. 2.16 a Crystal structure of at-PVA-Iodine complex, b the statistically disordered model of (a) (Translational model). The pairs of PVA-iodine couples (pink and purple) are translated 1/2 along the a axis to give the second pairs (green and blue). c The possible local structures imagined from the statistically disordered model (b). The large circles indicate the Iodine ions. The small circles are K ions. Reprinted from Ref [15] with permission of the American Chemical Society, 2020

The similar situation can be seen for PVA-iodine complex prepared by immersing the oriented PVA-d4 sample into 3M iodine-KI solution [16]. The regular packing structure, shown in Fig. 2.16a, which was proposed by analyzing the X-ray diffraction data, did not reproduce the observed WAND data at all (Fig. 2.17a). As similarly to the case of at-PVA itself, the introduction of packing disorder, as shown in Fig. 2.16b, could reproduce both of WAXD and WAND data of the complex consistently as shown in Fig. 2.17b.

304

2 Structure Analysis by Wide-Angle Neutron Diffraction Method

Fig. 2.17 Comparison of the observed equatorial line profiles of (a-1) X-ray and (a-2) neutron data with those calculated for the regular model shown in Fig. 2.16a. (b-1) and (b-2) are for the structurally disordered model shown in Fig. 2.16b. Reprinted from Ref [15] with permission of the American Chemical Society, 2020

These two cases are good examples to show the danger of the X-ray structure analysis using the small number of the observed diffraction peaks, and they show also the importance of the combined usage of the various kinds of experimental data to check the proposed model from the various points of view, in these cases the hybridization of X-ray and neutron diffraction data.

2.3 Crystal Structure Analysis by Neutron Diffraction Data

305

2.3.4 Deformed Electron Density Distribution The X-ray structure analysis gives us the information of structure factors F(k), from which the total electron density distribution ρ(r) can be estimated through the Fourier transform of F(k). (2.12) ρ(x) = (1/V ) Fobs X (k)e−ikx dk where k is a scattering vector and V is the volume of the unit cell. Before the chemical bonds are formed, the electron clouds distribute homogeneously around the individual atoms, but they are deformed once when the atoms are bonded to form the molecule. In other words, the originally spherical electron density distribution is deformed and electron density becomes preferentially high at the middle position of the bonds. The part of electron density concentrated to the bond is called the “bonded” or “deformed” electron density distribution. The information on the density distribution of the bonded electrons ρ(x) can be obtained by subtracting the nuclear density ρ o (x) of isolated atoms Δρ(x) = ρ(x) − ρo (x)

(2.13)

where the ρ o (x) is generally assumed to be spherical. The centers of electron density clouds of isolated atoms might be assumed approximately as the atomic nucleus positions. ρo (x) = (1/V )

Fcalc X (k)e−i kx dk

(2.14)

F calc X (k) is the structure factor calculated for the crystal structure by assuming that the atomic positions xX j and the thermal parameters U X j determined by the X-ray analysis (or the gravity centers of electron clouds) are equal to those of the nuclei where the atomic scattering factors f j X (k) of the isolated and non-interacting atoms without bondings are used: Fcalc X (k) =

 j

    f jX (k) exp i kx Xj exp i kU Xj

(2.15)

306

2 Structure Analysis by Wide-Angle Neutron Diffraction Method

Fig. 2.18 The bonded electron density distribution obtained by the difference Fourier transform based on the X-ray diffraction data analysis of PDCHD single crystal. Reprinted from Ref. [20] with permission of the American Chemical Society, 2018

The change of electron density distribution before and after the molecular formation, or the difference of the density distribution ρ(x) is calculated as Δρ(x) = ρ(x) − ρo (x) = (1/V )



 Fobs X (k) − Fcalc X (k) e−i kx dk

(2.16)

Unfortunately, because of the above-mentioned problem, the apparent nuclear positions are displaced from the true positions, the bonded electron density shows a curious pattern as shown in Fig. 2.18. The interatomic position is too short and the central electron clouds are also erased as illustrated in Fig. 2.19c. What we have to do is to know the true positions of nuclei by performing the neutron data analysis. The exact density distribution is obtained by the following equation.      x Δρ(x) = (1/V)k j Fobs X k j − Fcalc N k j e−i k j

(2.17)

where F calc N (kj ) is the structure factor calculated from the isolated-atom X-ray scattering factors f j X (k) combined with the parameters of nuclear positions xN j

2.3 Crystal Structure Analysis by Neutron Diffraction Data

307

Fig. 2.19 a The total electron density distribution derived by Fourier transform of X-ray-analyzed structure factors. b and c: the residual electron density distribution calculated by subtracting the spherical density distributions of the isolated atoms located (b) at the nuclear positions and (c) at the center of gravity of electron density picked up from (a). Reprinted from Ref. [20] with permission of the American Chemical Society, 2018

and thermal parameters U N j determined by neutron diffraction data analysis. This method is called the X-N method [17–19]. The actual example is seen for a giant single crystal of polydiacetylene [20]. The result of X-N method is shown in Fig. 2.20. The electron densities distribute at the central positions of each CC bonds. The density distribution along the CC triple bond is compared between the X-N method and the density functional theoretical calculation. The agreement between them is quite nice. As illustrated in Fig. 2.19b, the utilization of the positions of the carbon nuclei derived by the neutron data analysis gives the reasonable subtraction to result in the distribution shown in Fig. 2.20. We need to find the reason why the X-ray data analysis gives the apparently shorter C≡C distance from the different point of view. The atomic scattering factor consists of the contributions from the core and valence electron clouds. As shown in Fig. 2.21, the contribution of the outer-shell electrons or the valence electrons is relatively large in the low scattering angle region, while that of the core electron changes slowly in a wide angle range. The core electron contribution is overwhelmingly large in the higher angle region. Therefore, the asphericity shifts might be removed if enough higher order reflections are measured accurately. The whole electron density distribution ρ(x) is made by the analyses of the X-ray diffraction data collected in the lower angle range and the positions of the atomic cores or ρ o (x) are derived by the

308 Fig. 2.20 a The 2D map of the bonded electron density distribution of PDCHD chain calculated using the X-N method. b Comparison of the bonded electron density distribution along the C C≡C C bond of the skeletal chain between the experimental and DFT calculated results. Reprinted from Ref. [20] with permission of the American Chemical Society, 2018

2 Structure Analysis by Wide-Angle Neutron Diffraction Method

(a)

analysis in the higher angle range. Another problem is to separate the contribution of the atomic displacements due to the thermal vibrations from the effect of the deformed electron density. The former effect might be able to erase by performing the diffraction measurement at enough low temperature [21].

2.4 Structure Analysis by TOF Method

309

Fig. 2.21 The X-ray atomic scattering factor of neutral C atom separated to the contributions of the core and valence electrons. Reprinted from Ref. [20] with permission of the American Chemical Society, 2018

2.4 Structure Analysis by TOF Method 2.4.1 Principle of TOF Figure 2.22 shows the i-BIX system set in the beam line 03 of MLF (Materials and Life Science Experimental Facility), J-PARC, Tokai-mura, Japan. The goniometer head is set at the center of the instrument, around which many 2D detectors are set in the radial directions. The incident neutrons are pulses at 25 Hz (see Fig. 2.23). The pulse contains a wide range of neutron components of the different wavelengths. This can be understood by considering the Fourier transform of a pulse function: f (t) = A for t = −ta /2 ∼ ta /2 0 for t > ta /2, t < −ta /2

(2.18)

  The Fourier transform F(ω) = t=−∞∼∞ f (t)cos(2πtω)dt = A t=−ta/2∼ta/2 cos(2πtω)dt = (A/πω) sin(πt a ω). For t a = 10 s and 100 μs, for example, the wave components of angular frequency ω are distributed as shown in Fig. 2.24. In the actual neutron beam generation, the pulse width is in the range of 20 ~ 100 μs, corresponding to the wide range of 0 ~ 1 kHz frequency. These different neutron components are incident on the sample and scattered into all the directions. As mentioned above, the neutron with the different wavelength (and so frequency) has a different flying velocity, meaning the different arrival time at the detector, as illustrated in Fig. 2.23.

310

2 Structure Analysis by Wide-Angle Neutron Diffraction Method

Fig. 2.22 i-BIX neutron diffraction system in the MLF, J-PARC, Japan. (https://mlfinfo.jp/ja/ blmap.html)

40 msec

100 μsec

neutron pulse time

distance from generator to detector

arrival time

Fig. 2.23 Neutron pulses and the arrival times to the detector

The arrival time at the detector t = L 2 /λ where L 2 is the sample-to-detector distance (= 490 mm). However, in the actual experiment, the time = 0 is defined as the time of generation of the neutron pulses, and so the time necessary to reach the sample position or the time from the pulse source position to the sample position, L 1 , about 40 m, must be included. So, the time is given as

2.4 Structure Analysis by TOF Method

311

Fig. 2.24 Frequency distributions of a pulse with t a = 10 s and 100 μs

t = (L 1 + L 2 )/v

(2.19)

More detailed description is made for i-BIX or the Ibaraki Biological Crystal Diffractometer installed in the beam line 03 at the Material and Life Science Experimental Facility (MLF), J-PARC, Japan [7–10]. The pulsed neutron beam, which is generated from the fission decay (or spallation) of mercury by shooting the highly accelerated proton particles, is incident to the sample at 25 Hz through the collimator of 5 mmφ . The proton acceleration power is 200 kWatt ~ 1 MW. The wavelength range of the thus-generated neutron is 0.7 ~ 4 Å. Since the diffracted neutron satisfies the Bragg equation, we have such a relation. 2d sin(θ ) = λ(Å) = (h/m o )/v = 3956/v(m/s)

(2.20)

t = (L 1 + L 2 )/v = (L 1 + L 2 )2d sin(θ )/3956

(2.21)

When a certain detector set at a fixed angle collects the diffracted signals with the different d values, the arrival time of these signals is given by the above equation. L 1 + L 2 = 40.49 m and θ = 30° (for example) and then t = 5.118 ms for d = 0.5 Å and t = 10.236 ms for 1 Å signal. The 20 ~ 30 position-sensitive detectors were set around the sample. The detector was a scintillation counter composed of ZnS scintillator with the dopants Ag and 10 B2 O3 , and the emitted photons were collected using the photomultiplier, as mentioned above. The TOF measurement for one sample orientation was made for about several hrs by shooting the 183,600 proton pulses to the mercury target at 25 Hz. A sample is set on a goniometer head and the diffraction data are collected for the many different sample orientations at φ (0°, –90° and + 90°) and ω angles (every 25° in the range of –90° ~ + 85°). The total collection time was several days. The thus-collected data were processed using

312

2 Structure Analysis by Wide-Angle Neutron Diffraction Method

Fig. 2.25 Collection and data treatment of diffracted neutron signals

a software STARGazer developed there. Figure 2.25 shows the process of TOF data accumulation. The diffracted neutron signals are collected by an individual detector at every 1 μs timing, which is called an event data. The 400 event data are summed up together to give one histogram. Since the time of one neutron flight (or the total time-of-flight of one neutron) is 40 ms (corresponding to 25 Hz), the 100 histograms are produced in total for one detector for one pulse. If the total number of sample settings is 54, these histograms are accumulated for 183,600 pulses to make the intensity data I ij (x k , yk , TOFk ) where k = 1 ~ 400 histograms, i (sample setting) =

2.4 Structure Analysis by TOF Method

313

1 ~ 54, and j (detector number) = 1 ~ 20. As a result, the 400 × 54 × 20 = 432,000 histograms are collected in total, which are transferred to the data analysis. It must be noted that the flight time is shorter than 40 ms if we want to measure the diffracted signals without any overlap between the sets of the signals. For the diffraction of d = 5 Å, for example, t = 51.18 ms, and the signal overlaps with the next arrival set of diffracted signals (frame overlapping). In such a case we have to use the other detector of the different angle. When a single crystal is used as a sample, the thus obtained 2D histograms are used for the data analysis. At first, the peak search is made for all the histograms to know the diffraction peak positions (x, y) at the different TOF or the different wavelength of incident neutron (or the different diffraction angle), from which the unit cell parameters and hkl indices are estimated as well as the UB matrix. The integrated intensity of the individual diffraction spots is evaluated by summing up all the diffraction peaks at the same positions and same TOF after the background correction.

2.4.2 Data Collection by TOF Method It is useful for the structure analysis to take the 2D diffraction pattern of a uniaxially oriented polymer sample since we know many technics for analyzing this type of pattern. As mentioned above, in the TOF measurement, the wavelength distributes in a wide range. This means that the radius of Ewald sphere (= 1/λ) changes continuously in a range of effective wavelengths. As we already studied, an Ewald sphere of a radius 1/λ is drawn at the center position of the sample. The reciprocal lattice is set so that the origin crosses the sphere edge. So, the Ewald spheres corresponding to each neutron beam of the different wavelength λ are drawn as shown in Fig. 2.26a. In order to measure the reciprocal lattice points 001, 002, 003, and so on, the sample is tilted so that the 00l points cross the spheres of the different size. The tilting angle θ must satisfy the Bragg reflection law 2d 00l sin(θ) = λ. Since d 00l = c/l, sin(θ) = λ/(2d 00l ) = lλ/(2c). The Ewald sphere of the radius r = 1/λ = l/[2c sin(θ)] crosses the 00l reciprocal point. Since the λ and r change continuously in the TOF experiment, all the 00l reflections enter the same detector directed to the same angle 2θ. For example, sin(θ003 ) = 3λ003 /(2c) and sin(θ001 ) = λ001 /(2c). If λ003 = λ001 /3, then sin(θ003 ) = sin(θ001 ) or θ003 = θ001 . The wavelength λ changes continuously and all the 00l reflections enter the same detector directed to the same 2θ. In this way, we can measure a series of 00l reflections by a TOF method. Of course we have to notice that these 00l reflections arrive at the detector at different timings. As seen in Fig. 2.22, the i-BIX system has many 2D detectors surrounding the sample. If we happen to use only a detector D set at the angle φ = 139° from the direction of the incident neutron beam and if we want to measure the equatorial reflections (h00), the sample setting shown in Fig. 2.27 might be one idea. As shown in Fig. 2.22c, the c-axis of the oriented sample is tilted from the incident neutron

314

2 Structure Analysis by Wide-Angle Neutron Diffraction Method

(a)

(b) 003 002

00/

detecor

detector

001 neutron

neutron

sample

sample

Fig. 2.26 a The geometric relation of a series of 00l reflections crossing the Ewald spheres with the various radiuses. The sample is tilted by the angle θ and the neutron signals diffracted at the various 00l lattice planes direct to the same detector at the diffraction angle 2θ. b In the case of pulsed neutron beam, the Ewald sphere of a continuous radius is set (red zone). Then, many 00l are detected using one detector. The maximal radius of the sphere is determined by the shortest wavelength of the neutron components contained in the actual neutron pulse

direction by 69.5° (φ). If the h00 planes satisfy the Bragg’s condition and scattered beam goes to the detector (139°), 2θ = 180° – 139° = 41° and θ = 20.5°, which is equal to 90°–69.5°. That is to say, the neutron beam incident to the h00 planes is scattered at θ = 20.5° and it can enter the detector D in a back scattering mode. Since the sample is uniaxially oriented, a series of hk0 reflections enter the detector D, as shown in (b). In this experiment, we can know the arrival time of scattered neutron t and 2θ (= 180° – φ) from the detector position, then we can evaluate the d value using Eq. 2.21. This sample scatters the neutron beam into all the directions. Which detector is useful for the measurement of the hk2 layer-line reflections ? It is easy to consider by using one Ewald sphere. The sample is now at the position A and the hk0 reflection crosses the sphere as seen in (b). At this timing, the purple arrow is drawn from the sample center to the hk2 point crossing the sphere. The angle between the neutron incident direction (blue) and the arrow (purple) is 75°. The detector installed at this angle from the sample position is A. Since the wavelength is continuously distributed in the TIF experiment, all the hk2 and some hk1 reflections can be detected using this detector A.

2.4.3 Fiber Diffraction Pattern by TOF Method The oriented D4-HDPE sample was cut into rods, which were bundled together as shown in Fig. 2.7. The thus-obtained bundle was set horizontally (not vertically) on a goniometer head. This setting was to realize the situation shown in Fig. 2.27 in a

2.4 Structure Analysis by TOF Method

315

Fig. 2.27 a A detector D is accidentally set at the direction 139° from the incident neutron direction. The h00 reflections satisfying the Bragg’s law can be measured in a backscattering mode. b Since the sample is uniaxially oriented, all the hk0 reflections satisfy this condition. c The Bragg’s law for the forward- and back-scatterings. As seen in this figure, 2θ = 180° – φ. Depending on the detector direction the condition must be changed of course. d Sample setting. A bundle of oriented polyethylene rods was set on a goniometer head. The left-side pipe is a collimator

concrete way. In the experiments, we used many detectors set in the three dimension with the various orientations. The TOF patterns detected by the individual detectors are shown in Fig. 2.28. The reflections arrive at the detectors at the different timings. Figure 2.28 is the result of summing up these signals together. Since the crystallite orientation in the uniaxially oriented PE sample was not very high, the observed reflections are long. The pattern detected by the detector D contains only the hk0 reflections. The detector A collects the reflections hk2 and hk1 as seen in Fig. 2.27. The TOF measurements were repeated by changing the direction of the oriented sample to cover all the reciprocal space.

316

2 Structure Analysis by Wide-Angle Neutron Diffraction Method

(a)

(b)

Fig. 2.28 TOF diffraction measurement of a uniaxially oriented D4-HDPE sample by using the different detectors (A, B, C and D) installed at the various angles around the sample. a An illustration showing the hkl layer lines of the reciprocal lattice and the detectors. All the hk0 points cross the Ewald sphere of the continuous size (green color) and are detected by the detector D set at φ = 139°. b The TOF diffraction patterns detected by the various detectors. The A collects the hk1 and hk2 reflections as understood from (a)

Because the positions of detectors and the arrival time are known in the measurements, the detected reciprocal lattice spaces are known and the intensity distribution of the scattered neutron signals can be obtained finally. The thus-obtained 2D diffraction pattern is shown in Fig. 2.29, where the dimensions of the coordinates are 1/d. The background, which was mainly originated from the incoherent scatterings of the residual H atoms in the sample as well as the amorphous component, was erased using the WAND pattern of an amorphous polyethylene plate. The layer lines were

2.4 Structure Analysis by TOF Method

317

Fig. 2.29 a The TOF 2D neutron diffraction pattern measured for a uniaxially oriented deuterated PE sample. b The simulated pattern using the crystal structure of the orthorhombic PE (Fig. 1.134). Note that the coordinates are in the dimension of 1/d

Fig. 2.30 The 1D diffraction profiles deduced from Fig. 2.19, where the x axis was converted to 2θ by assuming the wavelength 1.034 Å. The calculated profiles are in good agreement with the observed ones

detected up to the 4th line. By assuming a suitable wavelength λ, the diffraction profiles along the individual layer lines are transformed to those expressed by the scattering angle 2θ. The obtained diffraction profiles along the individual layer lines are compared with those calculated by using a crystal structure of orthorhombic PE, as shown in Fig. 2.30.

318

2 Structure Analysis by Wide-Angle Neutron Diffraction Method

2.4.4 TOF Neutron Diffraction Data Analysis of Single Crystal The i-BIX of MLF, J-PARC (Tokai-mura, Japan) was built up originally for the purpose of the detailed crystal structure analysis of protein single crystals, in particular, those with the deuterated water molecules to clarify the positions of water molecules around the protein molecule. The TOF measurement of a single crystal is made in the way mentioned above. The positions and integrated intensities of the individual reflections are evaluated. The structure analysis is made using a software Shlex, for example. As an example, the structure analysis results of PDCHD single crystal are shown here, the X-ray analysis of which was already mentioned in Sect. 1.13.4 [20]. The diffraction pattern on a computer display is shown in Fig. 2.31, where the sharp reflections were detected clearly on the histograms of the different detectors. In this case, the large single crystal of about 2 mm × 0.5 mm × 5 mm size was set on the goniometer head and the data were measured for the 54 different sample orientations at φ (0°, –90° and + 90°) and ω angles (every 25° in the range of –90° ~ + 85°). The total collection time was about 6 days. The total number of the observed spots was 16,938, among which the unique diffractions was 3027 for F o > 2σ (F o ). The Rmerge factor for the same spots was 14.4%. The R factor of the final structure was 17.3% for the data of F o > 2σ(F o ) and 10.5% for all the observed diffraction spots. The thus-derived crystal structure gave us the information of the atomic nuclei coordinates which were combined together with the structure derived by the X-ray diffraction data analysis to know the deformed electron density distribution Δρ(x) (refer to Sect. 2.3.4).

Fig. 2.31 A single crystal of PDCHD with the size of ca. 2 mm × 0.5 mm × 5 mm. The diffraction patterns measured with the individual detectors are displayed on the computer screens. This single crystal is a little strained as known from the splitting of the reflection spots

References

319

References 1. D.L. Price, K. Scold, Introduction to neutron scattering. Methods Exp. Phys. 23-A, 1–97 (1986) 2. R-J. Roe, Methods of X-ray and Neutron Scattering in Polymer Science. Oxford Univ. Press. Inc. (2000) 3. C.C. Wilson, Single Crystal Neutron Diffraction from Molecular Materials (World Scientific, Singapore, 2000). 4. N. Niimura, Y. Karasawa, I. Tanaka, J. Miyahara, K. Akahashi, H. Saito, S. Koizumi, M. Hidaka, An imaging plate neutron detector. Nucl. Instr. Methods Phys. Res. A349, 521–525 (1994) 5. I. Tanaka, K. Kurihara, T. Chatake, N. Niimura, A high-performance neutron diffractometer for biological crystallography (BIX-3). J. Appl. Cryst. 35, 34–40 (2002) 6. N. Niimura, T. Chatake, A. Ostermann, K. Kurihara, I. Tanaka, High resolution neutron protein crystallography hydrogen and hydration in proteins. J. Z. Kristallogr. 218, 96–107 (2003) 7. K. Kusaka, T. Hosoya, T. Yamada, K. Tomoyori, T. Ohhara, M. Katagiri, K. Kurihara, I. Tanaka, N. Niimura, Evaluation of performance for IBARAKI biological crystal diffractometer iBIX with new detectors. J. Synchrotron Radiat. 20, 994–998 (2013) 8. K. Kusaka, T. Hosoya, I. Tanaka, N. Niimura, K. Kurihara, , T. Ohhara, current status of ibaraki biological crystal DiffractometeriBIX - several examples of the measurement. J. Phys. Conf. Series. 251, 012031 (2010) 9. K. Kusaka, T. Ohhara, I. Tanaka, N. Niimura, T. Ozeki, K. Kurihara, K. Aizawa, Y. Morii, M. Arai, K. Ebagata, Y. Takano, Peak overlapping and its De-convolution in TOF diffraction data from neutron biological diffractometer in J-PARC. Phys. B 385–386, 1062–1065 (2006) 10. T. Ohhara, K. Kusaka, T. Hosoya, K. Kurihara, K. Tomoyori, N. Niimura, I. Tanaka, J. Suzuki, T. Nakatani, T. Otomo, S. Matsuoka, K. Tomita, Y. Nishimaki, T. Ajima, S. Ryuuhuku, Development of data processing software for a new TOF single crystal neutron diffractometer at J-PARC. Nucl. Instrum. Methods Phys. Res. A. 600, 195–197 (2009) 11. Y. Nishiyama, N.P. Langan, H. Chanzy, Crystal structure and hydrogen-bonding system in cellulose Ibeta from synchrotron X-ray and neutron fiber diffraction. J. Am. Chem. Soc. 124, 9074–9082 (2002) 12. K. Tashiro, I. Tanaka, T. Ohhara, N. Niimura, S. Fujiwara, T. Kamae, Extraction of hydrogen atom positions in polyethylene crystal lattice from the wide-angle neutron diffraction data collected by 2-dimensional imaging plate system: a comparison with the X-ray and electron diffraction results. Macromolecules 37, 4109–4117 (2004) 13. K. Tashiro, M. Hanesaka, H. Yamamoto, K. Wasanasuk, P. Jayaratri, Y. Yoshizawa, I. Tanaka, N. Niimura, K. Kusaka, T. Hosoya, T. Ohhara, K. Kurihara, R. Kuroki, T. Tamada, S. Fujiwara, K. Katsube, K. Morikawa, Y. Komiya, T. Kitano, T. Nishu, T. Ozeki, Accurate structure analyses of polymer crystals on the basis of wide-angle X-ray and neutron diffractions. KobunshiRonbunshu 71, 508–526 (2014) 14. K. Tashiro, M. Hanesaka, T. Ohhara, T. Ozeki, T. Kitano, T. Nishu, K. Kurihara, T. Tamada, R. Kuroki, S. Fujiwara, I. Tanaka, N. Niimura, Structural refinement and extraction of hydrogen atomic positions in polyoxymethylene crystal based on the first successful measurements of 2-dimensional high-energy synchrotron X-ray diffraction and wide-angle neutron diffraction patterns of hydrogenated and deuterated species. Polym. J. 39, 1253–1273 (2007) 15. K. Tashiro, K. Kusaka, H. Yamamoto, M. Hanesaka, Introduction of disorder in the crystal structures of atactic poly(vinyl alcohol) and its iodine complex to solve a dilemma between X-ray and neutron diffraction data analyses, Macromolecules 53, 6656–6671 (2020) 16. K. Tashiro, H. Kitai, S.M. Saharin, A. Shimazu, T. Itou, Quantitative crystal structure analysis of poly(vinyl Alcohol)-Iodine complexes on the basis of 2D X-ray diffraction, raman spectra, and computer simulation techniques. Macromolecules 48, 2138–2148 (2015) 17. Accurate Molecular Structures. Their Determination and Importance edt. By A. Domenicano, I. Gargittai, I. Oxford Science Publ. (UK) (1991) 18. P. Coppens, X-ray Charge Densities and Chemical Bonding. Oxford University Press Inc. (UK) (1997)

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19. P. Coppens, Electron density from X-ray diffraction. Annu. Rev. Phys. Chem. 43, 663–692 (1992) 20. K. Tashiro, K. Kusaka, T. Hosoya, T. Ohhara, M. Hanesaka, Y. Yoshizawa, H. Yamamoto, N. Niimura, I. Tanaka, K. Kurihara, R. Kuroki, T. Tamada, Structure analysis and derivation of deformed electron density distribution of polydiacetylene giant single crystal by the combination of X-ray and neutron diffraction data. Macromolecules 51, 3911–3922 (2018) 21. B. Bagautdinov, H. Tanaka, C-H. K. Shih, S. Sugimoto, S. Sasaki, K. Tashiro, M. Takata, New Insight into the polymerization and structural mechanisms of polydiacetylene DCHD: an X-ray/MEM study. Acta Cryst. A67, C27–28 (2011)

Chapter 3

Structure Analysis by Electron Diffraction Method

Abstract Electron diffraction is different from the X-ray and neutron diffraction phenomena, the characteristic point of which is leaned at first. The principle of electron microscopy (TEM and SEM) is leaned by focusing on the diffraction phenomenon. Sometimes the crystal structure analysis based on the electron diffraction data gives the stereochemically curious results, the reason of which originates from the multiple scattering phenomenon. The comparison is made with the cases of X-ray and neutron diffraction analyses. Keyword Electron diffraction · Electron microscopy · Crystal structure analysis · Multi-scattering phenomenon

3.1 Principle of Electron Diffraction 3.1.1 Electron Wave Electron beams have both of the properties of particles and waves [1]. Dr. Young demonstrated the interference of light by performing a double-slit experiment. Later, one electron or plural electrons were used as a light source and the interference pattern of electrons was detected [2]. As shown in Fig. 3.1, one electron is incident to a double slit and reaches the screen. With the passage of time, the pattern on the screen changes gradually to show the stripe pattern after the long time. This kind of interference experiment was performed even for such a large molecule as fullerene, which is called the material wave [3]. In this way electron behaves as particle and wave. The energy of an electron is expressed as follows using the frequency ν, wavelength λ, velocity v, and mass m. De Broglie assumed the relation of hν = h(v/λ) = mv2 and then v = h/(mλ) or λ = h/(mv) = h/ p

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 K. Tashiro, Structural Science of Crystalline Polymers, https://doi.org/10.1007/978-981-15-9562-2_3

(3.1)

321

322

3 Structure Analysis by Electron Diffraction Method

slits detector electron

Fig. 3.1 Electron interference experiments. Reprinted from A. Tonomura et al., Am. J. Phys. 57, 117 (1989) with permission of the American Association of Physics Teachers, 1989

where h is a Planck constant and p = mv is a momentum. The λ is called the de Blogrie wavelength. By the application of voltage V, the kinetic energy is given as (1/2) mv2 = eV then v2 = 2eV /m

(3.2)

where e is an electron charge. By combining Eqs. 3.1 and 3.2, the wavelength λ is given as follows: [h/(mλ)]2 = 2eV /m λ = h/(2emV )1/2

(3.3)

The correction of relativistic theoretical effect changes Eq. 3.3 as follows:    λ(Å) = h/(em o V )1/2 1 − eV / 4m o c2   = (150.4/V )1/2 1 − 4.9 × 10−7 V

(3.4) (3.5)

where mo is the electron mass at the 0 speed. V is a voltage given in the unit “volt”. Using Eq. 3.5, λ = 0.061 Å for the applied voltage of V = 40 kV. Similarly, for V

3.1 Principle of Electron Diffraction

323

= 100 kV, λ = 0.037 Å. In this way the wavelength of electron wave is 1–2 order shorter than that of X-ray beam. The corresponding Ewald sphere’s radius is one order larger, and so the interpretation of electron diffraction is a little different from the case of X-ray diffraction. The concrete example will be shown later.

3.1.2 Electron Scattering When an electron is incident to an atom, they interact with each other through the electrostatic interactions between the electron charge and the potential field V (r) produced by the nucleus charge +Ze and the surrounding Z electrons with a minus charge [4, 5].  V (l) = eZ /r −

eρ(r )dvr /|l − r |

(3.6)

Here, l is the distance between the nucleus and the electron, and r is the distance between the electrons belonging to the atom and the nucleus. These electrons are distributed around the nucleus with the distribution function ρ(r). By solving the Schrödinger equation of the incident electron under the interactions with the atomic charges, the scattering amplitude of electron is given as follows. The wave vector k = 2π (s – so )/ λ and        2 φ(k) = 1/k Z − ρ(r) exp(i k · r)dvr = 1/k 2 [Z − f (k)]

(3.7)

where  f (k) =

ρ(r) exp(i k · r)dvr

is the atomic scattering factor, which is already known as the X-ray atomic scattering factor. The scattering intensity I(k) ~ φ(k)φ(k)* is given as  2    2   I (k) = Io 8π 2 me2 / h 2 1/r 2 |φ(k)|2 = Io 8π 2 me2 / h 2 1/r 2 |Z − f (k)|2 /k 4 (3.8) This equation is compared with the equation for the X-ray scattering. I (k)X-ray = Io e4 /(m 2 c4 )(1/r 2 )[1 + cos2 (2θ))/2]| f (k)|2

(3.9)

The polarization factor is lacking for the electron scattering. The electron scattering intensity is proportional to |φ(x)|2 instead of |f (k)|2 for the X-ray scattering.

324

3 Structure Analysis by Electron Diffraction Method

The interaction of electron beam with the atoms is overwhelmingly strong compared with that of X-ray beam. (This is a reason why we have to use an ultrathin sample for the electron diffraction experiment). The structure factor of electron for the crystal is given as       F(hkl) = 1/k 2 j Z j − f j (k) exp i k · r j

(3.10)

The atomic scattering factor for electron f j electron (k), shown in Eq. 3.7, is expressed as 

X-ray X-ray (k) /k 2 = 0.023934λ2 Z j − f j (k) / sin2 (θ) f jelectron (k) = Z j − f j (3.11) The inverse Fourier transform gives the potential distribution V (r) = (4π/v)h k l F(hkl) exp(−i k · r)

(3.12)

which is different from the electron density distribution in the case of X-ray scattering. The approximate expression of the atomic scattering factors for the electron diffraction is listed in Ref. [5]. The approximate equation is given as below. The cases of H, C, and O atoms are shown in Fig. 3.2 in comparison with the corresponding X-ray atomic scattering factors. Fig. 3.2 Atomic scattering factors of X-ray and electron waves (neutral atoms). The calculation was performed using Eq. 3.11. The curves calculated by Eq. 3.13 are used in general [5]

3.1 Principle of Electron Diffraction

f jelectron (k) =

325 n

  ai exp −bi (sin(θ)/λ)2

(3.13)

i=1

×10–4

a1

a2

H(Z = 1)

367

1269 2360

C(Z = 6) O(Z = 8)

a3

a4

a5

b1

1290

-

5608 37,913 135,557 377,229 -

1361 5482 12266 5971

-

3731 32,814 130,456 410,202 -

974

2039

2067 13,815 46,943

127,105 324,726

O−

2050 6280 11,700 10,300 2900

3920 26,400 88,000

271,000 918,000

O2−

421

2921 6910 2100 8520

6990

18,200 11,700 609

b2

5590

b3

29,600

b4

b5

115,000 377,000

Depending on the charge of the atom, O and O− atoms, for example, the curve of the scattering factor changes remarkably. In particular, in the low region of sin(θ)/λ, the effect of unscreened long-range Coulomb potential of the ionic charge on the nucleus causes the infinite divergence of electron scattering. The scattering of electron by an ionic atom is expressed approximately as follows [5]: f jelectron (k) =

n

  ai exp −bi (sin(θ)/λ)2 + k  Z /[sin(θ)/λ]2

(3.14)

i=1

where the first term [f o (e), Eq. 3.13] is for the neutral atom and k’ is a constant. Z is the ionic charge. The green curve in Fig. 3.3 shows the divergence due to the second term for oxygen atom.

Fig. 3.3 Atomic scattering factors of electron waves of oxygen atomic species [5]

326

3 Structure Analysis by Electron Diffraction Method

It must be noted that the above-mentioned equation is based on the kinetic theory. The electron scattering is affected remarkably by the so-called dynamical effect. The X-ray diffraction is treated mainly by using the interference of the secondary waves or the waves scattered by the incident (primary) X-ray beam, that is, the kinematical theory. On the other hand, the electrons interact strongly with the substance and the secondary waves generate the third-order waves, and the third-order waves generate the fourth-order waves, etc. The phenomenon is called the multi-scattering effect. As a result, the diffraction intensity of the secondary waves is modified, and the abovementioned equation cannot be utilized so simply. In this way, the electron diffraction phenomenon is needed to treat in more complicated way. This treatment is called the dynamical theory. The details are referred to in the textbooks [6, 7].

3.2 Electron Microscopy (TEM, SEM) 3.2.1 Transmission Electron Microscope (TEM) The wavelength λ of electron beam is expressed in Eqs. 3.3 and 3.4. Approximately, λ is proportional to V −1/2 . The refractive index n = λo /λ = [(V + )/V ]1/2 ∼ 1 + /(2V )

(3.15)

where  is an internal electric field in the substance. The value /(2V ) is in the order of 10–4 . The refractive index is controlled by changing the applied voltage. This property is applied to a lens of the electron microscope: the direction of the electron beam emitted from the filament is controlled by the voltage change. Using this technique, the sample image is enlarged by the electromagnetic lens. Before the description of the transmission electron microscope, it is better to remember the principle of a convex lens. As illustrated in Fig. 3.4, the sample image is enlarged by using a convex lens. The method to get the image is to draw the lines as indicated in red and green colors. The relation between the distances a and b and the focal distance f is expressed in the following equation: 1/ f = 1/a + 1/b This equation can be derived using a relation of similar triangles:   ( p/2) : p  /2 = a : b for the pink triangles and

  ( p/2) : p  /2 = f : (b − f ) for the green triangles.

(3.16)

3.2 Electron Microscopy (TEM, SEM)

327

Fig. 3.4 Convex lens

Therefore, a : b = f : (b – f ), resulting in the above Eq. 3.16. On the basis of Eq. 3.16, the clear and enlarged image can be observed with a convex lens of focal distance f by controlling the distances a and b. This principle can be applied to the electron microscopy, but the focal length f is changed by the control of the voltage applied to the coil. In addition, because the electron beam is incident to the sample, the diffraction of electrons occurs from the sample, in particular the crystalline lattice planes. That is to say, both the transmitted or not-diffracted electron beam and the diffracted electron beam must be taken into account for the practical application of the electron microscope [1, 2] (Fig. 3.5). As shown in Fig. 3.6, the diffracted electron beams are focused on the back focal point of the lens. The electron diffraction pattern can be recorded if the detector is put there. These diffracted beams are also focused on the screen to contribute to the imaging of the sample crystal (Fig. 3.6). As shown in Fig. 3.7a, if the diffracted beam components are blocked by the apertures, only the transmitted electron beam creates the image. This is called the bright field image. On the other hand, by cutting the transmitted beam and by allowing only some particular component of the diffraction signals to pass through, the dark field image is obtained, which is created by only the filtered diffraction component (Fig. 3.7b). In the actual measurement, the observation of the image and the electron diffraction pattern is controlled by using an intermediate lens. The focal distance of the intermediate lens is f int . The positions of the objective lens, the intermediate lens, and the eyepiece are fixed in the electron microscope. Let us see Fig. 3.8. (case 1) The transmitted and diffracted electrons from the actual sample A create the diff-A pattern on the focal position. The sample image B is created finally, which is zoomed into the image C and projected on the screen (detector) as the image D. By using Eq. 3.16, the geometrical relation is 1/a + 1/b = 1/ f int

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3 Structure Analysis by Electron Diffraction Method

Fig. 3.5 Transmission electron microscopy (JEOL JEM1010) with a CCD detector

sample objective lens transmitted beams

diffraction pattern

diffracted beams

focus

image observation sample image screen

Fig. 3.6 Electron diffraction and electron lens

3.2 Electron Microscopy (TEM, SEM)

Fig. 3.7 Bright and dark field images

Fig. 3.8 Imaging and diffraction

329

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3 Structure Analysis by Electron Diffraction Method

(case 2) The diff-A pattern is focused by the intermediate lens and the diff-B image is created, which is zoomed up by the eyepiece and recorded on the detector. The relation is  1/a  + 1/b = 1/ f int

By cutting the diffracted signals with the insertion of an aperture, only the diffracted image diff-A is projected onto the intermediate lens. Finally the diffraction image diff-C is detected on the screen. In case 2, if only the diffracted beam component of blue color is collected by shifting the aperture position, the diffraction pattern of blue-color component is recorded. This is called the selected (or restricted) field diffraction pattern. The similar technique is applied to the case 1, the selected (or restricted) dark field image is obtained (refer to Fig. 3.9). Detectors The image and diffraction pattern are detected using the 2D detectors. In the early days the films of fine AgCl powder were used, which were set to the metal frames individually and inserted to the screen position. After that, the imaging plates were started to be used as the 2D detector of higher sensitivity and wider intensity range. Recently the CCD detector is used for the 2D detector. The image can be viewed in a real time.

Fig. 3.9 TEM image and the diffraction pattern measured at the same position

3.2 Electron Microscopy (TEM, SEM)

331

3.2.2 Scanning Electron Microscope In general, a television screen is produced by scanning a beam spot in a zigzag mode over the whole region at a high speed. In the normal digital TV, the 1125 horizontal lines (the effective lines are 1080) are scanned in total. The total number of scanned images is 30 per second. This principle is utilized in the scanning electron microscope (SEM) [8]. As shown in Fig. 3.10, the electron beam of 1–100 nm size is incident on the sample surface in the high vacuum and the beam position is scanned along the x and then y axes electronically or magnetically. Once when the electron beam is incident, the secondary electron signals are emitted from the sample surface. These secondary electrons are detected on the screen which is operated synchronously with the electron beam scanning. The scattered secondary electron beams are zoomed-in with the usage of electromagnetic lens. The magnification is about 10 – 104 . (From the electron-irradiated sample surface, not only the secondary electrons but also the various beams are emitted such as the back-scattered electrons, the X-ray beam, Auger electrons, and so on. If the X-ray beam is used, then we can get the X-ray image.) The sample surface is charged by the electron beam incidence and the charges are accumulated with the exposure time. The SEM screen becomes brighter and finally saturated. In order to avoid the charging, the sample surface is coated in advance with a metal of 20–100 nm thickness (sputter coating). The metal used for this purpose is Au–Pd alloy, for example. During the vacuum evacuation process, the sample

electron

X-ray

Detector

Augerelectron

sample secondary electrons transmitted electron Fig. 3.10 Principle of SEM and the SEM images

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3 Structure Analysis by Electron Diffraction Method

mesh carbon-evaporated mesh

collodion-coated mesh

carbon-coated mesh

Fig. 3.11 Meshes for TEM experiment

is rotated in the various directions so that the surface is coated totally. Recently a tabletop SEM system started to appear commercially. The vacuum level is relatively low. The charging effect can be depressed by using such a low vacuum. The low vacuum system makes it possible to measure the sample containing water.

3.2.3 Sample Preparation for TEM The sample for the TEM measurement must be thin enough for the electrons to pass through the sample. The ideal thickness is in an order of several tens—hundreds nm. The thin films are prepared by the solution casting, the single crystal grown from the dilute solution, the vacuum evaporation on the amorphous supporting film such as carbon, polyvinyl formal, and collodion, which is coated on a mesh (grid) (see Fig. 3.11). The thus-prepared sample is set on a mesh and dried. The sample is stable on the thin supporting film. The film is set directly on the mesh. Very thin carbon (5 nm thickness) is evaporated onto the film to increase the stability against the electron beam. A powder sample must be set on the mesh with a collodion film.

3.3 Crystal Structure Analysis by ED 3.3.1 Polyethylene Single Crystal The TEM image and electron diffraction pattern of HDPE single crystals are shown in Fig. 3.12. As shown in Fig. 3.13, the different diffraction patterns must be collected from the various directions. For this purpose, the samples with the various orientations were prepared by the different methods [9]: (i) a single crystal grown from the dilute p-xylene solution, (ii) a single crystal grown epitaxially on the surface of benzoic acid, and (iii) a single crystal grown epitaxially on the KBr single crystal. The observed reflections are quite sharp, different from the broad X-ray reflections of the oriented bulk HDPE sample. The indexing of the observed reflections is made relatively easily (Fig. 3.12).

3.3 Crystal Structure Analysis by ED

333

Fig. 3.12 TEM image of the orthorhombic PE single crystal and its diffraction pattern

These electron diffraction patterns were measured with the electron beam accelerated at 100 kV. The wavelength of the electron wave is 0.037 Å. Similarly to the X-ray diffraction phenomenon, the electron diffraction is interpreted as the crossing of reciprocal lattice points with the Ewald sphere. However, the radius of the Ewald sphere (= 1/λ) is about 10 times large compared with the case of X-ray diffraction using a wavelength of 1–0.3 Å. As illustrated in Fig. 3.14, the Ewald sphere becomes almost flat and the reciprocal lattice points in a wide range cross the surface. In other words the reciprocal lattice plane can be assumed almost exactly the flat “plane”. Furthermore, we notice that only the equatorial plane crosses the sphere in the case of PE crystal, although the situation might be changed depending on the unit cell size.

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3 Structure Analysis by Electron Diffraction Method

Fig. 3.13 Electron diffraction patterns of PE single crystals grown in the various ways. Reprinted from Ref. [9] with permission of the American Chemical Society, 2004

In order to measure the hkl reflection planes with l = 0, the sample must be rotated so that the crossing with the Ewald sphere can occur. However, in the TEM instrument it is not very easy to rotate the sample and the rotation angle range is limited. The single crystal grown epitaxially on the inorganic or organic single crystal surface exhibits the different orientations of the crystals, as shown in Fig. 3.13. For example, for the single crystal grown on the benzoic acid crystal, the a*-axis is perpendicular to the surface. Since the a axial length is relatively longer than the c-axis, the 0kl, 1kl, and 2kl planes cross the sphere surface. In Fig. 3.13c the single crystal orients in the diagonal direction and the indexing is a little complicated. The unit cell parameters thus obtained are of the orthorhombic type; a = 7.46 Å, b = 4.98Å, c(fiber axis) = 2.55Å The total number of the detected reflections is 45. The integrated intensity was estimated for all of these reflections. The extinction rules of the reflections were relatively easily derived since the projected pattern of the hk0 reciprocal lattice plane is directly obtained. The space group is Pnam by referring to the X-ray diffraction data

3.3 Crystal Structure Analysis by ED

335

Fig. 3.14 Ewald sphere and electron diffraction

Fo

Fo-Fc

b

a Fig. 3.15 Crystal structure of PE derived by the TEM diffraction pattern analysis. The F o – F c map is also shown in the right side. Reprinted from Ref. [9] with permission of the American Chemical Society, 2004

also. The two zigzag chains were packed in the cell and the crystal structure shown in Fig. 1.134 was obtained. The R factor was 17.6%. As seen in Fig. 3.15, the H atomic peaks are detected though ambiguous. The F o – F c map gave clear peaks of the H atoms. The geometry of the molecular chain is estimated as follows: C–C = 1.506 ± 0.015 Å, C–H1 = 1.404 ± 0.109 Å, C–H2 = 1.247 ± 0.103 Å, C–C–C = 115.3° ± 1.4°, and H1 –C–H2 = 105.2° ± 12.6°. The setting angle of the zigzag chain is 44.0° ± 2.0°. Immediately we notice these geometries are not reasonable when compared with the general parameters. For example, the CCC angle 115.3° is much wider than the value 111.0° estimated by the X-ray data analysis. The C–C bond length is also

336

3 Structure Analysis by Electron Diffraction Method

Fig. 3.16 Electron diffraction patterns with and without multiple scattering effect. Reprinted from Ref. [9] with permission of the American Chemical Society, 2004

shorter than the standard value 1.54 Å. The reason comes from the abnormal intensity relation among the observed reflections. Figure 3.16 shows the electron diffraction patterns measured for the single crystals with the thicker and thinner thicknesses. The thinner single crystal gives the reflections of h00 and 0k0 with only the even numbers of h and k. On the other hand, the thicker crystal gives not only the even-number h00 and 0k0 reflections but also the odd-number reflections such as 100, 300, 010, and so on. This phenomenon comes from the multi-scattering phenomenon in which the electron beams reflected on the particular lattice planes are reflected again on the different lattice planes [10, 11]. As a result the relative intensity of the observed reflections is modified more or less. The observation of 100, 300, 010, reflections is

3.3 Crystal Structure Analysis by ED

337

Single crystal

Whisker by cation

from solution

polymerization

Fig. 3.17 Single crystal of POM. a a folded-chain crystal (FCC)and b an extended chain crystal (ECC). Reprinted from Ref. [12] with permission of the American Chemical Society, 2004

one typical evidence. The reflection intensities were estimated for the single crystals with the different thicknesses. As shown in Fig. 3.16, the relative intensity of 110 and 200 reflections is inversed with the change of the crystal thickness. The thicker sample shows the anomalous relation of the relative intensity due to the multiple scattering inside the crystal. In other words, the thinner crystal is better for avoiding the multiple scattering effects. Another technique to escape from the multi-scattering phenomenon is to use the higher acceleration voltage for the incident electron beam, which can pass the sample straightforwardly. The multi-scattering phenomenon is one of the dynamic scattering effects. The kinetic scattering theory is applied to the normal crystal structure analysis, as likely the case of X-ray structure analysis.

3.3.2 Polyoxymethylene Whisker Polyoxymethylene (POM) is obtained by the cationic polymerization reaction of trioxane. On the wall of the reaction vessel, whiskers or the needle-like single crystals are found. As shown in Fig. 3.17, the size of whisker is a few μm width and about 20 μm length. It is difficult to measure the X-ray diffraction pattern of such a tiny and thin whisker even using a synchrotron X-ray beam. The electron diffraction measurement may be one useful method. However, as shown in Fig. 3.18a and b, the POM is quite easily damaged by an irradiation of electron beam [12]. Then

338

3 Structure Analysis by Electron Diffraction Method

Fig. 3.18 a TEM image of POM whisker, b the POM whisker damaged by electron beam, c electron diffractions measured for the POM needle crystals with the various orientations. Reprinted from Ref. [12] with permission of the American Chemical Society, 2004

the electron beam power was reduced to appreciably low level. The thus-measured electron diffraction patterns are shown in Fig. 3.18c and d, where the orientation of the single crystal was changed by rotating around the needle axis. The indexing of the observed reflections was made by referring to the unit cell of the trigonal POM crystal. The reflection intensity was estimated by integrating the reflection spot. The direct method was applied and the thus-derived initial structure model was refined. The refined helical chain conformation is more or less deformed compared with that by the X-ray diffraction analysis. The reason may be the same as that mentioned for the PE single crystal analysis using the electron diffraction data.

3.3 Crystal Structure Analysis by ED

339

Fig. 3.19 Molecular conformation of POM obtained from a electron diffraction data analysis and b X-ray diffraction data analysis. Reprinted from Ref. [12] with permission of the American Chemical Society, 2004

Then, the structure derived by the electron diffraction data analysis was used as an initial model and refined furthermore using the X-ray diffraction data. The thus-derived structure is shown in Fig. 3.19.

References 1. L. Reimer, Transmission Electron Microscopy: Physics of Image Formation and Microanalysis, Springer (1989).; D.B. Williams, C.B. Carter, Transmission Electron Microscopy: A textbook for Materials Science. Springer (2009) 2. C. Jönsson, Electron diffraction at multiple slits. Am. J. Phys. 4, 4–11 (1974); A. Tonomura, J. Endo, T. Matusda, T. Kawasaki, H. Ezawa, Demonstration of single-electron buildup of an interference pattern. Am. J. Phys. 57, 117–120 (1989) 3. B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, A. Zeilinger, Matter-wave interferometer for large molecules. Phys. Rev. Lett. 88, 100404–1–100404–4 (2002)

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4. X. Zou, S. Hovmoller, P. Oleynikov, Electron Crystallography: Electron Microscopy and Electron Diffraction (Oxford University Press, Oxford, 2011) 5. L.-M. Peng, Electron atomic scattering factors and scattering potentials of crystals. Micron 30, 625–648 (1999) 6. Z.L. Wang, Elastic and Inelastic Scattering In Electron Diffraction And Imaging (Plenum Press, New York, 1995) 7. A.F. Moodie, J.M. Cowley, P. Goodman, Dynamical theory of electron diffraction. International Tables for Crystallograpy, Volume B, Chapter 5(2), 552–556 (2006) 8. L. Reimer, Scanning Electron Microscopy: Physics of Image Formation and Microanalysis (Springer-verlag, Belrin, 1998) 9. K. Tashiro, I. Tanaka, T. Ohhara, N. Niimura, S. Fujiwara, T. Kamae, Extraction of hydrogen atom positions in polyethylene crystal lattice from the wide-angle neutron diffraction data collected by 2-dimensional imaging plate system: a comparison with the X-ray and electron diffraction results. Macromolecules 37, 4109–4117 (2004) 10. D.L. Dorset, B. Moss, Crystal structure analysis of polyethylene with electron diffraction intensity data: deconvolution of multiple scattering effects. Polymer 24, 291–294 (1983) 11. D.L. Dorset, Structural Electron Crystallography (Plenum Press, New York, 1995) 12. K. Tashiro, T. Kamae, H. Asanaga, T. Oikawa, Structural analysis of polyoxymethylene whisker single crystal by the electron diffraction method. Macromolecules 37, 826–830 (2004)

Chapter 4

Small-angle X-ray Scattering Method

Abstract Small-angle X-ray scattering (SAXS) gives us the information of higher order structure of polymers in the several tens to hundreds nm scale. The principle of SAXS is leaned in terms of electron density difference and correlation function of density. The concrete equations of scattering intensity are derived for the various particles as well as the stacked lamellar structure. The treatment of lamellar stacking disorder of the second kind is leaned based on the concrete data analysis. The method of simultaneous measurement of wide-angle and small-angle X-ray scattering data is described, which is useful to know both of the structure information about the inner structure of crystallites as well as the aggregation structure of the crystalline and amorphous regions. Keywords Small-angle X-ray scattering · Higher order structure · Electron density difference · Correlation function · Lamellar stacking structure

4.1 Small-angle Scatterings and Hierarchical Structure As known from the Bragg equation (2d sin(θ) = λ), in order to know the structure of several thousands to several tens thousands Å scale, the X-ray scattering is needed to measure at a small-angle of 2θ range of 0.1°–0.01° for the incident X-ray beam of 1 Å wavelength. We call the X-ray or neutron scattering in this angle range the small-angle X-ray scattering (SAXS) or the small-angle neutron scattering (SANS). If the scattering angle is still low in the order of 0.001°, the detectable size is 60000 Å or 6 μm, which is comparable to the order of optical microscopic image. This SAXS measurement is called the ultra-small-angle X-ray scattering or USAXS. Polymer substance consists of the aggregation structure of crystalline and amorphous regions, which is called the higher order structure. By combining the SAXS measurement with the wide-angle X-ray scattering (or diffraction) (WAXS or WAXD) or wide-angle neutron diffraction (WAND) measurement, the hierarchical structure of the polymer substance can be studied in detail. Since the X-ray (or neutron) scattering occurs by the density fluctuation or density difference, the SAXS (SANS) measurement is becoming popular not only for the study of solid state but also for the study of

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 K. Tashiro, Structural Science of Crystalline Polymers, https://doi.org/10.1007/978-981-15-9562-2_4

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4 Small-angle X-ray Scattering Method

liquid and molten state. The SAXS study of a protein solution is important from the biological point of view. In the following sections the SAXS data will be treated mainly [1–7]. Of course the principle can be applied also to the SANS experiment although some difference may be present in the data treatment and so on.

4.2 Density Difference and SAXS 4.2.1 X-ray Scattering and Density Difference In Chap. 1, we studied the X-ray diffraction (or scattering) phenomenon. The structure factor F(k) is an amplitude of the scattered X-ray wave, and the scattering intensity I(k) is proportional to the square of the structure factor F(k) in the form I(k) ∝ F(k)F(k)*. The F(k) is a Fourier transform of the electron density ρ(x). Therefore, the scattering intensity I(k) per unit volume is expressed as below (V is a volume).   ∗ I (k) = (1/V )F [ρ(x)] · F ρ x        = (1/V ) ρ(x) exp(i kx)dx ρ x  exp −i kx  dx      = (1/V ) ρ(x) exp(i kx)dx ρ(x + y) exp(−i k(x + y))d y x  = x + y    = (1/V ) ρ(x)ρ(x + y)dx exp(−i k y)d y  = γ( y) exp(−i k y)d y (4.1)  I (k) =

γ(r) exp(−i kr)dr

(4.2)

Here, γ (r) is the autocorrelation function;  γ (r) = (1/V )

    ρ r  ρ r + r  dr 

(4.3)

In this way the scattering intensity is a Fourier transform of the autocorrelation function. Here |k| = k = (4π/λ)sin(θ). For example, if γ(r) is expressed as below, I(k) is given as γ(r ) = A exp(−|r |/ξ )

 2 I (k) = A/ 1 + k 2 ξ 2

(4.4)

  γ(r ) = exp −r 2 /ξ 2

  I (k) = (π)1/2 ξ exp −ξ 2 k 2 /4

(4.5)

4.2 Density Difference and SAXS

343

  γ(r ) = Aξ exp(−|r |/ξ )/r I (k) = A/ 1 + k 2 ξ 2

(4.6)

Equations 4.4 and 4.6 are named the Debye-Buche equation and Ornstein–Zernike equation, respectively. Let us consider the fluctuation of electron density. η(r) = ρ(r) − < ρ >

(4.7)

The statistical average is expressed using < >. The η(r) is a deviation of density from the averaged value. Inputting it into the correlation function γ(r), we have  γ(r) = (1/V )



    ρ r ρ r + r d r

       η r + < ρ > η r + r + < ρ > d r           = (1/V ) η r  η r + r  dr  + 2 < ρ > η r  d r  + < ρ >2 d r  = (1/V )

= γo (r)+ < ρ >2

(4.8) 

γo (r) = (1/V )

    η r  η r + r  dr

(4.9)

As shown in Eq. 4.2, the Fourier transform of γ(r) gives the scattering intensity I(k) as  I (k) = γo (r) exp(−i kr)d r+ < ρ >2 δ(k) (4.10) where δ(k) is a delta function. The γo (r) corresponds to the probability of the electron density at a positon distant by r from a certain point. Depending on the shape of the scattering body, γo (r) is different.

4.2.2 SAXS of Particle In the case of a dilute solution, in which the isolated particles are floating, Eq. 4.10 becomes as below. Here the average is made for all the orientation angles.  I (k) =
2 δ(k) >

(4.11)

The average for the angles θ and φ is made in the followin way: dr = dVr = r 2 sin(θ)dr dθdφ

(4.12)

344

4 Small-angle X-ray Scattering Method ez

z

θ x x

φ

r z

er

θ x

y

ex

y

φ

z

dzez

S

eθ y

dxex

eφ

dyey

S = dxex x dyey dV = S·dzez

ey

Fig. 4.1 a The Cartesian coordinate system and the polar coordinate system. b Both of these systems must be right-handed system. c Calculation of small volume dV

and  I (k) =


(4.13)

where the second constant term is ignored for simplicity. The k vector is parallel to the z-axis (Fig. 4.1), and so k · r = krcos(θ). By setting cos(θ) = u, du = d(cos(θ)) = -sin(θ)dθ. For θ = 0 ~ π, u = 1 ~ − 1. Then,  γo (r) exp(−i kr)r 2 sin(θ)dr dθdφ >    γo (r)r 2 exp(−ikr cos(θ)) sin(θ)dθdφdr =− 0∼R 0∼2π 0∼π    γo (r)r 2 exp(−ikru)dudφdr = 0∼R 0∼2π −1∼1    γo (r)r 2 [exp(−ikru)/(−ikr )] dudφdr = 0∼R 0∼2π −1∼1    γo (r)r 2 [exp(−ikru)/(−ikr )]1−1 dφdr = 4π γo (r)r 2 [sin(kr )/(kr )]dr =
= 4π I (k) = (1/V) < F [η(x)] · F η x 

 γo (r)r 2 [sin(kr )/(kr )]dr (4.25)

(Note) The Fourier transform given in Eq. 4.1 is F [ρ(x)] = F [η(x)+ < ρ >] = F [η(x)]+ < ρ > δ(k). The second term < ρ > δ(k) is not zero only at k = 0. If this term is ignored as mentioned in Eq. 4.13, Eq. 4.1 becomes Eq. 4.25. At first, let us see the first equation expressed using the Fourier transform F [η(x)]. As already mentioned, we know the concrete shape of F [η(x)].   ∗ > I (k) = (1/V ) < F [η(x)] · F η x  2       i kr 2   = (1/V )< η(r)e dr > = (1/V )η 4π

2 r sin(kr )/(kr )dr 2

0∼R

(4.26)   The integration of the term 0~R r 2 sin(kr)/(kr)dr = (1/k) 0~R rsin(kr)dr is made by utulizing the partial intergration; [rcos(kr)]’ = cos(kr) – kr sin(kr). By inputting this relation to the above equation, we can perform the integration as follows:      2 cos(kr ) − [r cos(kr )] dr r sin(kr )dr = 1/k (1/k) 0∼R   0∼R = 1/k 2 [sin(kr )/k − r cos(kr )]R0 

= [sin(k R) − k R cos(k R)]/k 3   = R 3 /3 × 3[sin(k R) − k R cos(k R)]/(k R)3 Therefore,  2    I (k) = (1/V ) 4πR 3 /3 η · 3 (sin(k R) − k R cos(k R))/(k R)3  2   V = 4πR 3 /3 = V η2 3(sin(k R) − k R cos(k R))/(k R)3

(4.27)

4.2 Density Difference and SAXS

351

This is the SAXS equation for an isolated sphere with a radius R. Another calculation of this equation is made starting from the autocorrelation funcion. We have to be careful for the treatment of γ o (r). This funtion looks apparently to be constant, in particular for the rigid and homogeneous sphere. For example, the following misunderstanding may be made, but this is a wrong calculation.  γo (r ) = (1/V )

    η r  η r + r  dr  = (1/V )η2

 I (k) = 4π η2

 dV = η2 (NG)

 r 2 [sin(kr )/(kr )]dr = 4π η2

r [sin(kr )/(k)]dr (NG)

The autocorrelation must be calculated in the following way. The density at the different positions shoud have a correlation with each other inside the rigid sphere. We assume the following correlation between η(r’) and η(r + r’) by putting η(r + r’) = η(r)exp(ikr’), then  γo (r) = (1/V )

    η r  η r + r  dr  = (1/V )



    η r  η(r) exp i kr  dr  (4.28)

The scattering intensity is expressed as         (1/V ) η r  η(r) exp i kr  dr  exp(−i kr)dr γo (r) exp(−i kr)dr =       = (1/V ) η r  exp i kr  dr  η(r) exp(−i kr)dr    (4.29) = (1/V )F η r  · F [η(r)]∗ 

I (k) =

This equation is the same as Eq. 4.25 and we have the scattering intensity equation of a sphere as given in Eq. 4.27. Figure 4.3 shows the I(k) plotted against k. It must be noted that the I(k) is plotted in a double logarithmic scale. The scattering intensity changes with alternately-chaging maximal and minimal peaks. In the k range ∼ 1/R, the slope is −4. In the Guiner region (k ∝   I (k) = I (0)/ 1 + ζ 2 k 2

exp(−r/ζ )/r

(4.32) (4.33)

The equation is called the Ornstein–Zernike equation. The ζ is the correlation length, which is estimated from the plot of I(k)−1 versus k 2 .

4.2.3.2

2-Phases

If there exists a system in which the two phases of different density repeat alternately, the Debye-Buche equation given below is useful. The correlation function is assumed to be γ(r) = exp(−ξ /r) and the correlation length ξ is obtained by the plot of I(k)−1/2 versus k 2 . 2  I (q) = I (0)/ 1 + ξ 2 q 2

(4.34)

4.2 Density Difference and SAXS

4.2.3.3

353

Surface

For the particles with surface area S and volume V, the correlation function is given as follows: γo (r ) = 1 − Sr/(4 V )

(4.35)

Then, the information of the surface is estimated by the so-called Porod equation. lim I (k)k 4 ∝ (S/V )Q/φ

k→∞

(4.36)

Here, Q is an invariant, which is a total integration of I(k)k 2 from k = 0 to ∞ (refer to Eq. 4.44). The S/V or the surface-to-volume ratio represents the degree of surface roughness.

4.2.3.4

Interaction Between Particles

The scattering profile originated from one sphere is expressed in Eq. 4.27, as reproduced here.  2 I (k) ∝ f (k) f (k)∗ ∝ (sin(k R) − k R cos(k R))/(k R)3

(4.37)

If the two spheres of the same electron density are coexistent at a certain distance in the system, the interference may occur between them.   2

f j (k) f m (k)∗ exp i kr j exp(−i kr m ) j=1 m=1   = 2 f (k) f (k)∗ + 2 f (k) f (k)∗ exp i kr  2

I (k) ∝

(4.38)

The position vector r = rj – rm is the difference vector between these two spheres. This pair of spheres is assumed to orient randomly in the space. The average over all the orientations gives the following equation:      I (k) = 2| f (k)|2 (1+ < exp(i kr  ) >) = 2| f (k)|2 1 + sin kr  / kr   2      ∝ (sin(k R) − k R cos(k R))/(k R)3 1 + sin kr  / kr  (4.39) This equation can be divided into two terms. The first term [(sin(kR) − kRcos(kR))/(qR)3 ]2 comes from the scattering factor of a sphere, which is now called the form factor, P(k). The second term [1 + sin(kr’)/(kr’)] originates from

354

4 Small-angle X-ray Scattering Method

the interference beween the spheres, which is called the structure factor, S(k). We have to be careful of the usage of the following terminologies. WAXD

SAXS

Form factor, f (k)

Atomic scattering factor

Structure factor, S(k)

Interfered scattering between the aggregated molecules (and unit cells)

Form factor, P(k) Structure factor, S(k)

Scattering by a particle (including a lamella) Interfered scattering between the aggregated particles (including the array of repeated lamellae)

The total intensity I(k) is a product of P(k) and S(k). I (k) = n p P(k)S(k)

(4.40)

The np is the number density of particles (the number per volume). If only one isolated particle exists in the system, there is no correlation and so S(k) = 1. Only the form factor contributes to the scattering intensity. In more general, the expression of the intensity was already given in the previous sections. For simplicity, by ignoring the coefficient,  2 I (k) = j <  Fj (k) > + < j m(m =j) Fj (k) Fm ∗ (k) expik( r j −r m ) >

(4.41)

When we have a system consisting of the monodispersed particles with the same F value, F j (k) = F m (k) = F(k), and then I (k) = n < |F(k)|2 > [1+ < j m(m =j) expi k( r j −r m ) >] = n P(k)S(k)

4.2.3.5

(4.42)

Stacked Lamellar Structure

In this case the interference between the lamellae is needed to consider. For the 1dimensional lamellar structure, the normalized correlation function g(r) is related to the intensity as  I (k) =  g(r ) cos(kr )dr   (4.43) g(r ) = η(ro )η(r + ro )dro / η(ro )η(ro )dro = k 2 Iobs (k) cos(kr )dk/Q   invariant Q = η(ro )η(ro )dro = k 2 Iobs (k)dk (4.44)

4.2 Density Difference and SAXS

355

The term I obs (k)k 2 is the Lorentz-factor-corrected observed intensity [8]. Q is an invariant, which is a total scattering power (per volume) and is equal to . If the volume fractions of the lamellar and amorphous regions are wc and wa , respectively, then < ρ > = ρc wc + ρa wa < η2 > = η2c wc + η2a wa = (ρc − < ρ >)2 wc + (ρa − < ρ >)2 wa = (ρc − ρa )2 wc wa = ρ 2 wc (1 − wc ) Therefore Q = Δρ 2 wc (1 − wc ) with Δρ = ρc − ρa

(4.45)

(Note) In the actual experiment, as already studied in the WAXD sections, the observed intensity must be corrected in order to discuss using the structure factor. The scattering intensity (I lamellae ) expected from Eq. 4.43 covers the whole surface of a small sphere of radius d* (= 1/d) which crosses the Ewald sphere; 4πd*2 = 4π(k/2π)2 = k 2 /π. The observed intensity (I obs ) is only a fraction of the whole scattering intensity; I obs = I lamellae /(k 2 /π). In other words, I lamellae ∝ k 2 I obs . Let us return to Eq. 4.43. We consider the meaning of the correlation function g(r). For example, the repetition of the square boxes along the r-axis is considered as shown in Fig. 4.4. The overlapping parts of the neighboring rectangular boxes give the non-zero contribution to the correlation function g(r). The case (a) is the full overlap of the two lamellae to give the maximal correlation. As the two lamellar distance is increased in the range of 0 < r < 200 Å, the correlation becomes lower [case (b)]. For the r longer than 300 Å the two lamellae do not overlap anymore, and the correlation is zero [case (c)]. When the distance r reaches 500 Å, the correlation starts again and the value increases to the maximal value. As a result, the g(r) curve is drawn as shown in Fig. 4.4d. In this way, the convolution of the two rectangular functions [η(r)] gives the triangular function [g(r)]. The concrete equations of g(r) are expressed in the following forms. In the region of 0 < r < d, In the region of d < r < L − d, In the region of L − d < r < L ,

g(r ) = −η2 (r − d) g(r ) = 0 g(r ) = η2 (r + d − L)

(4.46)

By changing the range of r, the g(r) must be shifted with keeping the same triangular shape. The Fourier transform of g(r) is calculated in the following way. Here, the cosine term in Eq. 4.43 is changed to the original exponential form.

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4 Small-angle X-ray Scattering Method

(b )

(a)

(c )

(d )

Fig. 4.4 Correlation between the lamellae

  I(k) ∝ g(r )eikr dr = (η2 )[

 (−r + d)eikr dr +

0∼d

(r + d − L)eikr dr ] + · · · L−d∼L

However, this equation consists of many terms and the actual calculation is quite difficult. It may be easier to consider that the stacked lamellar system is a series of rectangular functions [η(r)], which is a convolution between one rectangular function ηo (r) and a series of delta functions j δ(r-jL), as illustrated below: η(r) = ηo (r)ˆ[ j δ(r – jL)].

4.2 Density Difference and SAXS

357

The Fourier transform of convoluted two functions is the product of the Fourier transforms of the individual functions (F [fˆg] = F [f]× F [g]). In additon, we have the various theorems concerning the convolution [fˆg = gˆf (commutativity), fˆ(gˆh) = (fˆg)ˆh (associativity), fˆ(g + h) = fˆg + fˆh (distributivity), and f×g = F (f)ˆ F (g)]. Besides, as already seen in Fig. 4.4 (d), the convolution of the two rectangular functions ηo (r) is equal to the triangular function go (r). Therefore, Eq. 4.43 is rewritten as follows.  I (k) = F [g(r )] = F [η(r )∧ η(r  )] = F [ηo (r )∧ j δ(r − j L)]∧ [ηo (r  )∧ m δ(r  − m L)] = F [ηo (r )∧ ηo (r  )] × F [ j δ(r − j L)] × F [ m δ(r  − m L)]  2 = F [go (r )] × F [ j δ(r − j L)]

By calculating the Fourier transform of a triangular function, F [go (r)], and that of a series of delta functions, F [ j δ(r – jL)], we can obtain the concrete expression of I(k). A triangular function (η2 is ignored in Eq. 4.46. go is normalized to 1) is defined in the r range of –d ≤ r ≤ d as go (r ) = r + d = d −r

−d ≤ r ≤ 0 0≤r ≤d

We can derive the Fourier transform F [go (r)]:  F [go (r )] =

o

−d

⎤2 kd sin ⎢ 2 ⎥ ⎥ dr = d 2 ⎢ ⎣ kd ⎦ (4.47a) 2 ⎡



d

(r + d)eikr dr + o

(d − r)eikr

The Fourier tranform of a delta function δ(x – jL) is equal to exp(ijkL) where j = −N ∼ N (in total, 2N + 1). For a series of the delta functions, the following result is obtained. +N +N F [ +N j=−N δ(r − j L)] = j=−N F [δ(r − j L)] = j=−N exp(i jkL)

= exp(−ik L N ) + exp(−ik L(N − 1)) + · · · + exp(ik L(N − 1)) + exp(ik L N ) =

[eik L(N +1) − e−ik L N ] [eik L − 1]

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4 Small-angle X-ray Scattering Method

the numerator = exp(ik L(N + 1)) − exp(−ik L N ) = cos(k L(N + 1)) + i sin(k L(N + 1)) − cos(k L N ) + i sin(k L N ) = [cos(k L(N + 1)) − cos(k L N )] + i[sin(k L(N + 1)) + sin(k L N )]







kL k L(2N + 1) kL k L(2N + 1) sin ] + i[2 sin cos ] = [2 sin 2 2 2 2





k L(2N + 1) kL kL = 2i sin [cos − i sin ] 2 2 2

k L(2N + 1) −ik L/2 e = 2i sin 2 the denominator = eik L − 1 = cos(k L) + i sin(k L) − 1 



kL kL k L ik L/2 kL cos + i sin = 2i sin e = 2i sin 2 2 2 2 Therefore,



  kL k L(2N + 1) −ik L /sin F +N δ(r − j L) = e sin j=−N 2 2

(4.47b)

Finally, we have     +N ∗ I (k) = F [go (r )] × F +N j=−N δ(r − j L) × F j=−N δ(r − j L)



k L(2N + 1) kd sin sin 2 2

]2 = I (0)[ ]2 [ kd kL sin 2 2 By changing the total number of the delta functions (2N + 1) to N total ,



k L Ntotal kd sin sin 2 2 ]2 I (k) = I (0)[ ]2 [ kd kL sin 2 2

(4.48)

On the other hand, the same equation may be obtained by performing the Fourier transform of the original stacked lamellar structure.      I (k) = F [ηo (r )] × F +N F ∗ [ηo (r )] × F ∗ +N j=−N δ(r − j L) j=−N δ(r − j L)

Here the Fourier transform of ηo (r ) is given as

kd sin d/2 2 eikr dr = d[ ] F [ηo (r )] = kd −d/2 2 

4.2 Density Difference and SAXS

359

The square of F [ηo (r )] is equal to Eq. 4.47a. Therefore, we have the same equation as Eq. 4.48. The first term, [sin(kd/2)/(kd/2)]2 , is the form factor coming from the triangular function or the correlation between the two rectangular functions, and the second term is the structur factor originating from the lamellar stacking structure with long period L. For example, for the stacked lamellar structure with d = 40 Å, L = 100 Å and N total = 5, the I(k) curve is calculated as shown in Fig. 4.5 (a), where the form factor and the structure factor are also shown together. The long period peak appears at around k = 2π/L = 0.0628 Å−1 . In the actual analysis of the observed SAXS data, more realistic model of lamellar structure is used as shown in Fig. 4.5(b). The lamella is assumed to have the boundaries of gradually decreasing density. When we measure the SAXS intensity data I obs (k), we calculate the correlation function using the second equation of Eq. 4.43. The thus-obtained correlation function is shown in Fig. 4.5(c), from which the structure information of lamellar thickness, boundary zone size, long period, etc. are obtained concretely. In the case of oriented samples, the repetition of stacked lamellae gives the meridional scattering peaks, as shown in Fig. 4.19. If the lamellae are stacked with non-zero tilting angle measured from the drawing direction, these meridional scattering peaks are changed to the 4-point patterns. The details of the 2D SAXS patterns will be described in a later Sect. 4.3.

Fig. 4.5 a The structure factor, form factor and scattering intensity calculated for the stacked structure of rectangular lamellae as illustrated in Fig. 4.4. b An illustration of the stacked lamellae with deformed boundary zones, and c the corresponding correlation function, from which the various structural parameters are estimated

360

4 Small-angle X-ray Scattering Method

4.2.3.6

Fractal Structure

Typical fractal structures are seen in nature (Fig. 4.6). Polymer substances also sometimes form a fractal structure, a dendrimer, for example [9]. The fractal structure has the characteristic features: the partial shapes and the whole shape are similar, being named the self-similarity. The dimension of the fractal structure is not necessarily the integers. Let us see the examples. Figure 4.7a shows one of the quite famous fractal structures. One line segment is divided into three equal segments. The middle segment is copied and the two equivalent middle segments are prepared. These two segments are put in the middle part to form the triangle shape. At the second stage, the similar process is made for these initial four segments. As a result the four small triangle shapes are produced on the original four segmental lines. This process is repeated and the whole shape becomes complicated totally, but the local parts have the same shapes of the different sizes. Let us consider the relation between the number of segments (N) and the segmental length (L) for the above-drawn example (the Koch model). Before it, the normal linear segment (Fig. 4.7b) shows the following relation: (1) No. of segments (N ) = 1 the segmental length (L) = 1 (2) 3 1/3 (3) 9 1/9 (4) 27 1/27 On the other hand, for the Koch model case (Fig. 4.7a),

(a)

(b)

(c)

(d)

Fig. 4.6 Fractal structures. a Cabbage, b dendrimer, c fjord, and d ammonite

4.2 Density Difference and SAXS

(a)

361

(b)

Fig. 4.7 Fractal structures starting from the long rods. a Koch model b normal line

(1) No. of segments (N ) = 1 the segmental length (L) = 1 (2) 4 1/3 (3) 16 1/9 (4) 64 1/27 The plots of N versus L are shown in Fig. 4.8 in the linear and log scale. The N increases exponentially with the shortening of L. The double log plot gives the linear relation between N and L.

(a)

(b)

Fig. 4.8 Relation between the number of segments N and the segmental length L for the models shown in Fig. 4.7

362

4 Small-angle X-ray Scattering Method

log(N ) = −D log(L)

(4.49)

or N = (1/L) D

(4.50)

In the case of Fig. 4.7b, D = 1, corresponding to the usual 1-dimensional line. The N and 1/L are linearly proportional to each other. On the other hand, for the Koch curve, D = 1.26. This means the Koch curve does not behave in the simple 1D mode, but the behavior seems to be between the 1D and 2D modes. The similar situation can be seen for the 2D and 3D pictures, as illustrated in Fig. 4.9a, b. The D is called the fractal dimension. If the fractal occurs for the total body, it is called the mass fractal, while the fractal on the surface part is named the surface fractal.

(a)

(b)

Fig. 4.9 Examples of a 2D- and b 3D-fractal models

4.2 Density Difference and SAXS

363

The dimension of the fractal structure may be estimated from the SAXS data [10–13]. As mentioned already the scattering intensity I(k) is expressed using the Fourier transform of the correlation function.  I (k) ∝ g(r) exp(ikr)dr In the region of Rg −1 > k, the Guinier approximation is applied, as shown already in the previous section. In the region of 1/a > k > Rg −1 , where a is the radius of a particle, the correlation function is expressed as g(r ) ∝ r D−d exp(−r/ξ )

(4.51)

where d is the spatial dimension (d = 3 for the sphere, for example) and ξ is a measure of the size of cluster which scatters the X-ray beam. The Fourier transform gives the structure factor S(k) coming from the fractal structure as follows: (D−1)/2      Sfractal (k) = sin (D − 1) tan−1 (kξ ) / (D − 1)kξ 1 + k 2 ξ 2 ξ = [2/D(D + 1)]1/2 Rg

(4.52)

By introduction of the particle factor, the total scattering factor is expressed as Stotal (k) = Sfractal (k)Sparticle (k)

(4.53)

The k dependence of the scattering intensity I(k) is considered here. The total number of scattering units is N and that on the surface is N s . Depending on the region of the scattering vector k, the following relation is obtained. I (k) = N 2 S(k) ∝ N 2

for k < R −1 (R is the region of fractal aggregates)

∝ Ns (ka)−2Dm +Ds

for a −1 > k > R −1 (a is a particle size) (4.54)

For the aggregation with the smooth surface, Dm = 3 and Ds = 2 and so I(k) ∝ k −4 which is Porod’s law. For the fractal aggregataion, the slope changes in the form k −2Dm +Ds 4.2.3.7

Disordered Lamellar Stacking Structure

As shown in Fig. 4.5, the stacked lamellar structure shows the peaks corresponding to the long period between the neighboring lamellae. For the ideal structure, the

364

4 Small-angle X-ray Scattering Method

(a)

(b)

Fig. 4.10 a SAXS pattern of a surfactant and b the SAXS patterns measured for the various kinds of polyoxymetylene (POM) samples. Reprinted from Ref. [14] with permission of Elsevier, 2003

primary peak and the higher order peaks appear at the positions of k = k LP , 2k LP , 3k LP , and so on. Figure 4.10a is the case of the highly developed structure of surfactant composed of the long amphiphilic chain molecules. A series of peaks are located at the integer relations. Figure 4.10b is the case of POM crystallized from the melt [14]. The position of the first and second peaks is not in an integer relation of 1: 2. This noninteger relation of the several peaks comes from the stacking structure with the second kind of disorder. Hosemann et al. derived the equation of the scattering intensity for the system of such a disorder (paracrystal structure, see Sect. 1.15.9) [15]. Their theory is applied as shown below. In the equation, P(k) is the scattering factor of lamella (particle) and L(k) is the lattice (structure) factor of the stacking array, where the Δ is the difference of electron density between crystalline and amorphous phases, d is the thickness of main lamella, L is the lamellar periodicity or long period, g is a parameter of the lattice strain (Hosemann’s g factor) and defined as g = σL / L, and h = Lq/2π. σL is the standard error of Gaussian distribution for L. For the single lamellar stacking structure (Fig. 4.11a), 

(−d/2 ≤ x ≤ d/2) (4.55) 0 (x < −d/2, d/2 < x) ⎛ ⎞2 d/2 4 2 sin2 (dq/2) ⎜ ⎟ =⎝ cos(q x)d x ⎠ = (4.56) q2

ρ(x) = 2  ∞     P(q) =  ρ(x)e−iq x d x    −∞

−d/2

1 − exp(−4π 2 g 2 h 2 )  L(q) =  1 − exp(−2π 2 g 2 h 2 )]2 + 4 sin2 (π h) exp(−2π 2 g 2 h 2 ) =

sinh(q 2 σ L2 /2) cosh(q 2 σ L2 /2) − cos(Lq)

(4.57)

and I (q) = P(q)L(q) =

sinh(q 2 σ L2 /2) 4 2 sin2 (dq/2) × q2 cosh(q 2 σ L2 /2) − cos(Lq)

(4.58)

4.2 Density Difference and SAXS

365

Fig. 4.11 Two types of lamellar stacking structure. a Two-phase model and b inserted lamellar stacking model. Reprinted from Ref. [14] with permission of Elsevier, 2003

For the insertion model of the two lamellae of the different electron densities (Fig. 4.11b),  ∞ 2   2      2 1 sin(d1 q/2) 2 2 sin(d2 q/2) + exp(−i Lq/2) P(q) =  ρ(x)e−iq x d x  =  q q   −∞

= 4[ 21 sin2 (d1 q/2) + 2 1 2 cos(Lq/2)sin(d1 q/2)sin(d2 q/2) + 22 sin2 (d2 q/2)]/q 2

(4.59)

1 − exp(−4π 2 g 2 h 2 )  L(q) =  1 − exp(−2π 2 g 2 h 2 )]2 + 4sin2 (π h) exp(−2π 2 g 2 h 2 ) =

I (q) =

sinh(q 2 σ L2 /2) cosh(q 2 σ L2 /2) − cos(Lq)

4[ 21 sin2 (d1 q/2) + 2 1 2 cos(Lq/2) sin(d1 q/2) sin(d2 q/2) + 22 sin2 (d2 q/2)] sinh(q 2 σ L2 /2) q 2 [cosh(q 2 σ L2 /2) − cos(Lq)]

(4.60)  (4.61)

where Δ1 is the difference of electron density between the main crystalline phase and amorphous phase, Δ2 is the difference of electron density between the amorphous phase and the inserted crystalline lamella, which is sandwiched between main lamellae, d 1 is the thickness of main lamella, d 2 is the thickness of inserted lamella, L is the lamellar periodicity or long period. It was assumed an inserted lamella exists at the half position of the two neighboring original lamellae. The insertion model could reproduce the observed SAXS data well, as shown in Figs. 4.12 and 4.13 [14].

366

4 Small-angle X-ray Scattering Method

Fig. 4.12 Comparison of the observed SAXS profile of POM with the curve calculated using a two-phase model. Reprinted from Ref. [14] with permission of Elsevier, 2003

Fig. 4.13 Comparison of the observed SAXS profile of POM with the curve calculated using an insertion lamellar model. Reprinted from Ref. [14] with permission of Elsevier, 2003

4.3 2D-SAXS Patterns In general, the uniaxially oriented sample shows the two points scattering pattern along the meridional direction or the four points patterns. The three-dimensionally oriented sample shows the anisotropic SAXS patterns when the X-ray scattering measurement is made in the directions of through, edge and end directions as shown in Fig. 4.14a. This is the case of a doubly oriented sample of nylon 66 α form. In the X-ray diffraction pattern along the through direction, the two spots are observed along the draw direction. The spacing between the spot and the equatorial line gives the long period 118 Å of the stacked lamellae in the draw direction. The edge pattern shows the four points with the angle 40° from the draw direction, which shows the tilting angle of the lamellae. From all these data, the stacking structure of the tilted lamellae is derived as shown in Fig. 4.14b. On the other hand, the WAXD patterns measured in the three perpendicular directions can be interpreted well using the hydrogen-bonded sheets structure, in which the sheets are parallel to the rolled plane (Fig. 4.14c). The combination of the SAXS and WAXD data analysis shows the whole image of the three-dimensionally oriented structure of nylon 66 α form [16].

(b) (c)

Fig. 4.14 a b SAXS patterns measured for the doubly oriented nylon 66 α form along the 3 perpendicular directions. b The three dimensional lamellar model. c The WAXD pattern along the chain direction and the projected structure showing the hydrogen-bonded sheet structure. Reprinted from Ref. [16] with permission of Elsevier, 2003

(a)

4.3 2D-SAXS Patterns 367

368

4 Small-angle X-ray Scattering Method

4.3.1 Simulation of SAXS Pattern Let us see more details of the 2D-SAXS pattern. We need to study the relation between the structure and the corresponding Fourier transform pattern. Figure 4.15 shows the models of ball (sphere, circle), rectangular box, and extremely flat plate. The Fourier transform of these models gives the patterns shown in the right side of Fig. 4.15. In the case of a ball (or a circle), the series of rings are calculated and the cross section along the diameter direction gives the pattern predicted for a 3D ball as already shown in Sect. 4.2.2. The rectangular box gives the anisotropic 2D pattern, which contains the patterns predicted for the rectangular shape but the periods are different between the long and short axial directions. A flat plate gives extremely anisotropic profiles along the two axial directions. The aggregation of these particles shows the peaks corresponding to the spacing of these particles. Figure 4.16a shows the two balls systems with horizontal and tilt

Fig. 4.15 2D and 1D Fourier transforms calculated for a a ball, b a rectangular box and c a flat plate

4.3 2D-SAXS Patterns

369

arrangement, respectively. These structures are the convolutions of a ball shape (b) and the alignment line with a periodic structure (p); the function f = bˆp. The Fourier transform of f , F (f ), is expressed as a product of F (b) and F (p): F (f ) = F (b) × F (p). In fact, in the Fourier-transformed pattern, you can see a series of rings, which is a Fourier transform of a ball as already shown, and also the horizontal or tilted lines with a set of lines directing to the arrangement direction. The similar situation can be seen for the array of boxes (Fig. 4.16b). In the case of horizontal boxes aligned

(a)

(b)

Fig. 4.16 Fourier transforms of a the two balls system and b the two square boxes system

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4 Small-angle X-ray Scattering Method

1/h

1/[hcos( )]

Fig. 4.17 Fourier transforms of the variously stacked lamellar structure models. The h and d are the width and thickness of a small box, respectively.

along the tilting direction, the Fourier transform of a box is seen as indicated in a red box. The distance between the neighboring boxes, L, reflects on the period 1/L of a series of lines. These lines are perpendicular to the tilting line connecting the neighboring boxes. If the box itself is tilting, the Fourier-transformed box is detected in the tilting direction with the lines of 1/L spacing. When many boxes are arrayed in a particular direction, the Fourier-transformed pattern is dependent on the tilting direction of a box. The lines of short period are observed inside the red box, which correspond to the period between the neighboring boxes. It must be noted that, depending on the tilting direction of the box, the shape

4.3 2D-SAXS Patterns

371

1/h

h

1/h

d

1/D 1/L

L

D

1/d

1/L

1/L

1/D

 h

D



1/h



1/D 

L

1/d

1/L

d

Fig. 4.18 Fourier transforms of the variously stacked lamellar structures. The h and d are the width and thickness of a small box, respectively.

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4 Small-angle X-ray Scattering Method

(b)

1/h

(a)

1/h

h

1/d 1/L

1/d

d

1/L

L

L

(c) 1/h 1/D 1/h L

1/D

2/L

1/Lp

1/L

D Lp

Fig. 4.19 Relationship between the stacked lamellae and SAXS patterns

of the peaks is different. Besides, corresponding to the number of the boxes or the full length of the system, many lines of the short period are observed. From these calculation results, the SAXS patterns observed actually for the stacked lamellar structure can be interpreted in detail. You can pick up the detailed information from Figs. 4.17, 4.18 and 4.19. (The reflection width is inversely proportional to the thickness, but it is not exactly equal to each other. Rather the half-width of peak profile might be better for showing their relation as known from Eq. 1.105, Scherrer’s equation.) Figure 4.20 shows the SAXS data analysis made for the biaxially oriented LLDPE sample. The orientation of stacked lamellae changes depending on the incident position of the X-ray beam. The calculation of lamellar correlation function gives us the information of the lamellar thickness as well as the long period, etc. The correlation function can be calculated as follows:  C(x) ∝

I (k)k 2 cos(kx)dk

(4.62)

where I(k) is the intensity along the lamellar stacking direction. The k 2 is included for the Lorentz correction for the observed intensity.

4.3 2D-SAXS Patterns

373

flow axis

Model (a)

FFT

observed

(a) lamella

(b) (c) Line 1

(c)

(d)

(d) (e) (e) Line 2

(f)

(f)

Fig. 4.20 The observed SAXS patterns and their simulated patterns for the biaxially oriented LLDPE sample

As for the 2D-SAXS pattern measured for a uniaxially oriented fiber, the peak positions can be fitted with an ellipsoidal curve (Fig. 4.21) [17]. The mechanism speculated is due to the change of long period caused by the effect of the tensile stress T cos2 (φ) where φ is defined as the angle from the meridional direction. The long period is expressed as   L φ = L/ cos(φ) = A + B cos2 (φ) / cos(φ)

(4.63)

A = L φ (φ = 90°) = L E and B = L φ (φ = 0°) – L φ (φ = 90°) = L M – L E . In the small-angle region, this equation can be approximated as L φ = L E 2 tan2 (φ) + L M 2

(4.64)

Actually, the long period L φ evaluated from the observed SAXS pattern can be fitted nicely by this equation. The angle dependence of the long period is illustrated in Fig. 4.21, where the tilting angle changes with the change of the long period. Figure 4.21b corresponds to the above equation.

374

4 Small-angle X-ray Scattering Method

Fig. 4.21 a SAXS peak of the uniaxially oriented polymer sample shows the elliptical shape, which can be fitted using an equation of ellipsoid (b). c and d the fitting using Eqs. 4.65 [17]. Reprinted from Ref. [17] with permission of Wiley & Sons, inc., 2006

4.3.2 SAXS Simulation by MC Method The 2D-SAXS patterns can be calculated by assuming some structural model of the stacked lamellae. One convenient method is to perform the Fourier transformation using the commercial program (for example, Image J [18]). One example is shown in Fig. 4.20, where the SAXS patterns were measured at the various positions of the biaxially oriented LLDPE sample. The aggregation of lamellae was built up by drawing a picture using many small boxed, which was transferred to the program of Fast Fourier transformation (FFT). Another simulation method is to use the Metropolis Monte Carlo method [19]. In particular, the aggregation of many small particles is introduced and the positions of these particles are modified so that the observed SAXS pattern is reproduced as well as possible. For example, Fig. 4.22 shows the actual calculation result. The starting model consists of the randomly arranged aggregation of 6400 particles of 10 Å radius, which were packed in the cell of 2130 Å. In this calculation the collision of particles is avoided by introducing the judgement of collision and also by using the inter-particle interaction energy. The finally obtained structure is an image of stacked lamellar structure although the parallelism is not perfect. The simulated SAXS pattern is similar to the observed one. From the simulated lamellar image, the correlation between the neighboring lamellae can be calculated. The calculated curve is similar to that obtained from the observed SAXS data using Eq. 4.43.

4.4 SAXS Measurement System

(a) initial

final

375

(b) calculated

simulated

(c) correlation curve

Fig. 4.22 a initial and final models consisting of many particles used in the Monte Carlo simulation of the 2D-SAXS patterns, b the comparison between the observed and calculated SAXS patterns, and c the correlation curve which is almost the same as that obtained from the observed data. Reprinted from Ref. [19] with permission of the American Chemical Society, 2020

4.4 SAXS Measurement System 4.4.1 SAXS Instruments and Slit System Examples of SAXS instruments are shown in Figs. 4.23 and 4.24. The optical path is evacuated with a rotary pump to avoid the X-ray scattering by air. The X-ray beam size is reduced using a set of slits. The X-ray beam should be as brilliant as possible, which is attained with a confocal mirror in this example. The SAXS signal is detected using a 2D detector. The direct beam is stopped using a beam stopper set in front of the detector to avoid the damage of the detector. The so-called SAXS resolution is important, which is the lowest k (or q) value measurable in the actual system. The erase of parasitic scattering is needed, which can be made by reducing the size of incident X-ray beam. In this example a set of pin holes is used. This can be used since the X-ray beam is brilliant. When the X-ray beam is not very strong, the usage of pin hole is not very useful. The flat (or line) slit is used instead, which gives the X-ray beam of line shape.

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4 Small-angle X-ray Scattering Method

TM pump (10-7 Torr)

Rotation Anode

Rotary pump (10-3 Torr)

Confocal mirror (He gas)

slit

Sample stage

Beam stopper Vacuum pipe

Kapton film

2D detector

Kapton film

Rotary pump (10-3 Torr)

Fig. 4.23 SAXS measurement system (Rigaku Nano-Viewer®)

Fig. 4.24 SAXS measurement system (SPring-8 BL40B2)

Figures 4.25 and 4.26 show the several examples of the slit systems. (a)

Slit Camera: The X-ray beam size is reduced using a pair of slits (S1 and S2). The sharp slit blade gives a parasitic scattering, which can be erased by setting the third slit (S3).

4.4 SAXS Measurement System

377

(a) Slit System

parasitic scattering

S3

detector S2 S1 sample X-ray

Beam stopper

(b) Kratky U-slit System detector E B sample X-ray B

Beam stopper

B

E

X-ray

detector

B beam stopper

Fig. 4.25 SAXS slit systems

378

4 Small-angle X-ray Scattering Method S1

X-ray

S2

single crystal

sample

single crystal detector

Fig. 4.26 Bonse-Hart camera [20]

(b)

(c)

Kratky Camera: This system is powerful for reducing the parasitic scattering effectively [3]. The collimation of the X-ray beam is made using an edge slit (E) and the two blocks (B). The intensity of the scattered signal is high and the resolution is about 103 Å. Bonse-Hart Camera: This camera is to use the X-ray beam generated as a Bragg reflection from a single crystal. For example, a pair of Ge single crystals (S1) is set, and the incident X-ray beam is reflected on it to generate a monochromatic beam with a small divergence. The X-ray signals scattered from the sample enter again another set of Ge single crystals (S2) and detected finally with a detector. The S2 crystals are rotated to collect the scattering component of the different angles. The signal intensity is reduced remarkably (a synchrotron system is recommended) but the resolution is quite high to give the SAXS signal of 104 Å.

Different from the round pin hole, the SAXS data obtained using a line slit system need to be corrected for the slit function. The observed SAXS signal is a convolution of the true signal at a position r, I o (r), and the slit shape s(r): I(r) = I o (r − r)ˆs(r ). The Fourier transform gives F (I) = F (I o )F (s), and then the corrected SAXS data is obtained as F −1 [F (I)/F (s)]. The measurement of SAXS data needs a highly brilliant X-ray source. Figure 4.27 shows the several examples. The confocal mirror is good for getting the X-ray beam of high brilliance and small size. The small size of the X-ray beam is useful for taking off the parasitic scattering. Thin slit is used for decreasing the beam size to several tens to hundreds μm diameter, but the correction of slit shape is needed for the quantitative data analysis as mentioned above. The utilization of synchrotron radiation gives more easily an X-ray beam of submicron size. For this purpose, a pinhole is usually used. Fresnel zone plate (FZP), polycapillary X-ray lens, compound refractive X-ray lens (CRL), confocal mirror, and KB (Kirkpatrick-Baez) mirror are useful for the reduction of an X-ray beam size. The FZP is a grating composed of many alternate transparent and opaque rings [21]. These rings are made of Ta attached on SiN plate with about 25 nm width. The X-ray beam is incident to FZP and the diffraction occurs. The first-order diffraction beam is used as a focused beam with a size of about 1 μm. The capillary lens is to utilize a multiple internal reflection inside each capillary just like a waveguide [22]. The CRL is to use the refraction of X-ray beam when it passes a substance. Since the refractive index for X-ray beam is smaller than 1, the shape of lens is concave. By stacking many blocks, the focused beam is obtained. The KB mirror (Rh coated on fused silica) is similar to the Confocal mirror and it uses a pair

4.4 SAXS Measurement System

Fig. 4.27 The various mirror systems

379

380

4 Small-angle X-ray Scattering Method

of bent mirrors of 100 cm length. The X-ray beam is totally reflected on the surface and focused onto the sample position or the detector position. The scattered signal is absorbed and scattered by air. The vacuum system (a rotary pump) is used for purging the air. The vacuum of 10–3 Torr is enough. The vacuum pipe is used, which is made of aluminum or poly(vinyl chloride) resin. The popular window material of a pipe is Kapton film of suitable thickness (several hundreds μm). Sometimes, Kapton film is broken suddenly under the operation of evacuation. Carbon fiber reinforced film is tougher and used for window. The ideal case is to put all the equipment into the vacuum space. The air scattering is erased totally. The background must be erased, which is measured without a sample under the same condition as that of the sample.

4.4.2 Setting of WAXD Detector on the SAXS Instrument The simultaneous measurement of WAXD and SAXS is useful for tracing the structure change of a soft material, since the sample property is sensitively affected by the slight change in the preparation condition. Figure 4.28 shows two examples of the setting. One is to use an imaging plate with a small hole at the center position. The SAXS component can pass through the hole and reaches another detector set at a longer distance. The photon counting device with a central hole is more useful for this purpose. In Fig. 4.29, the flat panel or Pilatus 100 k is set so that the direct beam and SAXS component can pass just below the camera frame, where some parts of the equatorial and meridional scattering components are cut off. Figure 4.30 shows some examples of the simultaneously-measured WAXD and SAXS patterns of uniaxially-oriented samples using the systems shown in Figs. 4.28 and 4.29.

4.4.3 Samples and SAXS Data Treatments The samples for the SAXS measurement are basically the same as those used for the WAXD measurements. However, it may be better to pay the various attentions. (i)

(ii)

The degasification of solvent is needed to avoid the scattering from the bubbles contained in the sample. For the solid sample, the evacuation in the molten state is useful for this purpose. The scattering from the sample edge is strong. The X-ray beam size should be smaller than the sample width. The utilization of a microbeam X-ray is good for it. For example, the measurement of the SAXS pattern from a single filament of 5 μm diameter cross section is made using an X-ray beam of 1 μm diameter. Figure 4.31 shows the actually measured SAXS pattern from the carbon fiber monofilament. The streaky equatorial scattering does not come from the edge of the monofilament but it originates from the microvoids inside the fiber [23]. Another method is to match the density surrounding a polymer fiber by immersing the fiber into a liquid of the same density. The surface scattering from the fiber is reduced by this method [24].

4.4 SAXS Measurement System

381

(a) SAXS detector

WAXD detector with a central hole

SAXS detector

sample

Imaging Plate

(b) WAXD detector

SAXS detector

Photon counter with a central hole Fig. 4.28 a The simultaneous measurement system of WAXD and SAXS data. A small hole is opened on the central part of the imaging plate so that the SAXS signal can pass through. b The simultaneous measurement system of WAXD and SAXS data using photon counter. A central hole is opened on the WAXD detector (Advacam, Adva PIX Quad)

382

4 Small-angle X-ray Scattering Method

Fig. 4.29 The simultaneous measurement system of WAXD and SAXS data. The WAXD detector is set at the tilted angle

Fig. 4.30 The WAXD and SAXS data obtained using the simultaneous measurement systems: (a) an oriented HDPE (Fig. 4.28a), (b) an oriented HDPE (Fig. 4.28b) and (c) an oriented poly(tetramethylene terephthalate) (Fig. 4.29)

4.4 SAXS Measurement System

383

(b)

(a)

(c)

Fig. 4.31 SAXS pattern of a fiber. a The X-ray beam size is larger than the width of the fiber. The horizontal streaks are due to the edge scattering from the fiber. b The horizontal edge scattering is not generated because the sample width is larger than the X-ray beam size. c The actually-observed SAXS pattern from the carbon fiber

(iii)

(iv)

The scattering from the surface of the windows made of quartz, mica, Kapton, etc. must be avoided. If the scattering from these windows is relatively weak, the background subtraction is useful. The correction of the background must be made. The 2D-SAXS pattern measured for the unoriented sample is integrated to get the 1-dimensional scattering profile, which is named I obsd . If the effect of the dark current is serious, it must be erased from I obsd . I c obsd = I obsd – I dark . The sample thickness d and the measurement time t are taken into account to reduce I c obsd to I o obsd (I o obsd = I c obsd /(td)). The correction of absorption is made: the transmittance T sample is measured, and then I correct = I c obsd /T sample . The similar correction is made for the background data. The finally corrected intensity is expressed as follows:   I correct = (Iobsd − Idark )/ tdTsample − (Iback − Idark )/(tdTback )

(4.65)

384

4 Small-angle X-ray Scattering Method

(v)

When a slit of rectangular shape is used, the scattering profile is affected by the slit shape (the convolution between the slit shape and the scattering pattern). The slit correction is needed to make. The absolute scattering intensity is obtained by referring to the scattering intensity of the standard sample such as low-density polyethylene (Lupolen), glassy carbon, water, and so on. The standard sample must be stable even after the repeated X-ray scattering measurements and also stable chemically. The standard sample must be homogeneous in the thickness. The angle dependence of the scattering signal of the standard sample must be low. The scattering intensity from a sample Is (k) is given as follows:

(vi)

Is (k) = ε(Io /Ao )As ts ds Ts (d /d)s  = K As ts ds Ts (d /d)s

(4.66)

where ε is a detector efficiency, I o is an intensity of the incident X-ray (neutron) beam, Ao is an effective area of the incident beam, As is the area of the sample, t s is an exposure time, d s is the sample thickness, T s is the transmittance of the sample, (d /d)s is the differential scattering cross-section of the sample and  is the solid angle of one pixel of the detector and given as aL o /(L d )3 using the area a of one pixel, the sample-to-detector pixel distance (L o ) and the sample-to-detector distance L d . The I s (k) must be corrected for the dark current and background as already mentioned in Eq. 4.65. If the measurement condition is common to both the sample and standard substance, we can use the calibration constant K [= ε(I o /Ao ) ]. At first, I stand (k) of the standard substance is measured (including the background and dark-current corrections). The standard substance must be chosen properly, which gives a flat I stand (k) value in a wide k region. The (d /d)stand is given in the literature [25–30]: for example, 0.0165 cm-1 for distilled water (1.54 Å) and 36.63 cm-1 for glassy carbon (1.54 Å). By using the ratio of scattering length densities p = ( ρSANS )2 /( ρSAXS )2 , (d /d)SANS = p(d /d)SAXS is calculated. The calibration coefficient K is obtained using the thus-measured I stand (k) and the information Astand (k)t stand (k)d stand (k)T stand (k). The similar measurement is performed for the sample under the same condition, and (d /d)stand is obtained as follows (after the corrections of dark current and background): K = Istand (k)/[(d /d)stand Astand tstand dstand Tstand ] (d /d)s = Is (k)/[K As ts ds Ts ]

(4.67)

A crystalline sample might give the long period peak and it might be difficult to find the k region of flat intensity. One idea is to use the k range of Porod’s law, in which the plot of I(k)k 4 is almost constant. For the standard sample, [ lim Istand (k)k 4 /Astand tstand dstand Tstand ]/[ lim (d /d)stand k 4 ] = K (4.68) k→∞

k→∞

4.4 SAXS Measurement System

385

Even for a crystalline high-density polyethylene sample, the Porod’s plot in a high k range gives the horizontal curve, from which (d /d)s is estimated [29, 30].

4.4.4 Contrast Matching Contrast matching method is useful to extract the information of a solute in the solution, for example. In particular, this method is powerful for the small-angle neutron scattering (SANS), whose scattering intensity is given by almost the same equation as that of SAXS. Here, the principle of this method is described briefly about the SANS experiment [6, 31–35]. For the difference of the scattering length density ρ(r), the scattering intensity I(k) along the direction k is given as  ρ(r) exp(ikr)dr|2

I (k) = (1/V )|

(4.69)

V

  ρ(r) = i j ρij (r) = i j ρi (r) − ρj (r)

(4.70)

where ρj (r) is the density of the component j. Then  I (k) = (1/V )|

 ρ(r) exp(ikr)dr|2 = (1/V )|

V

i j ρij (r) exp(ikr)dr|2 V

(4.71) where ρij (r) = 0 for i = j. Let us see the case of three components (V = V 1 + V 2 + V 3 under the assumption of incompressive system).   I (k) = (1/V )| ρ12 (r) exp(ikr)dr + ρ23 (r) exp(ikr)dr  + ρ13 (r) exp(ikr)dr|2 = ρ12 2 S1212 + ρ13 2 S1313 + ρ23 2 S2323 + 2 ρ13 ρ23 S1323 + 2 ρ13 ρ12 S1312 + 2 ρ12 ρ23 S1223 Here,  S1212 = (1/V )|  exp(ikr 12 )dr12 |2 S1213 = (1/V )| exp(ikr 12 )dr12 exp(−ikr 13 )dr13 | By changing the H atoms in the component 3 into the D atoms, the scattering length density difference ρ13 (r) can be controlled to be zero (Fig. 4.32). Then the above equation becomes

386

4 Small-angle X-ray Scattering Method

Fig. 4.32 Contrast matching. The small red parts cannot be seen anymore by modifying the density of the background

I (k) = ρ12 2 S1212 + ρ23 2 S2323 + 2 ρ12 ρ23 S1223

(4.72)

In this way, we can extract the limited components of partial scattering functions (S) and derive the corresponding structural features. This method is called the contrast matching method. The contrast variation method is to change the contrast ρij continuously by controlling the scattering length density of the third component, for example. In a lucky case, all of the S components can be obtained by solving the simultaneous equations obtained for a series of samples with variously different density contrasts. In the case of two components system, we have I (k) = ρ12 2 S1212

(4.73)

As a simple example, let us consider the solvent, water. The scattering length density ρ(r) of water (r) is obtained by using the scattering lengths of H, D and O atoms as follows. ρ(r) = bi /v for the scattering length bi of the i−th atom and the molecular volume v. b(H) = −3.7390 × 10−15 m b(D) = 6.671 × 10−15 m b(O) = 5.803 × 10−15 m

b(H2 O) = −3.7390 × 2 + 5.803 = −1.675 × 10−15 m = −1.675 × 10−13 cm d = 1.0g/cm3 , M = 18.0g/mol, NA = 6.02 × 1023 mol−1 , v = M/(NA d) = 2.990 × 10−23 cm3   ρ(H2 O) = b/v = −1.675 × 10−13 cm/ 2.99 × 10−23 cm3 = −0.552 × 1010 cm−2

4.4 SAXS Measurement System

387

b(D2 O) = 6.671 × 2 + 5.803 = 19.172 × 10−13 cm d = 1.11g/cm3 , M = 20.03g/mol, NA = 6.02 × 1023 mol−1 , v = M/(NA d) = 2.998 × 10−23 cm3   ρ(D2 O) = b/v = 19.172 × 10−13 cm/ 2.998 × 10−23 cm3 = 6.395x1010 cm−2 For the mixture of H2 O ad D2 O, ρ(solvent) = ρo = ρ(H2 O)(1 − φ) + ρ(D2 O)φ where the volume fraction of D2 O is φ. For a polymer, the scattering length density is calculated using the information of the monomeric unit and the degree of polymerization. For the two components system consisting of polymer and water, the intensity I(0) at k = 0 is approximately given by  2 I (0) = Np ρp − ρo Vp 2 /V

(4.74)

where N p , V p and ρp are the number, volume, and scattering length density of the uniform polymer particles, respectively. V is the volume of the solution. By changing the ρo value of the H2 O and D2 O mixture, I(0)1/2 changes linearly with ρ (= ρp – ρo ). Find the minimal point of I(0), which is named the contrast matching point. This method is useful for the system containing the plural types of polymer component. We can erase the contribution of a particular polymer component and detect the signal of the remained polymer component. In some case of a polymer solid sample without any other component, the SAXS pattern cannot be detected even when both crystalline and amorphous regions coexist in the sample. The typical case is seen for isotactic polybutene-1 crystalline form II, as will be mentioned below. The reason is the almost equal density between the crystalline and amorphous regions [36].

4.5 Examples of SAXS Data Analysis of Polymers 4.5.1 Lamellar Insertion in Melt-Isothermal Crystallization of POM Polyoxymethylene (POM) shows a rapid crystallization when cooled from the melt. The structural evolution process during the crystallization phenomenon was studied by the rapid time-resolved measurements of WAXD and SAXS [14, 37]. At the same time, the infrared (IR) spectral technique was combined with them. The IR spectra of POM are sensitive to the morphology or the shape of crystallite [38, 39]. This characteristic feature can be utilized for the study of morphological change in the crystallization process. The IR crystalline bands can be classified to the two groups originated from the extended chain crystal (ECC) and folded chain crystal (FCC) morphologies. The details of the IR spectra will be described in the other chapter.

388

4 Small-angle X-ray Scattering Method

POM is relatively easily degraded at a high temperature. In order to avoid this problem, a copolymer of trioxane with small content of ethylene oxide (Duracon M90, Polyplastics Co. Ltd, Japan) was used without any loss of characteristic crystallization behavior of POM homopolymer itself, although the melting and crystallization temperatures are about 5 °C lower than those of the homopolymer. The isothermal crystallization experiment was performed by using a temperature jump cell made by ourselves (refer to Sect. 1.7.3.5). The sample was melted at 205 °C for several minutes and jumped to another window kept at Tc, where the temperature changing rate was 1000 °C/min. The temperature fluctuation was ± 0.1 °C. The infrared spectra were measured at every ca. 4 s in a rapid scanning mode. The time dependences of SAXS and WAXD were measured by using a synchrotron radiation as an X-ray source at the beam lines BL10C and BL15A, respectively, of the Photon Factory at the KEK (High-Energy Accelerator Research Organization) in Tsukuba, Japan. The wavelength of incident X-ray beam was 1.49 Å. The data were collected at every 3 s using PSPC detectors in both the SAXS and WAXD measurements. The SAXS profiles collected in the isothermal crystallization process at 130 °C are reproduced in Fig. 4.33a [37]. Around 11 s after the jump, a long period peak L 1 started to appear and increased in intensity with slight shift of the peak position toward higher k side, where k is a scattering vector defined as k = (4π/λ)sin(θ ) with the scattering angle 2θ. The peak had a shoulder at higher k side and this shoulder became weaker with the passage of time. At the same time a broad peak was also observed around k ~ 1.1 nm−1 , although it was too low in height and noisy. In the time region of 150 s another peak L 2 started to appear around k ~ 0.9 nm−1 and increased in intensity gradually. In parallel the intensity of the L 1 peak intensity decreased and became comparable to that of the L 2 peak at 1200 s. The long periods estimated for the L 1 and L 2 peaks at the terminal point of the experiment (1438 s) are 12.1 and 6.5 nm, respectively; the L 1 being almost twice the L 2 . This type of SAXS profile change was not detected when the isothermal crystallization was performed

Fig. 4.33 Time dependence of SAXS profiles of POM in the isothermal crystallization process at 130 and 150 °C. Reprinted from Ref. [37] with permission of Elsevier, 2003

4.5 Examples of SAXS Data Analysis of Polymers

130oC

389

150oC (100)

(100)

1196 sec 372

695 sec 372 198 165 126 102 69 36

126 51 36 27 18 9 10

12

14

K/

16

nm-1

18

20

10

12

14

K/

16

18

20

nm-1

Fig. 4.34 Time dependence of WAXD profile of POM in the isothermal crystallization process at 130 and 150 °C. Reprinted from Ref. [37] with permission of Elsevier, 2003

at 150 °C as shown in Fig. 4.33 b. Around 35 s after jump, a peak L 1 started to appear and increased in intensity with a slight shift of the peak position toward higher k side. The L 2 peak could not be observed here. A broad and weak peak was detected around k = 0.8 – 1.3 nm−1 in a later stage of crystallization. Figure 4.34 shows the temperature dependence of WAXD profiles measured in the isothermal crystallization at 130 and 150 °C from the molten state, respectively. The integrated intensity of the crystalline (100) reflection was estimated. The SAXS intensity and IR band intensity are plotted as shown in Fig. 4.35. These figures may be divided into several time regions. By comparing all these data, a structural evolution process may be deduced more concretely in the following way as illustrated in Fig. 4.36 (130 °C) and 4.37 (150 °C). < 130 °C > Time Region I Immediately after the sample was cooled steeply from the melt to 130 °C, the lamellae of FCC morphology with about 3.0 nm thickness started to appear and were stacked at an averaged long period of 14.5 nm. The intensity of the WAXD (100) reflection increased also steeply with time (Fig. 4.35a). Time Region II The long period of the stacked lamellae became shorter and reached almost the constant value, 12–13 nm. The lattice spacing estimated from the (100) reflection or the a-axial length of the unit cell was almost unchanged. Around 150 s the new lamellae of ca. 3.5 nm thickness started to appear at a long period of 6.3 nm. These new lamellae (daughter lamellae) locate between the already existing lamellae (mother lamellae). At the same time, the infrared bands intrinsic to ECC morphology

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4 Small-angle X-ray Scattering Method

(b) 150oC Temperature / o C

o

Temperature / C

(a) 130oC

Temeperature

I

II

III

WAXS

Intensity

Reduced Integrated Intensity

Intensity

I1 (L1) I2 (L2) FCC

ECC

0

I’

I’’

(100)

SAXS L1

IR

Intensity / a.u.

Intensity

IR

I WAXS

100

SAXS

Temperature

400 800 Time / sec

FCC

1200

0

200 400 Time / sec

600

Fig. 4.35 Time dependence of the various data of POM in the isothermal crystallization process at 130 and 150 °C. Reprinted from Ref. [37] with permission of Elsevier, 2003 14.5 nm 4.6 nm

3.0 nm

130oC

12.6 nm

II

12.1 nm

6.1 nm

6.3 nm 3.6 nm

3.5 nm

Melt Melt

4.4 nm

II II

III III

Fig. 4.36 Structural evolution of POM in the isothermal crystallization process at 130 °C. Reprinted from Ref. [37] with permission of Elsevier, 2003

started to increase in intensity gradually. These ECC bands may indicate the formation of taut tie chains passing through the adjacent lamellae. Figure 4.36 illustrates the structural evolution process. The taut tie chains of extended conformation are speculated to form a small bundle of cylindrical shape with 20 nm length and 2 nm radius.

4.5 Examples of SAXS Data Analysis of Polymers

391

Time Region III When the crystallization time was beyond 650 s, the SAXS intensities of the L 1 and L 2 peaks and the intensities of the FCC and ECC infrared bands changed their slopes slightly as seen in Fig. 4.35. In particular, the infrared intensity of ECC band increased more remarkably, suggesting an increase of the number of taut tie chains. The integrated intensity of the (100) reflection changed also but very slowly. All these findings suggest some additional change in the inner structure and/or the lamellar stacking mode.

Time Region I’ At 150 °C, the chain-folded lamellae of about 6 nm thickness appeared at first, which were staked at an averaged long period of 21 nm (Fig. 4.35b). This structural formation occurred more slowly and in a wider time region than 130 °C. The infrared FCC band appeared in parallel to the SAXS L 1 peak and the WAXD (100) reflection. Time Region I” The long period of stacked lamellae was shorter and reached the almost constant value of 15–16 nm. The WAXS (100) reflection intensity and the SAXS L 1 peak intensity increased gradually. But the infrared FCC band intensity was almost saturated in this time region (Fig. 4.35). The structural evolution is illustrated in Fig. 4.37. When the structural change occurring in the melt-isothermal crystallization process at a constant temperature (Tc) is compared with that traced in the slow and continuous cooling process from the melt, the temperature dependence (the latter) and the time dependence (the former) are in a systematic relation (Fig. 4.38). The structure attained after the passage of long time in the isothermal crystallization is equal to the structure detected at the corresponding temperature in the slow cooling process: the structure detected at Tc in the slow cooling process is not created instantly even when the molten sample is brought suddenly to Tc, but it is realized for the first time after passing through many successive stages observed in the nonisothermal crystallization process in the temperature region higher than Tc. In other words, the structural evolution in the isothermal crystallization is considered to consist of many elementary processes such as the generation of isolated lamellae in the melt, the stacking of these lamellae, the insertion of new and thinner lamellae in between the original lamellae, the extension of tie chain segments to form the ECC-like structure, etc.

150oC

6 nm

8 nm 21 nm

Melt Melt

I’ I’

16 nm

I’’ I’’

Fig. 4.37 Structural evolution of POM in the isothermal crystallization process at 150 °C. Reprinted from Ref. [37] with permission of Elsevier, 2003

392 Fig. 4.38 Relation of the structural evolution of POM in the isothermal crystallization and slow cooling processes. Reprinted from Ref. [37] with permission of Elsevier, 2003

4 Small-angle X-ray Scattering Method

temperature

130oC

time

50sec

200sec 140oC

150oC

time

100sec

180oC

4.5.2 Higher Order Structural Change and Phase Transition of it-Polybutene-1 Isotactic Polybutene-1 (it-PB) crystallizes at first to the crystalline form II from the melt. The form II transforms gradually to form I at room temperature [36]. The uniaxially oriented sample of form II is prepared by stretching the molten sample at a high temperature below the melting point (about 130 °C). Figure 4.39a shows the snapshots of 2D-WAXD and SAXS patterns of the stretched sample measured at 47 °C [36]. The WAXD pattern at the starting point showed clearly the existence of the highly oriented crystal form II, while the SAXS pattern gave almost no scattering signal. With the passage of time, the WAXD pattern started to change and the relative intensity of crystal form I diffraction peaks increased in parallel. At the same time the SAXS pattern started to appear and increased the intensity gradually. In the early stage of the phase transition, form II and amorphous phase are coexistent in the sample. In spite of the existence of the oriented form II as known from the WAXD data, the SAXS pattern could not be detected. This is because the electron density of crystalline form II (d II 0.89 g/cm3 ) and amorphous phase (d amorphous 0.87 g/ cm3 ) are close to each other. As the time passes the phase transition to the form I proceeds. The SAXS pattern increases its intensity gradually because of the higher density of form I (ρ I 0.97 g/cm3 ). By integrating the equatorial line of the WAXD pattern, the one-dimensional diffraction profile was obtained. The molar content of form I, X(I) was evaluated by analyzing the integrated intensity of the WAXD profiles.

4.5 Examples of SAXS Data Analysis of Polymers

(a)

393

(b) 211I

213II

200II 220II

300I 220I

Fig. 4.39 a WAXD and SAXS change in the isothermal crystallization of the oriented it-PB sample at 47 °C. b Relation between the invariant Q and the molar content of crystal form I. Reprinted from Ref. [36] with permission of Elsevier, 2016

On the other hand, the 1D-SAXS data obtained by scanning the 2D-SAXS patterns along the meridional line parallel to the draw axis were used for the calculation of the correlation function of the stacked lamellar structure. As already mentioned, the invariant Q is a measure of the electron density and the crystallinity (X c ) of the sample, which is related to the electron density difference Δρ between the crystal and amorphous regions as below (Eq. 4.45). Q = |Δρ|2 X c (1 − X c )

(4.75)

The Δρ is expressed as Δρ = ρc − ρamorphous = [X (I)ρI + X (II)ρII ] − ρamorphous = X (I)(ρI − ρII ) + ρII − ρamorphous ≈ X (I)Δρ(I, II) [Δρ(I, II) ≡ ρI − ρII ]

(4.76)

Therefore, we have Q = |Δρ(I, II)|2 X (I)2 X c (1 − X c ) ≈ K X (I)2

(4.77)

where K is a constant since the total degree of crystallinity X c is assumed not to change at such a low experimental temperature as −20 ∼ 60°C. This equation tells that Q is proportional to the relative content of form I crystal, X(I)2 . In fact, as shown in Fig. 4.39b, the time dependence of Q indicated by open circles and X(I)2 , solid circles, correspond quite well to each other.

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4 Small-angle X-ray Scattering Method

4.5.3 Change of Rg in the Crystallization from the Melt The size of random coil in the molten state is kept unchanged in the crystallization process, which was found for high-density polyethylene (HDPE) by the measurement of SANS in the crystallization process. The convenient way is to use the Guinier plot in the low k range. A small amount of deuterated (or hydrogenous) d-HDPE (or hHDPE) sample was mixed with hydrogenous (or deuterated) sample and the SANS measurement was made in the low k range. However, the d-HDPE and h-HDPE components did not mix very homogeneously, but some phase separation occurred in the crystallization process [40, 41]. A pair of d-HDPE and h-LLDPE (with 17 ethyl side groups per 1000 skeletal carbon atoms) can be crystallized perfectly homogeneously for the blend ratio of d/h = 0/100 ~ 100/0 even when the sample is slowly cooled from the melt [42]. The SANS data was treated using the following equations for the blend sample of H and D species [43, 44]. The observed scattering intensity I(k) was corrected for the estimation of the absolute cross section [d /d]coherent . The structure factor S(k) is   S(k) = [d /d]coherent / (bh − bd )2 /v2

(4.78)

v is a volume per monomeric unit and v = (vh vd )1/2 . vi is the volume of the i-component (i = h and d). On the other hand, this S(k) is expressed using the form factors as follows. 1/S(k) = 1/(vh φh Nh Ph ) + 1/(vd φd Nd Pd ) − 2χ/v

(4.79)

φi is the volume fraction. N i is the degree of polymerization (3340 for d-species and 2680 for h-species). The χ is the parameter to obey the following equation: χ = A + B/T (A = −1.63 × 10–3 and B = 0.79 for the (d/h) = (50/50) blend ratio, for example). This value is smaller than the critical χ value to cause the phase separation, indicating the perfect homogeneous mixing of d-HDPE and h-LLDPE. The form factor P is given as   P(u) = 2/u 2 [exp(−u) − 1 + u]

(4.80)

where u = (k Rgi )2 for the scattering vector k and Rgi is the radius of gyration of the i-th component. (Originally this equation is applied for the monodispersed Gaussian chain. Some correction is needed to apply it. Refer to the original paper for this treatment.) In the actual analysis, the observed plots of 1/S(k) vs u (= (kRgi )2 ) were reproduced by adjusting the Rgi values of the D and H chains (see Fig. 4.40). The thusestimated Rg (H) and Rg (D) are plotted against temperature, as shown in Fig. 4.41.

4.5 Examples of SAXS Data Analysis of Polymers

395

Fig. 4.40 Structure factor S(k) measured for the blend sample between D-HDPE and H-LLDPE (50:50) at the various temperatures. Reprinted from Ref. [44] with permission of Springer Nature, 1999

k2/10-4 A-2 Fig. 4.41 Temperature dependence of Rg estimated in the molten state and in the crystallized blend samples of D-HDPE and H-LLDPE. Reprinted from Ref. [44] with permission of Springer Nature, 1999

On the other hand, the Rg value at room temperature was obtained by performing the Guinier plot for the blend sample of h- and d-species with the dilute concentration of d-component 1~3%. The result is also plotted in Fig. 4.41. The Rg does not change almost even after passing the crystallization temperature. The polyethylene chains are entangled in the melt with the Rg value, which does not change even when the molecular chain changes the conformation to the planarzigzag form in the crystalline lamella. The aggregation state of the chains might be illustrated in Fig. 4.42. The chain fold occurs randomly (or only partially regularly) with keeping the Rg. This model may be similar to the Esterrungsmodell (solidification model) proposed by Fischer [45], but the situation is appreciably different from the present case. Their model was for the melt-quenched sample of h-HDPE and d-HDPE to avoid the partial phase separation, while the blend sample between d-HDPE and h-LLDPE does not show any phase separation and can be applied to the slow-crystallization phenomenon.

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4 Small-angle X-ray Scattering Method

Fig. 4.42 Illustration of structural change of a single chain of PE in the crystallization process from the melt. Reprinted from Ref. [44] with permission of Springer Nature, 1999

References 1. N. Kasai, M. Kakudo, X-Ray Diffraction by Macromolecules (Springer, Berlin, 2005). 2. R.-J. Roe, Methods of X-ray and Neutron Scattering in Polymer Science (Oxford University Press. Inc., 2000) 3. O. Glatter, O. Ktatky, Small-angle X-ray Scattering (Academic Press, London, 1982). 4. A. Guinier, G. Fournet, Small-angle Scattering of X-rays (Wiley, New York, 1955). 5. J.S. Higgins, H.C. Benoit, Polymers and Neutron Scattering. Oxford Series on Neutron Scattering in Condensed Matter, Series, vol. 8 (1997) 6. L.A. Feigin, D.I. Svergun, Structure Analysis by Small-angle X-Ray and Neutron Scattering (Plenum Press, New York, 1987). 7. T. Hashimoto, Principles and Applications of X-ray and Neutron Scatrering (in Japanese) (Kodansha, 2019) 8. M.J. Lorenzt, Buerger, The correction of X-ray diffraction Intensities for Lorentz and Polarization Factors. Physics 26, 637–642 (1940) 9. F.C. Moon, Chaotic and Fractal Dynamics: An Introduction for Applied Scientists and Engineers (Wiley-VCH, New York, 1992). 10. C.M. Sorensen, Light scattering by fractal aggregates: a review. Aerosol Sci. Technol. 35, 648–687 (2001) 11. W. Ruland, Apparent fractal dimensions obtained from small-angle scattering of carbon materials. Carbon 39, 323–324 (2001) 12. T.P. Rieker, M. H-Bischoff, F. Ehrburger-Dolle, Small-angle X-ray scattering study of the morphology of carbon black mass fractal aggregates in polymeric composites. Langumuir 16, 5588–5592 (2000) 13. T. Suzuki, A. Chiba, T. Yano, Interpretation of small-angle X-ray scattering from starch on the basis of fractals. Carbohyd. Polym. 34, 357–363 (1997)

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14. H. Hama, K. Tashiro, Structural changes in non-isothermal crystallization process of meltcooled polyoxymethylene [II] Evolution of lamellar stacking structure derived from SAXS and WAXS data analysis. Polymer 44, 2159–2168 (2003) 15. R. Hosemann, S.N. Bagchi, Direct Analysis of Diffraction by Matter (North-Holland Pub. Co., Amsterdam; Interscience Publ., New York, 1962) 16. K. Tashiro, S. Sasaki, Structural changes in the ordering process of polymers as studied by an organized combination of the various measurement techniques. Progress Polym. Sci. 28, 451–519 (2003) 17. N.S. Murthy, D.T. Grubb, Tilted lamellae in an affinely deformed 3D macrolattice and elliptical features in small-angle scattering. J. Polym. Sci., Polym. Phys. Ed. 44, 1277–1286 (2006) 18. W. S. Rasband, Image J, U. S. National Institutes of Health, Bethesda, MD, USA (1997– 2012). https://imagej.nih.gov/ij/ 19. D. Tahara, K. Tashiro, Metropolis monte carlo simulation of two-dimensional small-angle X-ray scattering patterns of oriented polymer materials. Macromolecules 53, 276–287 (2020) 20. P. Lesieur, T. Zemb, High resolution small-angle X-ray scattering of colloids, in Structure and Dynamics of Strongly Interacting Colloids and Supramolecular Aggregates in Solution, ed. By S. H. Chen, J. S. Huan, P. Tartaglia. NATO ASI Series (Series C: Mathematical and Physical Sciences), vol. 369 (Springer, Dordrecht, 1992) 21. V. Yurgens, F. Koch, M. Scheel, T. Weitkamp, C. David, K. Frieder, S. Mario, Measurement and compensation of misalignment in double-sided hard X-ray Fresnel zone plates, J. Synchrotron Rad. 27 (2020). https://doi.org/10.1107/S1600577520001757 22. Y.I. Dudchik, N.N. Kolchevsky, F.F. Komarov, Y. Kohmura, M. Awaji, Y. Suzuki, T. Ishikawa, Glass capillary X-ray lens: fabrication technique and ray tracing calculations. Nucl. Inst. Methods Phys. Res. A 454, 512–519 (2000) 23. A. Gupta, I.R. Harrison, J. Lahijani, Small-angle X-ray scattering in Carbon fibers. J. Appl. Cryst. 27, 627–636 (1994) 24. Y. Tsuji, J. Kojima, T. Kikutani, S. Sakurai, Analyses of higher-order structures in poly(Ethylene Terephthalate) fibers prepared with high-speed melt spinning by conducting small-angle X-ray scattering measurements for fibers immersed in non-solvent of which electron density was matched to that of fibers. J. Soc. Mater. 64, 11–17 (2015) 25. O. Kratky, I. Pilz, P. J. Schmitz, Absolute Intensity measurement of small angle X-ray scattering by means of a standard sample. J. Coll. Interface Sci. 1, 24 – 34 (1966) 26. I. Pilz, O. Kratky, Absolute Intensity measurement of small angle X-ray scattering by means of a standard sample, II. J. Coll. Interface Sci. 24, 211 – 218 (1967) 27. L. B. Shaffer, R. W. Hendricks, Calibration of polyethylene (lupolen) as a wavelengthindependent absolute intensity standard. J. Appl. Cryst. 7, 159 – 163 (1974) 28. F. Zang, J. Ilavsky, G. G. Long, J. P. G. Quintana, A. J. Allen, P. R. Jemian, Glassy Carbon as an Absolute Intensity Calibration Standard for Small-Angle Scattering. Metal. Mat. Trans. 41, 1151 – 1158 (2010) 29. R. Chen, Y. Men, Calibration of absolute scattering intensity in small angle X-ray scattering. Chin. J. Appl. Chem. 33, 774 – 779 (2016) (in Chinese) 30. C. A. Dreiss, K. S. Jack, A. P. Parker, On the absolute calibration of bench-top small-angle X-ray scattering instruments: a comparison of different standard methods. J. Appl. Cryst. 39, 32 – 38 (2006) 31. H. Endo, J. Allgaier, G. Gompper, B. Jakobs, M. Monkenbusch, D. Richter, T. Sottmann, R. Strey, Membrane decoration by amphiphilic block copolymers in biocontinuous microemulsions, Phys. Rev. Lett. 85, 102–105 (2000) 32. H. Endo, M. Mihailescu, M. Monkenbusch, J. Allgaier, G. Gompper, D. Richter, B. Jakobs, T. Sottmann, R. Strey, I. Grillo, Effect of amphiphilic block copolymers on the structure and phase behavior of oil–water-surfactant mixtures. J. Chem. Phys. 115, 580–600 (2001) 33. H. Endo, D. Schwahn, H. Cölfen, On the role of block copolymer additives for calcium carbonate crystallization: small-angle neutron scattering investigation by applying contrast variation. J. Chem. Phys. 120, 9410–9423 (2004)

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34. H. Endo, Study on multicomponent systems by means of contrast variation SANS. Phys. B 385–386, 682–684 (2006) 35. S. Miyazaki, H. Endo, T. Karina, K. Haraguchi, M. Shibayama, Gelation mechanism of poly(Nisopropylacrylamide)−clay nanocomposite gels. Macromolecules 40, 4287–4295 (2007) 36. J. Hu, K. Tashiro, Relation between higher-order structure and crystalline phase transition of oriented isotactic polybutene-1 investigated by temperature-dependent time-resolved simultaneous WAXD/SAXS measurements. Polymer 90, 165–177 (2016) 37. H. Hama, K. Tashiro, Structural changes in isothermal crystallization process of polyoxymethylene investigated by time-resolved FTIR, SAXS and WAXS measurements. Polymer 44, 6973–6988 (2003) 38. M. Shimomura, S. Iguchi, M. Kobayashi, Vibrational spectroscopic study on trigonal polyoxymethylene and polyoxymethylene-d2 crystals. Polymer 29, 351–357 (1988) 39. M. Kobayashi, M. Sakashita, Morphology dependent anomalous frequency shifts of infrared absorption bands of polymer crystals: Interpretation in terms of transition dipole–dipole coupling theory. J. Chem. Phys. 96, 748–760 (1992) 40. F.C. Stehling, E. Ergos, L. Mandelkern, Phase separation in n-hexatriacontane-nhexatriacontane-d74 and polyethylene-poly(ethylene-d4 ) systems. Macromolecules 4, 672–677 (1971) 41. J. Schelter, G.D. Wignall, D.G.H. Ballard, G.W. Longman, Small-angle neutron scattering studies of molecular clustering in mixtures of polyethylene and deuterated polyethylene. Polymer 18, 1111–1120 (1977) 42. K. Tashiro, Thermodynamic and kinetic aspects of cocrystallization and phase segregation phenomena of polyethylene blends between the D and H speceis as viewed from DSC, FTIR and synchrotron X-ray scattering. Acta Polymerica 46, 100–113 (1995) 43. K. Tashiro, K. Imanishi, M. Izuchi, M. Kobayashi, Y. Itoh, M. Imai, Y., Yamaguchi, M. Ohashi, R.S. Stein, Cocrystallization and phase segregation of polyethylene blends between the D and H species. 8. Small-angle neutron scattering study of the molten state and the structural relationship of chains between the melt and the Crystalline State. Macromolecules 28, 8484– 8490 (1995) 44. S. Sasaki, K. Tashiro, N. Gose, K. Imanishi, M. Izuchi, M. Kobayashi, M. Imai, M. Ohashi, Y. Yamaguchi, K. Ohyama, Spatial distribution of chain stems and chain folding mode in polyethylene lamellae as revealed by coupled information of DSC, FTIR, SANS and WANS. Polym. J. 31, 677–686 (1999) 45. E.W. Fischer, Chain conformation in the crystalline state by means of neutron scattering methods. Faraday Discuss. Chem. Soc. 68, 263–278 (1979)

Chapter 5

Structure Analysis by Vibrational Spectroscopy

Abstract Infrared and Raman spectroscopies are often used for the identification of chemical functional groups. These spectroscopic methods are quite powerful for the study of chain conformation, chain packing mode in the crystal lattice, and interatomic interactions. The methods how to know the IR/Raman activities, how to predict the number and polarization of the bands (factor group analysis), and how to calculate the vibrational frequencies and intensities of the bands (normal modes calculation) are learned in a concrete way. The IR and Raman spectra characteristic of polymer substances are also studied, which include the progression bands, the longitudinal acoustic modes, the critical sequences, etc., in relation to the structural transitions of polymers in the crystallization process, for example. The vibrational spectra are sensitive to the structural disorder included in the chain conformation as well as in the chain packing state, which will be known from the calculation of density of states function. The principles of the spectral measurements, the inner structure of spectrometers, and the various kinds of spectral measurement methods are learned. The preparation methods of the samples necessary for the IR and Raman spectral experiments are also studied in detail. Keywords Infrared spectra · Raman spectra · Factor group analysis · Normal modes treatment · Spectroscopic measurements The structural study of polymers is not only owned by the diffraction method but also performed by the vibrational spectroscopic technique. The IR and Raman spectroscopic methods are useful for the clarification of interatomic interactions, chain conformation, chain packing mode, identification of crystalline phases, thermal atomic motions, morphology, or even crystal shape. In this chapter, starting from the basics of vibrational spectroscopy, the application to polymer science is described in detail and concretely.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 K. Tashiro, Structural Science of Crystalline Polymers, https://doi.org/10.1007/978-981-15-9562-2_5

399

400

5 Structure Analysis by Vibrational Spectroscopy

5.1 Molecular Vibrations and Vibrational Spectra 5.1.1 Energy of Molecule and Infrared Spectra For example, let us consider a diatomic molecule A-B. Roughly speaking the energy of this molecule is expressed as a sum of the electronic energy E electron , translational energy E trans , rotational energy E rot, and vibrational energy E vib (see Fig. 5.1). E = E electron + E trans + E rot + E vib

(5.1)

E electron is the total sum of the electron–electron, electron-nuclei, and inter-nuclei interaction energies. The theoretical treatment of E electron will be mentioned in Chap. 6. E trans is the kinetic energy of the translational motion of the whole molecule [(1/2)mv2 ]. E rot is the energy of the rotational motion of the molecule around the center of gravity. E vib is the energy corresponding to the pure vibrational motion or the stretching motion of the bond length between the atoms A and B. By the irradiation of light into the molecule, the various types of the transition occur depending on the energy of the incident photon (ε = hv, where ν is the frequency of the incident photon). For the transition between the electronic levels, the light in the ultraviolet to visible region is needed. The irradiation of light in the infrared region causes the transition in the vibrational (and rotational) levels. The transition between the rotational levels is caused by the irradiation of far-infrared wave to microwave.

5.1.1.1

Vibrational Motion

The bond length r between the two atoms is expressed using the x-coordinate along the A-B axis. r = x1 − x2

(5.2)

Fig. 5.1 Energy levels of a molecule

Electronic excited state

Electronic ground state

photon

vibrational level

rotational level

5.1 Molecular Vibrations and Vibrational Spectra

401

The change of bond length r from the equilibrium distance r o is given as r = r − ro = x1 − x2

(5.3)

The energy corresponding to this change is E vib = (1/2)kr 2 = (1/2)k(x1 − x2 )2

(5.4)

where the harmonic vibration is assumed for simplicity. Newton’s equation of motion is given as   m 1 d2 x1 /dt 2  = force = −(dE vib /dx1 ) = −k(x1 − x2 ) m 2 d2 x2 /dt 2 = force = −(dE vib /dx2 ) = −k(x2 − x1 )

(5.5)

Replacing (x1 − x2 ) by r, we have   m d2 r/dt 2 = −kr

(5.6)

Here, m is a reduced mass and defined as 1/m = 1/m 1 + 1/m 2

(5.7)

Equation 5.6 indicates that the stretching vibration of a diatomic molecule can be treated as a vibrational motion of a harmonic oscillator. By assuming r = A cos(ωt + δ) = A cos(2π νt + δ)

(5.8)

in which ω is an angular frequency and ν is a vibrational frequency. [ω = 2π v; one cycle corresponds to the change of 360°or 2π. If the periodic vibration (stretching and contraction) occurs two times in second, ν = 2 cycles/s, and the total angle change ω = 360◦ × 2 = 720◦ or 4π/s.] By substituting Eq. 5.8 into Eq. 5.6, we have ω2 = 4π2 v2 = k/m, ω = (k/m)1/2 v = ω/(2π ) = (1/2π)(k/m)1/2

(5.9)

The Schrödinger equation for this vibrational motion is given as [1] Hˆ Φ = EΦ

(5.10)

The Hamiltonian Hˆ is an operator corresponding to the total energy or the sum of kinetic energy and potential energy. total energy H = (1/2)mv2 + (1/2)k(r )2

(5.11)

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5 Structure Analysis by Vibrational Spectroscopy

  Hamiltonian Hˆ = − 2 /2m (∂ 2 /∂r 2 ) + (1/2)k(r )2

(5.12)

where  = h/2π . Then, Eq. 5.10 becomes   − 2 /2m (∂ 2 Φ(r )/∂r 2 ) + (1/2)k(r )2 Φ(r ) = EΦ(r ) By using the relation ω = (k/m)1/2 , this equation is rewritten as   − 2 /2m (∂ 2 Φ(r )/∂r 2 ) + (1/2)mω2 (r )2 Φ(r ) = EΦ(r )

(5.13)

The solution (eigenvalue) of this equation is given as E vib (n) = (n + 1/2)hv = (n + 1/2)ω (n = 0, 1, 2, . . .)

(5.14)

The corresponding eigenfunctions are 1/4    exp −(r )2 /2b2 Φ0 (r ) = 1/πb2 1/4    (r/b) exp −(r )2 /2b2 Φ1 (r ) = 4/πb2       1/4 Φ2 (r ) = 4/πb2 (r/b)2 − (1/2) exp −(r )2 /2b2

(5.15)

where b = (/mω)1/2 Figure 5.2 shows the wave functions Φ of the various vibrational states. The probability of the molecule existing on each discrete energy level is given by ΦΦ ∗ = |Φ|2 . The n= 0 state has the energy (1/2)hv, called the zero-point energy. Because of this effect, the entropy is not zero and the molecule is vibrating even at T = 0 K. Fig. 5.2 Plot of Φn or the vibrational probability at the various excited states

potential energy

n=

n=

ro

r

5.1 Molecular Vibrations and Vibrational Spectra

403

By considering the Boltzmann distribution, N = g No exp(−E vib /k B T )

(5.16)

where N is the number of oscillators (vibrating molecule) with the energy E vib at a temperature T, and g and N o are the degeneracy and the total number of molecules, respectively. k B is the Boltzmann constant. The ratio N (n = 1)/N (n = 0) = exp(−hv/k B T ) is calculated by inputting the concrete values for the individual parameters. For example, at T = 300 K, for the vibrational frequency v˜ = 1000 cm−1 −34 −1 [wavenumber hv/k B T =  ν˜ = ν/c],  6.626 × 10 J sec ×1000 cm × 3 × 10 −23 10 cm/s/ 1.38 × 10 J/K × 300 K = 4.80. Then N (n = 1)/N (n = 0) = exp(−hv/k B T ) = exp(−4.80) = 0.0082

(5.17)

This result indicates that an overwhelmingly large number of vibrating molecules exists on the ground state. The difference in N becomes smaller for the oscillator with the lower frequency. For example, for the vibrational frequency v˜ = 100 cm−1 , N (n = 1)/N (n = 0) = exp(−0.480) = 0.6187

(5.18)

If the vibration occurs in the 3D system, the eigenvalue is E vib = (n + 3/2)hv

(5.19)

Fig. 5.3 Morse potential function and the derived force and force constant

Potential Energy (kcal) Force (kcal/Å) Force Constant (kcal/Å2)

for n = n x , n y and n z = 0, 1, 2, . . . and g = (n + 2)(n + 1)/2. For the 1D oscillator, g = 1. In the above treatment, the vibration is assumed to be harmonic with the parabolatype potential energy. If an asymmetric potential function such as Morse potential is used, the vibrational frequency is expressed in the following form (Fig. 5.3):

1.2

0.8

0.4

0.0

-5.0

0.0

5.0

10.0

Change of interatomic distance (Å)

404

5 Structure Analysis by Vibrational Spectroscopy

 2 E(r ) = Do 1 − exp(−α(r − ro )) ∼ E(ro ) + E  (ro )(r − ro ) + (1/2)E  (ro )(r − ro )2 + (1/6)E  (ro )(r − ro )3 + . . .

(5.20) where Do is the dissociation energy of the molecule. The second equation is developed as a power series. The fourth and higher terms give the anharmonic effect. E  (r o ) corresponds to the force constant and is expressed as k = E  (ro ) = 2α 2 Do . If the E (r) is taken into account, the apparent force constant k is expressed as k(r ) ∼ E  (ro ) + (1/3)E  (ro )(r − ro )

(5.21)

This equation indicates the force constant is changed with the interatomic distance r, although the potential energy is apparently expressed as E ∼ (1/2)k(r )(r − ro )2 . This treatment is called the quasi-harmonic approximation. The application of this approximation will be shown in a later section. The quantized vibrational energy is given as E vib = (n + 1/2)hv − (n + 1/2)2 hvx  1/2 v = (1/2π ) 2α 2 Do /m x = (hv)2 /4Do

(5.22)

The x is called the anharmonic constant.

5.1.1.2

Rotational Motion

The molecule is not only vibrating but also rotating around the molecular axis, the energy of which is also quantized. For the rotational motion of a diatomic molecule assumed as a rigid body, the total energy H = (1/2)I ω2 , where ω is an angular frequency and I is a moment of inertia defined as I = m 1r12 + m 2 r22

(5.23)

The r 1 is the distance of the atom (i = 1 and 2) measured from the center of gravity. As shown in Fig. 5.4, since m 1r1 = m 2 r2 and r = r1 + r2 , we have Fig. 5.4 Rotational motion of a diatomic molecule

m

r

r G

m

5.1 Molecular Vibrations and Vibrational Spectra

405

I = mr 2

(5.24)

The angular momentum P = I ω and so H = P 2 /(2I ). The corresponding Hamiltonian Hˆ is applied to the Schrödinger equation to give the eigenvalue for P2 as J(J+ 1) with J = 0, 1, 2, . . .The result is given as [2]   E rot = B J (J + 1) cm−1

J = 0, 1, 2, . . .

(5.25)

where the rotation constant    B = h/ 8π2 I c cm−1

(5.26)

The J state is degenerate by 2 J+ 1. The actual molecule is not rigid but the bond length is more or less stretched because of the rotational motion. If the effect of this centrifuge force on the bond length is taken into consideration, the rotation energy is given as E rot = B J (J + 1) − B  J 2 (J + 1)2

(5.27)

5.1.2 Transition (Perturbation Theory) A light is incident on a vibrating and/or rotating molecule. The interaction occurs between the dipole moment of the molecule and the electric vector of the light. The initial state of the molecule is expressed using a wave function ψn and the energy E n . By irradiation of light, the molecular state changes to a wave function ψm with the energy E m . The Hamiltonian of the molecule changes from Hˆ o to Hˆ o + Hˆ  . The time-dependent Schrödinger equation is



Hˆ o ψn = i(∂ψn /∂t)

(5.28a)

 Hˆ o + Hˆ  ψm = i(∂ψm /∂t)

(5.28b)

According to the perturbation theory, the state ψm is assumed to be constructed using the linear combination of the original states {ψ1 , ψ2 , . . . , ψn , . . .} (Fig. 5.5). ψm = k ck (t)ψk

(5.29)

The coefficient ck is the magnitude of the contribution of the state ψk to the state ψm . By inputting Eq. 5.29 into Eq. 5.28b, we have

406

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.5 An illustrative explanation of a perturbation theory

( Hˆ ◦ + Hˆ  ) k ck (t)ψk = i[∂( k ck (t)ψk )/∂t] or k ck (t) Hˆ ◦ ψk + k ck (t) Hˆ  ψk = i k (∂ck (t)/∂t)ψk + i k ck (t)(∂ψk /∂t) Seeing Eq. 5.28a, we find that the first term of the left side is equal to the second term of the right side. Then, k ck (t) Hˆ  ψk = i k (∂ck (t)/∂t)ψk

(5.30)

By applying ψm∗ to Eq. 5.30 from the left side and integrate the equation, we have the following equation.  k ck (t) Since have



ψm

∗

Hˆ  ψk dτ = i k (∂ck (t)/∂t)



ψm ∗ ψk dτ

(5.31)

ψm∗ ψk dτ = δmk (Kronecker δ function: 1 for m= k and 0 for m = k), we





k ck (t) = i(∂cm /∂t)

(5.32)







= ψm ∗ Hˆ  ψk dτ

(5.33)

Here

If the transition from the state n occurs in a short time t, only the state n among many k states in Eq. 5.29 may contribute to the state m in an approximation. For the

5.1 Molecular Vibrations and Vibrational Spectra

407

initial state n, cn (0) = 1 and cm (0) = 0 (m = n). Then, from Eq. 5.32, ∂cm /∂t = (1/i)

(5.34)

cm (t) ∼ (1/i)t

(5.35)

The probability of the transition is given as cm (t)cm (t)∗ ∼ t 2 2 /2

(5.36)

More concretely, the perturbation Hˆ  for a dipole moment μx in the x-directionpolarized electromagnetic field is taken into consideration. The vector potential Ax along the x direction is considered, which is related to the magnetic field (B = rot A) and electric field (E= -(dA/dt)/c for a plane wave). The magnetic radiation is incident on the particles of mass m and charge e (the mass is assumed to be common, for simplicity). The perturbation term of Hamiltonian is given as H˜  = (ie/cm)Aox j ∂/∂ x j

(5.37)

By referring to Eq.5.34, the coefficient cm (t) is obtained as follows.   cm (t) = i Aox /2c X mn vmn

  exp{2πi(vmn + v)t} − 1 /(vmn + v)   + exp{2πi(vmn − v)t} − 1 /(vmn − v) (5.38)

where   A x = Aox sin(2πvt) = Aox /2i [exp(2πivt) − exp(−2πivt)] vmn = (E m − E n )/ h





|X mn | = ψm j ex j ψn = ψm |μx |ψn  = transition dipole moment

(5.39) (5.40)

As a result, the transition probability of the n-to-m transition in a unit time is given as follows. 

    

2 2 sin2 π(v 2 cm (t)cm (t)∗ /t = π2 vmn 2 / c2 2 Ao x |X mn |2 vmn mn − v)t / (vmn − v) t 



2 2 (at v = v ) (5.41) = π2 vmn 2 / c2 2 Ao x |X mn |2 vmn mn

For the molecule with the normal vibration Q, the dipole moment can be developed as

408

5 Structure Analysis by Vibrational Spectroscopy

μx = μox + (∂μx /∂ Q)o Q + . . .

(5.42)

 |X mn | = m|μx |n = m μox n + (∂μx /∂ Q)o + . . . = μox δmn + μ + . . .

(5.43)

Then,

At this moment, we have the condition for the vibrating molecule to absorb the light and change the state from the ground state (n) to the excited state (m): the transition dipole is not equal to 0. μ = (∂μx /∂ Q)o = 0

(5.44)

Another term is .  =

ψm∗ Qψn dτ = 0 for m = n ± 1 = 0 for other cases

(5.45)

As long as the harmonic vibrational motion is assumed, the transition must occur from n to m = n ± 1 states or the next lower or higher energy level. This is the selection rule for the vibrational mode. Similarly, for the rotational motion, we can derive the following conditions. The wave function of the rotational motion of a rigid body molecule is given by the spherical function Y J M (θ, Φ) = nJM PJ |M| (cosθ )eiM Φ , where PJ |M| (cosθ ) is an associated Legendre function with the integers J and M, and nJM is a normalization constant. The components of a dipole moment vector are μx = μ sinθ cos, μy = μ sinθ sin, and μz = μ cosθ, and the corresponding transition dipole moment is given as below, where the incident light is assumed to have a polarization along the z axis. 

 X J,M,J  ,M  = ψ J  M  |μ|ψ J M = μ ∫ ∫ Y JM (θ, Φ)Y JM (θ, Φ) sin θ dΦdθ |M  |

= μ n J  M  n J M ∫ ∫ PJ 

 |M| (cos θ )e−i M Φ PJ (cos θ )ei MΦ sin θ dΦdθ

 |M  | |M| = μ n J  M  n J M ∫ e−i M Φ ei MΦ dΦ ∫ PJ  (x)PJ (x)dx

μ.

(x = cos θ )

(5.46)

The transition dipole moment X J,M,J  ,M  is not zero for the nonzero dipole moment μ = 0

(5.47)

The integrations in Eq. 5.46 are not zero for the following conditions. M = M  and

J = J ± 1

(5.48)

5.1 Molecular Vibrations and Vibrational Spectra

409

That is to say, for the observation of the rotational band, the molecule must be polar (Eq. 5.47). The transition occurs between the adjacent levels (Eq. 5.48).

5.1.3 Rotational Spectrum of Diatomic Molecule By combining Eqs. 5.25 and 5.48, we can predict the rotational spectrum as below. For the transition by absorption of light, J’= J+ 1. The frequency of an absorbed light ν should be hv = E J ’ − E J = B J ’ (J ’ + 1) − B J (J + 1) = B(J + 1)(J + 2) − B J (J + 1) = 2B(J + 1) v = 2B/ h, 4B/ h, 6B/ h, . . .

(5.49)

The rotational spectrum consists of many discrete bands with the spacing 2B/h as illustrated in Fig. 5.6. In the actual treatment, the frequency is converted to the wavenumber   ˜ + 1) v˜ cm−1 = v[cycle/ sec]/c[cm/ sec] = 2 B(J

(5.50)

  B˜ cm−1 = B[J]/(hc)

(5.51)

Figure 5.7 shows the far-infrared spectrum of CO gas. The CO molecule has a dipole moment and so the rotational spectrum can be observed. By reading the band positions, the spacing is about 3.863 cm−1 . Then 2 B˜ = 3.863 cm−1 For CO molecule, the reduced mass m is calculated as follows. The mass of one C atom, mC is 12 g/mol/N A = 12/6.02 × 10–23 g/atom = 1.99 × 10–23 g/atom. Similarly, for O atom, mO = 16/6.02 × 10–23 = 2.66 × 10–23 g/atom. So the reduced mass of CO is

Fig. 5.6 Rotational spectrum and transitions between the rotational levels

410

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.7 Pure rotational spectrum of CO gas

  m = 1/(1/m C + 1/m O ) = 1/ (1/1.99) × 1023 + (1/2.66) × 1023 = 1.139 × 10−23 g/molecule    Since B˜ = h/ 8π2 I c cm−1 and I= mr 2 , the C–O distance r is calculated as below.   B˜ = 3.863/2 cm−1

  = 6.626 × 10−34 (J s)/ 8 × 3.14 × 3.14 × 1.139 × 10−23 (g) × r 2 × 3 × 1010 (cm/s) .

Since 1 J = 1 Nm= 1 kgm2 /s2 = 107 gcm2 /s2 , then r = 1.13 × 10−8 cm = 1.13Å The CO bond is almost the triple bond and the C≡O distance is reasonable. In the actual spectrum, as seen in Fig. 5.7, the relative intensity of the bands changes depending on the J value. This can be interpreted by taking into account the Boltzmann distribution of the rotating molecules. In Eq. 5.16, N = No g exp(−E J /k B T ) = No (2J + 1) exp[−2B J (J + 1)/k B T ] where g = 2 J+ 1 is the degeneracy, and so the band intensity is Absorbance A ∼ v(2J + 1) exp[−2B J (J + 1)/k B T ] = 2B(J + 1)(2J + 1) exp[−2B J (J + 1)/k B T ]

(5.52)

5.1 Molecular Vibrations and Vibrational Spectra

411

(It must be noted here the integrated intensity of the absorption band is proportional to N or the number of the molecules to jump, the vibrational frequency ν and the probability |X J’J |2 .) As a trial, the B value is input to this equation and T = 300 K, then we have the absorbance A plotted against J. It must be careful about the unit conversion between ˜ B and B:   B[J]/k B T = hc B˜ cm−1 /k  BT = 6.626 × 10−27 gcm2 /s   × 3 × 1010 (cm/s) × 1.932cm−1 / 1.38 × 10−16 cm2 g s−2 K−1 × 300K = 0.00928 (dimensionless). Figure 5.8 shows the calculated result of A(J) at T = 300 K. This plot can be used for the estimation of the temperature of the gas. The intensity ratio of the neighboring bands is {(J + 2)(2J + 3) exp[−2B(J + 1)(J + 2)/k B T ]} A J +1 = {(J + 1)(2J + 1) exp[−2B J (J + 1)/k B T ]} AJ = [(2J + 3)(J + 2)/(2J + 1)(J + 1)] exp[−4B(J + 1)/k B T ] Then   ln A J +1 /A J − ln[(2J + 3)(J + 2)/(2J + 1)(J + 1)] = −4B(J + 1)/k B T (5.53) Fig. 5.8 Calculated relative absorbance A of the rotational bands of CO gas plotted against J

412

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.9 Logarithmic plot of absorbance ratio against J + 1 for CO gas. The contributions of the two terms in Eq. 5.53 are also plotted

  The plot of ln A J +1 /A J − ln[(2J + 3)(J + 2)/(2J + 1)(J + 1)] against (J+ 1) gives the slope 4B/(k B T ) and so the temperature T. In the case of Fig 5.8, the plot is made as shown in Fig 5.9. The temperature estimated from the slope is 300 K. (This is only a demonstration of the method how to evaluate the temperature from the spectral data. In this example, we used the absorbance calculated with Eq. 5.52, and so the result gives of course the same temperature.) In the case of a nonlinear polyatomic molecule, the molecule can rotate around the three perpendicular axes (symmetric or asymmetric top). The three different moments of inertia are defined for the latter case (I a = I b = I c ). The energy of the top is expressed as E = (1/2)Ia ωa2 + (1/2)Ib ωb2 + (1/2)Ic ωc2 = Pa2 /(2Ia ) + Pb2 /(2Ib ) + Pc2 /(2Ic )

(5.54)

Pi is an angular momentum around the i-axis. For the symmetric top (benzene, CHCl3 ) with I a = I b = I c , the energy is expressed as     E = Pa2 + Pb2 /(2I⊥ ) + Pc2 / 2I//     2 = P 2 /(2I⊥ ) + 1/ 2I// − 1/(2I⊥ ) P//



P 2 = Pa2 + Pb2 + Pc2



(5.55)

The solution of quantum mechanical calculation gives the energy levels as     E J,K = J (J + 1)2 /(2I⊥ ) + 1/ 2I// − 1/(2I⊥ ) K 2 2 = B J (J + 1) + (A − B)K 2

5.1 Molecular Vibrations and Vibrational Spectra

413

J = 0, 1, 2, . . . K = 0, ±1, . . . , ±J

(5.56)

The degeneracy is 2(2J + 1) for K = 0 and (2 J + 1) for K = 0. The selection rule for the IR absorption is J = 0, ± 1 and K = 0. For the Raman spectra, K = 0 and J = 0, ± 2 (for K = 0) and J = 0, ± 1, ± 2 for K = 0. The case of I a = I b = I c , the molecule is a sphere top (CCl4 ). Since A = B, the energy is expressed as E J = J (J + 1)2 /(2I ) = B J (J + 1)

(5.57)

The degeneracy is (2 J + 1)2 . The spherical top molecule does not have polarity and a pure IR (or microwave) rotational spectrum cannot be detected. The vibrotational spectrum can be observed.

5.1.4 Vibration–Rotation Spectra of Diatomic Molecule As seen in Fig. 5.1, the molecules staying on the vibrational ground level (n = 0) are excited and transform to the vibrationally excited state of n = 1, where the transition from n = 0 to n = 2 does not occur as long as the harmonic vibration is assumed. However, we cannot ignore the existence of the rotational energy levels. These rotational levels are located on each vibrational level, and the energy difference between the neighboring levels is not very large, meaning that the irradiation of infrared beam causes the simultaneous transition of the vibrational and rotational levels. The optically active conditions are used for both the vibrational and rotational transitions. Therefore, the following equation is obtained. At 300 K, the vibrational transition occurs preferentially from n = 0 to n = 1. The rotational transitions occur from J to J ± 1. n=0→1 J → J +1 hν = E vib,rot (n = 1, J + 1) − E vib,rot (n = 0, J ) = [3hν0 /2 + B1 (J + 1)(J + 2)] − [hνo /2 + Bo J (J + 1)] = hνo + (B1 − Bo )J 2 + (3B1 − Bo )J + 2B1

(5.58)

n=0→1 J → J −1 hν = E vib,rot (n = 1, J − 1) − E vib,rot (n = 0, J ) = [3hνo /2 + B1 (J − 1)J ] − [hνo /2 + Bo J (J + 1)] = hνo + (B1 − Bo )J 2 − (B1 + Bo )

(5.59)

414

R branch

5 Structure Analysis by Vibrational Spectroscopy

J=3 P branch J=2

n=1

R branch

P branch

J=3-4 J=1-2 J=2-3 J=0-1

J=1 J=0

J=2-1 J=4-3 J=1-0 J=3-2

J=3 J=2 J=1 J=0

n=0

o

higher frequency

Fig. 5.10 Vibration–rotation transitions predicted for HCl gas

The subscript “n” of the rotational constant Bn is to show the rotation at the vibrational level n = 1 and 0. The bond length is slightly different at these two vibrational states and so the B constant is also different from each other. For simplicity, these B values are assumed at first to be the same for HCl gas. Then we have the following simpler equations: ν˜ = ν˜ o + 2 B˜ J + 2 B˜ (J = J to J + 1)

(5.60)

ν˜ = ν˜ o − 2 B˜ J (J = J to J − 1)

(5.61)

The predicted vibrational–rotational spectrum of HCl molecule is shown in Fig. 5.10. Many bands are positioned at the spacing 2 B˜ around the center at ν˜ = ν˜ o . That is to say, the rotational spectrum discussed in the above section is realized at the right and left sides with the center of vibrational frequency v0 . However, Eqs. 5.60 and 5.61 are the too simplified equations. In the actual spectrum, the band spacing changes depending on the J value since the J 2 terms are not neglected in Eqs. 5.58 and 5.59. The actually observed spectrum of a mixed gas of HCl and DCl is shown in Fig. 5.11. The spectrum is measured using a gas cell of 15 cm length with a pair of CaF2 plate windows. DCl was generated by adding D2 O into the conc. HCl aqueous solution. The gases existing in the space above the aqueous solution are transferred into the cell evacuated in advance (Fig. 5.12). The IR bands of HCl appeared in the range of 3200–2600 cm−1 , while those of DCl in the 2200–1900 cm−1 . Table 5.1 summarizes the wavenumbers of the bands observed for HCl and DCl gases. The careful observation clarified that each band is split into two components with the intensity ratio of 3:1, which originate from the coexistence of the isotopic species, HCl35 and HCl37 (and DCl35 and DCl37 ) at the ratio of 3:1. By fitting the observed peak positions to Eqs. 6.58 and 6.59 without ignoring the J 2 terms with the nonlinear

5.1 Molecular Vibrations and Vibrational Spectra

415

Fig. 5.11 Infrared spectra of HCl and DCl mixed gas

least squares method (see Fig. 5.13), the structure parameters were obtained as follows: For the HCl35 bands, v˜ o = 2886.2 cm−1 , B˜ o = 10.33 cm−1 , and B˜ 1 = 10.02 cm−1     k = 479.36N/m, ro H − Cl35 = 1.292Å and r1 H − Cl35 = 1.312Å For the HCl37 bands, ν˜ o = 2884.0 cm−1 , B˜ o = 10.32 cm−1 , and B˜ 1 = 10.02 cm−1     k = 479.34 N/m, ro H−Cl37 = 1.293Å and r1 H−Cl37 = 1.312 Å For the DCl35 bands, ν˜ o = 2090.7 cm−1 , B˜ o = 5.354 cm−1 , and B˜ 1 = 5.243 cm−1

416

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.12 Gas cell used for the IR measurement of HCl gas. A gas cell is evacuated at first. The HCl gas in the space above the HCl aqueous solution bottle is transferred into the gas cell. For the windows of the gas cell, a pair of CaF2 single crystal plates are used, which transmits the IR beam

    k = 489.2 N/m, ro D−Cl35 = 1.289Å and r1 D−Cl35 = 1.299 Å For the DCl37 bands, ν˜ o = 2088.2 cm−1 , B˜ o = 5.308 cm−1 , and B˜ 1 = 5.204 cm−1     k = 489.6 N/m, ro D − Cl37 = 1.291Å and r1 D − Cl37 = 1.304Å In the calculation, it must be careful to adjust the units of parameters and constants. For example, the reduced mass should be for one molecule, not 1 mol molecules:   m HCl35 = 1/(NA /1 + NA /35) = 1.620 × 10−24 g,   and m HCl37 = 1/(NA /1 + NA /37) = 1.622 × 10−24 g,

5.1 Molecular Vibrations and Vibrational Spectra

417

Table 5.1 Band positions observed for HCl35 , HCl37 , DCl35 , and DCl37 gases J → J± 1

m

HCl35

HCl37

DCl35

DCl37

12 → 13

13

3098.1

3095.6

2209.0

2206.7

11 → 12

12

3086.0

3083.6

2201.3

2199.0

10 → 11

11

3073.2

3070.7

2193.7

2190.8

9 → 10

10

3059.7

3057.3

2185.7

2183.0

8→9

9

3045.4

3042.7

2177.7

2174.6

7→8

8

3030.4

3028.0

2168.7

2166.0

6→7

7

3014.8

3012.4

2160.0

2157.1

5→6

6

2998.4

2996.0

2150.3

2148.1

4→5

5

2981.7

2978.7

2141.0

2138.5

3→4

4

2963.7

2961.3

2132.0

2129.0

2→3

3

2944.3

2943.3

2121.3

2119.2

1→2

2

2926.3

2923.9

2111.3

2109.0

0→1

1

2906.2

2904.4

2101.5

2098.8

1→0

−1

2865.4

2863.1

2080.5

2077.3

2→1

−2

2843.9

2841.7

2068.8

2066.7

3→2

−3

2821.9

2819.5

2057.6

2055.3

4→3

−4

2799.3

2797.3

2046.3

2044.4

5→4

−5

2776.1

2774.0

2034.2

2032.2

6→5

−6

2752.3

2750.0

2022.9

2020.4

7→6

−7

2728.1

2725.7

2010.9

2008.4

8→7

−8

2703.3

2701.4

1997.3

1996.3

9→8

−9

2678.0

2676.3

1986.7

1988.6

10 → 9

−10

2652.3

2649.6

1972.7

11 → 10

−11

2626.0

2623.8

1960.7

12 → 11

−12

2599.3

2597.5

1947.3

13 → 12

−13

2572.2

2570.0

1936.0

  where N A is the Avogadro number. Similarly, m DCl35 = 3.153 × 10−24 g and   m DCl37 = 3.162 × 10−24 g. Do not use the mass of a chlorine atom 35.5 g/mol as seen often in the textbook, which is an average between the molecular weights of two isotopic species, Cl35 and Cl37 , existing at the ratio of 3:1 in nature. By comparing the thus-obtained parameters, the isotopic effect on the bond length and force constant is almost nothing. √ Deuteration of HCl shifts the central position of the band branches by about 1/ 2, which is due to the change of the reduced mass from m/10–24 = 1.62 to 3.15.

418

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.13 Curve fitting of the vibrational frequency observed for HCl35 molecule on the basis of Eqs. 5.62 and 5.63

************************ Actual data analysis of IR spectra of HCl and DCl gases For the J → J + 1 transition, the variable J in Eq. 5.58 is substituted by m = J + 1 (J = 0, 1, 2, . . . , m = 1, 2, 3, . . .). The equation is changed to     ν˜ = ν˜ o + B˜ 1 − B˜ o m 2 + B˜ 1 + B˜ o m

(5.62)

Similarly, for the J – J-1 transition, Eq. 5.59 is rewritten as below for the m = −J (m = −1, −2, . . .).     ν˜ = ν˜ o + B˜ 1 − B˜ o m 2 + B˜ 1 + B˜ o m

(5.63)

In this way, the equation has the common form of y = a + bx + cx 2 . The least squares calculation is made as follows. The total error of the observed band positions is  2  2 Φ = i yiobs − yicalc = i yiobs − a − bxi − cxi2

(5.64)

The Φ must be minimal for the parameters a, b and c. That is to say, ∂Φ/∂a = 0, ∂Φ/∂b = 0 and ∂Φ/∂c = 0

(5.65)

  ⎤ i  yiobs − a − bxi − cxi2  = 0 i  yiobs − a − bxi − cxi2 xi = 0 ⎦ i yiobs − a − bxi − cxi2 xi2 = 0

(5.66)

5.1 Molecular Vibrations and Vibrational Spectra

419

These three simultaneous equations are combined to solve the unknown parameters a, b, and c.   ⎤ a N + ( i xi )b +  i xi2 c =  i yiobs 2 3 obs ⎦ a(  i xi2)+  i xi3 b +  i xi4 c = i yiobs xi2 a i xi + i xi b + i xi c = i yi xi

(5.67)

where N is the total number of the data points. By inputting the observed values of yi obs (peak frequency ν) and xi (= m, m = ±1, ±2, equations, we    can  ..) to these ˜ ˜ ˜ ˜ get the parameters a, b, and c or the values of ν˜ 0 , B1 + Bo , and B1 − Bo . The force constant k and bond lengths r o and r 1 are estimated from v˜ 0 , B˜ 0 , and B˜ 1 . That   1/2 is, k = 4π2 c2 ν˜ 2o m and r = h/ 8π2 mc B˜ . One example of the curve fitting is shown in Fig. 5.13 for the IR spectrum of HCl35 gas. If the vibrational frequency data (yi ) have some error (standard deviation σy ), the thus obtained parameters a, b, and c have the corresponding errors. are  expressed in  Since   these  parameters     the form of ( i xi ), i xi2 , i x13 , i xi4 , i yiobs , i yiobs xi , and i yiobs xi2 (see Eq. 5.67), the standard deviations σa , σb , and σc can be is  σ/N). For  example, a = expressed using  σy  (the  standard   error      f ( i xi ), i xi2 , i xi3 , i xi4 , i yiobs , i yiobs xi , i yiobs xi2 , and then, in an approximate expression, σa ∼ (∂ f /∂ y)σy . The coefficient is obtained and then σa can be calculated if the standard deviation σy is known. ***************************************

5.1.5 Vibrational Spectra of Polyatomic Molecules In the case of a polyatomic molecule, the change in dipole moment can occur even for the nonpolar molecule. For example, CO2 linear molecule has the 3 × 3 = 9 degrees of freedom since each atom has the freedom along the x-,  The  y-, and z-axes. nine freedom consists of the 4 degrees of vibrational freedom vs , vas , δy , δx , the 2 degrees of rotational freedom (Rx and Ry ), and the 3 pure translational modes (T x , T y, and T z ). These modes are illustrated in Fig. 5.14. The rotation around the z-axis is not meaningful since the molecule does not change the shape. The vibration– rotation-coupled IR spectrum is detected for the anti-symmetric stretching mode (νas ) and bending mode (δ). The anti-symmetric stretching vibrational mode (νas ) induces the dipole moment which changes the direction alternately with time. The CO2 molecule is polar during this vibrational motion and, as a result, the rotation spectrum becomes active. Therefore, the coupling between the νas vibrational mode and the rotational mode can occur and gives the fine spectrum consisting of many

420

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.14 Molecular motions of CO2 molecule

rotational band components. Figure 5.15 is the actually observed spectrum in the νas frequency region. The band splitting is too small, and the band is apparently broad. The higher resolution (0.5 cm−1 ) can detect clearly a series of bands corresponding to the rotational bands. The symmetric stretching mode (νs ) is infrared inactive since the dipole moment is always zero and cannot be detected in the IR spectrum. The bending mode (δ) is also polar and can couple with the rotational mode to give the fine structure of the spectrum. The δ mode is doubly degenerated (δx and δy ).

resolution 0.5cm -1 CO2 gas

transmission

(a)

2400

2350

2300

resolution 2 cm -1 (b)

2400

2350

2300

wavenumber / cm -1

Fig. 5.15 Vibration–rotation IR spectrum of CO2 gas in the frequency region of C-O anti-symmetric stretching mode: a high resolution and b low resolution

5.1 Molecular Vibrations and Vibrational Spectra

421

5.1.6 Raman Spectral Activity A laser beam is incident to a molecule, then the photon is scattered from the molecule to all directions. The scattered signal with the same frequency as that of the incident laser is called the Rayleigh scattering. The scattering components with different frequencies from that of the incident beam are the Raman scattering (Fig. 5.16). The signals with a lower frequency than the incident laser are called the Stokes Raman bands, and those with the higher frequency are the anti-Stokes Raman bands. The Raman scattering occurs by the vibrating electron clouds induced by the electric field of the incident light [3, 4].

5.1.6.1

Classical Treatment (Energy Change)

Electric vector E = E o cos(2π vincident t) of an incident light generates the vibrating polarization P. P = α E = α E o cos(2πvincident t)

(5.68)

Since the molecule vibrates at the frequency ν, the polarizability tensor α is also changed with the frequency ν. Therefore,   P = α o + α  cos(2πνt) E o cos(2πνincident t) = α o E o cos(2πνincident t) + α  cos(2πνt)E o cos(2πvincident t)    = α o E o cos(2πνincident t) + α  E o /2 cos 2π(νincident − ν)t + cos 2π(νincident + ν)t

(5.69)

(a)

scattering

Rayleigh scattering

(b)

Raman scattering

laser

sample

Stokes anti-Stokes o

high

low frequency

Fig. 5.16 a Light scattering, b Stokes and anti-Stokes Raman spectra and Rayleigh scattering

422

5 Structure Analysis by Vibrational Spectroscopy

The first term corresponds to the Rayleigh scattering. The second term with the frequency (νincident − ν) corresponds to Stokes Raman scattering and the third one with the frequency (νincident + ν) to anti-Stokes Raman scattering. The Raman peak position is shifted from the Rayleigh scattering position by the molecular vibrational frequency ν. The Raman active condition is the transition polarizability tensor α  = 0.

5.1.6.2

Interpretation Including Phonon Vector

The story is essentially the same as Sect. 5.1.6.1, but here we introduce the phonon and photon vectors. In the solid sample, the phonons propagate through the lattice, the phonon vector of which is expressed here as q. The incident direction of photons is given as kincident . As mentioned above, the incident photon interacts with the vibrating molecule. The induced polarization P(r, t) is a function of the position r and time t. Using the polarizability tensor α (kincident , ωincident , Q), which is related to the normal vibration of the molecule Q(r, t). P(r, t, Q) = P(kincident , ωincident , Q) cos(kincident·r − ωincident t) = α(kincident , ωincident , Q)E(kincident , ωincident ) cos(kincident·r − ωincident t) (5.70) The normal mode vector is Q(r, t) = Q(q, ω) cos(q · r − ωt)

(5.71)

for the angular frequency of the normal vibration ω. Using Q vector, the α is expanded as α(kincident , ωincident , Q) ≈ α o (kincident , ωincident ) + (∂α/∂ Q)0 Q(r, t)

(5.72)

Then, P(r, t, Q) = α o (kincident , ωincident )E(kincident , ωincident ) cos(kincident·r − ωincident t) + (∂α/∂ Q)0 Q(r, t)E(kincident , ωincident ) cos(kincident ·r − ωincident t) = α o (kincident , ωincident )E(kincident , ωincident ) cos(kincident·r − ωincident t) + (∂α/∂ Q)0 Q(q, ω)E(kincident , ωincident ) cos(kincident·r − ωincident t) cos(q · r − ωt) (5.73) The second term corresponds to the induced polarization and is rewritten as (∂α/∂ Q)o Q(q, ω)E{cos[(kincident − q) · r − (ωincident − ω)t] + cos[(kincident + q) · r − (ωincident + ω)t]}/2

(5.74)

5.1 Molecular Vibrations and Vibrational Spectra

423

kStokes = kincident - q

kanti-Stokes = kincident + q

kStokes

kanti-Stokes

kincident

kincident q

q

Fig. 5.17 Collision of phonon and photon

The first component indicates the Stokes Raman scattering and the second one the anti-Stokes Raman scattering. The vector of the scattered photon is defined as kscatter , which is equal to Stokes Raman

kStokes = kincident − q, ωStokes = ωincident − ω

(5.75)

anti-Stokes Raman kanti−Stokes = kincident + q, ωanti−Stokes = ωincident + ω (5.76) The frequency changes were already mentioned in Sect. 5.1.6.1. The phonon vector of the normal mode vibration q is related to the photon vector of the incident and scattered lights as given in Eqs. 5.75 and 5.76. They indicate the transfer of momentum between photon (k) and phonon (q) in the inelastic collision (Fig. 5.17).

5.1.6.3

Quantum Mechanical Treatment

An incident photon is scattered by hitting the charged particle(s) of a molecule. As shown in Fig. 5.18, the molecule of the ground state n transforms to the excited state n’. The scattering intensity I is expressed as   I (s, es ) = 16π4 vs4 /c4 [αs (ei , es )]2 I (si , ei )

(5.77)

Here, the incident direction of light is si and its electric vector is ei . The scattered direction of light is s and the electric vector is es . The frequency νs and polarizability αs (ei , es ) for the Rayleigh scattering are νo and αo (ei , es ), respectively, and νo ± ν and α’s (ei , es ) for the Raman scattering. The transition polarizability tensor α’s of the Raman scattering is expressed as below according to the second-order perturbation theory. In this theory, the transition occurs from the state n to the state n’ under the perturbation or by the irradiation of light, where the perturbed state is assumed to be a combination of the various intermediate states m at the different contributions.

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5 Structure Analysis by Vibrational Spectroscopy

ei

es

scattered light

s

si incident light n

→

(m1, m2, m3,…)

→

n’

m100 m10

transitions

m3 m2 m1 scattered light

incident laser n’

n

Fig. 5.18 a Scattering of light and b the transitions of the molecule levels

αs  (ei , es ) =



  n|μs |m m|μi |n  m

 n|μi |m m|μs |n  + [h(νm − νn − νo )] [h(νm − νn  + νo )]

(5.78)

This equation is called the Kramers-Heisenberg-Dirac dispersion equation. Here, μi is the i-component of the electric dipole moment (i = x, y or z). The integral part  is m|μs |n = φm ∗ μs φn dτ (5.79) In this way, the Raman scattering is the two-photon process in which the incident and emitted photons are related to the change of the molecule from the state n to the state n’ through the intermediate states m. As seen in Eq. 5.78, if the frequency νo of the incident laser light is close to the νm – νn or the energy difference between the initial state n and the intermediate state m, the denominator becomes infinitely zero, and so the polarizability tensor and the Raman scattering intensity are increased remarkably. This is called the resonance Raman scattering. When the laser light frequency νo is in the region of the absorption maximum of the UV–Visible absorption spectrum, the true resonance Raman scattering occurs (Fig. 5.19). The scattering intensity is enhanced by about 106 times compared with the normal Raman scattering case. If the laser frequency is near the foot part of the absorption curve, the resonance becomes weaker, which is called the preset resonance Raman scattering. The resonance Raman scattering method is useful to identify the image of the intermediate state related directly to the transition from the state n to the particular intermediate state m, which is not the hypothetical state (as treated in the perturbation theory)

5.1 Molecular Vibrations and Vibrational Spectra Fig. 5.19 Resonance Raman scatterings and UV–Vis spectrum.

425

UV-Vis absorption spectrum

preset resonance true resonance

frequency

incident laser frequency

but the real excited state giving the maximal peak in the UV–Vis spectrum. This resonance Raman technique can be applied practically to the characterization of a color paint on a fine-art picture. The resonance spectrum collected from a part of the picture tells us the substance of the paint color, from which the appraisement of the picture can be made.

5.1.6.4

Raman Spectra of Rotating Molecule

The Raman activity occurs when the transition polarizability tensor is not zero. This condition gives the following selection rule for the rotational motion of a diatomic molecule. J = 0 and ± 2

(5.80)

(5.81) The Raman scattering intensity is given as  2 I ∝ (νo − ν)4 N J αi j E o2

(5.82)

where νo and E o are the frequency and electric field strength of an incident laser light, respectively, and αij  is the ij-component of the transition polarizability tensor. N J is the population factor.

426

5 Structure Analysis by Vibrational Spectroscopy

  N J = N g J (2J + 1) exp − E˜ J hc/k B T /Q r ot   Q r ot = J g J (2J + 1) exp − E˜ J hc/k B T = rotational partition function (5.83) In these equations, (2 J+ 1) is the degeneracy coming from the angular momentum quantum number M = -J, -J + 1, …., J-1, J. The gJ is the degeneracy due to the nuclear spin effect and is different depending on the spin:  O2 , g J = 1 for J = odd, 0 for J = even   for the spin I = 1/2 1 H2 , g J = 3 for J = odd, 1 for J = even   for the spin I = 1 14 N2 , g J = 3 for J = odd, 6 for J = even

for the spin I = 0

16

The term N J /gJ is calculated as follows.   N J /g J ∝ (2J + 1) exp − E˜ J hc/k B T     = (2J + 1) exp − E˜ J cm−1 × 1.440(cmK)/T (K)

Fig. 5.20 a Boltzmann factor, b N J and c Raman spectrum predicted for 14 N2 gas

(5.84)

5.1 Molecular Vibrations and Vibrational Spectra

427

For 14 N2 gas, for example, the rotational energy E˜ J = 10.44J (J + 1)cm−1 . At T = 300 K, the value N J /gJ plotted against J is shown in Fig. 5.20a. By taking into account the contribution gJ , the N J value changes as shown in Fig. 5.20b. In the actual spectrum, not only the N J but also the other factors such as vibrational frequency, etc., affect the spectral profile.

5.1.7 Group Vibrations In general, polyatomic molecule vibrates in the various modes, the total number of which is 3 N – 6. The atoms belonging to a functional group vibrate together (group vibration) and the characteristic vibrational frequency is similar to each other among the similar functional groups of the different molecules. For example, CH3 CH2 COOH and CH3 CH2 CH3 show the bands of CH3 and CH2 groups at similar frequency positions although some frequency shifts might occur depending on the environmental difference of the groups. In other words, the identification of a particular functional group can be made by checking the corresponding group vibrational bands. A table showing the relationship between the position of a characteristic band and the type of a characteristic functional group is proposed in various ways, one of the most famous and useful tables is Colthup’s chart (Fig. 5.21) [5, 6]. The spectral data collection is also useful for the estimation of functional groups in the unidentified molecule: SDBS web (spectral database of organic compounds by Japan national institute of advanced industrial science and technology), NIST standard reference database, Spectral database of Internet chemistry.com, and so on [7].

Fig. 5.21 Colthup’s IR chart (parts). Reprinted from Ref. [5] with permission of the Optical Society of America, 1950

428

5 Structure Analysis by Vibrational Spectroscopy

5.2 Spectrometers 5.2.1 IR Spectrometer 5.2.1.1

IR Spectrometer (General)

A spectrometer is composed of a light source, a monochromator, and a detector. Depending on the spectral range to study, these components are different. For the IR spectrometer, SiC (globar lamp) is used popularly in the range of wavelength λ = 2 − 30 μm (or the wavenumber ν˜ = 1/λ = 5000−330 cm−1 ). The temperature necessary for the IR light emission is about 1000– 1600 o C. The emitted IR light follows the behavior of the so-called black body radiation. A Tungsten filament also emits the IR light but the cover glass absorbs the IR components and so can be used as the near-IR light source (5000–40,000 cm−1 ). For the far-IR region (400–10 cm−1 ) the high-pressure mercury lamp is used. The stronger IR beam is better for the measurement of the weak bands, in which the light source must be cooled by the water cycling system. In an extreme case, the synchrotron IR source is useful for this point. The IR light in the wide wavelength range is separated into components by using a monochromator. The light is dispersed by the monochromator. The typical one is a prism and a grating. Prism for mid-IR light is made of such an alkaline halide as NaCl, KBr, CaF2 , etc. These halides are water absorbable and hard to maintain transparency. Recently, the grating monochromator is rather often used. The various types of diffraction grating are commercialized. One example of a reflection-type diffraction grating is shown in Fig. 5.22a. Aluminum is coated on the glass plate surface by vacuum evaporation. The grooves are scribed on the aluminum surface at a constant spacing d. In a special case, the surface is processed into a blaze shape [blazed (diffraction) grating]. When the angles of incident and reflected lights are defined as α and β, respectively, the optical path differences are dα = d sin(α) and dβ = d sin(β), respectively. The incident beam II arrives at

Fig. 5.22 a Diffraction grating, b blazed grating

5.2 Spectrometers

429

Fig. 5.23 Thermocouple

the grating surface at a little late timing than the incident beam I. The reflected beam I goes to the detector a little later than the reflected beam II. Here, the angle β is defined as negative if the reflected light is in the side opposite to the angle α with respect to the normal line of the reflection plate. Then the brightest condition of the reflected light is. dα + dβ = d[sin(|α|) − sin(|β|)] = mλ

(5.85)

where λ is the wavelength of the incident light and m = 0, 1, 2, … If |α| = |β|, m = 0 since the sign of these two angles is opposite, indicating that only the normal reflection occurs and no separation of light components is made. By changing the angles α and β, the different wavelength components can be obtained. In the special case of a blazed grating, the blaze angle is θblaze . As known from Fig. 5.22b, 2θblaze = |α| − |β| (for the reflection on the blade surface, |α| − θblaze = |β| + θblaze ). Inputting this equation to |β|, the light is dispersed by changing the angle α. If α = 0°, Eq. 5.85 results in the following equation: d sin(2θblaze ) = mλ

(5.86)

where m is the order of the diffraction. The resolution of the grating is given as λ/λ = m N . Here, N is the total number of grooves and λ is the smallest resolvable wavelength difference. For the infrared light region of λ= 3~ 10μm, N = 100 ~ 500 grooves/mm. The IR detector used in a classic spectrometer is a thermocouple, which is a pair of different two types of metal wires with the ends connected together. As shown in Fig. 5.23, these two metal wires are subjected to different temperatures, then a voltage is generated (Seebeck effect). For example, a copper-constantan thermocouple generates about several hundreds μV then the spectrum is recorded for 100 °C difference. The IR light component incident to the thermocouple heats the thermocouple and the voltage is detected by a monitor to know the photon energy. The recent IR spectrometer uses DTGS or MCT detector. The DTGS is a deuterated-triglycine sulfate crystal and exhibits the pyroelectric property: the current is generated by heating or cooling. The wavelength region is actually infinitely wide (Fig. 5.24). The thermocouple is also useful in a wide region of wavelength. The working frequency is limited and useful for the signal frequency 10~5 Hz. The photoconductive MCT detector is made of mercury (M), cadmium (Cd), and telluride (Te). The sensitivity is high but the detector must be cooled to liquid nitrogen temperature (77 K). The most effective signal frequency is 500~10,000 Hz, making it possible to

430

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.24 IR detectors a effective wavelength range, b effective frequency range of a driving mirror

perform the rapid-scanning measurement of IR spectra. The measurable frequency region is limited, higher than 600~400 cm−1 . A Ge bolometer is highly sensitive though it is needed to cool down to a liquid He temperature (4 K). It is used for the far-IR spectral measurement in a low wavenumber region (15~2000μm wavelength, 700~5 cm−1 ).

5.2.1.2

Dispersion-Type IR Spectrometer

The dispersion-type spectrometers are more or less classical compared with the modern Fourier transform-type infrared spectrometer. However, it is important and necessary to understand the principle of the dispersion type spectrometers. There are two types of dispersion-type spectrometer. One is the single-beamtype and another is the double-beam-type. Figure 5.25 shows these two types. In the single-beam-type spectrometer, the light emitted from the source is incident to the sample, and monochromatized by the prism or diffraction grating, and enters the detector, then the spectrum is recorded. The measurements are made twice with and without the sample. The background spectrum is obtained without the sample, which is now Bo (ν). The spectrum obtained with the sample contains both of the components, B(ν) and Bo (ν). The transmission or transmittance, T %, is given by T % = B(ν)/Bo (ν) × 100

(5.87)

5.2 Spectrometers

431

(a) Single beam mode Light source

monochromator

sample

detector

amplifier

recorder

(b) Double beam mode Light source

sample

rotating sector

monochromator

detector

amplifier

servo motor

recorder

Fig. 5.25 a Single-beam-type spectrometer and b double-beam-type spectrometer

Absorbance A is defined as A = log[Bo (ν)/B(ν)] = log(100/T %)

(5.88)

In an ideal case, the calculation of T % or A erases the contribution of background and so the pure spectrum consisting of only the sample signal should be obtained. However, the intensity of incident light may fluctuate sometimes or periodically, and Bo (ν) is not always necessarily the same between the timings of Bo (ν) and B(ν) measurements. Therefore, the background signals (CO2 bands, water vapor bands, etc., in Fig. 5.26) overlap with the sample spectrum. This is one of the serious problems in the single-beam-type system. One merit of the single-beam-type is the

resolution 2 cm -1

air CO2

H2O as,

7000

6000

5000

as

H 2O

CO2

s

4000 3000 wavenumber / cm -1

2000

1000

Fig. 5.26 FTIR spectrum of the background component Bo (ν), originating from an air atmosphere containing CO2 and H2 O gases as well as the signals of the system

432

5 Structure Analysis by Vibrational Spectroscopy

relatively high brightness of the incident light since the light is not divided into two, different from the double-beam-type case. Figure 5.25b shows the block diagram of the double-beam-type spectrometer. The light is divided into two light components. One light passes the sample. Another light passes the air (or the reference). These two lights enter the detector alternately by using a rotating sector, which is rotated at about 10 Hz (Fig. 5.27). At first the sample is set on a sample stage. The sample absorbs an incident IR beam partially. The intensity of the detected light is weaker for the sample side (B) than that of the reference side (Bo ); B < Bo . Then, in order to balance the intensity of B and Bo (B = Bo ), the intensity Bo is reduced to B by inserting an optical comb into the reference path by using a servomotor. The monochromator is rotated to the next frequency position. If no absorption of light occurs in the sample side at this new frequency position, the sample side (B) becomes brighter than the reference path (Bo ) because the optical comb remains still at the same position (B > Bo ). In order to increase the brightness of the reference side (Bo ) and equalizes B and Bo , the optical comb goes outside (B = Bo ). In this way, the intensity unbalance between the sample (B) and reference (Bo ) paths becomes zero at all the frequencies of the light by changing the position of the optical comb. The movement of the optical comb is recorded on a chart to draw the spectrum. This principle is called the optical balance method. The double-beam-type IR spectrometer is stable in intensity even when the light source is fluctuated since the ratio B/Bo is calculated instantaneously. A ratio-type spectrometer is the digitization of the double-beam path system. The electric signals B and Bo are digitized and the ratio B/Bo is calculated at each moment. The rotating sector to separate the optical paths of the sample and the background (a) Optical Comb

intensity

Movement Δx light

Movement Δx

(b) Rotating Sector intensity

rotation Fig. 5.27 a Optical comb in the double beam optical system and b the rotating sector

5.2 Spectrometers

433

is still used, but the optical comb is not needed anymore. In the above-mentioned system, the movement of the optical comb is not necessarily reproduced perfectly, and the signal detection is not enough quick. In the ratio-type system, the detection of the digitized signals is fast and the ratio B/Bo is recorded more accurately than the optical comb system. The response is also faster than the latter. The digitized data are easily treated using the software.

5.2.1.3

Fourier Transform IR Spectrometer

The inside of an FTIR spectrometer is illustrated in Fig. 5.28. The whole system is filled with N2 gas or dried air. In the far IR region, the system is evacuated to purge water vapor since many rotational bands of water molecules overlap the bands of a sample. A light source, for example, SiC globar emits IR components of a wide range of frequencies. The spectrum of the light source is expressed as B(ν). The emitted light is incident into a Michelson interferometer and splits into two paths by a beam splitter (BS). The BS is a KBr plate covered with Ge (5000–400 cm−1 ), a CsI plate with Ge evaporated on the surface (be careful of the deliquescence, 5000–200 cm−1 ), or a Mylar film (100–30 cm−1 , 25 micrometer thickness). These split lights are reflected on the driving and fixed mirrors, and they go back to the BS part and are recombined again. In such a sense, the BS is called also the beam combiner (BC). < Michelson Interferometer> Figure 5.29 shows a Michelson’s interferometer. The amplitudes of the two split light waves are expressed as follows:

B

IR source detector

laser

He-Ne laser voice coil

beam splitter

A

detector

driving mirror

beam splitter IF

driving mirror fixed mirror Sample chamber

sample

Fig. 5.28 An FTIR spectrometer. The right picture is the FTS7000 spectrometer (Digilab)

434

5 Structure Analysis by Vibrational Spectroscopy

fixed mirror

optical path x2

optical path x1 Light source B( )

BS

driving mirror

detector Fig. 5.29 Michelson interferometer

A1 = (A/2) cos[2πνt − 2π(2x1 /λ)]

(5.89a)

A2 = (A/2) cos[2πνt − 2π(2x2 /λ)]

(5.89b)

where ν is the frequency of the light and the x 1 and x 2 are the distances between the mirrors and BS. The intensity of the recombined light waves is written as below (optical paths are 2x 1 and 2x 2 ): A = A1 + A2 = (A/2){cos[2πνt − 2π(2x1 /λ)] + cos[2πνt − 2π (2x2 /λ)]} = A cos[2πνt − 2π(x1 + x2 )/λ)] cos[2π(x1 − x2 )/λ]}(See Note.) (5.90) The time-averaged intensity of the interfered light I is given as I = A2 < cos2 [2πνt − π(x1 + x2 )/λ)] > cos2 [2π(x1 − x2 )/λ]       = A2 /2 cos2 [2π(x1 − x2 )/λ] = A2 /2 cos2 [πx/λ] = A2 /4 [cos(2πx/λ) + 1]

x = 2(x1 − x2 ) = optical path difference ****************** (Note) cos(A − B1) + cos(A − B2) = [cos A cos B1 + sin A sin B1] + [cos A cos B2 + sin A sin B2]

(5.91) (5.92)

5.2 Spectrometers

435

= cos A(cos B1 + cos B2) + sin A(sin B1 + sin B2) = 2 cos A cos[(B1 + B2)/2] cos[(B1 − B2)/2] + 2 sin A sin[(B1 + B2)/2] cos[(B1 − B2)/2] = 2{cos A cos[(B1 + B2)/2] + sin A sin[(B1 + B2)/2]} cos[(B1 − B2)/2] = 2 cos[A − (B1 + B2)/2] cos[(B1 − B2)/2] = 2 cos[2πνt − 2π(x1 + x2 )/λ] cos[2π(x1 − x2 )/λ]

****************** In this way, the intensity of the recombined lights is a function of the optical path difference x. One mirror of the interferometer is driven at a constant speed and the interfered signal is collected at the various x positions. In a more general case, the continuous light source spectrum B(ν) is used instead of only two beams of a fixed frequency (A1 and A2 ). Since λ = c/v, Eq. 5.91 is rewritten as  I (x) =

∞

−∞



∞

B(ν)[cos(2πνx/c) + 1]dν = 2

B(ν)[cos(2πνx/c) + 1]dν

0

(5.93) The second term I (0) = 2 0∼∞ B(ν)dx is the direct current component and the first term is the alternate current component and is called the interferogram F(x). 

∞

F(x) ≡ [I (x) − I (0)]/2 =

B(ν) cos(2πνx/c)dν

(5.94)

0

The inverse Fourier gives the light source spectrum B(ν). 

∞

B(ν) =

F(x) cos(−2πνx/c)dx

(5.95)

0

As shown in Fig. 5.30, by measuring the interferogram F(x) for both the air background and the sample (+ air) and by calculating the corresponding Bo (ν) and Bs (ν), respectively, we can calculate the IR spectrum as follows: T %(ν) = Bs (ν)/Bo (ν) × 100

(5.96)

5.2.2 How to Control FTIR Spectrometer In the actual measurement, the interferometer must be controlled and the interferogram is saved as digitized data. The concrete mechanism is shown in Figs. 5.28 and 5.31. The driving mirror of the interferometer gives the optical path length difference x (box A in Figs. 5.28 and 5.31). Another small mirror is set on the backside

436

5 Structure Analysis by Vibrational Spectroscopy

F(x) of sample + air

F(x) of air

FT

FT

Bs( ν)

Bo( ν)

Bs/Bo

Fig. 5.30 Principle for getting the IR spectrum from the interferograms measured for the air and sample (+air) starting point

Fixed mirror

Interferogram of white light

Driving mirror

collection points

voice coil

IR beam

white light He-Ne laser

B

Interferogram of laser light

B Interferogram

photo diode

A

A

B IR detector

Digitized data

C

Fig. 5.31 Collection of interferogram data

of the main driving mirror (box B). Using this small mirror, a small interferometer is constructed and the laser light is used as a light source. The interferogram of this small interferometer is a cosine wave. At the same time, a white lamp with the continuous frequency distribution is incident to this small interferometer, giving

5.2 Spectrometers

437

a pulse. The pulse is used for the starting trigger of the data collection. The main interferogram is detected at the timing of the zero points of the cosine signal. The thus-collected interferogram is an assemblage of the digitized data points. When these data are used for the Fourier transform calculation of the interferogram, the obtained interferogram does not have necessarily the ideal shape. We need to correct the calculated interferogram.

5.2.2.1

Censored Data Points Effect

Originally, the Fourier transform must be made in the range of x = 0 ~ ∞. However, the actually collected data is in the limited range from −x max to x max or the amplitude of the driving mirror movement. The interferogram contains many ripples coming from this censored data points effect. In such a case, the introduction of a window function is useful for erasing these ripples (apodization). As shown in Fig. 5.32, the window function is a kind of weight. The simplest case is the horizontal line, inside which the weight is unity but the weight is 0 outside. The Fourier transform calculation is made as ∞ B(ν) =

F(x)R(x) cos(−2πνx/c)dx = F(F)∧ F(R)

(5.97)

0

(a)

(c)

(b) ideal

actual

Fig. 5.32 Apodization of interferogram. a The effect of window function R(x), b the censored data of interferogram and the application of R(x), and c representative window functions and their FT patterns

438

5 Structure Analysis by Vibrational Spectroscopy IR source

BS detector

driving mirror

fixed mirror

Fig. 5.33 High-resolution FTIR spectrometer using cat’s eyes

This is a convolution of Fourier transforms between F(x) and R(x). Depending on the shape of R (x), the obtained shape of B(ν) is different. It is important to note that the half width of the bands is modified depending on R(x) (see Fig. 5.32c).

5.2.2.2

Resolution Power

The resolution power of the spectrum, ˜ν is dependent on xmax ˜ν = 1/xmax

(5.98)

For example, for the measurement of an ultra-fine structure of the rotational spectra of a gas, the resolution of an order 0.001 cm−1 is required sometimes. In such a case, x max = 10 m is needed. The driving mirror is moved on the long rail of 10 m length. As shown in Fig. 5.33, sometimes the cat’s eye mirror is used, and the light is reflected many times to get a long distance. One example is a Fourier transform IR spectrometer for the detection of the gas components of the cosmos.

5.2.2.3

Highest Wavenumber

The highest wavenumber ν˜ max of the spectrum is dependent on the sampling interval h : ν˜ max < 1/(2h). As shown in Fig. 5.34, the sampling of the interferogram is made at an interval h. The Fourier transform gives the spectrum B(ν) in the range of 0 ~ 1/h. If the h value is too large, the repeated B(ν)s overlap with each other. Then the maximal frequency νmax should be equal to 1/(2h). As shown in Fig. 5.31, if the sampling is made using the zero points of the small interferogram of He–Ne laser (λ = 0.6328 μm), which locate at every λ/2 position, then, h = λ/2 = 0.3164 μm. As a result, νmax = 15, 802cm−1 .

5.2 Spectrometers Fig. 5.34 Data sampling and obtained spectrum B(ν)

5.2.2.4 (a)

(b)

(c)

439

Interferogram

Merits of FTIR Spectrometer

One reciprocating scan of a driving mirror gives the interferogram, from which the spectrum is obtained. If the driving speed of the mirror is in the order of msec (the movement frequency of the driving mirror 1000 Hz), for example, a series of time-resolved spectra will be obtained. The so-called time-resolved spectral measurement can be made relatively easily. The data saving time is almost determinative of the total time. (The actually available driving speed is dependent also on the resolution power as well as the data saving rate.) Since the whole spectrum can be obtained in one scanning of the driving mirror, as mentioned above, the S/N ratio becomes overwhelmingly higher than the case of dispersion-type spectrometer when compared in the same measurement time. For example, the m wavenumber points are recorded by scanning a grating monochromator in the dispersion-type spectrometer. During this measurement, the m spectra can be obtained by the FTIR spectrometer. This means that the S/N ratio of the latter system is higher by (m)1/2 . For example, for the spectral measurement in the region of 4000–400 cm−1 with an interval 2 cm−1 , m = 3600/2= 1800 points. The S/N ratio is higher by (1800)1/2 = 42 (Fellgett advantage). Since the FTIR spectrometer is a single beam type, the optical throughput is high, which is defined as (light source area) × (usable solid angle of the light). Compared with the dispersion-type spectrometer, the optical throughput is about 100 times higher for the FTIR spectrometer. The absorbance axis is more accurate than that of the dispersion-type spectrometer (Jacquinot advantage). It must be noted that the absorbance 3, corresponding to the transmission of 0.1%, may be a limit of the accuracy.

440

5 Structure Analysis by Vibrational Spectroscopy

(d)

The data collection is controlled by using a He–Ne laser. Therefore the sampling points are exact and so the wavenumber of the spectrum is accurate (Connes advantage).

5.2.2.5 (i)

Time-Resolved Measurement

Rapid-scanning type

As mentioned already, the FTIR spectrometer is convenient for the measurement of the spectra at a high time resolution by scanning the driving mirror at a high driving speed. Roughly speaking, several tens spectra can be measured in a second. However, the time resolution is dependent on the resolution power. You must check the catalogue when you purchase an FTIR spectrometer for this purpose. The 4– 8 cm−1 resolution is not recommended for the quantitative analysis of the spectra. (ii)

Step-scanning type

As mentioned above, spectrum B(ν) is obtained by the Fourier transform of an interferogram F(x). The F(x) is the interfered signal intensity at an optical path difference x. As illustrated in Fig. 5.35, instead of changing the x at a high speed, the data collection is repeated quickly in the time region (t 1 , t 2 ,…, t max ) at the constant x = x 1 position of the mirror and a series of F(t, x 1 ) is saved as F(t1 , x1 ), F(t2 , x1 ), . . . , F(tmax , x1 ). Then, the mirror position is changed to x 2 , and the data collection is made again to obtain a set of data F(t 1 , x 2 ), F(t 2 , x 2 ),…, F(t max , x 2 ). This process is repeated at x= x 1 to x max . Finally, we have a series of interferogram signals {F(t i , x i )}. The thus-collected data are arranged as a matrix shown below. If these data are traced along the individual row at a constant time t i , they are exactly the data points of an interferogram expressed as a function of the optical path difference

F F

Fmax

x F F

Fmax

t t

tmax

x F F

x

t t

x

Fmax

t t

tmax

tmax

x

x

time

Fig. 5.35 A principle of step-scanning-type time-resolved measurement of FTIR spectra. The cyclic structural change must be repeated in the perfectly same way at each x position

5.2 Spectrometers

441

x i : {F(ti , x1 ), F(ti , x2 ), . . . , F(ti , xmax )}. The Fourie transform of this row data at a time t i gives the spectrum B(ν,t i ). In this way, we can get a set of time-resolved spectra at t= t 1 , t 2 ,…. x1

x2

x3

…

x max

FT

t1

F(t 1 ,x 1 )

F(t 1 , x 2 )

F(t 1 ,x 3 )

…

F(t 1 ,x max )

B(ν, t 1 )

t2

F(t 2 ,x 1 )

F(t 2 , x 2 )

F(t 2 ,x 3 )

…

F(t 2 ,x max )

B(ν, t 2 )

t3

F(t 3 ,x 1 )

F(t 3 , x 2 )

F(t 3 , x3 )

…

F(t 3 ,x max )

B(ν, t 3 )

…

…

…

…

…

…

…

t max

F(t max ,x 1 )

F(t max , x 2 )

F(t max , x 3 )

…

F(t max , x max )

B(ν, t max )

This method is called the step-scan method. The sample state is changed in the period of t 1 ~t max and this change is repeated many times at x = x 1 to x max . As understood from the above matrix, the sample state change must be repeated perfectly at the various x i values. That is to say, the step-scan method can be applied only to the perfectly reversible phenomenon. Several papers were reported using this method: for example, the time-resolved measurement of the IR spectra in the cyclic stretching and contracting process of a polymer film. However, the polymer deformation is not necessarily perfectly reversible. It is always attended more or less by a fatigue phenomenon and so the structural change is not necessarily the same at all the steps of the different x i . The application of electric voltage to a liquid crystal cell might be better for this type of measurement. In this way, we need to check the applicability of this method with a deep care. For the irreversibly occurring phenomenon, the rapid scanning method is more useful and safer although the time resolution is not very high (several sec to several msec interval), compared with the step scanning experiment (several msec to several μsec interval).

5.2.2.6

Miniaturization of FTIR Spectrometer

Recently, the FTIR system is being miniaturized into a scale of several cm or less. The so-called MEMS, or Micro Electro Mechanical System, aims to produce a quite small kit of the instrument with almost the same functionality as that of the usual scale system. The MEMS FTIR spectrometer is now being developed. The principle is to use a small Michelson interferometer. As shown in Fig. 5.36 [8], the key point is to have a small driving mirror of 1.6 mm2 . The mirror is translationally moved upand downward alternately using a MEMS circuit and spring system. The amplitude of the driving mirror is about 100μm at the frequency 5.2 kHz. Although the resolution power is low, about 30 cm−1 , the time resolution is 1 ms. The spectral range is 4000–400 cm−1 . Another idea is to use the ATR principle. As shown in Fig. 5.37, the evanescent waves are emitted from a polysilicon heater and propagated through the total reflection inside a Si internal element [9]. A sample is contacted with this element to measure the ATR spectrum using a polysilicon thermocouple detector. The whole working space size is about 15 × 15 mm2 .

442

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.36 One example of a miniaturized FTIR system. A small mirror moves up- and downward as shown by arrows. The total size is about 10 cm width × 20 cm length. The IR detector is cooled by a Peltier effect. Reprinted from Ref. [8] with the permission of the International Society for Optics and Photonics, 2007 Vgroove

Sample surface

Vgroove

Top wafer

Detector Heater Recess

Absorber Bottom Wafer

Fig. 5.37 A miniatured ATR FTIR instrument. Reprinted from Ref. [9] with the permission of Elsevier, 2008

5.2 Spectrometers

443

5.2.3 Raman Spectrometer 5.2.3.1

Dispersion Type

Figure 5.38 shows a dispersion-type Raman spectrometer, which consists of a laser, a sample room, a monochromator, and a detector. (i)

Laser

Laser is an excitation light source. The typical gas laser is He–Ne ion laser (λ = 632.8 nm), Ar ion laser (λ = 514.5, 488.0 nm), Krypton ion laser (647.1 nm), etc. Nd: YAG laser is one of the solid lasers and has a wavelength 1.064 μm. As a semiconductor laser, GaN (0.4 μm), InGaAsP (1.0–2.1 μm), etc. is popular. A laser providing a wide range of wavelength is a dye laser with the wavelength 0.38~1.0 μm. For example, a dye stilbene emits laser of 390–435 nm, coumarin 2 for 460– 515 nm, and a rhodamine 6G for 570–640 nm. The selection of the laser component used for the Raman experiment is made using a monochromator. The laser beam is focused on the sample position using a lens. The too sharp focusing of the beam may cause the sample damage, in particular, for the polymer materials. (ii)

Diffraction grating monochromator

The Raman scattering signals are dispersed using a diffraction grating monochromator. Depending on the spectral resolution power, the number of monochromators is different. Usually, a single monochromator is used. However, for the higher resolution, the double or triple monochromator is used (Fig. 5.39). They are useful for erase of a strong Rayleigh scattering. These monochromators are controlled to rotate synchronously. The angle of the monochromator should be stable irrespective of the fluctuation of the environmental condition. For this purpose, the temperature of the monochromator must be controlled.

grating

lens laser

slit

sample

Fig. 5.38 Dispersion-type Raman spectrometer

detector

444

5 Structure Analysis by Vibrational Spectroscopy

mirror detector

Raman scattering Fig. 5.39 A Triple monochromator system for Raman measurement (Japan Spectroscopic Company, NRS2100 spectrometer)

(iii)

Detector

A photomultiplier is often used as a detector of Raman signals. The strong light may damage the detector easily. The recent Raman spectrometer uses a multichannel detector (or a linear array detector) in a certain range of wavenumbers. The wellknown detector is a CCD detector. Cooling of the detector is useful for the reduction of noise. Of course, the strong Rayleigh scattering must be erased using a monochromator and/or an optical filter. For the measurement of a low-frequency region of several cm−1 , the slit width must be controlled carefully. For Ar ion laser, the usage of iodine vapor is told to be useful for the reduction of Rayleigh scattering, the frequency of which is almost equal to that of absorption maximum of iodine vapor (532 nm). (iv)

Scattering angle

The Raman scattering occurs in all directions. Collection angle can be changed; 90° scattering, back scattering, 180° scattering, and so on. The relation between the incident photon vector and the phonon vector is important for the detection of the phonon dispersion (see Fig. 5.17).

5.2 Spectrometers

445

Fig. 5.40 Miniature Raman spectrometer using an optical fiber

optical fiber

sample laser

CCD

(v)

Mobile Raman spectrometer

A miniature Raman spectrometer is becoming popular for the Raman measurement for any sample at any place. An optical fiber is used for the irradiation of an excitation laser and as a carrier of the scattered signals. The 180°scattering signals are utilized mainly (Fig. 5.40). Refer to Sect. 5.15.2.

5.2.3.2

FT-Raman Spectrometer

Quite often, the Raman scattering measurement of polymer materials is difficult to perform in a good condition. One of the most serious causes is the simultaneous emission of strong fluorescence from the impurities contained in the sample. If polymer materials are colored more or less by an impurity or by being annealed at high temperature, they generate strong fluorescence, which makes it impossible to measure the Raman spectra any more. Long irradiation of a laser beam to the sample may reduce the florescence gradually (bleaching). The purification of the sample may improve the situation better, but not very hopeful. The Raman scattering occurs more quickly than the emission of fluorescence and so the opening of the gate for a picosecond may allow the path of only Raman signals, but the technique is difficult to set up. The anti-Stokes Raman components do not contain the fluorescence and so the Raman spectrum might be obtained safely, but the signals are overwhelmingly weaker than the normal Stokes signals (see Fig. 5.16). One of the useful methods is the utilization of the laser of a long-wavelength or low-energy. Krypton laser (647.1 nm) or near-IR laser (1.064 μm) is used for this purpose. The near-IR laser induces the excitation of only the vibrational energy levels (because of too low energy to induce the transition of the electronic level) and so the generation of the fluorescence can be avoided. However, since the scattering intensity is proportional to 1/λ4 , the Raman signals are quite weak. In order to detect these quite weak Raman signals, a Fourier transform spectrometer is introduced. As already mentioned, the Fourier transform technique can detect the signals more effectively. At the same time, the utilization of a semiconductor detector (such as Ge, Si, or PbS detector) increases the S/N ratio of the detected signals. The combination of a near-IR laser, an FT-type monochromator, and a highly sensitive Ge detector makes it possible

446

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.41 FT Raman spectrometer. Reprinted from Ref. [10] with the permission of the American Chemical Society, 1986

to measure the Raman spectra of the polymer materials with quite low fluorescence background. Figure 5.41 illustrates one example of the FT-Raman spectrometer [10]. Figure 5.42 shows the Raman spectra of poly-p-phenylene terephthalamide fiber (Kevlar® ) measured using an Ar-ion laser and a Nd:YAG laser. Fig. 5.42 Raman spectra of Kevlar fiber. A Ar-ion laser (dispersion-type spectrometer) B Nd:YAG laser (FT-Raman). Reprinted from Ref. [10] with the permission of the American Chemical Society, 1986

5.3 Sample Preparation for IR Spectroscopy

447

5.3 Sample Preparation for IR Spectroscopy 5.3.1 Films The most popular type of polymer sample used for the IR spectral measurement is a film (refer to Sect. 1.7).

5.3.1.1

Cast Film from Solution

A polymer is dissolved into a solvent, which should not react with the polymer sample. The highly volatile solvent is easier for this purpose. The solution is gently dropped on a glass plate and spread to proper size using a spatula. This glass plate is set on the horizontal table and covered with a petri dish in a small space so that the solvent evaporates slowly. After a while, a film is formed on the slide. In some time, the solvent remains even after the film is prepared apparently completely. The film is hard to peel off the slide even by gentle pulling. The strong pulling may cause orientation of the film. Another good way is to immerse the slide glass into water. Of course, the film must not be dissolved in water. The water penetrates into the space between the glass and the film, and then the film floats from the glass spontaneously. The thus isolated film is scooped by a paper frame or a metal holder. The water is wiped away from the film gently and the film is dried up. The dry film is set on a paper frame shown in Fig. 5.43. The aluminum foil with a hole at a center is pasted on the paper frame and the film is fixed on the aluminum foil. This method is to avoid the IR signals from the burr of the cut paper. At the same time, the control of the hole size is important to avoid the leakage of IR beam from the gap between the film and the frame (or Aluminum foil) hole. This is a serious problem since the beam leakage leads to incorrect absorbance. As shown in Fig. 5.43, the saturated band should show the 0% transmission, but the IR beam leakage lifts up the baseline paper

Transmission

100%

I0

I0 Al I

0% T = I/I0 = 0/Io

T = I/Io

A = -log(T) = ∞

A = -log(T) gap

film holder

Fig. 5.43 Effect of IR beam leakage on the spectral profile

sample film

448

5 Structure Analysis by Vibrational Spectroscopy

and the transmission is different from 0%. The IR beam leakage must be avoided carefully for the correct spectral measurement. Cast of the film on KBr plate is also convenient, which can be directly used for the IR measurement. However, it must be checked whether the film does not interact with KBr to form the complex or the polymer chains are preferentially oriented on the KBr plate. Kevlar film is difficult to prepare. Kevlar is dissolved into sulfonic acid at room temperature. The solution is spread on the glass slide and immersed into water bath to give a coagulated unoriented film. The solution is sandwiched between a pair of slide glasses, and one glass is moved horizontally to give the oriented liquid crystalline solution, which is immersed immediately into a water bath to get an oriented film.

5.3.1.2

Cast from Melt

The sample is sandwiched by a pair of slide glasses and put on a hot plate heated above the melting point of the sample. Heating at too high temperature causes the thermal degradation of the sample. The molten sample is pressed gently to make a thin film. Then the slide glasses are taken out of the hot plate and cooled to room temperature (or quenched into an ice-water bath or liquid nitrogen vessel). The film is peeled off the glass carefully. Sometimes the film is hard to peel off the glass. In such a case, the sample is sandwiched by a pair of polytetrafluoroethylene (Teflon® ) or Kapton® film, and the sample is melted between a pair of slide glasses. The thuspressed film is easy to peel off. The film thickness is several μm to several hundreds μm depending on the experimental purpose. A hot roller or a hot press is useful for the preparation of the melt-pressed film. The rolled film might be more or less oriented. The usage of microtome is another method for the preparation of a thin film, the size of which is small and suitable for the IR microscopic measurement (see a later section). For example, as shown in Fig. 5.44, a small sample is put in a microcapsule of gelatin, and methyl methacrylate (MMA) monomer liquid is filled in the capsule [11]. MMA monomer is polymerized by adding an initiator. The thusobtained PMMA rod is sliced to get a thin film of the sample. This sliced sample with the PMMA frame is used for the IR microscopic measurement. (a) microcapsule + sample

(b) fill MMA monomer

(c) polymerization (d) cut out the sample

Fig. 5.44 Preparation of a small film using PMMA resin [11]

5.3 Sample Preparation for IR Spectroscopy

449

5.3.2 Oriented Film A cast film is stretched carefully using a stretcher or a pair of pliers or by hands. In a case the sample is heated on a hot plate or in an oil bath and then stretched. Shearing the film between a pair of glass slides causes the chain orientation to some extent. Rolling the film is another way to get the oriented film. The plane orientation might occur in this case. If the sample is too thin to roll, the sample is sandwiched by a thin aluminum plate and roll the aluminum plate itself. The sandwiched sample is also stretched together. A single fiber of several μm width is wound around a KBr plate in parallel. The resultant plate of parallel fiber bundle may be too thick to get an IR spectrum of suitable intensity. The ATR method might be better for it.

5.3.3 Powder Powder sample is ground by a mortar. For a polymer powder, static electric charges are generated by the friction, and the powder may spatter. In such a case, mixing of KBr powder with a polymer powder is one idea to escape from it. KBr is not needed to take away from the sample because it is transparent for IR up to 400 cm−1 . The mixture of sample and KBr is strongly pressed under a high pressure to produce the KBr disk. If KBr is not needed, it can be dissolved away by washing by water. Long grinding of the sample may cause the phase transition by the shearing effect. For example, PVDF form II sample is transformed into form I by grinding. The diffusion reflection method might be good for the IR spectral measurement of the powder sample. Grinding of a powder sample with KBr powder may cause the chemical reaction or ion exchange in the sample. In such a case, Nujol mulling method is better. A liquid paraffin is used in stead of KBr powder. The small amount of paraffin liquid is blended with a sample powder and mixed homogeneously by a mortar grinding. Of course, the IR bands coming from the paraffin must be reminded in the data analysis.

5.3.4 Liquid Viscous polymer liquid is sandwiched between a pair of KBr plates and the IR spectrum is measured directly. If the KBr plates are small, wrap the plates with an aluminum foil and open a window for IR transmission and set it to the thick paper frame as shown in Fig. 5.45. More quantitative measurement is to use a so-called liquid cell (Fig. 5.46). A metal spacer of a homogeneous thickness is put on a KBr plate and drop a liquid onto it. The pair of KBr plates are set to a metal holder and fastened with screw bolts. The thickness of the sample is controlled by changing the spacer thickness. In some liquid cell, the spacer is permanently fixed on the KBr plate with an adhesive. A small hole is opened on a KBr plate, from which a liquid sample can be injected using a needle. The window materials can be replaced with the other kind of single-crystal plates depending on the sample. If the sample is an aqueous solution, CaF2 , KRS5,

450

5 Structure Analysis by Vibrational Spectroscopy KBr plate

IR

Al foil

windows

liquid

Fig. 5.45 Preparation of KBr plate cell for a liquid sample

Fig. 5.46 Liquid cell with a constant thickness

screw volt Metal holder KBr spacer KBr Metal holder

ZnSe, etc. may be used as a window. Be careful of the interference fringe bands of the IR spectrum (Fig. 5.47). These fringes can be used for the estimation of the thickness of the sample since the following equation is applied: refractive index in vacuum in substance

wavelength

frequency

velocity

no (= 1)

λo

ν

c (= λo ν)

n

λ

ν

v = c/n(= νλ)

d

path 1 path 0

n Fig. 5.47 Generation of fringes

5.3 Sample Preparation for IR Spectroscopy

451 (a)

(b)

-.47 -.48

small hole

nuts

spacer

KBr plate

-.49 -.5 -.51 -.52

KBr plate

holder

-.53 2900

2800

2700

2600

2500

2400

Wavenumber / cm-1

liquid sample

Fig. 5.48 Fringe bands generated from an empty liquid cell

Since v = c/n,νλ = νλo /n and so λ = λo /n. Path 0 Path 1

ko = number of waves in the space d → ko = d/λ = dn/λo k1 = 3d/λ = 3dn/λ0

The difference in the number of waves  = k1 − ko = 2dn/λo. Therefore, d =  · λo /(2n)

(5.99)

By counting the number of fringes, we can estimate the thickness d by inputting the refractive index n. (The wavelength in the air is assumed to be equal to that in vacuum.) In the actual IR spectrum, the fringes appear at the different positions of the wavenumbers. Equation 5.99 is a little modified as below (Fig. 5.48). By using the above-mentioned equations at a different wavelength λ, we have the following relations. The number of waves of wavelength λ1 included in the path 2d 1 = 2d/λ1 = 2nd/λo1 The number of waves of wavelength λ2 included in the path 2d 2 = 2d/λ2 = 2nd/λo2 Then, the number of fringes included between the two positions (λo 1 and λo 2 ) is  = 1 − 2 .     = 1 − 2 = 2nd/λo1 − 2nd/λo2 = 2nd λo2 − λo1 /λo2 λo1     Using the wavenumbers ν˜ 1 = 1/λo1 and ν˜ 2 = 1/λo2 , we have the following equation:    = 2nd(˜ν1 − ν˜ 2 ), d = / 2n(˜ν1 − ν˜ 2 )

(5.100)

452

5 Structure Analysis by Vibrational Spectroscopy

Absorbance

(a) PE film homogeneous thickness fringes

(b) PE film inhomogeneous thickness

Wavenumber / cm-1 Fig. 5.49 IR spectra of polyethylene film of a flat thickness

By applying this equation to the empty liquid cell shown in Fig. 5.48, the cell thickness d = 112.5 μm, where n = 1 for air and the nine fringes are included in the range of ν˜ 1 = 2800 cm−1 ∼ ν˜ 2 = 2400 cm−1 . The fringes are detected also for a film with a smooth surface. One example is shown in Fig. 5.49. In this case, five fringes are  of ν˜ 1 =  included in the range 800 cm−1 ∼ ν˜ 2 = 1200 cm−1 , and so d = 5/ 2 × 1.5 × 400 cm−1 = 42 μm. Here, n= 1.5 is assumed (the n value changes quite slightly from 1.515 (25000 cm–1 ) to 1.490 (6250 cm–1 )).

5.4 Key Points for Spectral Measurements 5.4.1 IR Spectra (1)

Background

In the FTIR spectral measurement, the background spectrum is measured at first and then the sample spectrum is measured. If the sample spectral measurement is made after a long time passage since the background measurement, the atmosphere environment might be different from that in the background measurement. As a result, the IR bands of water vapor and CO2 gas cannot be erased perfectly from the sample spectrum (see Fig. 5.26). Before the spectral measurement, the sample box is kept for a while to stabilize the atmospheric condition and then the measurement is started. By repeating this process, the background spectrum is also stabilized to the same condition. A more ideal method is to purge the sample box by dry N2 gas or even to put the system in vacuum. This is in particular important for the far-infrared spectral measurement.

5.4 Key Points for Spectral Measurements

(2)

453

Leakage of IR beam

This is already described in the previous Sect. 5.3.1. (3)

Windows

Many times, we need to use a single-crystal plate to support the sample. The most popular plate is KBr single crystal. The others are single crystals of NaCl, ZnSe, CaF2 , Si, Ge, diamond, etc. The transmission of IR beam is limited to the wavenumber region shown in Fig. 5.50. Silica glass is convenient and cheap, but the transmission range is limited to the high-frequency region. Si, Ge, and diamond are useful since they are chemically stable, but they show several characteristic bands in the fingerprint region. The window surface should be smooth to avoid the scattering of the beam. If not, the baseline becomes extremely tilted. Polishing up the crystal surface is difficult. For example, a KBr plate is polished at first by a sandpaper set on a smooth glass plate and then polished further by using a soft cloth with MeOH solvent (or a little bit of water). (4)

Energy Check

Compared with the dispersion-type spectrometer, the FTIR spectrometer is bright, meaning that the effective usage of the incident IR beam can be made. Therefore, a sample of small size can be measured at a relatively high S/N ratio. The spectrum of low energy is difficult to erase the water and CO2 bands coming from the unbalance of the background. We need to find the sample position giving the highest energy of the interferogram. Since the IR light is focused on the sample position, the sample position must be adjusted carefully to catch the focused light (see Fig. 5.51). Of course, it is indispensable to adjust the orientation of the fixed and driving mirrors of interferometer so that the interferogram energy becomes maximal. In some spectrometers, the corner cube mirrors are used, by which the parallelism of the beams is guaranteed (Fig. 5.52).

transmission

Si KBr ZnSe diamond

SiO2 glass

Wavenumber / cm-1 Fig. 5.50 IR spectra of the various plates

454

5 Structure Analysis by Vibrational Spectroscopy

F(x)

max

Moving distance x Fig. 5.51 Interferogram and IR beam focused on the sample position

(a) Flat 2D corner

(c)

(b) 3D corner mirror

BS

Fig. 5.52 a, b Corner reflections and c Michelson’s interferometer using corner cube mirrors

5.4.2 Raman Spectra (1)

Optical system optimization

Raman signals are overwhelmingly weaker than the incident laser beam intensity. The optical path must be adjusted precisely. The synchronization of slits is important. The diffraction-grating-type monochromator must be stable against the fluctuation by the various effects (temperature, vibration, etc.). The angle of the monochromator must be precisely controlled, which determines the absolute frequency position of Rayleigh scattering peak. The calibration of the peak frequency is indispensable before the start of the experiment. The natural emission of the laser beam, Neon lamp, etc. are useful as the standard frequencies. Table 5.2 shows some examples of

5.4 Key Points for Spectral Measurements

455

Table 5.2 Wavenumber of Ne lamp [A. R. Striganov et al., Tables of Spectral Lines of Neutral and Ionized Atoms, IFI/Plenum Data Corp., New York, 1968; S. B. Kim et al., Appl. Spectr., 40, 412 (1986)] in air (cm−1 )

Raman shift (488.0 nm = 0.0 cm−1 )

Raman shift (514.5 nm = 0.0 cm−1 )

19383.97

1108.42

51.17

19262.92

1229.47

172.22

18694.37

1798.02

740.77

18576.14

1916.25

859.00

18475.24

2017.15

959.90

emission light peaks from Ne lamp. The Raman band from silicon plate is used as a standard, the peak position of which is 520.7 cm−1 . In the low-frequency region, BiO4 powder gives many peaks which are convenient to check the peak positions. (2)

Samples

The measurement of Raman spectra is relatively easy for the usual solid and liquid samples. The powder sample should be tightly packed so that the density of the scatterer is increased. The thin film is stacked together tightly to increase the sample thickness. The liquid sample is packed in a liquid cell and the laser is incident along the long axis of the cell. The colored sample tends to emit the fluorescence, which may obscure the weak Raman signals. For a gas sample, the gas cell of long path is used. Colored samples are easily damaged by a strong laser beam. The incident laser is defocused slightly by controlling the lens-sample distance. For example, the PVAiodine complex is easily damaged by a green laser beam. As the irradiation time is increased, the Raman bands of I3 – and I5 – ion species change their relative intensity because of the chemical decomposition of iodine ions. In order to avoid the decomposition by an incident laser, the sample must be moved continuously during the measurement so that the laser-irradiated position can be changed.

5.5 Various IR Spectral Measurement Methods 5.5.1 Transmission Spectra Transmission spectral measurement is the most standard and basically important. Many users look to preferentially utilize the ATR method, but the transmission spectra are the best for the quantitative analysis. The points to notice in the measurement of the transmission spectra are summarized below, though some points were already described in the previous section. (1)

The leakage of IR light modifies the exact absorbance

456

5 Structure Analysis by Vibrational Spectroscopy

(2)

Good balance between the spectra of sample and background to eliminate the air band components Smooth surface causes the interference fringes Check of maximal energy for the transmitted light (interferogram) Resolution of bands.

(3) (4) (5)

5.5.2 Polarized IR Spectra The spatial orientation of polymer chain, crystal lattice or local group can be estimated by measuring the polarized IR spectra. The polarized IR beam can be produced by passing the IR beam through the polarizer. The polarizer for the IR beam is usually a wire grid polarizer, which is made of a parallel array of thin metal wires produced by scribing the surface of a single-crystal plate of KRS5, CaF2 , BaF2 , ZnSe, Ge, and so on. The space of lines is narrower than the IR wavelength. When the electromagnetic wave is incident to the wire grid, the electric field of the IR beam causes the forced vibration of the free electrons inside the metal wire, from which the secondary scattering waves are emitted as the perpendicularly polarized (p) IR beam, as illustrated in Fig. 5.53. The parallel polarization (s) component is reflected by the metal wire. The polarizer made of poly(vinyl alcohol)-iodine complex cannot be used for the IR spectral measurement since it absorbs the IR beam strongly. When you try to measure the polarized IR spectra, you need to measure the background using the same polarization as that of the sample measurement. For the measurement of a parallel-polarized spectrum, that is, the polarized spectrum with the IR electric vector parallel to the chain axis, the background must be measured at first using the parallel polarizer. The polarizer is kept at the same setting and the spectrum of the sample is measured. The sample is taken out, and the polarizer is now rotated by 90°, and then the background is measured. Then the sample is set again without any change in the direction, and the perpendicularly polarized spectrum is measured. If you use the air background without polarizer, the spectrum contains the bands characteristic of the polarizer material itself, as shown in Fig. 5.54. Fig. 5.53 Wire-grid polarizer (Do not touch the surface of the polarizer. Polishing up the surface is nonsense!)

s polarized wave

p polarized wave

wire grid

p polarized wave

5.5 Various IR Spectral Measurement Methods

Absorbance

δ(CH2)

457

non-polarizer

r(CH2)

background = air

background = polarizer

Wavenumber / cm-1 Fig. 5.54 Polarized IR spectra of an oriented polyethylene film. When the background spectrum is measured without any polarizer, the additional bands coming from the polarizer material are observed. The background must be measured necessarily with the same polarizer as that of the sample. Many water bands in the 1400–1800 cm−1 are detected in these spectra

The electric vector E of the polarized IR beam interacts with the transition dipole  2 moment μ in the form of E · μ = E 2 μ2 cos2 (θ ), where θ is the angle between the vectors E and μ . Therefore the strongly polarized band has the transition dipole μ parallel to the electric vector E of the incident IR beam. In the polarized spectra of an oriented polyethylene film shown in Fig. 5.54, the CH2 bending (δ) and CH2 rocking (γ) bands detected at 1450 and 730 cm-1 , respectively, are strong for the perpendicular electric field with respect to the chain axis. These polarization characters can be understood from the vibrational mode and the generated transition dipole moment of the CH2 unit, as illustrated in Fig. 5.55. However, we need to recognize that such an interpretation based on only the local modes is not necessarily correct. We need to consider the phase relations between the vibrational motions of the neighboring CH2 units. A more detailed analysis must be made for the vibrational modes of a planar-zigzag PE chain, as will be described in a later section.

νs(CH2)

νas(CH2)

δ(CH2)

ω(CH2)

Fig. 5.55 Local transition dipole moments (red color) for the various normal modes (blue color) of CH2 unit of polyethylene zigzag chain

458

5 Structure Analysis by Vibrational Spectroscopy

5.5.3 Reflection Spectra When light is incident into an interface between the two substances, as shown in Fig. 5.56, the light passes into the substance 2 and, at the same time, it reflects toward the original substance 1, where the refractive indices of these substances are n1 and n2 (n1 > n2 ), respectively. They obey Snell’s law n 1 · sin(φ1 ) = n 2 · sin(φ2 )

(5.101)

The ratio between the incident and reflected lights or the reflectance R (the square of the reflection coefficient) is dependent on the polarization of the light. The s wave is the light with the electric vector perpendicular to the reflection plane. The p wave is the light with the electric vector within the reflection plane. The Rs and Rp are given as



n 1 cos(φ1 ) − n 2 cos(φ2 ) 2



Rs =

n 1 cos(φ1 ) + n 2 cos(φ2 )



n 1 cos(φ2 ) − n 2 cos(φ1 ) 2

R p =

n cos(φ ) + n cos(φ )

1

2

2

(5.102)

1

For the natural or unpolarized light   R = Rs + R p /2

(5.103)

The transmittance T s and T p are given as Ts = 1 − Rs , T p = 1 − R p

p

incidence

s

reflection

φ1 φ1 n1 n2 φ2 transmission

Fig. 5.56 Reflection and refraction of light at the interface between the two substances

(5.104)

5.5 Various IR Spectral Measurement Methods

5.5.3.1

459

Normal Reflection

The IR beam is incident on the surface of the sample and the reflectivity is measured (see Fig. 5.57). This is called the normal reflection spectroscopy or the specular spectroscopy or the external reflection spectroscopy. This measurement is used often for the single crystal of organic or inorganic substance since the thin film suitable for the transmission spectral measurement is difficult to prepare. The sample absorbs the IR beam to some extent and the refractive index must be expressed as a complex: n*= n – ik, where n is a refractive index and k is an absorption coefficient. The absorption spectrum measures the k term. For the perpendicularly incident IR beam, φ = 0o and Eq. 5.102 is expressed below, where the polarization (s and p) is ignored for simplicity.



∗

2 2

n 1 − n ∗2 2 (n 1 − ik1 ) − (n 2 − ik2 ) 2

= (n 1 − n 2 ) + (k1 − k2 )



R= ∗ =

(n − ik ) + (n − ik )

n 1 + n ∗2

(n 1 + n 2 )2 + (k1 + k2 )2 1 1 2 2 (5.105a) If the substance 2 is air, n2 = 1 and k 2 = 0, and so Eq. 5.105 is changed to R=

(n 1 − 1)2 + k1 2 (n 1 + 1)2 + k1 2

(5.105b)

When the spectrum is measured, it is deformed more or less. This is because the refractive index shows the abnormal dispersion at the absorption peak frequency. Therefore, in order to compare the absorption spectrum with the transmission spectrum, we need to convert the reflection spectrum to the absorption spectrum k(ν) using a Kramers-Kronig (KK) relation. In the actual case, the thus-calculated spectrum does not necessarily have the normal band shape because of the many factors.  n(νi ) = 1 + (2/π)P 0

IR

∞

  νk(ν)/ ν 2 − νi2 dν

(5.106)

IR

n1

n1

n2

n2

Fig. 5.57 a Reflection on the substance 1 b Reflection at the interface between the substances 1 and 2

460

5 Structure Analysis by Vibrational Spectroscopy



∞

k(νi ) = (2νi /π)P 0

  n(ν)/ ν 2 − νi2 dν

(5.107)

where P is a Cauchy’s principal value. One example of the optical equipment is shown in Fig. 5.58a. A pair of flat mirrors are set so that the IR beam is incident on a sample surface and the reflected IR beam goes toward the direction of the detector. The reflection spectra obtained for thick PE and PTFE plates are shown in Fig.5.58b and c, respectively. The band shapes are almost equal to those of the abnormally dispersed refractive index. By performing the KK conversion, the IR spectra similar to the transmission spectra are obtained, but they are more or less deformed from the originally expected transmission spectra. In the case of a sheet of thin film, the sample is set on the mirror plane (see the right figure of Fig. 5.58a). The spectrum is that of the hybridization of the transmission and reflection components. The KK conversion did not give the spectrum of the normal pattern.

5.5.3.2

ATR

When the IR beam is incident from substance 1 toward sample 2, reflection and refraction occur in sample 2. The refraction angle is φ2 . If an angle φ1 in Snell’s law is selected (=φ*1 ) so that the refraction angle φ2 = 90°, no refraction occurs into sample 2 and the incident IR beam is reflected totally. This phenomenon is called the total reflection phenomenon.   n 1 · sin φ∗1 = n 2 · sin(90o ) = n 2

then

  sin φ∗1 = n 2 /n 1 (< 1)

(5.108)

However, some components of the incident IR beam near the boundary penetrate into sample 2, which is called the evanescent wave (Fig. 5.59). The reflected IR component is absorbed to some extent by sample 2. The reflectivity R(ν) is attenuated by the amount of the characteristic spectral profile of sample 2. This method is the attenuated total reflection spectroscopy (ATR). The penetration depth of the IR beam into sample 2 is expressed using the following equation:   1/2  dp = 1/ 2π˜νn 1 sin2 φ1 − (n 2 /n 1 )2

(5.109)

By changing the refractive index n1 of substance 1 and/or the incident angle of the light, the penetration depth is changed (Fig. 5.60). The depth changes also depending on the wavenumbers. Therefore, the evaluation of the relative intensity between the bands at the different wavenumbers must be careful by taking this situation into account. The ATR method is quite easy for the measurement of IR spectra since the sample is only attached to substance 1. However, it must be remembered strongly that, as

5.5 Various IR Spectral Measurement Methods

461

(a) sample

mirror sample

IR mirror

(b) PE Reflection spectrum Observed transmission spectrum

KK-converted transmission

(c) PTFE Reflection spectrum Observed transmission spectrum

KK-converted transmission

Fig. 5.58 a An equipment for the IR reflection spectral measurement. b The actually observed reflection spectra of PE and the transmission spectrum obtained by the KK conversion in comparison with the observed transmission spectrum. c The case of PTFE sheet

462

5 Structure Analysis by Vibrational Spectroscopy

 n1 n2 Fig. 5.59 ATR. Evanescent waves penetrate into the substance 2

Fig. 5.60 Penetration depth of evanescent wave in the ATR spectral measurement

already mentioned above, the R(ν) spectrum may be deformed more or less because of the contribution of abnormal dispersion of the refractive index. This occurs more seriously when the band has a large absorption intensity (i.e., for the strong band). Figure 5.61 shows the polarized ATR spectra of poly(vinylidene fluoride) form I film obtained by the solution casting method. By changing the incident angle φ1 , the s- and p-polarized spectra change their profiles remarkably. Besides, the peak position and relative intensity of the individual bands are different from those of the transmission spectrum.

5.5 Various IR Spectral Measurement Methods

463

Fig. 5.61 Polarized ATR spectra of unoriented PVDF film (crystal form I) depending on the incident angle of IR beam

The equipment used for the ATR spectral measurement are listed as follows. (i)

Internal Reflection Element

Substance 1 must have a refractive index higher than that of the sample to be measured. Besides, substance 1 must be transparent for the IR beam ideally. The substance for the ATR method is called the internal reflection element (IRE). The popular IRE is diamond (n = 2.42), ZnSe (n = 2.43), KRS5 (n = 2.38), Ge (n = 4.01), and so on. The shape of IRE is usually trapezoidal (Fig. 5.62). The incident angle is fixed to 45°. Another IRE prism is of a hemisphere shape. This makes it possible to change the incident angle φ1 arbitrarily (Fig. 5.62).

trapezoidal prism

Fig. 5.62 Internal reflection elements

hemisphere prism

464

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.63 Actual setting for ATR spectral measurement

pressure

Al (+ rubber)

sample

IR beam

IRE

(ii)

Background

In many ATR equipment, the IRE is set at the lower side, and the sample is put on the IRE, as illustrated in Fig. 5.63. The ATR spectrum is obtained by measuring the background at first and then the sample. The background is an aluminum foil with a rubber plate wrapped inside. The foil is in contact with an IRE and pressed by a pressure applicator. The usage of rubber plate between the foil and pressure applicator end is recommended to press the foil to the IRE prism completely as well as a cushion for the high pressure. The press of sample to the IRE is also in the same way: the rubber plate is inserted between the sample and the IRE prism. Too strong pressure might break the element easily. A torque screwdriver (a pressure applicator) is useful to control the pressure applied to the sample so that the pressure is reproducible always even when the sample is exchanged (the more details in a later section). Do not forget to wipe the IRE surface before setting the sample (gently). (iii)

Polarizer

The polarization measurement of ATR spectra is useful for the study of molecular orientation of the sample surface. The electric vector of the polarizer is set horizontally (E s ) or perpendicularly (E p ) to the IR spectrometer. The horizontal setting of the electric vector is equal to the polarization of the s wave. The perpendicularly polarized beam contains the electric vector components parallel and perpendicular to the surface (Fig. 5.64). Therefore, the interpretation of the spectra needs some attention. Fig. 5.64 Polarized ATR measurement

incidence

reflection

Ep

p=h+v h

Ep

Es

n1 n2

v Es s

5.5 Various IR Spectral Measurement Methods

465

Fig. 5.65 Multiple scatterings of incident IR by small particles

By irradiating the polarized IR beam from the two perpendicular directions, the 3D orientation of the molecules on the sample surface can be derived [12–14]. (iv)

Pressure Applicator

The gap between the sample and IRE is important. The sample is pressed by the pressure applicator (with a torque screwdriver). The torque is checked to know whether the too strong pressure is applied to the sample or not and whether the pressure is almost the same among the various samples. Insertion of an aluminum-wrapped rubber plate between the sample and the applicator is better for the soft press of the sample, as already mentioned. The insertion of liquid between IRE and sample for filling the space gap is not necessarily good, since the refractive index difference is changed and the IR bands of the liquid overlap the sample bands.

5.5.4 Diffusion Refection Spectrum When the IR beam is incident to the powder sample, the IR beam transmits through the small particles and/or it is reflected on the particle surface (Fig. 5.65). In this way, the IR beam is scattered repeatedly. As a result, the thus-obtained spectrum is close to the transmission spectrum. The relation between the absorbance A and the diffusion reflection intensity R is given as follows, which is called the Kubelka–Munk equation [15, 16]: A/S = (1 − r )2 /(2r ) = f (r )

(5.110)

where r = R(sample)/R(reference) and R is the reflectivity. The reference is a standard sample like KBr powder. S is the scattering coefficient. The logarithm of f (r) is log f (r ) = log(A) − log(S)

(5.111)

466

5 Structure Analysis by Vibrational Spectroscopy

powder

IR Fig. 5.66 Diffusion reflection spectral measurement

Therefore, the logarithm of the diffusion reflection spectrum is equal to the logarithm of absorbance A with a baseline shift log(S). The diffusion reflection spectrum is easy to measure and useful for the study of the adsorption state or reaction process of molecules on the solid particles, the identification of the extracted components of the chromatograph, etc. But, if the particle is not very fine, the normal reflection contributes too much and the spectrum may be deformed more or less. Sometimes the sample is mixed with KBr powder and ground well to get the fine powder, but the powder sample itself can be also used as it is (Fig. 5.66).

5.5.5 IR-RAS As indicated in Fig. 5.67, when the IR beam is incident to a metal surface at a high angle of about 80°, the electric field intensity on the metal becomes very strong. In particular, the p wave component is enhanced remarkably compared with the s wave. If a sample is adsorbed on the metal plate surface, the IR beam is absorbed by the sample. Since the p wave component is strong, the vibrational modes with the transition dipoles parallel to the normal to the metal surface are absorbed preferentially.

Fig. 5.67 a and b IR-RAS measurement, c Transmission spectral measurement

5.5 Various IR Spectral Measurement Methods

467

Fig. 5.68 RAS (right) and transmission spectra measured for the VDF-TrFE copolymer film prepared by a vacuum evaporation made for a a short time and b a relatively long time. The structural images derived from these data are illustrated also. Reprinted from Ref. [17] with permission of Elsevier, 1994

In other words, the molecular orientation or the functional group orientation can be investigated. This method is called the infrared reflection absorption spectroscopy (IR-RAS). It is said that this method allows us to detect the molecular layer of several tens Å thickness on the metal surface. If the IR beam can pass through the metal, the vibrational modes with the transition dipole moments parallel to the metal surface absorb the IR beam since the electric vector of the IR beam is parallel to the metal surface. In this way, the combination of the IR-RAS and transmission spectroscopy is useful to know the molecular orientation on the metal surface. For example, Fig. 5.68 shows the case of VDF-TrFE copolymer evaporated on a Si plate surface [17]. The IR-RAS spectrum and the transmission spectrum measured for the thin film are shown in Fig. 5.68a. Figure 5.68b is the case of the thick film prepared by a long time evaporation. In the RAS spectrum of the thin film, the bands of the symmetry species B1 are observed with high intensity. In the transmission spectrum of the thin film, the A1 and B2 modes are detected. The B1 mode has a transition dipole moment parallel to the chain axis, and the A1 and B2 modes have the dipole moments perpendicular to the chain axis. Therefore, the chains are considered to stand up vertically along the normal to the substrate surface. On the other hand, the thick film gives the bands of all the symmetry species in both the transmission and RAS spectra, indicating the low orientation of the chains. By annealing this thick film above the phase transition point to the high-temperature phase, the patterns change

468

5 Structure Analysis by Vibrational Spectroscopy

25oC as-cast Poling, dipoles ⊥ surface High-temperature gauche phase melt

Low-temperature phase (transzigzag chains ⊥ surface)

Fig. 5.69 Temperature-dependent RAS spectra of VDF-TrFE copolymer film measured in the poling, heating and cooling processes. Reprinted from Ref. [17] with permission of Elsevier, 1994

to those observed for the thin film. It means that the molecular chains are re-oriented preferentially in the direction normal to the surface by heating at a high temperature. The heating of the metal plate makes it possible to measure the temperature dependence of IR-RAS spectra. Figure 5.69 shows one example, where a VDF-TrFE copolymer film was prepared by a solution casting on a Si plate [17]. (i) At first, the film was subjected to a high voltage with a corona poling technique. (ii) Then the sample was heated to the temperature region of the paraelectric phase (~125 °C). (iii) Next, the sample was melted above the melting point, and (iv) cooled to room temperature. During this process, the IR-RAS spectra were measured. The poling treatment induced the preferential orientation of the CF2 dipoles along the normal to the Si plate, as known from the increment of the A1 bands at 1270 and 840 cm−1 or the vibrational bands with the transition dipoles parallel to the CF2 dipoles (or the b axis). The B1 bands (μ’ // c) are weaker oppositely. These spectral data indicate that the zigzag chains are laid on the plate with the CF2 dipoles perpendicular to the surface. At

5.5 Various IR Spectral Measurement Methods

469

125 °C, these bands characteristic of the trans-zigzag chain conformation disappeared and the spectral profile changed to that of the conformationally disordered gauche form (high-temperature phase). In the cooling process from the melt, the spectrum became at first similar to that at 125 °C in the heating process. The spectrum changed finally to that of the low-temperature phase (or the ferroelectric phase of the zigzag conformation) at room temperature. Here, the B1 bands increased in intensity, and the A1 bands became weaker. These observations indicate the vertical orientation of the zigzag chains on the surface.

5.5.6 IR Microscope The IR spectra of a tiny sample are measured using an IR microscope. The IR beam is focused on the sample by the Cassegrain mirrors. The IR beam passes through the sample to give the transmission spectra. In another way, the IR beam is reflected on the sample using an IRE prism to give the ATR spectra. The imaging measurement can be made with the 1D- or 2D-arrays of the detectors. Scanning along one line of the sample takes a long time. Figure 5.70 shows the 2D-IR imaging system with 2D array detectors. This is an example of the Varian Fast-Image-IR rapid-scanningtype FTIR microscope, which is a combination of an FTIR spectrometer and an IR

Fig. 5.70 A 2D IR microscope (Varian Fast-Image-IR rapid-scanning-type FTIR microscope, which is the combination of the FTIR spectrometer Excalibur and UMA 600 IR microscope. An IR beam enters a cat’s eye and focused on the sample position through the Cassegrain mirrors. 2D FPA detector can measure 32 × 32 IR spectra at the same time. By scanning the sample horizontally the wider range of spectra can be collected, from which the 2D image of a particular band intensity can be drawn. Reprinted from Ref. [19] with permission of the American Chemical Society, 2016

470

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.71 a Polarized IR spectra of a spherulite of it-PB-1 form II, b 2D images of the intensity distributions of the polarized bands with the perpendicular and parallel polarizations. These intensity distributions can be interpreted by assuming that the a (and b)-axis orients along the radial direction of the spherulite and the c-axis along the lateral direction as shown in (c). [Reprinted from Ref. [19] with permission of the American Chemical Society, 2016.] d This interpretation is supported by the X-ray measurements at the various positions of the same spherulite. Reprinted from Ref. [20] with permission of Springer Nature, 2019

5.5 Various IR Spectral Measurement Methods

471

microscope [18]. The 1024 IR spectra are measured in a range of 175 × 175 μm2 area using 32× 32 focal plane array (FPA) detectors at a 5. 5 μm/spectrum spatial resolution at 2 cm−1 spectral resolution. By moving the sample stage along the x- and y-axes, the 2 × 2 zone images are obtained, which are combined together to give the large image of about 350 × 350 μm2 area. The data collection time is about 5 min for 1 zone by accumulating 128-scanned spectra. The collected spectra are saved in the 3D matrix of 32 rows × 32 columns × wavenumber axis (4000–890 cm−1 ). By using a polarizer, the polarized IR spectra can be measured, from which the 2D image can be drawn by taking into account the orientation of functional groups. Figure 5.71 shows the 2D IR images obtained for a spherulite of isotactic poly(butene-1) form II [19, 20]. The polarized IR beam is incident to the spherulite as illustrated in Fig. 5.70. The IR image obtained for the 1217 cm−1 band of the parallel polarization shows the maximal intensity along the lateral direction, while that of the perpendicularly polarized 905 cm−1 band shows the maximal intensity in the direction perpendicular to that of the 1217 cm−1 band. As a result, the a (b)- and c-axes orient as shown in Fig. 5.71c. This interpretation was supported by the synchrotron microbeam X-ray scattering measurements at the various positions of the same spherulite, as shown in (d).

The spatial resolution is expressed as d = gλ/(2NA)

(5.112)

where g is a coefficient to take the effect of a tilted light incidence into account, NA = n sin(u) is an aperture size of a Cassegrain mirror (n is the refractive index of the medium between Cassegrain and sample and u is the outermost angle against the light path), and λ is a wavelength. For example, g = 1.22 and NA = 0.6, and so d = 1.22λ/(2 × 0.6) = 1.02λ. The spatial resolution is close to the wavelength of IR beam. Since IR beam is in the order of 1000 cm−1 or λ = 10 μm, d ∼ 10.2 μm. The IR microscope can distinguish the local image at this resolution. In the case of ATR method, the resolution is in an order of 1 ∼ 2 μm. The front part of the ATR IRE is round and the contacting part with the sample surface is much smaller, 1 ∼ 2 μm. In the transmission spectral measurement using an IR microscope, the sample thickness is important. The thickness problem is the same as the normal transmission IR measurement. The small size and thick size are quite different concepts to be distinguished.

472

5 Structure Analysis by Vibrational Spectroscopy

5.6 Various Methods of Raman Spectral Measurements 5.6.1 Polarized Raman Spectra Laser as a Raman excitation beam is originally polarized. The scattered Raman signals are separated into the polarized components using an analyzer (or polarizer). It must be noticed at first that the data treatment of the polarized Raman spectra is different in principle from that of the polarized IR bands. The latter is related to the direction of the transition dipoles. The Raman spectra are related to the polarizability tensors. P  = α E

(5.113)

where P’ is the polarization of the Raman bands, E is the electric vector of the incident laser light, and α’ is the transition polarizability related to the Raman spectra. ⎞ ⎛    ⎞⎛ ⎞ αx x αx y αx z Px Ex ⎟ ⎝ Py ⎠ = ⎜ ⎝ αyx αyy αyz ⎠⎝ E y ⎠ Ez Pz αzx αzy αzz ⎛

(5.114)

In principle, α’ is the symmetric second-rank tensor, and αi j = αji . The x, y, and z are based on the Cartesian coordinate system.

5.6.2 Liquid Sample The molecules in a liquid rotate freely and the α’ij is averaged with respect to the orientational angles of the molecule [3, 4]. The polarizability tensor based on the molecular coordinate x and that based on the experimental coordinate system X has the following relation (Fig. 5.72): X = Tx The electric vectors Ex and EX and the polarizations P’x and P’X are related, respectively, as follows: Fig. 5.72 Polarized Raman spectral measurement of a liquid sample

5.6 Various Methods of Raman Spectral Measurements

EX = T Ex

and

P X = T P x

473

(5.115)

Since P X and EX are related by α XX , and P x and Ex are by α xx , we have the following relation: P X = T P x = T α x x E x = T α x x T t E X = α X X E X α X X = T α x x T t

(5.116)

where “t” indicates a transpose matrix. As well known, the second-rank tensors can be expressed as the ellipsoidal body, which is diagonalized to the principal axes. ⎞ αx x 0 0 = ⎝ 0 αyy 0 ⎠ 0 0 αzz ⎛

α x x

(5.117)

In this case, the transformation becomes simpler α X X = αx x TX2 x + αyy TX2y + αzz TX2z αY Y = αx x TY2x + αyy TY2y + αzz TY2z α Z Z = αx x TZ2x + αyy TZ2y + αzz TZ2z α X Y = αx x TX x TY x + αyy TX y TY y + αzz TX z TY z ..........

(5.118)

The invariant quantity is   α = αx x + αyy + αzz /3

(5.119)

The anisotropic term is γ 2 =

 2  2  2   αx x − αyy + αyy − αzz + αzz − αx x + 2 αx2y + αyz2 + αzx2 /2 (5.120)

For the freely rotating molecules, the averaged values are given as below: < αx x >=< αyy >=< αzz >=

  1/2 45α 2 + 4γ  2 /45

< αx y >=< αyz >=< αzx >= γ /(15)1/2

(5.121)

The 90° Raman scattering intensity I i of the normal mode Qi is      2 Ii ∝ gi (νo − νi )4 45α2 i + 4γi / 45 1 − exp(−hνi /k B T )

(5.122)

474

(a)

5 Structure Analysis by Vibrational Spectroscopy

(b)

Fig. 5.73 a Polarized Raman spectra of CCl4 liquid. b The zoomed-up band of the totally symmetric C–Cl stretching mode

νo and νi are the vibrational frequencies of the incident laser and the scattered Raman signal, respectively, and gi is the degeneracy of the i mode. The depolariazion ratio ρ is defined as        ρ = Ix y /Ix x = I⊥ /I// = γ2 /15 / 45α2 + 4γ2 /45 = 3γ2 / 45α2 + 4γ2 (5.123) For CCl4 molecule with the point group symmetry T d , the totally symmetric possesses the vibration (the C–Cl stretching mode of the A1 symmetry species)   transition polarizability tensor components of αx x + αyy + αzz = 3α and γ’ = 0. Therefore, ρ = 0. In fact, the observed Raman peak at 460 cm−1 shows this situation clearly (Fig. 5.73). For the asymmetric vibrational modes or the modes belonging to the E or T symmetry species, α’xx and α’xy ,… are contained and so 0 < ρ < 0.75. In the case of general molecules, the totally symmetric vibrational modes show 0 ≤ ρ < 0.75, and for the other modes, ρ = 0.75 because of α’= 0. It is noted that the totally symmetric stretching band of CCl4 molecules consists of the five components, which corresponds to the modes of such different isotopic molecules as C(35 Cl)4 , C(35 Cl)3 (37 Cl), and so on, as shown in Fig. 5.73b.

5.6.3 Oriented Solid Sample Polarized Raman spectra are obtained by setting the directions of the electric field vectors of the incident and scattered lights. As shown in Fig. 5.74, the polarized laser with electric field vector EX is incident along the Z axis and the scattered light with electric field vector EZ is collected by the detector set along the Y-direction. In this

5.6 Various Methods of Raman Spectral Measurements

475

-Z(XZ)Y incident direction

scattered direction

electric field

electric field

X EZ Y

EX Z Fig. 5.74 Polarized Raman spectral measurement of a solid sample

case, we express the scattering geometry as follows (Porto’s symbol). As a result, the Raman signal of the transient polarizability component α’XZ can be detected.

5.6.3.1

Uniaxially Oriented Sample

Depending on the orientation of a crystal, the Raman signals obtained are different. Table 5.3 shows the case of a uniaxially oriented sample. As an example, the uniaxially oriented PE sample was used for the measurement of the polarized Raman spectra. The measurement geometry is shown in this Table 5.3. The observed Raman bands are different in the relative intensity depending on the scattering geometry, as seen in Fig. 5.75. For example, the 1132 cm−1 band is strong in intensity for the ZZ-component and weak for the ZX-component, being able to be assigned to the A1g symmetry species (the more detailed description of the symmetry species will be made in a later section). On the other hand, the 1062 cm−1 band appears strongly for the ZX and weak for ZZ and XX, being the band of the symmetry species B3g or B2g . For the band components related to the axes perpendicular to the orientation direction, the measurement in the XY geometry contains the components of aa, bb, and ab. Unfortunately, the distinction between the components related to the a- and b-axes cannot be made as long as the uniaxially oriented sample is used.

5.6.3.2

3D-Oriented Sample

The separation of these bands can be made by measuring the polarized Raman spectra of the 3D oriented sample. The 3D-oriented sample is prepared by rolling or by a plain strain compression method [21]. In many cases, these methods give the twinned structure, or the crystallites orient into the two different directions with the common

476

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.75 (a) Raman spectra of an unoriented PE sample, and (b) the polarized Raman spectra of a uniaxially-oriented PE sample. νs , νas , δ, w, t, and r: symmetric stretching, anti-symmetric stretching, bending, wagging, twisting, and rocking modes of CH2 units, νs (CC) and νas (CC): symmetric and anti-symmetric CC stretching modes

5.6 Various Methods of Raman Spectral Measurements

477

Table 5.3 Polarized Raman spectral components of uniaxially oriented sample

ij

(a)

(b)

(c)

Equatorial profile

38o

(200)

c b a edge

207 Å

(020)

c = 2.51Å

(110)

c

through triclinic

through

b (100)T

(200)

(110)

b a

(010)T

edge

a* through

b

b’

a = 7.47Å

a’

edge end

a

b = 4.97Å

edge

c

a

end

Fig. 5.76 3D-oriented polyethylene sample prepared by the plane strain compression method. a Xray diffraction patterns measured along the three mutually perpendicular directons, from which the orientation of the orthorhombic unit cell was estimated as illustrated in (b). The SAXD data revealed the orientation of stacked lamellae. Small amount of triclinic form is included in the sample as indicated by an arrow. c The crystal structure of orthorhombic polyethylene

478

5 Structure Analysis by Vibrational Spectroscopy

boundary. The plain strain compression method was applied to the high-density polyethylene (HDPE) sample. The sample was melted and then compressed into the metal block. The sample was extruded from one exit to give the oriented sample. The X-ray diffraction patterns were measured using the X-ray beam incident into the 3 mutually perpendicular directions (through, end and edge). In particular, the end pattern tells us the direction of the a- and b-axes in the cross section of the sample. As illustrated in Fig. 5.76, the a-axis of the orthorhombic PE crystals is oriented along the through plane. The triclinic crystals are also contained a little bit as known from the weak 100 reflection peak, which can be neglected in a good approximation. The Raman spectral measurement was made in the back scattering mode. In Fig. 5.77, the electric field vectors are indicated in the individual cases and the actually obtained polarized Raman spectra of the 3D-oriented HDPE sample are given. The band assignments are given in Table 5.13. Here the correlation splitting occurs for the bands in principle. Then, the observed Raman band is needed to be assigned to the two species (for example, Ag and B3g ). The A1g band at 1132 cm–1 [CC symmetric stretching] appears strongly for the (cc), (aa) and (bb) modes. The B1g band at 1294 cm–1 is observed in the spectra of (bc) component [t(CH2 ) mode]. It must be noted that the spectra of (aa), (bb) and (ab) polarizability and those of (bc) and (ac) are relatively similar to each other. One reason may come from the

y(b) a(cc)a

a(cb)a

a(bb)a

x(a) through

z(c) b(ca)b

b(aa)b

edge end

c(ab)c Fig. 5.77 a Geometrical settings of the 3D-oriented polyethylene sample for the polarized Raman spectral measurement. b Polarized Raman spectra of the 3D-oriented polyethylene sample in the frequency region of 1500 – 1000 cm–1 . c Polarized Raman spectra of the 3D-oriented polyethylene sample in the frequency region of 2750 – 3000 cm–1

5.6 Various Methods of Raman Spectral Measurements

Fig. 5.77 (continued)

479

480

Fig. 5.77 (continued)

5 Structure Analysis by Vibrational Spectroscopy

5.6 Various Methods of Raman Spectral Measurements

481

relatively low degree of orientation, which is not enough for the perfect separation between the ac and bc polarizations, and also between the aa and ab. Another factor might be due to the weak anisotropic property of the polarizability tensor in the ab plane of the PE crystal.

5.6.4 Resonance Raman Spectra As mentioned already, the transition polarizability is expressed by the KHD equation shown below. In this case, the transition occurs from the level n to n via many intermediate states m. 

    n|μs |m m|μi |n  n|μ |m |n m|μ i s + αs  (ei , es ) = (5.124) [h(νm − νn − νo )] [h(νm − νn  + νo )] m If the frequency νo of the incident laser beam is close to the difference of the energy levels vm −vn , the first term becomes infinitely large. As already mentioned, this is the resonance Raman phenomenon. The Raman scattering intensity is increased by about 106 times from the normal Raman scattering intensity (in the actual case, the finite lifetime (Γ ) of the states m causes the breadth of the transition, which is introduced to the above equation in the form (vm − vn ± vo + iΓ ) instead of (vm − vn ± vo ). In the resonance condition, the interaction between the incident laser light and the electron excited state is important. Another important interaction occurs between the two electronically excited states through the coupling with the normal mode vibration (vibration-electron state interactions). The resonance Raman phenomenon occurs for the normal modes which are strongly related to the electronically conjugated structures. Figure 5.78 is the UV–visible absorption spectrum of polydiacetylene compound, 1, 4-di (2, 3, 5, 6-tetrafluoro-4-butylphenyl) diacetylene (FDAC) [22]. This polymer is obtained by the photo-induced solid-state polymerization reaction of the monomer single crystal. The absorption maximum is detected at around 550– 600 nm wavelength. The band at 2200 cm−1 corresponds to the C≡C stretching mode. As well known, the C≡C bond strength is weaker with an increase of the electronic conjugation length. This phenomenon should be detected as a lower frequency shift of the C≡C stretching band (This can be known by solving the Schrödinger equation. The concrete exercise will be given in Chap. 6). But, in the actual polymerization reaction, the peak position was observed to shift rather toward the higher frequency side. In the resonance Raman spectral measurement using a laser light of a fixed frequency, the Raman band corresponding to a particular conjugation length is detected mainly. Then, the observed peak position shift may be interpreted in the following way: the polydiacetylene species produced in the solid matrix of the monomer molecules is highly tensioned in the earlier stage of the reaction by the external stress coming from the surrounding monomer molecules. As the polymer

482

5 Structure Analysis by Vibrational Spectroscopy

F

F

F

F

F

F

C C C C F

F

Fig. 5.78 UV–Vis spectra and the resonance Raman spectra of electronically conjugated polydiacetylene compound, FDAC. The UV–Vis spectrum changed with the increase of irradiation time. The ν(C≡C) band shifted toward the higher frequency side as the polymerization reaction proceeded under the irradiation of UV light.

content is increased, this type of tension is relaxed and so the peak position is shifted to the higher frequency side. Refer to Sect. 8.2.10. As already mentioned, the resonance Raman spectra are enhanced by 106 times the normal Raman spectra. In other words, a small amount of a colored substance within a polymer sample causes strong Raman bands. For example, the thermal anneal of vinyl polymer such as poly(vinyl chloride), poly(vinylidene fluoride), etc. gives the bands at around 1600–1500 cm−1 region, which is considered to come from the conjugated polyacetylene sequence generated by the thermally induced chemical reaction [23]. 

−(CH2 CHCl)n− −→ −CH2 − CH = CH − CH = CH − CH = CH − .. − CH = CH − CH = CH− . . . CH2 CHCl−

5.6 Various Methods of Raman Spectral Measurements

metal surface

483

molecules surface plasmon

air Evanescent light

Raman

laser

prism Fig. 5.79 The Evanescent light generated on the prism surface is enhanced by a metal plate to produce the surface plasmon, which causes the strong enhancement of Raman scattering of the molecules (blue ball) adsorbed on the metal surface

Fig. 5.80 a Electrode cell for the SERS measurement. b SERS spectra of 4 vinyl pyridine on the silver electrode surface. Reprinted from Ref. [24] with permission of the American Chemical Society, 1990

5.6.5 Raman Spectra of Surface (SERS) The normal Raman spectral measurement of a thin film is difficult because of the weak scattering power from not enough thickness. The Raman spectra of a thin film are measured by utilizing the principle of the surface plasmon resonance. Usually, the

484

5 Structure Analysis by Vibrational Spectroscopy

gap between glass and air is used. When the laser light is incident into the glass, the total reflection occurs on the glass surface contacting with air because the refractive index relation is n(glass) > n(air). Some portions of the electric field penetrate into the air zone (Evanescent wave). If a thin metal exists there, the Evanescent wave resonates with the metal surface and the electrons of the metal are forced to vibrate, which propagates along the metal surface. This is called the surface plasmon. As illustrated in Fig. 5.79, if organic molecules are adsorbed on the metal surface, the enhanced electric field causes strong Raman signals. The most typical metal is Au and Ag. Instead of air, the liquid sample is filled in the gap for the measurement of the Raman spectra of the liquid molecules (Fig. 5.79). SERS A silver electrode system is used, for example, and a laser beam is incident on the silver surface. Some voltage is applied to the electrode. Pyridine molecules adsorbed on the silver surface scatter the Raman signals quite strongly. This phenomenon is Fig. 5.81 Raman spectra of 4VP on the Ag sol surface. Reprinted from Ref. [24] with permission of the American Chemical Society, 1990

5.6 Various Methods of Raman Spectral Measurements Fig. 5.82 Time dependence of SERS pattern of 4VP adsorbed on the silver electrode. Reprinted from Ref. [24] with permission of the American Chemical Society, 1990

Fig. 5.83 Schematic illustration of the polymerization reaction of 4VP on the silver electrode under the application of electric voltage. Reprinted from Ref. [24] with permission of the American Chemical Society, 1990

485

486

5 Structure Analysis by Vibrational Spectroscopy

called the surface-enhanced Raman scattering (SERS). The principle of SERS is said to come from the surface plasmon resonance, as mentioned above. By repeating the oxidation and reduction of the silver surface by the application of the alternate electric voltage, the nanoscale particles of silver are created on the electrode surface, which is useful for the amplification of SERS intensity. For the same reason, the silver sols are used instead of the silver electrode system. The silver nitrate is reduced by hydroxylamine or BH4 to give the silver nanoparticle of 10–30 nm. (The silver surface is covered with the negatively charged phosphate.) The point to get the strong Raman signal is to prepare the stable silver colloid. As an example of the application of SERS method, a polymerization reaction of 4vinyl pyridine on the silver electrode surface is shown here [24]. The electrochemical cell is shown in Fig. 5.80a. 4-Vinyl pyridine is dissolved in water. Raman spectrum of 4VP monomer is shown there also. The SERS spectrum in the early time is similar to that of the solution. The spectrum changed to that of poly(4-vinylpyridine) after the passage of several hours. On the other hand, the 4VP polymerizes on the silver electrode surface. The 4VP on the silver collide surface did not polymerize even after the long time passage (Fig. 5.81). Figure 5.82 shows the time dependence of the SERS spectra under the application of the various electric voltages. As the voltage was higher the spectral change became faster. The C=C stretching band of the vinyl group at 1627 cm−1 decreased in intensity and the polymer bands increased the intensity instead. From these experimental data, the polymerization mechanism on the silver surface is illustrated in Fig. 5.83. The 4VP molecules on the electrode surface may be attacked at first by a proton radical (H). The thus-prepared 4VP monomer radical attacks the neighboring 4VP monomers stepwise to create the poly(4-vinyl pyridine) chains on the surface.

5.6.6 Raman Microscope A laser beam is narrowed by a lens to a size of 1μm radius. The back-scattered Raman component is collected with the same lens and detected using a CCD detector (Fig. 5.84). If the sample is scanned at a constant step along the x- and y-directions, the plot of a particular band intensity gives the 2D map. However, the stepwise scan takes a long time. The 1D array of laser irradiation can collect the Raman spectra on line, accelerating the collection speed [25]. The line-focused laser beam is generated using a cylinder lens and irradiated on the sample, from which a set of Raman spectra is obtained online. Some attention must be paid to the polarized Raman measurement. Since the direction of the laser beam is bent by the objective lens, the electric vector direction is also bent. This causes the mixing of the different polarization components of the band. The laser brilliance is increased by focusing. The sample is more easily damaged by such a powerful laser beam. For example, the Raman measurement of the colored sample might result in different spectra from those without any damage.

5.7 Normal Modes and Symmetry (Factor Group Analysis)

487

mirror

monochromator

laser beam splitter objective lens sample Fig. 5.84 Schematic illustration of a microscopic Raman spectrometer

5.7 Normal Modes and Symmetry (Factor Group Analysis) So far, we have learned the methods on how to measure the IR and Raman spectra of polymer substances. But, we need to know how to analyze the spectral data and how to interpret the analyzed results from the structural point of view.

5.7.1 Factor Group Analysis of Molecules For example, Fig. 5.85 shows the normal modes of a water molecule. The total degree of freedom of a water molecule is 3 atoms× 3 (x, y, z) = 9. The three translational motions occur along the x-, y- and z-axes: T x , T y , and T z, respectively. The rotation of the whole molecule occurs around the x-, y- and z-axes: Rx , Ry, and Rz, respectively. As a result, the number of the normal vibrational modes is 9 – 3T – 3R = 3, which correspond to the symmetric stretching of OH bonds [νs (OH)], the anti-symmetric stretching of OH bonds [νas (OH)], and the HOH angle bending mode [δ(HOH)]. Fig. 5.85 Various modes of a water molecule

Tz

Tx

+ +

Ry

+ -

νs (OH)

νas(OH)

+ Rz

Rx -

z

Ty

+ δ (HOH)

y

x

488

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.86 Symmetry relation in the anti-symmetric OH stretching mode of water molecule

The IR activity is related to the non-zero transition dipole moment. The symmetry treatment of the dipole moment is the same as that of the translational mode [26]. For example, let us see the symmetry of the normal mode νas (OH). As shown in Fig. 5.86, arrow #1 is moved by symmetry operation Cˆ 2 to arrow #1 . By multiplying −1 to the #1 arrow, we have arrow #2. In other words, the phase relation between arrows #1 and 2 is π or −1[= exp(iπ)] with respect to the symmetry operation Cˆ 2 . (#2) = −Cˆ 2 (#1) = exp(iπ)Cˆ 2 (#1) Similarly, the (yz) mirror plane σˆ 1 moves arrow #1 to the opposite arrow #1 , which must be inversed by multiplying −1 to get arrow #2. The phase relation is π. (#2) = −σˆ 1 (#1) = exp(iπ)σˆ 1 (#1) The (xz) mirror plane σˆ 2 keeps arrows #1 and #2 to the original ones, and the phase angle is 0 or the coefficient value is +1. (#1) = 1 × σˆ 2 (#1) = exp(0)σˆ 2 (#1) Summarizing the phase relation for the νas (OH) mode, we have σ1 σ2 E C2 +1 −1 −1 +1 exp(0) exp(iπ) exp(iπ) exp(0) By referring to the character table of C 2v (Table 5.4), these phase relations are Table 5.4 Character table of the point group C 2v C 2v

E

C2

σ1

σ2

Vibrational mode

T (IR)

α (Raman)

Rotation

A1

+1

+1

+1

+1

νs (OH), δ

z

xx, yy, zz

–

B1

+1

−1

– 1

+1

νas (OH)

x

xz

Ry

A2

+1

+1

−1

−1

–

–

xy

Rz

B2

+1

−1

+1

−1

–

y

yz

Rx

5.7 Normal Modes and Symmetry (Factor Group Analysis)

489

found to correspond to the symmetry species B1 . We can say that the νas (OH) vibrational mode belongs to the symmetry species B1 . The total vector produced by summing the vectors #1 and #2 directs to the x-axis; T  x . The symmetry operation applied to T  x tells us that it shows the same phase relation as that of νas (OH) mode. The symbol x in the character table is related to the symmetry species B1 . In other words, by looking at the column of the x, y, or z, we can predict the IR active symmetry species. In a similar consideration, the Raman polarizability tensor component xz belongs to the B1 species. The symmetry species of the second-rank tensor can be predicted by the direct product (⊗) of the two species. For example, the x belongs to the B1 and the z belongs to the A1 , then B1 ⊗ A1 gives the characters as below.

A1 B1 A1 ⊗ B1

E +1 +1 +1

C2 +1 −1 −1

σ1 +1 −1 −1

σ2 +1 z +1 x +1 x z

As known from the characters (+1, −1, −1,+ 1), the tensor component xz belongs to the symmetry species B1 . By applying these considerations to all the normal modes, we have the total result shown in Table 5.4. The rotational mode Ry belongs to the B1 species as understood from the phase relation between the two arrows with respect to the various symmetry operations. (The rotation axis y is parallel to the outer product of x and z.) The symmetry species of the overtone mode and combination mode can be predicted also using the direct product. The overtones belong always to the totally symmetric symmetry species A1 because A1 ⊗ A1 = B1 ⊗ B1 = A2 ⊗ A2 = B2 ⊗ B2 = A1 . The combination mode is given as B1 ⊗ B2 = A2 , for example. When we try to assign the combination bands in the near-IR region, not only the peak position, which might be approximately the sum of the two normal mode frequencies (3340 cm–1 = 1740 + 1600 cm–1 ), but also the polarization must be checked carefully. Number of normal modes The number of the degree of freedom belonging to the symmetry species  can be predicted using the following equation: n  = (1/N ) j cj n j t j

(5.125)

where cj  is the character of the symmetry element Rj of the symmetry species , nj is the number of atoms that do not move even by the symmetry operation Rj , and t j is the trace of the symmetry operation matrix Rj . For example, C 2 and σ 1 matrices are expressed as

490

5 Structure Analysis by Vibrational Spectroscopy

⎛

−1 C2 = ⎝ 0 0

0 −1 0

⎛ ⎞ 0 −1 0 ⎠ σ1 = ⎝ 0 1 0

0 1 0

⎞ 0 0⎠ 1

(5.126)

The traces of these matrices are −1 (= summation of the diagonal components, “−1”+ “–1”+ “1”) and 1, respectively. N is the total number of symmetry elements. Let us use Eq. 5.125 for the water molecule with the point group C 2v . N= 4. The operation C 2 exchanges the hydrogen atoms. The O atom does not move. So, n= 1. The operation E does not move all the atoms and so n= 3. Applying Eq. 5.125 gives the following calculation: n A1 = [9 × 1 + (−1) × 1 + 1 × 1 + 1 × 3]/4 = 12/4 = 3 Subtraction of the translation and rotation freedoms from the total number n , the number of the vibrational modes is given as below. Since T z belongs to the A1 species, the number of the vibrational modes is n A1 – T z = 3 – 1= 2 (Table 5.5). Finally, we obtain vibration = 2 A1 + B1

(5.127)

The local symmetry coordinates (S), which express the normal mode images using a combination of internal displacement coordinates, are created by the projection operator. ˆ S  = (1/N ) j cj RR j

(5.128)

ˆ is the symmetry operation and Rj is the j-th internal displacement coordiwhere R nate. For example, for the OH stretching coordinate r 1 , the operation Rˆ = Eˆ Table 5.5 Factor group analysis of H2 O molecule C2

σ1

σ2

nΓ

C 2v

E

T

R

A1

+1

+1

+1

+1

3

z

-

2

B1

+1

–1

–1

+1

3

x

Ry

1

A2

+1

+1

–1

–1

1

Rz

0

B2

+1

–1

+1

–1

2

Rx

0

n

3

1

1

3

t

3

–1

1

1

n×t

9

–1

1

3

y

Vibration

5.7 Normal Modes and Symmetry (Factor Group Analysis)

491

transforms r 1 to r 1 and Rˆ = Cˆ 2 transforms r 1 to r 2 and so on. Using Eq. 5.128,

  ˆ 1 + 1 × Cˆ 2 r1 + 1 × σˆ 1 r1 + 1 × σˆ 2 r1 S1A1 = (1/4) 1 × Er = (1/4)[r1 + r2 + r2 + r1 ] = (r1 + r2 )/2, which is normalized to S1A1

= (r1 + r2 )/21/2

(5.129)

Similarly,   ˆ + 1 × Cˆ 2 φ + 1 × σˆ 1 φ + 1 × σˆ 2 φ S2A1 = (1/4) 1 × Eφ = (1/4)[φ + φ + φ + φ] = φ

(5.130)

  ˆ 1 − 1 × Cˆ 2 r1 − 1 × σˆ 1 r1 + 1 × σˆ 2 r1 S3B1 = (1/4) 1 × Er = (1/4)[r1 − r2 − r2 + r1 ] = (r1 − r2 )/2, which is normalized to S3B1 = (r1 − r2 )/21/2

(5.131)

The S 1 and S 3 correspond to the symmetric and anti-symmetric OH stretching modes, respectively, and S 2 is the bond angle bending mode. In this way, a set of S 1 , S 2 and S 3 is separated (or reduced) into the two groups, A1 and B1 . In the actual IR spectrum of water, we usually call the observed bands using only one local symmetry coordinate such as the symmetric stretching mode and so on. However, the actual bands do not correspond to these local modes simply. We have to express the vibrational modes using the combination of these local symmetry coordinates or the normal coordinates. The actually calculated normal modes are expressed as a linear combination of the local modes belonging to the same symmetry species. Q 1 (A1 ) = 0.9S1 + 0.1S2 ,

Q 2 (B1 ) = 0.1S1 + 0.9S2 , and Q 3 (A1 ) = S3 (5.132)

The coefficients indicate the degrees of the contribution of the local symmetry coordinates. Therefore, we have to express in the following way: the band observed at 3800 cm−1 is assigned to the normal mode Q1 , which originates mainly from the symmetric OH stretching mode coupled slightly with the HOH bending mode. It must be emphasized again that such a coupling among the various local symmetry coordinates occurs among the coordinates belonging to the same symmetry species.

492

5 Structure Analysis by Vibrational Spectroscopy

As mentioned above, the S 1 and S 2 belong to the A1 symmetry species. The Q3 is the pure anti-symmetric stretching mode since only S 3 belongs to the symmetry species B1 and no coupling occurs between the local coordinates belonging to the same symmetry species B1 . Normal Modes (CO3 2− )

(2)

Another example is CO3 2− ion. The point group symmetry is D3h . The character table is shown in Table 5.6. The phase angles 0 and π give, respectively, the coefficient 1(= exp(i0)) and − 1(= exp(iπ)). The rotation around the C 3 -axis is related by the phase angle 120°= 2π/3; ε = exp(i2π/3) = cos(2π/3) + i sin(2π/3). As seen in the character table of the point group C 3 , the character of the E symmetry species is 2-dimensional and the vibrational modes of the phase angle 2π/3 are doubly degenerated. The coefficients are usually given as 1 + 1 = 2 and ε + ε∗ = 2 cos(2π/3) = −1 as seen below. The trace of Cˆ 3 matrix is 1 + ε + ε∗ = 0. Species

E

C3

C32

A

1

1

1

E

1

ε

ε*

1

ε*

ε

E

C3

C32

A

1

1

1

E

2

−1

−1

The number n is calculated using Eq. (5.125) as follows: 

n A1 = (1/12)[12 × 1 + 2 × 0 × 1 + 3 × (−2) × 1 + 4 × 1 + 2 × (−2) × 1 + 3 × 2 × 1] = 1 

n A2 = (1/12)[12 × 1 + 2 × 0 × 1 + 3 × (−2) × (−1) + 4 × 1 + 2 × (−2) × 1 + 3 × 2 × (−1)] = 1 

n E = (1/12)[12 × 2 + 2 × 0 × (−1) + 3 × (−2) × (0) + 4 × 2 + 2 × (−2) × (−1) + 3 × 2 × (0)] = 3

Table 5.6 Character table for CO3 2− ion D3h

E

2C 3

3C 2

σh 2S 3

3σv

nΓ

T, R

A1 ’

+1

+1

+1

+1

+1

+1

1

A2’

+1

+1

−1

+1

+1

−1

1

E’

+2

−1

0

+2

−1

0

3

A1 ”

+1

+1

+1

−1

−1

−1

0

A2 ”

+1

+1

−1

−1

−1

+1

2

z

E”

1

(Rx , Ry )

+2

−1

0

−2

1

0

n

4

1

2

4

1

2

Trace

3

0

−1

1

−2

1

12

0

−2

4

−2

2

n x trace

α’

nvibration

x 2 + y2

1

Rx

0

(x, y)

(x 2 -y2 ,

xy)

2 0 1

(xz, yz)

0

5.7 Normal Modes and Symmetry (Factor Group Analysis)

493



n A1 = (1/12)[12 × 1 + 2 × 0 × 1 + 3 × (−2) × 1 + 4 × (−1) + 2 × (−2) × (−1) + 3 × 2 × (−1)] = 0 

n A2 = (1/12)[12 × 1 + 2 × 0 × 1 + 3 × (−2) × (−1) + 4 × (−1) + 2 × (−2) × (−1) + 3 × 2 × 1] = 2 

n E = (1/12)[12 × 2 + 2 × 0 × (−1) + 3 × (−2) × (0) + 4 × (−2) + 2 × (−2) × (1) + 3 × 2 × (0)] = 1

The final result is Γ = A1 + A2 + 3E  + 2 A2 + E 

(5.133)

Since the E mode is doubly regenerated, the total number of the degree of freedom is 12, which is equal to the total degree of freedom = 3× 4 atoms. The translational mode is also degenerated and expressed as (T x , T y ) or (x, y) in Table 5.6. The pure rotational mode is also in the same way. The subtraction of these translational (A2 ” and E’) and rotational (A2 ’ and E”) modes results in Γvibration = A1 + 2E  + A2

(5.134)

As seen in the columns of T and α’ in Table 5.6, the A1 ’ mode is Raman active, A2 ” is IR active and E’ are active for both Raman and IR. The local coordinates are created now. Starting from r 1 (CO bond stretching), Eq. 5.128 gives   A’ ˆ 1 + 2 × 1 × Cˆ 3 r1 + 1 × 3 × Cˆ 2 r1 + 1 × σˆ h r1 + 2 × 1 × Sˆ 3 r1 + 3 × 1 × σˆ v r1 S1 1 = (1/12) 1 × Er   = (1/12) (r1 ) + (r2 + r3 ) + (r1 + r3 + r2 ) + (r1 ) + (r2 + r3 ) + (r1 + r3 + r2 ) = (4r1 + 4r2 + 4r3 )/12 = (r1 + r2 + r3 )/3

(5.135)

This mode corresponds to the totally symmetric CO bond stretching mode (see Fig. 5.87).

Fig. 5.87 Normal modes of CO3 2− ion

494

5 Structure Analysis by Vibrational Spectroscopy

   ˆ 1 + 2 × (−1) × Cˆ 3 r1 + 3 × 0 × Cˆ 2 r1 + 2 × σˆ h r1 + 2 × (−1) × Sˆ 3 r1 + 3 × 0 × σˆ v r1 S2E = (1/12) 2 × Er = (1/12)[(2r1 ) − (r2 + r3 ) + (2r1 ) − (r2 + r3 )] = (2r1 − r2 − r3 )/6

(5.136)

For r 2 , ’

S3E = (2r2 − r1 − r3 )/6

(5.137)

The form is the same as that of S 2 E’ (Eq. 5.136). This mode is doubly degenerated. However, we have to be careful of one point. The application of Eq. 5.128 can be made also to r 3 . The S 4 E’ generated for r 3 is ’

S4E = (2r3 − r1 − r2 )/6 which is not independent of the other two coordinates, S 2 E’ and S 3 E’ , because S 4 E’ = −(S 1 E’ + S 2 E’ )/2. The bond angle deformation φ1 is also used for the creation of the local coordinates. A’

S5 1 = (φ1 + φ1 + φ3 )/3

(5.138)

This coordinate is always 0 since the total sum of these bond angles is constant, 2π: φ1 + φ1 + φ3 = 2π

or

φ1 + φ1 + φ3 = 0

(5.139)

Such a coordinate is named the redundancy. For the E’ modes, ’

S6E = (2φ1 − φ2 − φ3 )/6 ’ S7E = (2φ2 − φ1 − φ3 )/6 ∗

(5.140)

These two coordinates are doubly degenerated. S8E = (2φ3 − φ1 − φ2 )/6 is expressed using S 6 E’ and S 7 E’ . As another vibrational modes we have to consider the atomic displacements along the z axis (the normal to the molecular plane) or the out-of-plane modes. The details are skipped here. One mode belongs to the A2 ” symmetry species, IR-active (see Fig. 5.87). The other types of the out-of-plane modes are related to the pure rotational motions. As already mentioned, the normal coordinates are not equal to these local symmetry coordinates. The local coordinates belonging to the same symmetry species are coupled together more or less. All the normal coordinates of CO3 2− ion are shown in Fig. 5.87. The normal coordinate of the species A1  is equal to the local symmetry coordinate S 1 or the totally symmetric stretching mode and no coupling occurs with the other coordinates. The normal mode of the species A2  (outof-plane mode) is also pure and not coupled with the others. The local symmetry coordinates belonging to the E  species are coupled together at the various ratios to

5.7 Normal Modes and Symmetry (Factor Group Analysis)

495

give the normal coordinates. The degree of contribution of these local coordinates is calculated with the normal coordinate calculation.

5.7.2 Factor Group Analysis of 1D Crystal Lattice In general, a crystal contains atoms of Avogadro number. Since the 3 degrees of freedom is assigned to one atom, we can predict the infinitely large number of freedom for the crystal, indicating the IR and Raman spectra might be tremendously complicated. This is not true. The principle for the vibrational spectroscopic activity, as mentioned before, can be applied to the crystal lattice. As already learned, the transition dipole moment should not be zero for the IR active modes. The same can be said also for the Raman activity (the transition polarizability). If the phase relation of the vibrational modes occurring in the unit cells is random, the total sum of the transition dipole moments should be zero. Only the modes with the phase angle 0 between the neighboring unit cells keep the transition dipole moment in total and can be observed in the IR and Raman spectra. The analysis of the vibrational modes on the basis of this concept is called the factor group analysis. In this analysis, the translational operation between the neighboring unit cells is ignored and only the symmetries within the unit cell are taken into account because the phase relationship among the many unit cells is the same with each other. We can predict the IR and Raman active modes by using the factor group analysis. Let us see one example of 1D polyethylene chain of the planar-zigzag conformation [27, 28]. Refer to the description in Chap. 1 about the 1D space group of this polymer. It must be noted that the unit cell contains 2 CH2 units, both of which possess the C 2 (x)-axis as seen in Fig. 5.88. We have to count these two axes in the calculation of the normal modes. Different from the point group symmetry, the symmetry elements do not need to pass through the center of gravity of the unit cell. Now let us calculate

Fig. 5.88 Symmetry elements in the planar-zigzag polyethylene chain

496

5 Structure Analysis by Vibrational Spectroscopy

Table 5.7 Character table of D2h and the factor group analysis of an isolated PE chain σ(xy)

D2h

E

C2 s (z)

C 2 (y)

C 2 (x)

i

Ag

1

1

1

1

1

1

B1g

1

−1

1

−1

1

B2g

1

−1

−1

1

1

B3g

1

1

−1

−1

Au

1

1

1

B1u

1

−1

1

B2u

1

−1

−1

B3u

1

1

n

6

0

Trace

3 18

n × trace

σ(xz)

σg (yz)

n

nvib

1

1

T

R

3

3

−1

1

−1

2

2

−1

−1

1

1

1

1

1

−1

−1

3

2

1

−1

−1

−1

−1

1

1

−1

−1

1

−1

1

Ty

3

2

1

−1

1

1

−1

Tx

3

2

−1

−1

−1

−1

1

1

Tz

2

1

0

2

0

6

2

0

−1

−1

−1

−3

1

1

1

0

0

−2

0

6

2

0

Rz

the number of the IR and Raman active modes. The factor group is D2h as already studied in the X-ray section. The character table is shown below (Table 5.7). For the total degree of freedom, n  = 3Ag + 2B1g + B2g + 3B3g + Au + 3B1u + 3B2u + 2B3u

(5.141)

For the vibrational modes, n vibration = 3Ag + 2B1g + B2g + 2B3g + Au + 2B1u + 2B2u + B3u

(5.142)

The 4 gerade species (Ag , B1g , B2g , B3g ) are Raman active and the 3 ungerade species (B1u , B2u , B3u ) are IR active. The local symmetry coordinates are created in a similar way to the method mentioned before. For example, the CH stretching mode r 1 is used to get the local symmetry coordinate S B2u .  s ˆ 1 + (−1) × Cˆ 2 z r1 + (−1) × Cˆ 2y r1 + (1) × Cˆ 2x r1 S B2u = (1) × Er  + (−1) × ˆir1 + 1 × σx y r1 + 1 × σx z r1 + (−1) × σgyz r1 /8 = (1/8)[r1 − r4 − r3 + r2 − r4 + r1 + r2 − r3 ] = [(r1 + r2 )/2 − (r3 + r4 )/2]/2

(5.143)

This equation represents the symmetric stretching mode of CH2 units (r 1 + r 2 , for example), but the phase relation between the neighboring CH2 units is π: when one CH2 unit shows the symmetric stretching mode of CH bonds, the next CH2 unit shows the symmetric contracting mode of CH bonds. As a result, the transition dipole orients along the x-axis direction. On the other hand, for the local coordinate of the symmetric stretching mode belonging to the Ag species with the 0 phase angle is S Ag = [(r 1 + r 2 )/2 + (r 3 + r 4 )/2]/2, the transition dipole

5.7 Normal Modes and Symmetry (Factor Group Analysis)

497

moments of the first and second terms are cancelled to become 0, IR inactive. In a similar way, we can predict the symmetry species of the individual normal modes, as indicated in Table 5.8. The phase angle in this table is indicated with respect to the C 2 s (z) operation. Instead of making the local symmetry coordinate as mentioned above, the simpler investigation of the symmetry species can be made as follows. Let us focus on the bond stretching r 1 . The other bond stretching r 2 , r 3 , and r 4 are generated by the symmetry operations of C 2x , C 2y and i, respectively. For example, for the anti-symmetric stretching mode [νas (CH2 )], there are two sets of combinations between the anti-symmetric stretching modes (r1 − r2 ) and (r3 − r4 ), where the definition of these coordinates is shown in Fig. 5.88. S1 = (r1 − r2 ) + (r3 − r4 ) S2 = (r1 − r2 ) − (r3 − r4 ) In this case, the r 2 is generated by C 2 (x) with the opposite sign. Similarly, the r 3 is generated from r 1 by C 2 (y) with plus sign for the mode S 1 and minus sign for the mode S 2 . As shown in Fig. 5.89, these signs correspond to the signs of the symmetry Table 5.8 Symmetry species and local vibrational modes of PE chain Vibrational mode

Symbol

Phase relation between the neighboring CH2 units (with respect to C 2 s (z))

Symmetry species

Symmetric CH2 stretching

νs (CH2 )

0

Ag

π

B2u

Anti-symmetric CH2 stretching

νas (CH2 )

0

B3g

π

B1u

0

Ag

π

B2u

0

B3u

π

B1g

0

B3g

π

B1u

0

Au

CH2 bending

δ (CH2 )

CH2 wagging

ωCH2 )

CH2 rocking

γ(CH2 )

CH2 twisting Whole motion of CH2 units

t(CH2 )

π

B2g

νs (CC)

0

Ag

π (T x )

B2u

νas (CC)

0 (T z )

B3u

π

B1g

0 (Rz )

B3g

π (T y )

B1u

5 Structure Analysis by Vibrational Spectroscopy

∇

-

2

2(

∇

∇

+ +

1

)

+ -

3

) ∇

2(

+

( ) 4

∇

498

+

4

B1u B3g

Fig. 5.89 Phase relations among the CH stretching coordinates

species B1u and B3g , respectively, as shown in Table 5.8. The sign of C 2 s (z) is – and +, respectively, or the phase π and 0 with respect to this operation. For the skeletal CC stretching modes, we can draw the red and green arrows as shown in Fig. 5.90. In the case of ν s (CC), both of the CC bonds are extended and the vector sums are directed to the x-directions as indicated by blue arrows (Note that the CCC bond angles are also changed in this vibrational mode). The corresponding symmetry species is Ag . If the blue arrows direct in the same direction, the motion corresponds to the translational mode T x , which belongs to the B2u species. In the ν as (CC) case, one CC bond is stretched (green arrow) and another CC bond is contracted (red arrow). The vector sum orients toward the z-axis but in the opposite directions (blue arrow). In this case, the phase angle is π with respect to the C 2 s operation. The mode belongs to the B1g species. The 0 phase corresponds to the translation along the z-axis, T z (B3u species). The T y and Rz can be drawn using the + and – movements in the y-direction, which belong to B1u and B3g species, respectively.

Fig. 5.90 Skeletal modes of planar-zigzag polyethylene chain

5.7 Normal Modes and Symmetry (Factor Group Analysis)

499

5.7.3 Factor Group Analysis of 3D Crystal Lattice The orthorhombic polyethylene crystal is focusd as an example [28, 29]. As shown in Fig. 5.91, the space group Pnam (P21 /n21 /a21 /m) is isomorphous to the point group (factor group) D2h . The character table is shown in Table 5.9. The CH2 units are on the ab-plane. The two zigzag chains are contained in the cell. Therefore, the total number of atoms in the cell is 12. The number n of nonmovable atoms is 12 for the symmetry operations E and ab-plane mirror. Here, it must be noted that the two CH2 units are laid on one mirror positioned at z = 1/4. Another 2 CH2 units are on the mirror at z = 3/4. Then, 12 atoms are not moved by the mirror symmetry. n = 0 for all the other operations. By considering the traces of these operations, the numbers of the motions belonging to the individual symmetry species are calculated using Eq. 5.125. The result is shown in Table 5.9. By subtracting the translational modes, the number of vibrational modes is calculated also. In addition to the local group vibrations, the librational and translational lattice modes are possible for the crystal lattice in which the whole chains vibrate as the rigid bodies. The explanation about these lattice modes is given in a later section. The final result of the factor group analysis is generally presented in the following manner.  = 6Ag (Raman, α’aa , α’bb , α’cc ) + 3B1g (Raman, α’bc ) + 3B2g (Raman, α’ac ) + 6B3g (Raman, α’ab ) + 3Au (inactive) + 5B1u (IR, μ’a ) + 5B2u (IR, μ’c ) + 2B3u (IR, μ’b )

Fig. 5.91 Crystal structure and space group symmetry of orthorhombic polyethylene

500

5 Structure Analysis by Vibrational Spectroscopy

Table 5.9 Factor group analysis of the orthorhombic PE crystal σg (bc) σg (ac) σ (ab) nΓ

D2h

E

C 2 s (a)

C 2 s (b)

C 2 s (c)

i

Ag

1

1

1

1

1

1

1

1

6

a 2 , b2 , 6 c2

B1g

1

1

−1

−1

1

1

−1

−1

3

bc

nvibration

3

B2g

1 −1

1

−1

1 −1

1

−1

3

ca

3

B3g

1 −1

−1

1

1 −1

−1

1

6

ab

6

Au

1

1

1

1

−1 −1

−1

−1

3

3(0)

B1u

1

1

−1

−1

−1 −1

1

1

6

a

5

B2u

1 −1

1

−1

−1

1

−1

1

6

c

5

B3u

1 −1

−1

1

−1

1

1

−1

3

b

2

12

n Trace n× trace

12

0

0

0

0

0

0

3 −1

−1

−1

−3

1

1

1

0

0

0

0

0

12

36

0

5.7.4 Symmetry Relation Between the Isolated Chain and the 3D Crystal Lattice Polyethylene chains in the crystal lattice do not have a high symmetry as that of the isolated chain (D2h ). The factor group of a chain is C 2h , which is called the site symmetry and only the E, i, σ (ab), and C 2 s (c) symmetry elements exist. These symmetry elements are found commonly in the factor group of an isolated PE chain (D2h ; E, i, σ (x y) , and C 2 s (z)) and also in the crystal lattice (D2h ; E, i, σ (ab) and C 2 s (c)). By focusing on these common symmetry elements, we can correlate the symmetry species as follows (Fig. 5.92 and Table 5.10). How can we interpret this correlation? For example, the CH2 rocking mode of the B1u symmetry species is considered here. The total transition dipole of the isolated chain directs to the y-axis (Fig. 5.92a). The symmetry relation of this vibrational mode itself is kept even in the lattice, as long as the symmetry elements of E, i, σ(ab) and C 2 s (c) are concerned with. In the crystal lattice, these two chains are distinguished by the phase relation of the other symmetry elements. For example, let us see the σg (bc)-plane. The thick blue arrows, which originate from the dipole moments of the single chain as indicated by blue arrows in (a), are related with this glide plane, but the phase relation must be 0 and π. In the case of (b-1), these two blue arrows are in the 0 phase relation with respect to σg (bc)-plane, while the case of (b-2) is π. These two vibrational modes of the unit cell are different in the vibrational frequency because the intermolecular interactions are different between these two modes. As a result, the IR band (B1u ) of the isolated chain splits into two (B1u and B2u ). In this way, we can check the connection of B1u (single chain) to the B1u and B2u through the site symmetry (Bu ). [It is noted here the phase relation between the two dipole moments is necessarily 0 or π because the phase difference between the neighboring unit cells is always 0 for the IR active mode. The phase relation 0 between the two

5.7 Normal Modes and Symmetry (Factor Group Analysis)

501

Fig. 5.92 CH2 rocking modes in the crystal lattice of the orthorhombic PE. a The transition dipole of the individual rocking mode and the total sum of these dipoles in the chain. b The two chains are connected by the σg (bc)-plane. The total dipole moments are related between the neighboring chains with the phase angle o or π. As a result, the total dipole is parallel to the b-axis or the a-axis as shown in red arrows

arrows results in the 0 phase angle of the unit cell (0 + 0 = 0, see Fig. 5.92(b-1)). Similarly, the π phase relation results in also 0 since π + π = 2π (Fig. 5.92(b-2)). In this example, the σg (bc) is used for the explanation of the phase relation between the two arrows. We can use the other symmetry operation also for checking the phase relation between the arrows, but the 0 and π might be changed depending on the type of symmetry operation. For example, (b-1) has a phase relation π for the C 2 s (a) operation and 0 in the (b-2) case. This type of band splitting is called the correlation splitting or the Davidov splitting. The 720 and 730 cm−1 bands observed in the actual IR spectrum correspond to this type of splitting. The similar splitting is observed for all the bands, although the splitting width depends on the bands. Since the intermolecular interactions cause such a band splitting, the splitting width changes with the change of the unit cell size caused by temperature and/or external stress. The unit cell size changes also with the degree of side group branching, resulting in the different splitting width, as shown in Fig. 5.93 [30].

502

5 Structure Analysis by Vibrational Spectroscopy

Table 5.10 Comparison of the factor groups of an isolated molecular chain, a chain at a lattice site and the whole crystal of orthorhombic polyethylene D2h

E

C2s(a)

C2s(b)

C2s(c)

i

Ag

1

1

1

1

1

B1g

1

1

-1

-1

B2g

1

-1

1

B3g

1

-1

Au

1

1

B1u

1

1

-1

B2u

1

-1

1

B3u

1

-1

-1

1

C2h

E

C2s(c)

i

-1

Ag

1

1

1

1

1

-1

Bg

1

-1

1

-1

-1

-1

1

Au

1

1

-1

-1

-1

-1

-1

Bu

1

-1

-1

1

g(bc)

g(ac)

(ab)

1

1

1

1

1

-1

-1

1

-1

-1

1

1

1

1

-1

-1

-1

-1

1

1

-1

-1

1

-1

1

-1

1

1

-1



(ab)

D2h

D 2h

C 2h

Ag

Ag

Ag

Bg

B1g B2g

B1g B2g

B3g

B3g Au B1u

Au

Au

B2u

Bu

B1u B2u

B3u

B3u

The Raman spectral data can be analyzed in a similar way to that of the IR spectra. But, the Raman spectra are so complicated due to the other factors. For example, the CH2 stretching region and the CH2 bending region are so complicated, which are originated from the vibrational coupling between the normal modes and overtones, which is named the Fermi resonance. The details are explained in a later section. A similar analysis is made for it-PP crystal [31, 32]. An isolated molecular chain takes the (3/1) helical conformation, whose symmetry is expressed as 31 (threefold screw symmetry). The factor group is C 3 . The space group of the crystal is P21 /c, which corresponds to the factor group C 2h [33, 34]. As seen in Fig. 5.94, the molecular chains are located at the general positions and no symmetry can be found for the molecular chains. That is, the site symmetry is C 1 . The factor group analysis is made for the isolated chain as shown in Table 5.11. The total number of atoms in a repeating period is 27. The degree of freedom is 3 × 27 = 81. For an isolated chain, vibration = 25A + 26E. Similarly, the factor group analysis made for the crystal lattice is vibration = 81Ag + 80Bg + 80 Au + 79Bu . The symmetry correlation is obtained as below.

5.7 Normal Modes and Symmetry (Factor Group Analysis)

503

Fig. 5.93 a Splitting of the CH2 rocking bands detected for the orthorhombic PE crystal with the different degree of branching. b c Temperature dependence of r(CH2 ) band splitting in the crystallization process from the melt. The blend sample between LLDPE and DHDPE shows the smaller band splitting due to the cocrystallization between D and H species. Reprinted from Ref. [30] with permission of the American Chemical Society, 1992 Table 5.11 Factor group analyses made for an isolated chain and the crystal structure (α2 form) of it-PP (a) For an isolated (3/1) helix C3

E

Cs3

Cs32

n

nvibration

A

1

1

1

27

z, Rz

x 2 + y2 , z 2

E (ε= e2pi/3 )

1 1

ε ε*

ε* ε

27 27

(x, y)

(x 2 – y2 , xy)

26

(yz,xz)

26

σg (ac)

nΓ

n

27

0

0

Trace

3

0

0

25

(b) For the crystal lattice C s 2 (b)

C 2h

E

Ag

1

1

1

1

81

Bg

1

−1

1

−1

81

Rc

Au

1

1

−1

−1

81

b

80

Bu

1

−1

−1

1

81

a, c

79

n

108

0

0

0

−1

−3

1

0

0

0

Trace

3

n × trace

324

i

nvibration a2 , b2 , c2 ,ac

81

ab, bc

80

504

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.94 Crystal structure of it-PP. Reprinted from Ref. [31] with permission of Springer Nature, 1992 Isolated chain

Site symmetry

A

Ag (Raman)

A E

Space group

Bg (Raman) Au (IR) Bu (IR)

The vibrational band of an isolated chain splits into four bands in the crystal, among which the two bands are Raman active and the other two bands are IR active. That is, the band splits into two in the actual spectra. This splitting is called correlation splitting. The regular conformation of (3/1) helix is deformed when the chains are put into the crystal lattice. As a result, the band positions shift more or less since the local geometry is different among the three monomeric units in the repeating period. This is the site group splitting of the bands due to the lowered symmetry. The doubly degenerated bands (E) split into two due to the loss of the 31 helical axis. These bands are split furthermore into four due to the correlation splitting, as mentioned above. In other words, the E band splits into eight bands (4 Raman and 4 IR). In the actually observed spectra, the CH3 groups are rotating at room temperature. By cooling down the sample to a low temperature (–165 °C), the CH3 groups stop the rotation, resulting in the clear observation of the above-mentioned band splitting (Fig. 5.95). In general, if the number of the asymmetric units is n, the vibrational band is expected to split into the n bands, although the activity is dependent on the space group, as exemplified in the case of it-PP crystal.

5.7 Normal Modes and Symmetry (Factor Group Analysis)

A

Isolated chain

A

Site group

Crystal

505

E

IR, Raman active

A

Au

Bu

Au, Bu

IR active

Ag

Bg

Ag, Bg

Raman active

Fig. 5.95 Symmetry correlation and the observed far-IR spectra of it-PP. At low temperature the CH3 rotational band splits to the multiple bands. Reprinted from Ref. [31] with permission of Springer nature, 1992

The above-mentioned factor group analysis is applied to the primitive cell with the translational symmetries between the unit cells. If the Bravais lattice is the complex cell, the translationally equivalent units are included in the cell. In such a case, we need to perform the factor group analysis for the smaller primitive lattice. For example, poly(vinylidene fluoride) (PVDF) form I takes the space group symmetry of Cm2m, which is the C-centered lattice, as shown in Fig. 5.96 [35, 36].

506

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.96 Zigzag conformation and packing mode of molecular chains of PVDF form I

The factor group analysis is made as shown below. For the isolated PVDF chain, σ(xy)

σ(xz)

n

C 2v

E

C 2 (x)

A1

1

1

1

1

6

A2

1

1

−1

−1

2

B1

1

−1

−1

1

4

z

B2

1

−1

1

−1

6

y, Rz

n

6

2

6

2

Trace

3

−1

1

1

18

−2

6

2

n × trace

nvibration x

x 2 , y2 , z 2

5

yz

2

xz

3

yx

4

 = 5A1 + (2 A2 ) + 3B1 + 4B2 For the primitive lattice, C 2v

E

C 2 (b)

A1

1

1

A2

1

B1

1

B2

1

−1

σ(ab)

σ(bc) 1

1

–1

−1

2

−1

−1

1

4

c

1

−1

6

a

6

2

6

2

Trace

3

−1

1

1

18

−2

6

2

6

nvibration

1

n n × trace

n b

 = 5A1 + (2 A2 ) + 3B1 + 5B2

a 2 , b2 , c 2

5

ac

2

bc

3

ab

5

5.7 Normal Modes and Symmetry (Factor Group Analysis)

507

The correlation between the isolated chain, the site symmetry, and the space group is drawn as follows. Isolated Chain

Site Group

Space Group

C2v

C2v

C2v

A1

A1

A1

A2

A2

A2

B1

B1

B1

B2

B2

B2

(P)

The vibrational band does not split and keeps the polarization property even after being packed in the unit cell, although the band position might be shifted more or less because of the intermolecular interactions [37, 38]. The actual IR (and Raman) spectra of PVDF form I are appreciably more complicated than those predicted from the factor group analysis for such a simple regular structure. Many weak but sharp crystalline bands are detected, which are considered to come from the irregular packing mode of the conformationally-disordered chains, which activate the originally-inactive vibrational modes [37]. Refer to Sect. 5.10.

5.7.5 External Lattice Vibration The molecular vibrations in the crystal lattice are classified roughly into two categories. One is the internal vibrational modes such as CH2 stretching mode, CCC bending mode, and so on. Another one is the external vibrational modes, which are the vibrations of the rigid molecules at the lattice sites. These molecules librate (rotate with a small amplitude) and translate, during which the center of gravity of the unit cell is kept. At first, let us consider the small molecules. Since one rigid molecule takes the three translational motions and the three rotational motions, the total degree of freedom is 6N for the N molecules included in the unit cell. The three translational motions are the pure translations with the vibrational frequencies 0. Therefore, 6 N- 3 modes are there as the external vibrational modes. In the case of polymer chain, the rotational mode is one, that is, the librational motion around the chain axis, and one chain molecule experiences four rigid-body motions in total (T x , T y , T z, and Rz ). For the N polymer chains in the unit cell, the total degree of freedom is 4N – 3 for the external lattice vibrational modes of the chains. For example, in the case of the orthorhombic polyethylene crystal, N is 2 and the external modes are 4× 2 − 3 = 5. These modes are illustrated in Fig. 5.97 with

508

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.97 Lattice vibrations in the orthorhombic polyethylene crystal

the symmetry species. The translational modes L(Ta )B2u The librational modes

L(Rco )Ag

L(Tb )B1u L(Tc )Au L(Rcπ )B3g

The IR active lattice modes are L(T b ) and L(T a ), which are detected at 76 and 59 cm−1 , respectively. The Raman active modes are L(Rc o ) and L(Rc π ), which are observed at 137 and 108 cm−1 , respectively. The assignment to the lattice vibrations can be made by taking into account the following behaviors; (i) The temperature dependence of the vibrational frequency is in general larger than the internal modes. (ii) These modes appear at the lowest frequency region in general.

5.8 Normal Coordinates Calculations 5.8.1 Infinitely Repeated 1D Crystal Lattice Figure 5.98 shows the 1D crystal lattice, in which the two atoms are contained in the unit cell of the spacing ao = 2a. These atoms are connected by the springs of the force constant k and vibrate along the x-axis. (Do not confuse about the symbol k between the force constant used here and the phonon vector used in the other chapters). The atom i at the equilibrium position X i is vibrating with the displacement x i . The equations of motion are given below:     m 1 d2 x1 /dt 2 = k(x2 − x1 ) − k x1 − x2’

(5.144)

5.8 Normal Coordinates Calculations

509

    m 2 d2 x2 /dt 2 = −k(x2 − x1 ) + k x1’’ − x2

(5.145)

where mi is the mass of the i-th atom. If we take a snapshot of this vibrating system, and the displacement is drawn upward and downward by rotating 90°, the displacement forms a wave with the wavelength λphonon . In the example shown in Fig 5.98, λphonon = 8a. The phase angle between the equivalent atoms is 2π/(λphonon /a0 ) = 2π/[8a/(2a)] = π/2. In more general, the wave vector q is defined as q = 2π/λphonon and then the phase angle between the neighboring equivalent atoms (open circle, for example) is given as φ = q · Xi

(5.146)

If X i = 2a and q = 2π/λphonon = 2π/(8a), then φ = [2π/(8a)] · 2a = π/2, as already mentioned. When the ◯ and ● atoms are equivalent to each other, the phase angle between the equivalent atoms should be φ = π/4. Now, the displacement x i of the vibrating atom i is given as x1 = A1 exp[2πivt + iq X 1 ], x2 = A2 exp[2πivt + iq X 2 ],

(5.147)

where Ai is the amplitude and ν is the vibrational frequency. The coordinates are X 1 = 0, X 2 = a, X 2  = -a, X 1 = 2a,… Inputting Eq. 5.147, Eqs. 5.144 and 5.145 result in the following equation, where cos(qa) = [exp(iqa) + exp(−iqa)]/2 is used. Fig. 5.98 Vibrating 1D lattice a 2 atomic model, b the vibrating mode, and c frequency-phase angle dispersion curve

510

5 Structure Analysis by Vibrational Spectroscopy



2k − 4π2 m 1 ν2 , −2k cos(aq) −2k cos(aq), 2k − 4π2 m 2 ν2



A1 A2



  0 = D· A= =0 0

(5.148)

For the condition that the amplitudes A1 and A2 are not zero at the same time, we have | D| = 0

(5.149)

The solution of this equation gives  1/2 4π2 v2 = k[(1/m 1 ) + (1/m 2 )] ± k [(1/m 1 ) + (1/m 2 )]2 − 4 sin2 (qa)/(m 1 m 2 ) (5.150) As shown in Fig. 5.98c, the vibrational frequency ν is a function of q and changes remarkably depending on the force constant k and the atomic masses m1 and m2 . The q = 0 corresponds to the in-phase vibration of all the atoms, which is called the gamma point (). The point of ν = 0 at q = 0 in the acoustic branch corresponds to the translational mode, where all the atoms vibrate with the same amplitude (A1 = A2 ) in the same direction in phase. In the dispersion curve of ν2 the moving directions of the atoms are opposite at q = 0 (A1 = –A2 ). If the atoms have the positive and negative charges, the dipole moment changes at the frequency ν2 , becoming optically active and can be detected in the IR spectral measurement. In the case of m 1 = m 2 = m, Eq. 5.150 becomes    aq  k 1 sin and A1 = A2 π m 2    aq  k 1 cos and A1 = −A2 ν2 = π m 2 ν1 =

The curves ν1 and ν2 are coincident with each other at q = π/(2a). The whole dispersion curve has a period of 2π/a, corresponding to the Brillouin zone for the infinitely long chain with the period √ a. The slope of the acoustic phonon curve (ν1 ) near q = 0, (dv1 /dq)o = (a/2π ) k/m corresponding to a so-called group velocity and related to the Young’s modulus of the system.

5.8.2 Finite 1D Crystal Lattice So far we discuss the infinitely long 1D crystal. Let us see now the molecular chain of a finite length. The typical case is polyethylene for the former case and the n-paraffin for the latter case. As illustrated in Fig. 5.99, each spring with a green ball corresponds to one oscillator, which is oscillating at the frequency vo = (ko /m)1/2 /2π. Here, one oscillator corresponds to the vibrational mode of an isolated oscillating group, for example, CH2 stretching mode, rocking mode, and so on. These oscillators are

5.8 Normal Coordinates Calculations

511

Fig. 5.99 Simply coupled oscillating motions

combined together by the springs of force constant k (or the covalent bond) to form the finite chain. We call this type of treatment the simply coupled oscillator model [39–46]. Of course, as another method we can solve the vibrational modes for the system consisting of many atoms gathered together in the unit cell (or even in the whole system) by solving Newton’s equation of motion. The concrete explanation is made in a later section (the GF matrix method or the lattice dynamics). In the present section, the coupling of oscillators is treated. The number of mass points is n, the interatomic force constant is k and the mass of the points is m. In this case, we need to consider the condition at the end parts of the finite chain, which is assumed as the interactions between the wall and the end groups. The force constant representing this interaction is c. The atomic displacement of the i-th atom is indicated as x i . The equations are given as follows [47]:   m d2 x1 /dt 2 = −c(x1 − x0 ) + k(x2 − x1 )   m d2 x2 /dt 2 = −k(x2 − x1 ) + k(x3 − x2 ) ............  2  m d xn−1 /dt 2 = −k(xn−1 − xn−2 ) + k(xn − xn−1 )   m d2 xn /dt 2 = −k(xn − xn−1 ) + c(xn+1 − xn )

(5.151)

512

5 Structure Analysis by Vibrational Spectroscopy

In the case of (i) free ends, c = 0 and x 0 and x n+1 can be ignored. In the case of (ii) fixed ends, the c is assumed to be equal to k and x 0 and x n+1 are equal to zero. In the case of (iii) one free end and one fixed end, these two conditions (c = 0 and c = k) are introduced at the same time. If the atomic displacement takes the form xi = xio exp[i(2πνt + φ)], then we have the following set of equations: (λ − α)x1 + x2 = 0, x1 + (λ − 2)x2 + x3 = 0, ..., xn−1 + (λ − β)xn = 0

(5.152)

where α = β = 1 for (i), α = β = 2 for (ii), and α = 1 and β = 2 for (iii), and λ = v2 /vo2 and νo = (k/m)1/2 /2π (it must be noticed that one isolated oscillator vibrates at the frequency νo ). Since all the x i are not zero simultaneously, the secular equation Dn = 0 is obtained as follows: (i) Free ends

(5.153a)

(ii) Fixed ends

(5.153b)

(iii) One free end and one fixed end

(5.153c)

5.8 Normal Coordinates Calculations

513

By solving the equation Dn = 0, then the eigenvalues λ or the vibrational frequencies ν can be obtained for these three cases. The solution by utilizing the recurrence equations of Dn , Dn-1, etc. is given in the reference [39]. For example, in the case (ii), Dn = (λ − 2)Dn−1 − Dn−2 . Concrete numerical calculations can be made using the software “Mathematica® ”. Figure 5.100 was obtained by the latter method. As shown in this figure, the vibrational frequency changes with the phase angle between the adjacement mass points φ.

Fig. 5.100 Vibrational frequencies plotted against s and φ calculated for the 1D lattice of n = 10 under the different conditions about the end groups

Fig. 5.101 Vibrational frequencies plotted against the phase angle φ calculated for the 1D lattice of n = 6, 9 and 10 under the various conditions. All of these three types of simply-coupled oscillator chains give the same dispersion curves when plotted against the phase angle φ

514

5 Structure Analysis by Vibrational Spectroscopy

The phase angles φ are given as follows, where n is the number of oscillators. Free ends φ = (s − 1)π/n Fixed ends φ = sπ/(n + 1) Fixed end and free end φ = 2sπ/(2n + 1)

(5.154)

where s = 1, 2, … n. When the frequency is plotted against s, the curves are different depending on the boundary conditions. On the other hand, when the vibrational frequency is plotted against the phase angle φ, the dispersion curve is common to all of these three different conditions. This situation can be applied to all the chains of the various n values. Figure 5.101 shows the plot of the vibrational frequency and phase angle for the chains of n = 6, 9 and 10 with the different boundary conditions. In other words, the dispersion curves can be applied to the various chains of different lengths and different end group conditions.

5.8.3 GF Matrix Method (Isolated Molecule) 5.8.3.1

Based on Internal Displacement Coordinates

At first, let us review Lagrange’s equation of motion, which is equivalent to Newton’s equation of motion [48]. But the complicated equation of motion can be derived relatively easily. The motion of the system can be described using the kinetic energy 1/2 T and the potential energy V. The generalized coordinate is qj = m j xj and the     1/2 corresponding velocity is q˙j = m j ∂ xj /dt = ∂qj /∂t . The kinetic energy T =   2 2 (1/2) m j dxj /dt = (1/2) q˙j2 . The potential energy V = (1/2) kij xi − xj =  2 1/2 1/2 (1/2) kij qi /m i − qj /m j . For Lagrangian L = T − V , Lagrange equation of motion is expressed as   d ∂ L/∂ q˙j /dt − ∂ L/∂qj = 0

(5.155)

Now let us consider the molecular vibration. The Cartesian coordinate of the i-th atom is X i = [X i , Y i , Z i ]. The displacement of the i-th atom is X i . The kinetic energy T is expressed as • •  2 T = (1/2) j m j dX j /dt = (1/2) X t M X

(5.156)

5.8 Normal Coordinates Calculations

515

The M is the matrix of the atomic mass m. The “t” indicates the transpose of the vector X. •

 X t = [dX 1 /dt, dY1 /dt, dZ 1 /dt, dX 2 /dt, dY2 /dt, . . . . . . , dZ n /dt] ⎛

m1 ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ M=⎜ 0 ⎜ ⎜ 0 ⎜ ⎝ 0

... 0 0 ...... 0 0 m1 0 . . . 0 0 m2 . . . . . . . . . 0 ... 0 0 0 m2 ...... ... 0 0 0 ...... mn

0 m1 0 0 0

(5.157)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(5.158)

As will be mentioned later, X can be expressed in terms of the internal displacement coordinates R using the B matrix. R = BX

(5.159)

The R is the change of bond length, bond angle, torsional angle, and so on. The momentum P X and P R are defined as follows: •

•

P X = M X,  X = M −1 P X ,

P R = B P X,

P X = Bt P R

(5.160)

The kinetic energy is expressed as • •  t   T = (1/2) X t M X = (1/2) M −1 P X M M −1 P X t  = (1/2) P tX M −1 P X = (1/2) B t P R M −1 B t P R

= (1/2) P tR B M −1 B t P R = (1/2) P Rt G P R

(5.161)

Here G = B M −1 B t

(5.162)

is named the G matrix. Using the components of G matrix, gij , we have   2 2 + (g22 )PR2 + . . . + (g12 )PR1 PR2 + . . . T = (1/2) (g11 )PR1

(5.163)

516

5 Structure Analysis by Vibrational Spectroscopy •

In order to apply Lagrange equation of motion,  R is used instead of P R . The kinetic energy is now expressed as follows, where D is the still unknown matrix. •

•

T = (1/2)R t DR

(5.164)

The conjugated coordinate of R is •

•

P R = (∂ T /∂R) = DR

(5.165)

Then •

R = D−1 P R

(5.166)

Therefore, • • t    T = (1/2)R t DR = (1/2) D−1 P R D D−1 P R = (1/2) P tR D−1 P R (5.167)

By comparing two equations of T, Eqs. 5.161 and 5.167, we can get the following relation: D−1 = G

or

 −1 D = G −1 = B t M B −1

(5.168)

In conclusion, •

•

T = (1/2)R t G −1 R

(5.169)

The potential energy V is given as V = (1/2)F11 R12 + F12 R1 R2 + . . . + (1/2)F22 R22 + F23 R2 R3 + . . . + (1/2)Fnn Rn2 ⎞⎛ ⎞ ⎛ . . . . . . F1n R1 F11 F12 F13 ⎟ ⎜ ⎜ F21 F22 F23 . . . . F2n ⎟ ⎟⎜ R2 ⎟ = (1/2)[R1 , R2 , . . .]⎜ ⎠ ⎠ ⎝ ⎝ ............ . . . . . . Fnn Rn Fn1 Fn2 Fn3 = (1/2)Rt F R R where FR is the F matrix consisting of the force constants.

(5.170)

5.8 Normal Coordinates Calculations

517

˙ In this way, we have now the two terms T and V in the form of R and  R. Using Lagrange equation of motion, we can get the following equation:  •  d ∂ L/∂ R j /dt − ∂ L/∂ Rj = 0

(5.171)

G −1  R¨ + F R R = 0

(5.172)

The normal coordinates Q are related to R. R = L Q

(5.173)

G −1 L Q¨ + F R L Q = 0

(5.174)

As already mentioned, Q changes with the vibrational frequency ν. Q = Q o exp[i(2πvt + φ)]

(5.175)

Q¨ = −4πv2 Q

(5.176)

  −4π2 ν 2 G −1 L + F R L Q = 0

(5.177)

4π2 ν 2 G −1 L = F R L 4π2 ν 2 L = G F R L

(5.178)

Then, Eq. 5.174 becomes

Here let us assume that the number of answers of the vibrational frequency is n (the total degrees of freedom), and put the term 4π2 ν2 = λ (eigenvalue). Then we have λ1 L 1 = G F R L 1 , λ2 L 2 = G F R L 2 , ...,

λ1 L n = G F R L n

Here Li is the columnar vector corresponding to the eigenvalue λi and is called the eigenvector. The L, Λ and GFR L are related as follows, L = [L 1 , L 2 , ..., L n ], ⎡

λ1 ⎢0 ⎢ =⎢ . ⎣ .. 0

0 λ2 .. .

0 0 .. .

⎤ ⎥ ⎥ ⎥, ⎦

· · · λn

G F R L = [G F R L 1 , G F R L 2 , ..., G F R L n ] = [λ1 L 1 , λ2 L 2 , . . . , λn L n ] = L

518

5 Structure Analysis by Vibrational Spectroscopy

It must be noted that the newly-defined vector L covers all the eigenvectors (Li ) as shown above, though the apparent form of the equation is the same as Eq. 5.178. For the non-zero eigenvectors, we have the following secular equation. |G F R − Λ| = 0 (5.179)



G F R − 4π2 ν2 E = 0

The T and V energies are given by a series of the normal coordinates Q as   t ˙ t E Q˙ T = (1/2) Q˙ 2l + Q˙ 22 + . . . + Q˙ 2n = (1/2) Q˙ Q˙ = (1/2) Q

(5.180)

  V = (1/2) λ1 Q 21 + λ2 Q 22 + . . . + λn Q 2n = (1/2) Q t Λ Q

(5.181)

Therefore, •

•

˙ t G −1 (L Q) ˙ = (1/2) Q ˙ L t G −1 L Q˙ T = (1/2) R t G −1  R = (1/2)(L Q) t

˙ t E Q˙ = (1/2) Q

(5.182)

V = (1/2)Rt F R R = (1/2)(L Q)t F R (L Q) = (1/2) Q t L t F R L Q = (1/2) Q t Λ Q

(5.183) L t G −1 L = E and L t F R L = Λ

(5.184)

(Symmetry Blocking) Previously, we learned the local symmetry coordinates which belong to the separated symmetry species. The internal displacement coordinates blocked out into the symmetry species are expressed as a vector Rs . Rs = UR

(5.185)

For example, the local symmetry coordinates are assumed to belong to the two species a and b,  U=

 Ua 0 . 0 Ub

(5.186)

5.8 Normal Coordinates Calculations

519

The kinetic energy T is also blocked out as follows. Since R = U −1 Rs = U Rs , t

•



•

−1

•

T = (1/2) R G  R = (1/2) U  Rs t

•t

t

•

t G

−1

•t



•



U  Rs t

•

= (1/2) Rs U G −1 U t  Rs = (1/2) Rs G −1 s  Rs −1 t t G −1 = U G U or G = U GU s s t    V = (1/2)Rt F R R = (1/2) U t Rs F R U t Rs = (1/2)Rts U F R U t Rs = (1/2)Rts F Rs Rs

(5.187)

(5.188)

F Rs = U F R U t By substituting U of Eq. 5.186 into these equations,  GS =

  t      0 Ua 0 U a GU ta G sa 0 Ua 0 G = = 0 Ub 0 U tb 0 U b GU tb 0 G sb     0 F Rsa 0 U a F R U ta = Fs = 0 U b F R U tb 0 F Rsb

(5.189)

(5.190)

Similarly, B matrix is also blocked out as Bs = UB. The eigenvalues or the vibrational frequencies can be solved for each GFR matrix belonging to each symmetry species.

5.8.3.2

Based on the Orthogonal Coordinates

Instead of R, we use X and X = L x Q

(5.191)

R = BX = B L x Q = L Q

(5.192)

• •  2 T = (1/2) j m j dX j /dt = (1/2) X t M X

(5.193)

Then,

As a result,

V = (1/2)Rt F R R = (1/2)(BX)t F R (BX) = (1/2)X t B t F R BX (5.194)

520

5 Structure Analysis by Vibrational Spectroscopy

By inputting Eq. 5.191 into Eq. 5.193, •

•

•

•

•

•

T = (1/2) X t M X = (1/2) Q t L tx M L x Q = (1/2) Q t E Q

(5.195)

and so L tx M L x = E

(5.196)

L x L tx = M −1

(5.197)

By inputting X m = M 1/2 X = M 1/2 L x Q = L xm Q,  T = (1/2) M

−1/2



• Xm

t

 M M

−1/2



• Xm



•

•

= (1/2) X tm  X m

(5.198)

  t  V = (1/2) M −1/2 X m B t F R B M −1/2 X m = (1/2)X tm M −1/2 B t F R B M −1/2 X m = (1/2)X tm Dx X m

(5.199)

where Dx = M −1/2 B t F R B M −1/2

(5.200)

which is named the dynamical matrix. By using Lagrange equation, the secular equation is derived as follows: | Dx − Λ| = 0

(5.201)

Dx L xm = L xm Λ

(5.202)

How to estimate the vibrational modes (i)

Expression using the Internal Displacement Coordinates

The equation R = L Q is concretely written as R1 = L 11 Q 1 + L 12 Q 2 + . . . R2 = L 21 Q 1 + L 22 Q 2 + . . . ............

(5.203)

5.8 Normal Coordinates Calculations

521

Oppositely, Q = L −1 R     Q 1 = L −1 11 R1 + L −1 12 R2 + . . .     Q 2 = L −1 21 R1 + L −1 22 R2 + . . .

(5.204)

By knowing the L vector, we can estimate the concrete contribution of individual internal displacement coordinates Ri to the normal mode Qj . By dissolving Eq. 5.178 (G F R L = LΛ), the L vector is obtained. Since L t G −1 L = E (5.184), L t = L −1 G

or

L −1 = L t G −1

(5.205)

The equation X = L x Q indicates that X 1 = (L x )11 Q 1 + (L x )12 Q 2 + . . . . . . X 2 = (L x )21 Q 1 + (L x )22 Q 2 + . . . . . . ............

(5.206)

In the opposite form, Q = L −1 x X and so     X 1 + L −1 X 2 + . . . Q 1 = L −1  x−1 11  x−1 12 Q 2 = L x 21 X 1 + L x 22 X 2 + . . . ............ As seen in Eqs. 5.162 and 5.168, G −1 = −1 t −1 M B G . Since BLx = L,



Bt

−1

L x = B −1 L = M −1 B t G −1 L

5.8.3.3

(5.207)

M B −1 and so B −1 =

(5.208)

Based on the Dynamical Matrix

As seen in Eqs. 5.200 and 5.202, Dx L xm = L xm Λ By solving the equation of Dx , we can know the Lxm vector, which is equal to M 1/2 Lx or the mass-adjusted Lx vector. The drawing of the vibration of H atom gives the appreciably long arrow, while that of C atom is shorter since the mass is larger. This problem is reduced by taking the mass difference into account. The Lx is obtained as L x = M −1/2 L xm .

522

5.8.3.4

5 Structure Analysis by Vibrational Spectroscopy

Potential Energy Distribution

The potential energy V is given as   V = (1/2) λ1 Q 21 + λ2 Q22 + . . . + λn Q 2n = (1/2) Q t Λ Q

(5.209)

where Λ = [λk ]. The eigenvalue λk is related to FR and L in the following form. L t F R L = Λ (5.184) or λk = i j L ik L jk (FR )ij ∼ L 2ik (FR )ii

(5.210)

i

where L ik is the i-th L vector component corresponding to the eigenvalue λk . The potential energy term of the k-th normal mode is.   Vk = (1/2)λk Q 2k ∼ (1/2) i L 2ik (FR )ii Q 2k

(5.211)

Therefore, the contribution of the i-th internal displacement coordinate Ri to the potential energy component is expressed approximately as PEDik = L 2ik (FR )ii / i L 2ik (FR )ii = L 2ik (FR )ii /λk

(5.212)

which is called the potential energy distribution of the i-th internal coordinate to the k-th normal mode.

5.8.3.5

Summary of Normal Modes Calculation

Displacement coordinates R

T

V

• • (1/2) R t G −1  R

G = B M −1 B t

(1/2)Rt F R R

Equation for the eigenvalues

L vector







G F R − 4π 2 v 2 E = 0

L t G −1 L = E

G F R L = LΛ

Lt FR L = ΛL

X

(1/2) X t M X

(1/2)X t B t F R BX



M − Bt FR B = 0

X m

•t • (1/2) X m  X m

(1/2)X tm Dx X m

| Dx − Λ| = 0

Dx = M −1/2 B t F R B M −1/2

Dx L xm = L xm Λ

5.8.3.6

•

•

Lx Lxm

Examples of Normal Modes Treatment (Water)

The internal displacement coordinates and the Cartesian displacement coordinates are defined as shown in Fig 5.102.

5.8 Normal Coordinates Calculations

523

Fig. 5.102 Coordinates of water molecule

⎛

⎞ x0 ⎜ y0 ⎟ ⎜ ⎟ ⎜ z ⎟ ⎜ 0⎟ ⎜ x ⎟ ⎜ 1⎟ ⎜ ⎟ X = ⎜ y1 ⎟ ⎜ ⎟ ⎜ z 1 ⎟ ⎜ ⎟ ⎜ x2 ⎟ ⎜ ⎟ ⎝ y2 ⎠

⎛

⎞ r1 R = ⎝ r2 ⎠ φ

⎛

⎞

0 ⎠ Xo = ⎝ 0 r0 cos(φ/2) ⎛ ⎞ −ro sin(φ/2) ⎠ X1 = ⎝ 0 0 ⎛ ⎞ ro sin(φ/2) ⎠ X2 = ⎝ 0 ⎛

e01

0

⎞ ⎛ ⎞ − sin(φ/2) sin(φ/2) ⎜ ⎟ ⎜ ⎟ = ⎝0 ⎠ e02 = ⎝ 0 ⎠ − cos(φ/2) − cos(φ/2)

z 2

524

5 Structure Analysis by Vibrational Spectroscopy ⎞ ⎞ ⎛ − cos(φ/2) − cos(φ/2) ⎟ ⎟ ⎜ ⎜ = ⎝0 ⎠ p02 = ⎝ 0 ⎠ sin(φ/2) − sin(φ/2) ⎛

p01

⎛

⎞ ⎛ r1 B11 B12 B13 B14 B15 B16 B17 B18 ⎜ ⎟ ⎜ R = ⎝ r2 ⎠ = ⎝ B21 B22 B23 B24 B25 B26 B27 B28 φ B31 B32 B33 B34 B35 B36 B37 B38

⎞ x0 ⎜ y0 ⎟ ⎟ ⎜ ⎟ ⎜ z 0 ⎟ ⎞⎜ ⎟ ⎜ B19 ⎜ x 1⎟ ⎟ ⎟⎜ ⎜ B29 ⎠⎜ y1 ⎟ ⎟ ⎟ ⎜ B39 ⎜ z 1 ⎟ ⎟ ⎜ ⎜ x2 ⎟ ⎟ ⎜ ⎝ y2 ⎠ ⎛

z 2

The calculation of the B matrix components is made in the following ways. r

(i)

r12 = (X 1 − X 0 )2 Then, r0 r1 = (X 1 − X 0 )t X 1 − (X 1 − X 0 )t X 0 Using the unit vectors eij ,     r1 = (X 1 − X 0 )t /r0 X 1 − (X 1 − X 0 )t /r 0 X 0 = et01 X 1 − et01 X 0 ⎡ ⎤ ⎡ ⎤ x1 x0 = [− sin(φ/2), 0, − cos(φ/2)]⎣ y1 ⎦ − [− sin(φ/2), 0, − cos(φ/2)]⎣ y0 ⎦. z 1 z 0 (5.213) That is to say, B11 = sin(φ/2), B12 = 0, B13 = cos(φ/2), B14 = − sin(φ/2), B15 = 0, B16 = − cos(φ/2), B17 = 0, B18 = 0, B19 = 0 Similarly, B21 = − sin(φ/2), B22 = 0, B23 = cos(φ/2), B24 = 0, B25 = 0, B26 = 0, B27 = sin(φ/2), B28 = 0, B29 = − cos(φ/2) (ii)

φ r1r2 cos(φ) = (X 1 − X 0 )(X 2 − X 0 )

Differentiation of this equation gives the following equation:

5.8 Normal Coordinates Calculations

525

r1r02 cos(φ) + r01 r2 cos(φ) − r01r02 sin(φ)φ = X 1 (X 2 − X 0 ) + X 2 (X 1 − X 0 ) − (X 1 + X 2 )X 0 + 2X 0 X 0 We use the following various vectors defined in Fig. 5.102, r1 = et01 X 1 − et01 X 0 , r 2 = et02 X 2 − et02 X 0 po1 = [e01 cos(φ) − e02 ]/ sin(φ)





 p02 = [−e02 cos(φ) + e01 ]/ sin(φ), p01 = p02 = 1 (Be careful of the direction of po2 .

(5.214) Finally, we get the following equation:       φ = − p01 /r01 + p02 /r02 X 0 + p01 /r01 X 1 − p02 /r02 X 2

(5.215)

Inputting the concrete values to these vectors, we have, B31 = 0, B32 = 0, B33 = −2 sin(φ/2)/r0 B34 = − cos(φ/2)/r0 , B35 = 0, B36 = sin(φ/2)/r0 , B37 = cos(φ/2)/r0 , B38 = 0, B39 = sin(φ/2)/r0 By inputting s = sin(φ/2) and c = cos(φ/2), the final result is ⎛

s 0 B = ⎝ −s 0 0 0

c c −2s/r0

−s 0 −c/r0

−c 0 s/r0

0 0 0

0 s c/r0

0 0 0

⎞ 0 −c ⎠ s/r0

(5.216)

The G matrix (= BM −1 Bt ) is calculated as below. The dimension of G is (3 × 9)(9 × 9)(9 × 3) = (3 × 3). The atomic masses of H and O atoms are mH1 = mH2 = mH and mO , respectively. ⎛

s G = ⎝ −s 0 ⎛ −1 mO ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜ × ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎝0 0

0 0 0 0 m −1 O 0 0 0 0 0 0 0

c c −2s/r0

−s 0 −c/r0

0 0 m −1 O 0 0 0 0 0 0

0 0 0 0 m −1 H1 0 0 0 0

0 0 0 m −1 H1 0 0 0 0 0

0 0 0

0 0 0 0 0 m −1 H1 0 0 0

−c 0 s/r0

0 0 0 0 0 0 m −1 H2 0 0

0 0 0 0 0 0 0 m −1 H2 0

⎞ 0 −c ⎠ s/r0 ⎞ ⎛ ⎞ s −s 0 0 ⎜ ⎟ 0 0 0 ⎟ ⎟ ⎜ 0 ⎟ ⎜ ⎟ c −2s/r0 ⎟ 0 ⎟ ⎜ c ⎟ ⎜ 0 ⎟ 0 −c/r0 ⎟ ⎟ ⎜ −s ⎟ ⎟ ⎜ ⎟ 0 ⎟×⎜ 0 0 0 ⎟ ⎟ ⎜ ⎟ 0 ⎟ ⎜ −c 0 s/r0 ⎟ ⎟ ⎜ ⎟ ⎜ 0 ⎟ s c/r0 ⎟ ⎟ ⎜ 0 ⎟ ⎠ 0 ⎠ ⎝ 0 0 0 −1 m H2 0 −c s/r0

0 s c/r0

0 0 0

526

5 Structure Analysis by Vibrational Spectroscopy

⎞ μO cos(φ) −μO sin(φ)/ro μH + μO = ⎝ μO cos(φ) μH + μO −μO sin(φ)/ro   ⎠ (5.217) −μO sin(φ)/ro −μO sin(φ)/ro 2[μH + μO (1 − cos(φ))]/ ro2 ⎛

where μO = 1/m o and μH = 1/m H . The potential energy V is assumed to be expressed as follows: V = (1/2)K r (r1 )2 + (1/2)K r (r2 )2 + (1/2)Hφ (φ)2 + Frr (r1 )(r2 ) + Frφ (r1 )(φ) + Frφ (r2 )(φ) ⎛ ⎞⎛ ⎞ K r Frr Frφ r1 = (1/2)[r1 , r2 , φ]⎝ Frr K r Frφ ⎠⎝ r2 ⎠ = (1/2)Rt F R R (5.218) Frφ Frφ Hφ φ The concrete values are input to these matrixes G and FR . m O = 16.0g/mol = 2.66 × 10−23 g/atom, m H = 1.0g/mol = 0.166 × 10−23 g/atom μO = 0.376 × 1023 g−1 , μH = 6.020 × 1023 g−1 ◦

φ = 104.5 , ro = 0.957Å, K r = 8.453mdyn/Å, Hφ = 0.694mdynÅ/rad2 , Frr = −0.105mdyn/Å, Frφ = 0.161mdyn/rad

⎞ μO cos(φ) −μO sin(φ)/ro μO + μ H ⎠ G = ⎝ μO cos(φ) μO + μ H −μO sin(φ)/ro 2 −μO sin(φ)/ro −μO sin(φ)/ro 2[(1 − cos(φ))μO + μ H ]/ro ⎛ ⎞ 6.396 −0.094 −0.380 = ⎝ −0.094 (5.219) 6.396 −0.380 ⎠ × 1023 −0.380 −0.380 14.173 ⎛ ⎞ 8.453 −0.105 0.161 F R = ⎝ −0.105 (5.220) 8.453 0.161 ⎠ 0.161 0.161 0.694 ⎛

The GFR matrix is calculated as ⎛

54.01 G F R = ⎝ −1.527 −0.890

−1.527 54.01 −0.890

⎞ 0.751 0.751 ⎠ 9.714

(5.221)

It must be noted here that G and FR are symmetric matrices, but the GFR is not so. The eigenvalues obtained by solving the secular equation are

5.8 Normal Coordinates Calculations

527

λ = 55.541, 52.456, 9.7450

  x1023 g−1 mdyn/Å = 1028 /s2

= 4π2 ν2 = 4π2 c2 ν˜ 2 ν˜ = (λ)1/2 /(2πc) = 530.79(λ)1/2 cm−1 [1302.9(λ)1/2 cm−1 for the automic mass unit of g/mol] = 3955.8, 3844.3, 1657.0 cm−1

(5.222)

(Symmetry Block) The 3 × 3 GFR matrix is easily solved to get the eigenvalues. However, it is better to separate it to the blocks of the symmetry species. As already mentioned, the following U matrix is useful for this purpose. G s = U GU t FR = U FRU t In the present case of water molecule, the following U matrix is used (refer to the Sect. 5.7.1): S1 = (r1 + r2 )/21/2 symmetry species A1 S2 = φ symmetry species A1 S3 = (r1 − r2 )/21/2 symmetry species B1 √ ⎛ ⎞ ⎛ √ ⎞⎛ ⎞ S1 1/ 2 1/ 2 0 r1 ⎝ S2 ⎠ = ⎝ 0 0√ 1 ⎠⎝ r2 ⎠ = UR √ S3 φ 1/ 2 −1/ 2 0 The G and FR matrices are blocked out as follows: G s = U GU t √ ⎞ − 2μO sin(φ)/ro 0 μO (1 √+ cos(φ)) + μH ⎠ = ⎝ − 2μO sin(φ)/ro 2[(1 − cos(φ))μO + μH ]/ro2 0 0 0 (1 − cos(φ))μO + μH (5.223) √ ⎛ ⎞ 2Frφ 0 K√r + Frr ⎠ F Rs = U F R U t = ⎝ 2Frφ (5.224) Hφ 0 0 0 K r − Frr ⎛

Then,  A1 = G sA1 F Rs

√ √   − 2μO sin(φ)/ro K√r + Frr 2Frφ μO (1 √+ cos(φ)) + μH − 2μO sin(φ)/ro 2[(1 − cos(φ))μO + μH ]/ro2 Hφ 2Frφ

528

5 Structure Analysis by Vibrational Spectroscopy

 =

6.302 −0.538 −0.538 14.172



8.348 0.228 0.228 0.694



 =

52.486 1.064 −1.260 9.713

 (5.225)

B1 = [(1 − cos(φ))μO + μH ][K r − Frr ] = 6.490 × 8.558 = 55.541 G sB1 F Rs (5.226)

By inputting the numerical values to the parameters and solve the equation |Gs F Rs − Λ| = 0 to get the eigenvalues. The answer is the same as above, but the assignment to the symmetry species is clear. 3844.3, 1657.0cm−1 3955.8 cm−1

A1 B1

The L vectors are useful to know the vibrational modes in a concrete manner. As an example, the 2×2 matrix D is treated here. For the two eigenvalues, we have the following equations. DL 1 = λ1 L 1 

ab cd



l11 l21



DL 2 = λ2 L 2 

= λ1

l11 l21



 and

ab cd



l12 l22



 = λ2

l12 l22

 (5.227a)

By combining together, DL = L 

ab cd



l11 l12 l21 l22



 =

l11 l12 l21 l22



λ1 0 0 λ2

 (5.227b)

The secular equation is given as below, and the two eigenvalues are obtained.



%

a − λ b

  &

= 0, λ = (a + d) ± (a + d)2 − 4(ad − cb) 1/2 /2

c d − λ

By inputting λ1 into the above equation,  For λ1 ,

ab c d



l11 l21



 = λ1

l11 l21



al11 + bl21 = λ1l11 cl11 + dl21 = λ1l11

(a − λ1 )l11 + bl21 = 0, (a − λ1 )l11 = −bl21 then l11 = −b and l21 = a − λ1 .

5.8 Normal Coordinates Calculations

529

However, this is only the ratio between l11 and l 21 . The normalization is needed, which is 2 2 + l21 = G 11 L L t = G or l11

(5.228)

For the A1 symmetry species of water molecule,  A1 G sA1 F Rs

=

52.486 1.064 −1.260 9.713

 (5.229)

For λ1 = 52.456, l11 = −34.323l21 . After normalization as l11 2 + l 21 2 = g11 = 6.302, we have l 11 = 2.509 and l 21 = –0.074. Similarly, for λ2 = 9.745, l12 = −0.094 and l22 = 3.763. The contribution of the local modes S 1 and S 2 to the normal modes Q1 and Q2 is remarkably different between these two modes, but they are coupled together more or less.

5.8.4 Lattice Dynamics of Crystals The optically active vibrational modes are those with phase difference 0 between the unit cells, as already described before. If the unit cell contains several asymmetric units, the phase relation must be taken into account between these asymmetric units with keeping the 0 inter-unit-cell phase angle. Figure 5.103 shows the examples. The phase angle between the neighboring cells is commonly zero. But, the green and red units experience the displacements (a) in the opposite directions and (b) in the same direction. The latter corresponds to the pure translational mode with the vibrational frequency zero. The Cartesian coordinate system is now considered [49, 50]. The unit cell contains the m asymmetric units. The atomic displacement coordinate X ◦ (i, m, k) is for the i-th atom belonging to m-th asymmetric unit in the k-th unit cell. A set of the vectors X ◦ (i, m, k) included in the m-th asymmetric unit is now expressed as X ◦ (m, k). The vibrational mode or the set of atomic displacements is common to all the asymmetric units, but the phase relation between them is different, as illustrated in Fig. 5.103. By expressing the phase relation as c(m) R, , which is a coefficient in the character table of a factor group  concerning the symmetry operation (R) connecting the 1st and m-th asymmetric units, we have the following relation between X ◦ (1, k) and X ◦ (m, k).

530

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.103 Optically-active modes, where the phase angle between the neighboring unit cells is zero. The 21 screw symmetry is used here. (a) the phase relation between the two asymmetric units or c(2)21 is (a) +1 and (b) -1.

X ◦ (m, k) = X ◦ (m, 0) = c(m) R, X ◦ (1, 0)

(5.230)

The internal displacement vector R(l) of the asymmetric unit l in the unit cell is also related to R(1) with the phase relation. R(l) = c(m) R, R(1)

(5.231)

5.8 Normal Coordinates Calculations

531

The two vectors R and X o (m, k) are connected by the B matrix as R(l) = m k B ◦ (m, k)X ◦ (m, k)

(5.232)

Then, using Eq. 5.230, R(l) = [ m k c(m) R, B ◦ (m, k)X ◦ (1, 0) = B  X ◦ (1, 0)

(5.233)

where, B  = m k c(m) R, B ◦ (m, k)

(5.234)

It must be noted here that the B matrix [Bo (m,k)] is based on the m-th asymmetric unit itself, as shown below. But, in the actual numerical calculation, the B matrix components must be calculated on the basis of the coordinate system of the 1st asymmetric unit, B(m,k), which can be transformed to Bo (m,k) as follows. B ◦ (m, k) = B(m, k)[ A(m) ⊗ E] where [ A(m) ⊗ E] is the direct product of A(m) and identity matrix E.

On the other hand, the force constant matrix is produced as follows. …

ΔR(1)

ΔR(2) ΔR(3)

……….

ΔR(1)

… FR(α) c(β)RΓFR(β) c(γ)RΓFR(γ) ………..

ΔR(2)

… c(β)*RΓFR(−β) FR(α) c(β)RΓFR(β) c(γ)RΓFR(γ) ….

…..

…………………

(5.235)

532

5 Structure Analysis by Vibrational Spectroscopy

  F R =F R (α) + c(β)R F R (β) + c(β)R F R (−β)   + c(γ )R F R (γ ) + c(γ )R F R (−γ ) + . . .  R  R t = F R (0) + cm F R (m) + cm F R (m)

(5.236)

Here, FR (0) indicates the force constant matrix showing the interactions within the basic unit, and FR (m) is the matrix to show the interactions between the first and the m-th units. The vibrational frequency and the L vectors are calculated as follows: DX = M −1/2 B t F R B  M −1/2 Dx L xm = L xm Λ

(5.237)

5.8.5 Application of Lattice Dynamics to Polyethylene Single Chain (1D Crystal) 5.8.5.1

Translational Units

Here, only the skeletal atoms (or the united atom CH2 ) are considered. The two units are included in the repeating period. The torsional modes are neglected. The internal displacement coordinates are R = R1 , R2 , 1 , and 2 . The atomic positions are as follows, where the z-axis is parallel to the chain axis. The x-axis is in the zigzag plane and perpendicular to the z-axis. X t1 = (−0.436, 0, 0), X t2 = (0.436, 0, 1.27), X t3 = (−0.436, 0, 2.54), X t4 = (0.436, 0, 3.81), . . . The internal displacement coordinates are defined as shown in Fig. 5.104.

X

y



R1

 1

4(2’)

2

Z R2 3(1’)

Fig. 5.104 Definition of internal coordinates of 2 methylene units of planar-zigzag polymethylene chain

5.8 Normal Coordinates Calculations

533

R

Atom

Atom

R1

1

2

R2

2

3

Atom

1

2

1

3

2

3

2

4

The B matrix is calculated by applying the s vectors. Bond stretching

r = −e12 X 1 + e12 X 2

e

e12 = (X 2 − X 1 )/r12

Bond angle deformation

  φ = − p01 /r01 + p02 /r02 X 0 +     p01 /r01 X 1 − p02 /r02 X 2

0 e02

p01  p02

1

2

p01 = [e01 cos(φ) − e02 ]/ sin(φ)

e01

Internal rotation

p02 = [−e02 cos(φ) + e01 ]/ sin(φ) 4

p 234 = r23×r34

3 

1

2 p123 = r12×r23

τ = u 123 p123 X 1   − (u 123 − w321 ) p123 − w234 p234 X 2   + (u 432 − w234 ) p234 − w321 p123 X 3 −u 432 p234 X 4    u i jk = 1/ ri j sin φi jk      wi jk = cos φi jk / ri j sin φi jk cos(τ ) = p123 p234

The unit vectors are calculated in the following ways. R1 = R12 = −e12 X 1 + e12 X 2 R2 = R23 = −e23 X 2 + e23 X 3       1 = 213 = − p21 /Rcc + p23 /Rcc X 2 + p21 /Rcc X 1 − p23 /Rcc X 3 ⎞ ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ ⎛ 0.566 −0.566 0.824 0.824 ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎜ e12 = ⎝ 0.000 ⎠ e23 = ⎝ 0.000 ⎠ p21 = ⎝ 0.000 ⎠ p23 = ⎝ 0.000 ⎠ 0.824 0.824 −0.566 −0.566

534

5 Structure Analysis by Vibrational Spectroscopy

We need to note that the B matrix does not consist of only the α unit but also the β unit. The Bo α and Bo β are summed up by taking the phase difference into account. Since the phase angle for the optically active modes is 0 and the translational unit is used for the α and β units, Xα = Xβ . At the same time, the transformation matrix A = E.

The B matrix is obtained finally as follows: ⎛

B

x1

R1  0.566 R2 ⎜ ⎜  0.566 ⎝ 1.070 1 1.070 2

y1

z1

x2

0.000  0.824 0.566 0.000 0.824 0.566 0.000 0.000  1.070 0.000 0.000  1.070

y2

z2

⎞ 0.000 0.824 0.000  0.824 ⎟ ⎟ 0.000 0.000 ⎠ 0.000 0.000

Similarly, the FR matrix consists also of the α and β terms as shown below. The potential energy of the α unit is expressed simply as follows:     2V = K R R12 + R22 + H 21 + 22 + FRR (R1 R2 + R2 R1 ) The FR matrix is given as

For F RR term, it is the interaction between the neighboring CC bonds, the potential energy is expressed as F RR R1 R2 and F RR R2 R1 . The latter should be equal to FRR R2 R1 . The FR matrix of the α unit is given by summing up FRα and FRβ : F R = F Rα + F Rβ .

5.8 Normal Coordinates Calculations

535

R1

⎛

R2

1

R1 K R 2FRR R2 ⎜ ⎜ 2FRR K R ⎝ 0 0 1 0 0 2

FR 

2

⎞ 0 0 0 0 ⎟ ⎟ H 0 ⎠ 0 H

Usually, the FR matrix is a sum of the A matrices corresponding to each force constant. The reason to use A matrices is the convenience to refine the force constants using the same A matrices. F R = f 1 A1 + f 2 A2 + . . .

(5.238)

For example, ⎛

K√r + Frr F R = ⎝ 2Frφ 0

√ √ ⎞ ⎞ ⎛ ⎛ ⎞ 0 2Frφ 20 100 √0 ⎠ = K r ⎝ 0 0 0 ⎠ + Frφ ⎝ 2 0 0 ⎠ + . . . .. Hφ 0 0 K r − Frr 0 0 0 001

= K r A1 + Frφ A2 + . . .

(5.239)

The concrete values of the force constants are K R = 3.6 mdyn/Å, FRR = 0.15 mdyn/Å and H = 1.35 mdynÅ/rad2 FR is calculated as follows: R1 R1 R2

FR 

1 2

3.6 0.3 0 0

R2

2

1

0.3 0 3.6 0 0 1.35 0 0

0 0 0 1.35

The atomic mass matrix is given as follows: X1

M

X1 Y1 Z1 X2 Y2 Z2

m 0 0 0 0 0

m = 14 g/mol or 2.33 × 10−23 g/atom

Y1 0 m 0 0 0 0

Z 1 X 2 Y2 0 0 m 0 0 0

0 0 0 m 0 0

0 0 0 0 m 0

Z2 0 0 0 0 0 m

536

5 Structure Analysis by Vibrational Spectroscopy

The dynamical matrix DX is calculated by using the equation: D X = M −1/2 B t F R B M −1/2 , DX L Xm = L Xm The thus-obtained vibrational frequencies and the Lxm vectors are given as below for the optically active vibrational modes. The two carbon atoms are contained in the unit cell and the total degree of freedom is 2 × 3 = 6. The three translational motions and one rotational motion around the z-axis have the zero frequencies (No definition of torsional mode results in the curious rotational mode with 0 frequency). The two vibrational modes correspond to the symmetric and anti-symmetric skeletal stretching modes (see Fig. 5.105). λ

ν(cm-1 )

LY 1

L Z1

LX2

LY 2

LZ2

Modes

0.807

1170.3

0.180

0.000

0.058

– 0.180

0.000

– 0.058

νs (CC)

0.640

1042.3

– 0.058

0.000

0.180

0.058

0.000

– 0.180

νas (CC)

0.000

0.000

0.000

0.267

0.000

0.000

0.000

0.000

T y ° + Rz °

0.000

0.000

0.205

0.000

0.000

0.205

0.000

0.000

T x°

0.000

0.000

0.000

0.000

0.000

0.000

0.267

0.000

T y ° + Rz °

0.000

0.000

0.000

0.000

0.189

0.000

0.000

0.189

T z°

L X1

Fig. 5.105 Vibrational modes of a planar-zigzag polymethylene chain

5.8 Normal Coordinates Calculations

5.8.5.2

537

One Repeating Unit

One CH2 unit (actually one C atom) is taken into account. The other C atoms along the chain axis are created by the symmetry operation of 21 screw rotation (see Fig. 5.106). This symmetry operation is needed to consider for the construction of the B matrix.

The B matrix consists of the α, β, γ, … terms. The Bα , Bβ ,.. are summed up by taking the phase difference into account, where c(m) is the coefficient of the C 2 s (z) symmetry operation. B  = m c(m) B o (m) The Bo (m) is based on the Cartesian coordinate system of the m-th unit itself. In the actual calculation shown above, the B matrix components are calculated on the basis of the Cartesian coordinate system of the first unit. The B matrix based on the first unit is B(m), then B o (m, k) = B(m, k)[ A(m) ⊗ E] This operation affects the case of transformation from the 1-st unit to the second unit (β term). The B matrix components of the 1st and 3rd units do not change. The transformation matrix A(2) is given as below. ⎛

⎞ −1 0 0 A(2) = ⎝ 0 −1 0 ⎠ 0 0 1 X 2 ( )

y

R

 1

Z R 3 ( )

Fig. 5.106 Definition of internal coordinates of one methylene unit of planar-zigzag polymethylene chain

538

5 Structure Analysis by Vibrational Spectroscopy

By applying this operation, for the case of c = + 1 we have the following B matrix.

B = Boα + cBoβ + c2 Boγ + . . .

(5.240)

The B(c = + 1) matrix is finally obtained as follows: x B(c = +1) =

y

z

R −1.132 0.000 0.000 2.140 0.000 0.000

Similarly, the B(c = −1) matrix is obtained as follows: x B(c = −1) =

R

y

z

0.000 0.000 −1.648 0.000 0.000 0.000

Here, the symmetry species having the coefficient c= + 1 and c= –1 are separated into two groups. D2h

E

Ag B1g B2g B3g

C 2 s (z)

i

σ (xy)

σ g (yz)

σ (xz)

C 2(x)

C 2 (y)

1

1

1

1

1

1

1

1

1

−1

1

−1

1

−1

−1

1

1

1

−1

−1

1

−1

1

−1

1

−1

−1

1

1

1

−1

−1

Au

1

1

1

1

−1

−1

−1

−1

B1u

1

−1

1

−1

−1

1

1

−1

B2u

1

1

−1

−1

−1

1

−1

1

B3u

1

−1

−1

1

−1

−1

1

1

c = +1 Ag , B3g , Au , B3u c 1 B1g , B2g , B1u , B2u

5.8 Normal Coordinates Calculations

539

In the present calculation, only the skeletal CC bond stretching modes are included. The symmetric CC stretching mode belongs to the Ag species and the anti-symmetric CC stretching mode to the B2g species. A similar consideration is made for the force constant matrix.

The FR matrix of the α unit is given by summing up the FRα and FRβ : FR = FRα + cFRβ + cFt Rβ . For c = + 1 R F R (c = +1) =

R K R + 2FR R 0 0 H

For c = − 1 F R (c = −1) =

R

R K R − 2FRR 0 0 H

The atomic mass matrix is given as follows:

M=

X Y Z

X Y Z m 0 0 0 m 0 0 0 m

m = 2.33 × 10−23 g/atom The dynamical matrix DX is calculated by using the equation: DX = M −1/2 B  F R B  M −1/2 t

The thus-obtained vibrational frequencies and the Lxm vectors are given as below for the optically active vibrational modes. The answer is the same as that mentioned for the treatment of the translational unit.

540

c=+1

c = −1

5.8.5.3

5 Structure Analysis by Vibrational Spectroscopy λ

ν(cm−1 )

0.807

1170.4

0.000

0.000

0.000

0.000

0.640

1042.3

0.000

0.000

0.000

0.000

L X1

LY 1

L Z1

Modes

0.180

0.000

0.058

νs (CC)

0.000

0.000

0.189

T z°

0.000

0.267

0.000

R° z

– 0.058

0.000

0.180

νas (CC)

0.189

0.000

0.000

T °x

0.000

0.267

0.000

T °y

Polymethylene Chain

Figure 5.107 shows the PE zigzag chain with both C and H atoms. The two CH2 units are contained in the repeating period with the 21 screw axis. The Cartesian coordinates of one CH2 unit are given below. The calculation is made by using only one CH2 unit (α). The internal coordinates are defined for the first unit. X C (1, 1)t = X(1) = [−0.436, 0.0, 0.0] X H (3, 1)t = X(3) = [−1.071, −0.90, 0.0] X H (2, 2)t = X(5) = [1.071, −0.90, 0.0] X C (1, 3)t = X(7) = [−0.436, 0.0, 2.538]

X H (2, 1)t = X C (1, 2)t = X H (3, 2)t = X C (1, 4)t =

X(2) = [−1.071, 0.90, 0.0] X(4) = [0.436, 0.0, 1.269] X(6) = [1.071, 0.90, 0.0] X(10) = [0.436, 0.0, 3.807]

(Note) The numbering of atoms must be made in the same order among the CH2 units repeated along the chain. The C atom of the δ CH2 unit has a number 10, not the number 8, since the two H atoms are included in the γ CH2 unit, although they are not used in the calculation. The H atom number 5 (= 2 ) and 6 (= 3 ) are related to the H atom numbers 2 and 3 by the 21 screw axis, respectively. The B matrix is constructed by combining the components Bα , Bβ , Bγ , and Bδ by taking the symmetry operations into account. Bδ is needed for the introduction of the CC torsional mode. Fig. 5.107 Definition of internal coordinates of one methylene unit of planar-zigzag polyethylene chain

5.8 Normal Coordinates Calculations

5.8.5.4

541

Force Constants

The potential function is given as a set of interactions between the internal displacement coordinates. The coefficients are the force constants. For a polyethylene single chain, the following equation is used for the normal modes treatment:       V = (1/2)K R R 2 + (1/2)K r r12 + r22 + FCH (r1 r2 ) + FCC RR ’       + (1/2)Hφ φ12 + φ22 + φ32 + φ42 + (1/2)Hθ θ 2 + (1/2)H 2 + FCC,HCC (Rφ1 + Rφ2 + Rφ3 + Rφ4 )   + FCC,CCC R + R ’  + f HCC,HCC (φ1 φ2 + φ3 φ4 )     + f HCC,HCC φ’2 φ4 + φ’1 φ3 + f CCC,CCC ’  + f HCC,CCC (φ1 + φ2 + φ’3 + φ’4 + f HCC,HCC (t) (φ1 φ3 + φ2 φ4 ) ’ (t) + f HCC,HCC (g) (φ1 φ4 + φ2 φ3 ) + f HCC,HCC (φ1 φ1’ + φ2 φ2’ ) ’ (g) + f HCC,HCC (φ1 φ2’ + φ2 φ1’ ) 

+ f HCC,HCC (t) (φ1 φ4’ + φ2 φ3’ ) 

+ f HCC,HCC (g) (φ1 φ3’ + φ2 φ4’ ) + (1/2)T (τ )2

(5.241)

The numerical values of the force constants are expressed in the following way. The parentheses represent the common atoms for the two internal coordinates. The expression CH, CH (C) indicates the interaction between the CH bonds with the common C atom, that is, the interaction between r 1 and r 2 (see Fig. 5.108). Similarly, HCC, HCC(CC) indicates the interaction between φ1 and φ2 with the common CC bond.

Fig. 5.108 Internal coordinates of planar-zigzag polyethylene chain necessary for the definition of force constants

542

5 Structure Analysis by Vibrational Spectroscopy

Force constant Atoms (Common) Value KR

CC

4.279 mdyn/Å

f HCC, HCC

HCC, HCC(CH)

0.081 mdynÅ/rad2

Kr

CH

4.441 mdyn/Å

f CCC, CCC

CCC, CCC (CC)

0.087 mdynÅ/rad2

F CH

CH, CH (C)

0.029 mdyn/Å

f HCC, CCC

HCC, CCC, −0.057 CC mdynÅ/rad2

F CC

CC, CC (C)

0.138 mdyn/Å

f HCC, HCC (t)

HCC, HCC(CC)t

0.091 mdynÅ/rad2

Hφ

HCC

0.692 mdynÅ/rad2

f HCC, HCC (g)

HCC, HCC(CC)g

0.005 mdynÅ/rad2

Hθ

HCH

0.500 mdynÅ/rad2

f  HCC, HCC (t)

HCC, HCC(C)t

−0.008 mdynÅ/rad2

H

CCC

1.130 mdynÅ/rad2

f  HCC, HCC (g)

HCC, HCC(C)g

0.009 mdynÅ/rad2

F CC, HCC

CC, HCC(CC)

0.281 mdyn/rad

f ’HCC, HCC (t)

HCC, HCC

−0.015 mdynÅ/rad2

F CC, CCC

CC, CCC (CC)

0.404 mdyn/rad

f ’HCC, HCC (g)

HCC, HCC

−0.013 mdynÅ/rad2

f HCC, HCC

HCC, HCC (CC)

−0.001mdynÅ/rad2

T

CCCC

0.095 mdynÅ/rad2

These numerical values are determined so that the observed frequencies are reproduced as well as possible. In general, the parameters of the valence force filed (VFF) force constants are not very much universal for the various groups of the various compounds although the values are in similar orders.

In this case, the potential energy is assumed to consist of the diagonal terms of the bond stretching and bond angle deformation as well as the nonbonded interatomic repulsions between the terminal atoms [51, 52].     V = (1/2)K r12 + r22 + (1/2)Hr1or2o (φ)2 + (1/2)F q 2   + K ’ r1o r1 + r2o r2 + H ’r1or2o φ + F ’ qo q (5.242) where the terms of the first order are contained also since the q is a function of r 1 , r2, and φ, and they are not independent of each other (Fig. 5.109). The q is now developed with respect to the terms of r 1 , r 2, and φ up to their second orders. q = sr1 + sr2 + tro φ + (t 2 /2qo )(r1 )2 + (t 2 /2qo )(r2 )2 − (sro3 /qo2 )(φ)2 − (t 2 /qo )(r1 )(r2 ) + (tsro /qo )(r1 )(φ) + (tsro /qo )(r2 )(φ) t = ro sin(φ)/qo

5.8 Normal Coordinates Calculations

543

Fig. 5.109 Interatomic repulsion expressed using the coordinate q

s = 2ro sin2 (φ/2)/qo

(5.243)

where r 1o = r 2o = r o is assumed for simplicity. By inputting q into Eq. 5.242, V is expressed in the various cross terms between the internal displacement coordinates, where the higher order terms are ignored.     V = (1/2) K + 2Fs 2 + 2F ’ t 2 (r1 )2 + (1/2) K + 2Fs 2 + 2F ’ t 2 (r2 )2 +   (1/2) H − Fs 2 + F ’ t 2 (φ)2 + . . . The comparison is made between the Urey-Bradley force constants and the Valence force constants: K R = K + 2Fs 2 + 2F ’ t 2 Hφ = H − Fs 2 + F ’ t 2 etc. The force constants are a little complicated since the consideration of the cross terms representing the interactions between the various internal coordinates is needed for the good reproduction of the observed IR and Raman vibrational frequencies. Among the various force fields, the two representative sets have been explained here: the valence force field and the Urey-Bradley force field [51, 52]. Recently, the energy calculation gives conveniently the force constants. The second derivative of the potential energy, ∂ 2 V /∂x∂y is called the Hessian matrix component, which corresponds to the F R matrix component (The Hessian matrix is obtained by the numerical calculation of the potential energy. Be careful of the calculation errors of the second derivatives.) In many cases, the Hessian matrix derived from the quantum chemical calculation gives the appreciably higher vibrational frequencies and the socalled fitting parameter is introduced, which reduces the answer by applying a suitable coefficient (for example, the quantum-chemically calculated force constants give the C = O stretching mode at 2200 cm−1 , which is reduced to 1750 cm−1 by multiplying the coefficient 0.8 without any deep reasoning !). In the classical mechanics treatment, the potential functions are assumed in various ways so that the various physical quantities are reproduced for many compounds. For example, COMPASS force field is said to be excellent for this purpose [53]. A detailed discussion will be made in a later section (energy calculation).

544

5 Structure Analysis by Vibrational Spectroscopy

5.8.6 Application of Lattice Dynamics to Orthorhombic PE 3D Crystal As seen in the crystal structure of orthorhombic PE crystal, the four CH2 units are contained. One CH2 unit is assumed to be one sphere or the so-called united atom (Fig. 5.110 [54]). The orthogonal unit cell parameters are a = 7.40 Å, b = 4.93 Å, and c = 2.54 Å. The fractional coordinates of the basic CH2 unit (#1) are given as follows. x(1, 1, k = 0)t = (x1 , y1 , z 1 ) = (−0.0380, −0.0649, −0.2500) The other units are generated by using the transformation matrices and the translational vectors.

(a)

(b)

Fig. 5.110 (a) Atomic positions and symmetry relations in the orthorhombic polyethylene crystal. The atomic displacements indicated by the solid-lines are based on the coordinate system of #1. The atomic displacements indicated by the broken-lines are those based on the individual asymmetric units, as converted by the corresponding transformation matrices from those of the unit #1 system. (b) The definitions of the intramolecular- and intermolecular-internal coordinates of the unit #1. Reprinted from Ref. [54] with permission of the America Chemical Society, 1978

5.8 Normal Coordinates Calculations

545

ˆ x(1, 1, k = 0)t = (x1 , y1 , z 1 ) = Ex(1, 1, k = 0) + (0, 0, 0) s

x(2, 1, k = 0)t = (x2 , y2 , z 2 ) = Cˆ 2 (c)x(1, 1, k = 0) + (0, 0, 0.5) x(3, 1, k = 0)t = (x3 , y3 , z 3 ) = σˆ g (ac)x(1, 1, k = 0) + (0.5, 0.5, 0) x(4, 1, k = 0)t = (x4 , y4 , z 4 ) = σˆ g (bc)x(1, 1, k = 0) + (0.5, 0.5, 0.5)

⎛

1 Eˆ = ⎝ 0 0 ⎛ 1 σˆ g (ac) = ⎝ 0 0

⎞ 00 1 0⎠ 01 ⎞ 0 0 −1 0 ⎠ 0 1

⎛

⎞ −1 0 0 Cˆ 2 (c) = ⎝ 0 −1 0 ⎠ 0 0 1 ⎛ ⎞ −1 0 0 σˆ g (bc) = ⎝ 0 1 0 ⎠ 0 01 s

The development of equations necessary for the calculation of B and FR matrices is the same as that mentioned above. The coefficients of the symmetry species are as follows: Ag , B3u B1g , B2u B2g , B1u B3g , Au

E C2s (c) σg (bc) σg (ac) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

The definition of the internal coordinates is shown in Fig. 5.110. The B(m, k) matrices based on the first unit are given as shown in Table 5.12. For example, BAg is created in the following way (refer to Eq. 5.235): s

B Ag = B α + 1× B β σˆ 2 (c) × E + 1× B γ σˆ g (ac) × E + 1× B δ σˆ g (bc) × E ⎛ r

B

A5



x1

⎜  0.7350 ⎜ 1.4335 ⎜ ⎜ 0 ⎜ p1 ⎜ 0 p2 0.2493 p3  1.4987 p4  1.4987

y1

 0.8371 1.6324 0  0.8847  1.9028 0 0

z1 ⎞

0 0 0 0 0 0 0

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

(5.244)

−0.3676

0.3585

−0.5312

0.8981

0.1246

−0.7493

−0.7493

r

φ

τ

p1

p2

p3

p4

−0.5886

−0.5886

−0.9514

−0.4430

0.4664

0.4082

−0.4186

−0.3032

0.3032

0.2816

0

0

−0.3643

−0.8306

0.7493

0.7493

−0.1246

−0.8981

0.5312

−0.7170

0.3676

0.5886

0.5886

0.9514

0.4430

−0.4664

−0.8164

0.4186

0.3032

−0.3032

−0.2816

0

0

0

0.8306

0

0

0

0

0.5312

0.3585

0

Bγ z 1

x1

y1

Bβ

z 1

x1

y1

Bα

x1

0

0

0

0

−0.4664

0.4082

0

y1

0

0

0

0

0

0.3643

0

z 1

0

0

0

0

−0.5312

0

0

x1

Bδ

0

0

0

0

0.4664

0

0

y1

0

0

0

0

0

0

0

z 1

Table 5.12 B matrix of orthorhombic polyethylene crystal based on the coordinate system of the first unit. Reprinted from Ref. [54] with permission of the America Chemical Society, 1978

546 5 Structure Analysis by Vibrational Spectroscopy

5.8 Normal Coordinates Calculations

547

Similarly,   Ag F R = F(11) + m c(m) F(1m) + F t (1m)

(5.245)

The force constants used here were K r = 3.948 mdyn/Å, F rr = 0.231 mdyn/Å, F rφ = 0.216 mdyn/rad, Hφ = 1.523 mdyn Å/rad2 , and T τ = 0.095 mdyn Å/rad2 . The intermolecular interatomic interactions were calculated using Buckingham-type potential function V ij = Bexp(-Cr ij ) – A/r ij 6 . The corresponding force constant is calculated as P = d2 V ij /dr ij 2 by inputting the corresponding atomic distance. Using the thus-obtained FR Ag and BAg and M matrices, the Dx Ag is calculated using Eq. 5.237, from which the eigenvalues or vibrational frequencies are obtained.

548

5 Structure Analysis by Vibrational Spectroscopy

5.8.7 How to Adjust the Calculated Frequencies to the Observed Ones The calculated vibrational frequencies are not necessarily in good agreement with the observed data. It may come from the accuracy of the atomic coordinates, which is determined, for example, by the X-ray structure analysis or by the energy calculation. Another serious problem is the reasonableness of the force constants of the FR matrix. The initial values of the force constants might be quoted from the literature, the rough empirical estimations, the calculation of Hessian matrix (the second derivatives of the potential energy as a function of atomic coordinates), and so on. The refinement of the force constants may result in the better reproduction of the observed spectral data. One way is to modify the force constants by a trial and error method. Another method is to use the Jacobian matrix. As already known, the force constant matrix FR and the eigenvalues λ are related by the following equations: L t F R L = , F R = FRn An and λk = 4π2 ν2k = i j L ik L jk ( FRn An )ij (5.246) The differentiation of νk with respect to the force constant (F R )ij is calculated as below: 8π 2 vk vk = i j L ik L jk ( FRn An )ij

(5.247)

or      Jkn = (vk /FRn ) = 1/8π 2 vk i j L ik L jk ( An )ij = 1/8π 2 vk L t AL kn (5.248) vk = vkobs − vkcalc = n Jkn FRn = J t F R

(5.249)

The contribution of the modified force constant to the change of the calculated frequency is known from the J (Jacobian) values. At first, focus on the large J value and change the corresponding force constant. Or, by solving the equations of the plural ν values, the change of force constants can be obtained by solving the equations. The force constant adjustment is not a numerical computation game. What we have to be careful of the calculated results of the normal modes is to remind several important and common sense points: (i) the calculated frequencies are close to the observed values, (ii) the force constants are in the reasonable range obtained for the other molecules with the similar chemical structures, (iii) the 3D structure of the targeted molecule (polymer chain, molecular crystal, polymer crystal, etc.) is reasonable or not.

5.9 Vibrational Frequency-Phase Angle Dispersion Curves

549

5.9 Vibrational Frequency-Phase Angle Dispersion Curves So far the normal modes with the phase difference 0 between the neighboring unit cells have been treated. The phase difference between the original unit cell (X = 0) and the unit cell at the coordinate X is given as 2πkX. The displacement coordinates with the phase difference taken into account are expressed as the summation of these terms [55] (note: the symbol showing the wave vector is now k, equivalent to q given in Eq. 5.146: R(k) = j Rj e2πi kX j , Rj = k R(k)e−2πi kX j

(5.250)

X(k) = j X j e2πik X j , X j = k X(k)e−2πi kX j

(5.251)

The R(k) and X(k) are the Fourier transforms of Rj and X j , respectively. Since Rj = m B m X j+m and X j+m = X j + X m , R(k) = j Rj e2πi kX j = j m B m X j+m e2πi kX j = j m B m e−2πi kX m X j+m e2πi kX j+m = m B m e−2πi kX m j X j+m e2πi kX j+m = B(k)X(k)

(5.252)

B(k) = m B m e−2πi kX m

(5.253)

Similarly, the potential energy V is V =(1/2) j m Rtj F Rjm Rm    t    =(1/2) j m k R(k)e−2πi kX j F Rjm k R k e−2πik X m

(5.254)

The interaction FRjm is related only to the translational vector between the j-th and m-th cells, which is set to n = j – m. Then, X m = X j-n = X j – X n , and so   t     V = (1/2) j n k R(k)e−2πi kXj F Rn k R k e−2πi k X j −X n  t        = (1/2) k k j R(k)e−2πi kX j n F Rn e2πi k X n j R k e−2πi k X j  t           = (1/2) k k j R(k)e−2πi kX j n F Rn e−2πi kX n e2πi k −k Xn j R k e2πi k X j    ∗  = (1/2) k k R(k)t F R (k)R k n e2πi k−k X n

Since (1/N ) n e2πi (k −k) X n = 1 (k= k’) and 0 (k = k’), 

∗

V = (1/2) k R(k)t F R (k)R(k)

(5.255)

550

5 Structure Analysis by Vibrational Spectroscopy

F R (k) = j F Rj e−2πi kX j

(5.256)

The dynamical matrix is as below. ∗

DX (k) = M −1/2 B(k)t F R (k)B(k)M −1/2 DX L xm = L xm Λ

(5.257)

5.9.1 Application to 1D Lattice This system was already discussed in the previous section. The following matrices are created for the 1D model consisting of 2 mass points (m1 and m2 ) with the spacing a (period 2a). R=

r1 and X1 X1

M=

X2

r1 r2

f 0

r1 =

X2

m1 0 0 m2 r1

FR =

r2 , FR = f

r2

0 f

B= r1

0 0

r1 r2

r2 =

X1

X1 −

X1 X2 X1 −1 1 0 0 −1 1

X2 0 0

−1 1 0 −1

0 −4πika e 0

X2

r2

0 0

B(k) = B α + B β e−4πika + . . . = F R (k) = F R =

X2 −

+

0 1

f 0 0 f

DX(k) = M −1/2 B t *F R B M −1/2 = =

1/m 1 1/2 0 0 1/m 2 1/2

−1 e4πika 1 −1

f 0 0 f

1 −1 −4πika −1 e

1/m 1 1/2 0 0 1/m 2 1/2

2 f /m 1 − f /(m 1 m 2 )1/2 1 + e 4πika 1/2 −4πika − f /(m 1 m 2 ) 1+e 2 f /m 2

(5.258)

The secular equation | D x − Λ E| = 0, then  1/2 λ = 4π v2 = f (1/m 1 + 1/m 2 ) ± f (1/m 1 + 1/m 2 )2 − 4 sin2 (2π ka)/(m 1 m 2 ) (5.259) Compare this equation with Eq. 5.150, where q = 2πk and k = f .

5.9 Vibrational Frequency-Phase Angle Dispersion Curves

551

5.9.2 Application to Polyethylene Single Chain As already shown before, only the skeletal atoms (or the united atom CH2 ) are considered. The two units are included in the repeating period. The torsional modes are neglected. The internal displacement coordinates R = R1 , R2 , 1 and 2 . The B matrix was already calculated.

The phase difference between the first and second unit cells is δ = 2πkc with the c value 2.54 Å. On the basis of Eq. 5.253 B matrix is created as e−

(δ)

=

−0.566 0 0 −0.566e − 0.535(1 + e − ) 0 1.07e − 0

−0.824 0.824e − −0. 368(1 − e 0

−

)

+⋯

0.566 0.566 −1.070 −0. 535(1 + e

−

0 0 0 ) 0

0.824 −0. 824 0 −0.368(1 − e − )

When δ = 0, this matrix is coincident with that obtained for the translational units case (Sect. 5.8.5.1). Similarly, the FR (k) is obtained as

(5.261)

552

5 Structure Analysis by Vibrational Spectroscopy

By setting up the dynamical matrix Dx (k) = M −1/2 B t F R B M −1/2 = B * F R B M −1/2 , the eigenvalues are obtained as a function of the phase angle M δ = 2πkc. By using the same force constants as those mentioned above (m = 14 g/mol, K R = 3.6 mdyn/Å, F RR = 0.15 mdyn/Å and Hφ = 1.35 mdynÅ/rad2 ), the vibrational frequency-phase angle relation obtained for the single PE chain is shown in Fig. 5.110(a), where the phase angle (δ) is between the neighboring unit cells. The curves are symmetrical at the boundary of δ = π. Usually the curves are plotted in the range of δ = 0 to π. At δ = 0, the two optically-active modes have the nonzero vibrational frequencies, corresponding to the symmetric and anti-symmetric CC stretching modes. With the change of δ, the vibrational frequencies change and are coincident with each other at δ = π. The vibrational frequencies of the other 4 modes are zero at δ = 0, corresponding to the pure translational modes (T °x , T °y and T °z ) and the pure rotational mode (R°z ), as pointed out in Fig. 5.105. As the δ changes, the frequencies of these modes become non-zero, because of the contribution of the skeletal CCC bond angle deformation mode (In the present calculation, a skeletal torsional mode is not taken into account). These two modes are coincident at δ = π. The similar calculation was performed with one methylene unit as a repeating unit by taking into account the C 2 s (z) symmetry (Sect. 5.8.5.2). The B(δ) and FR (δ) are expressed as below. −1/2

' ( ( ' ( 0 0 0 −0.566 0 −0.824 −0.566 0 0.824 −iδ B(δ) = e−i2δ + e + 0.535 0 0.368 0.535 0 −0.368 1.070 0 0   (  ' 0 −0.824 1 − e−iδ  −0.566 1 + e−iδ = 0.535 + 1.070e−iδ + 0.535e−i2δ 0 −0.368 1 − e−i2δ ( ' ( ' ' ' ( ( FRR 0 −iδ FRR 0 iδ K R + 2FRR cos(δ) 0 KR 0 + FR (δ) = e + e = 0 H 0 0 0 0 0 H '

For δ = 0, these matrices are the same as those shown in Sect. 5.8.5.2. Using the same values for the parameters, the dispersion curves were obtained as shown in Fig. 5.110 (b). The assignments of the vibrational modes are indicated there. We have to notice that the phase angle is between the adjacent methylene units. The symmetric CC stretching mode [νs (CC)] appears at δ = 0 (c = eiδ = +1, Sect. 5.8.5.2), while the anti-symmetric CC stretching mode [νas (CC)] at δ = π (c = eiδ = –1, Sect. 5.8.5.2). By folding back these curves at δ = π/2, we get the same dispersion curves calculated for the translational unit (Fig. 5.110 (a)). As you understand already, the combination of the phase angles δ1 (between the CH2 units #1 and #2) and δ2 (between the CH2 units #2 and #1’) gives the phase angle δ defined between the neighboring unit cells: δ = δ1 + δ2 . The actual calculation of the dispersion curves of an isolated polyethylene chain and the crystal lattice of the orthorhombic polyethylene crystal are reported in the literatures [56–58]. See Fig. 5.112. The band assignment is shown in Table 5.13. It

5.9 Vibrational Frequency-Phase Angle Dispersion Curves

(a)

553

(b)

(c)

Fig. 5.111 Vibrational frequency-phase angle dispersion curves calculated for an isolated polymethylene chain. The phase angle is (a) between the adjacent unit cells and (b) between the adjacent CH2 units. (c) The dispersion curves (b) are folded back at δ = π/2, giving the same dispersion curves as (a)

554

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.112 Vibrational frequency-phase angle dispersion curves calculated for the orthorhombic polyethylene crystal [56–58]. The branches: 1: CH2 symmetric stretching, 2: CH2 bending, 3: CH2 wagging, 4: CC stretching, 5: skeletal bending, 6: CH2 anti-symmetric stretching, 7: CH2 rocking, 8: CH2 twisting, 9: skeletal torsion [Reprinted from Refs. [58] with permission of the American Institute of Physics, 1967]

B1u

Au

2874

1489

749

76

2851

1473

731

73

59

-

2919

1053

-

2919

1184

1051

-

1308

171

137

1061

1127

1131

1295

1164

1168

1408

1437

1418

1370

2845

2848

B1g

2899

2883

Ag

Calcd/cm−1

Obsd/cm−1

Species

B2u

1175

δ(CH2 ) Translational

1050

109

νs (CH2 ) r(CH2 )

720

νas (CH2 ) B3u

1463

Translational

2851

2919

ω(CH2 ) t(CH2 )

1131 108

νas (CC)

1168

ω(CH2 ) t(CH2 )

1441

2848

2883

Librational

νs (CC)

B3g

1061

r(CH2 )

1295

1370

δ(CH2 )

B2g

νas (CH2 )

Obsd/cm−1

νs (CH2 )

Species

Modes

Table 5.13 IR and Raman bands assignments of orthorhombic polyethylene crystal [58]

1059

1175

105

737

1479

2877

2917

138

1127

1164

1455

2838

2904

1051

1303

1413

Calcd/cm−1 Modes

t(CH2 )

ω(CH2 )

Translational

r(CH2 )

δ(CH2 )

νas (CH2 )

νas (CH2 )

Librational

νs (CC)

r(CH2 )

δ(CH2 )

νs (CH2 )

νas (CH2 )

νas (CC)

t(CH2 )

ω(CH2 )

5.9 Vibrational Frequency-Phase Angle Dispersion Curves 555

556

5 Structure Analysis by Vibrational Spectroscopy

is noted that the band splitting due to the correlation between the two chains in the unit cell are observed also for the dispersion curves. The factor group D2h is applied to the vibrational modes at the phase angle 0, while the factor group of the non-zero phase angle becomes a lower C 2h group, which is called the subgroup.

5.9.3 How to Use the Dispersion Curves The vibrational frequency changes with the phase angle. The IR and Raman active modes are limited to several phase angles. The vibrational frequencies predicted for the IR and Raman active modes of an (M/N) helical chain are detected at the phase angle of the repeating unit = 0 and θ (for IR) and 0, θ, and 2θ (for the Raman), where θ = 2π × N /M. For the (2/1) chain of PE, θ = 2π × 1/2 = π . Then, IR and Raman bands at δ = 0, π, and 2π (= 0) are active. In the case of POM (9/5) helix, θ = 2π × 5/9 = 10π/9. The IR active modes are at phase angles 0 and 10π/9 [59–61]. Both angles correspond to 2π for the nine repeating units contained in the translational period. The modes of the phase difference 0 are in general the totally symmetric A modes with the parallel IR polarization, and those of θ are of the doubly generated E 1 modes with the perpendicular IR polarization. The A, E 1 , and E 2 modes are Raman active. In the case of crystal lattice with the polytype phenomenon, the observed bands can be interpreted using the dispersion curves. A well-known example is seen for the SiC crystal [62]. The C and Si have the orbitals of sp3 type, and the C and Si are linked together by the covalent bonds to form the layer but with variously different arrangements. The basic layer structure is A, B, and C (Fig. 5.113). In the layers A and B, the layer orientation does not change, while it is reoriented by 60° in the C layer. The various arrangements of these layers result in the various types of layer stacking structure, which is called the polytype (The layer itself has the same structure but the stacking mode is different). SiC is known to show many polymorphs, among which more than 250 polytypes are reported. For example, the regular repetition of A and B gives the ..ABABABA.. structure, which is called the 2H structure (H: the hexagonal lattice). The repetition of A, B and C like …ABCABCABC… is seen for 3C structure (C: cubic). The case of 4H structure is shown in Figure 5.113, which has the repetition of ABCB layers. Correspondingly, the Raman bands are observed differently for these various polytype structures, which are found to fit well to the dispersion curves of the simple 1H structure (…AAAAA…). By folding the dispersion curves at the positions of the possible repeating period (the zone-folded mode) the detectable band positions can be predicted since they become Raman active. In Fig. 5.113, the 6H, 15R, 21R are the 6 layers in the repeating period of the hexagonal lattice, 15 layers in the rhombohedral lattice (R), etc. Depending on the layer number the phase angle observing the Raman bands is different. The dispersion curves are LO and LA in the 111 direction, which were calculated for the 3C structure. The LO and LA are the longitudinal optical and acoustic modes, the explanation of which is made in a later section. For the 21R layer structure, the 21/3 = 7 boundaries

5.9 Vibrational Frequency-Phase Angle Dispersion Curves

(a)

557

(b)

(c)

Fig. 5.113 a Raman spectra of 6H, 15R and 21R SiC crystal. b Dispersion curves of LO and LA modes along the 111 direction of 3C SiC. c The stacking model of 4H structure with ABCB layer structure. Reprinted from Ref. [62] with permission of the American Physical Society, 2000

are the folding points; the Raman active positions are 0, 2/7, 4/7 and 6/7 of the zone width. The dispersion curves are observed by the coherent inelastic scattering measurement of the neutron scattering. The usage of fully deuterated sample is useful for this purpose. The force constants or the parameters of the potential energy functions can be modified by adjusting the calculated dispersion curves to these observed data. Another method is to measure the IR and/or Raman progression bands of the oligomers with finite chain length (for example, a series of n-alkanes), which will be described in a later section. The dispersion curves are used for the prediction of the physical properties of the crystals. The three acoustic modes exist, which start from the 0 positions corresponding to the pure translational modes. The slope of the acoustic dispersion curve corresponds to Young’s modulus as already mentioned. In the case of PE, Young’s modulus was estimated along the a, b, and c axes on the basis of the dispersion curves

558

5 Structure Analysis by Vibrational Spectroscopy

observed by the neutron experiments. The LAM (longitudinal acoustic mode) bands observed in the low-frequency Raman spectra of n-alkane molecules with finite length fit well to the LAM dispersion curve along the chain axis [63]. However, the effect of the end groups must be taken into account for the quantitative comparison with the dispersion curve calculated for the polyethylene chain, as will be discussed in a later section. The density of states is calculated from the dispersion curves calculated at the various points of the Brillouin zone or the reciprocal lattice. The thus-obtained density of state is g(ν) (see Sect. 5.10). According to Einstein model or Debye model, the thermodynamic functions like entropy S, enthalpy H, free energy (F or G), etc. are expressed using the density of state as below. For example, under the harmonic vibrational approximation,        S = (kB /N ) j k hvj (k)/2kB T coth hvj (k)/2kB T − ln 2 sinh hvj (k)/2kB T

(5.262)

where the summation is made for all the frequency-phase angle dispersion curves νj (k). N is the number of phonons included in the calculation of these dispersion curves. For the continuous density of sates g(ν),  S = −kB

      g(v) ln 1 − e−hv/kB T + hv/ kB T 1 − ehv/kB T dv

(5.263)

Kobayashi et al. calculated the dispersion curves of the orthorhombic and triclinic polyethylene crystals at the various Brillouin zones and estimated the density of states, from which Helmholtz free energies were calculated at the various temperatures and compared between these two phases [64–66]. As shown in Fig. 5.114, the result is that the free energy of the orthorhombic phase is lower than that of the triclinic phase at room temperature. The lattice energy or the chain packing energy is always lower for the triclinic phase than the orthorhombic phase. The contribution of the vibrational entropy term reverses the stability of these two phases. This type of discussion is quite important for the investigation of the phase transition of the crystalline phases. If the crystal phases do not show any structural disorder, the entropy term comes from the vibrational entropy only. The stability of the phase is determined by the free energy which is the combination of enthalpy H originating from the lattice energy (or the molecular packing energy) and vibrational entropy S vib . G(or F) = H (U ) − Svib T

(5.264)

The calculation of H and S vib is made once when the packing energy and the vibrational frequency-phase angle dispersion curves are calculated as mentioned above.

5.9 Vibrational Frequency-Phase Angle Dispersion Curves

(a)

559

(b)

Vibrational frequency distribution

orthorhombic

monoclinic

Wavenumber/cm-1

(c)

(d)

Fig. 5.114 a Vibrational frequency distribution functions calculated for the orthorhombic and triclinic PE crystals, b crystal structures of the various crystal forms of PE, c the comparison of the thermodynamic functions between the calculated and observed values of orthorhombic PE, and d the comparison of the calculated thermodynamic functions between the orthorhombic and triclinic PE crystals. Reprinted from Ref. [64] with permission of the American Chemical Society, 1975

560

5 Structure Analysis by Vibrational Spectroscopy

If there is any structural disorder in the crystal, which will contribute additionally to the entropy term and so the stability might be inversed in a case. Notes (Brillouin zone) In Fig. 5.98, for example, the dispersion curves are shown in the q range of –π/2a ~ + π/2a. These curves are repeated infinitely by extending the q range. The first zone of –π/2a < q < + π/2a is basically important and called the 1st Brillouin zone. The extended zones are the second, third, … zones. The points of the Brillouin zone are named the  point (q = 0), , , etc. depending on the q values (In the actual case the 3D Brillouin zones must be considered).

5.10 Analysis of Vibrational Spectra Characteristic of Polymers 5.10.1 Disorder and Vibrational Spectra The regular crystal lattice exhibits the IR and Raman spectra obeying the rules originating from the factor group treatment. Some bands are symmetrically forbidden and some other bands are optically active. However, when the crystal is disordered by any cause, the forbidden bands are optically active and the spectra become more or less complicated. The amorphous phase is a typical case where no regular structure exists in the long range. The prediction of a vibrational spectrum of such disordered structure is made by calculating the density of phonon g(ν).When the degree of disorder from the originally regular structure is not very high, the g(ν) is estimated by integrating the dispersion curves of the original structure. The integration is made as illustrated in Fig. 5.115. The number of phonons belonging to the 1cm-1 span on the dispersion curves is counted. Of course, the density of phonons does not give the

Fig. 5.115 Vibrational frequency-phase angle dispersion curve and the corresponding distribution g(ν)

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

561

intensity information. The spectrum due to the defect is expressed as I (v) = D M 2 g(v)

(5.265)

where D is the defect concentration, M is the transition dipole moment induced by the perturbation due to the defect, and g(ν) is the density of states of the perfect lattice. In more general cases, the calculation of the dispersion curves becomes hard. The density of states g(ν) is estimated by using the method proposed by Dean [67] and Zerbi [68]. The number of phonons at the eigenvalue ν is calculated. One example is presented here about the 1D chain. In the previous section, the equation of vibrations of two atoms repeated alternately along the infinitely long chain was derived as below.     m 1 d2 x1 /dt 2 = k(x2 − x1 ) − k x1 − x2

(5.266)

    m 2 d2 x2 /dt 2 = −k(x2 − x1 ) + k x1 − x2

(5.267)

By putting x1 = A1 exp[2πivt + iq X 1 ] and so on, the following sequential equation is obtained (ω = 2π ν): 

 (k12 + k2 1 ) − m 1 ω2 x1 + k12 x2 + k2 1 x2 = 0

(5.268)

Here, the force constant k ij is distinguished by giving the related atomic numbers. If we assume the atoms 1, 2, 1 ,… are different species, Eq. 5.268 changes in the following way:    ki,i+1 + ki−1,i − m i ω2 xi + ki,i+1 xi+1 + ki−1,1 xi−1 = 0

(5.269)

The atomic displacement coordinate x i is replaced with ui = mi 1/2 x i , then Eq. 5.269 is modified as        1/2 1/2 1/2 1/2 ki,i+1 + ki−1,i /m i − ω2 u i + ki,i+1 /m i m i+1 u i+1 + ki−1,i /m i−1 m i u i−1 = 0

(5.270) or   αi − ω2 u i + (βi+1 )u i+1 + (βi )u i−1 = 0 β0 = βN = 0 (i = 1, 2, . . . , N )

(5.271)

The definition of α i and β i are known by comparing these two equations. This equation is a generalized form of the equation derived for the 1D chain of the regularly repeated two atoms. By setting the mass and force constant of individual atoms

562

5 Structure Analysis by Vibrational Spectroscopy

randomly, we can solve the equation of the vibrational problem for the disordered chain. The prediction of g(ω) is made by knowing the eigenvalues of Eq. 5.271. However, how can we solve the Eq. 5.271 with the tremendously large N? The negative eigenvalue theorem is utilized for this purpose [67]. The following partition matrix M is used, which corresponds to the determinant of the secular equation related to Eq. 5.271. ⎛

α1 ⎜β t ⎜ 1 ⎜ M=⎜ ⎜ ⎝ 0

⎞ β2 0 ⎟ α2 β3 ⎟ ⎟ ...... ⎟ ⎟ ⎠ ...... βN tαN

(5.272)

The negative eigenvalue theorem is expressed as below. The matrix X of a system of dimension N is divided into the m blocks with the matrices Ai and Bi as shown below, where Ai is a square matrix of dimension di (i = 1 ~ m), Bi is of dimension d i-1 × d i , and d 1 + d 2 + … + d m = N. ⎛

A1 B 2 ⎜ B t A2 B 2 1 X =⎜ ⎝ ··· 0

0

⎞ ⎟ ⎟ ⎠

(5.273)

B tm Am

If the number of real negative eigenvalues of the matrix X is expressed as η(X), then, the number of the negative eigenvalues of the matrix X – xE is calculated using the following equation: η(X − x E) = i=1∼m η(U i ) U i = Ai − x E i − B it U −1 i−1 B i U 1 = A1 − x E 1

(5.274)

The X − xE is the matrix having the negative eigenvalue smaller than the critical eigenvalue x. For the above-mentioned 1D vibrating system, the negative eigenvalue theorem is expressed as   η M − ω2 E = i=1∼N η(h i ) = number of negative h i s where h i = αi − ω2 − βi2 / h i−1 (i = 2, 3, . . . , N ) h 1 = α1 − ω2

(5.275)

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

563

This theorem was applied to the matrix M, and the resultant g(ω) is shown in Fig. 5.116. The randomly generated chains consisted of 8000 atoms, where the atomic

Fig. 5.116 Squared frequency distribution g(ω2 ) calculated for a chain consisting of heavy and light atoms. The cL is the fraction of the light atoms. The mass ratio is 2:1. The total number of atoms is 8000. Reprinted from Ref. [67] with permission of American Physical Society, 1972

564

5 Structure Analysis by Vibrational Spectroscopy

mass ratio was 2:1 and the equal force constants were used. The percentage of light atoms was varied from 5 to 95%. In the case of cL = 0.05, the main peak corresponds to the edges of the dispersion curves of the heavy atom chain. On the other hand, the peak detected at ω2 = 1.0 (cL = 0.95) corresponds to the edge of the dispersion curve of the light atom chain. By the introduction of a random arrangement of light and heavy atoms along the chain, the many peaks are detected. For example, peak A corresponds to the local vibration of ..HHHLHHH.. structure. Peak G corresponds to the structure of ..HHLLHH… and F to the local structure ..HHLHLHH… Let us see the case of conformationally disordered polymer. The introduction of conformational defects is made by using the B matrix introduced in Sect. 5.8. As already mentioned, the internal displacement coordinates and the Cartesian displacement coordinates are connected by the B matrix. The components of B matrix are calculated by referring to the table of S vectors. The atomic positions are generated from those of the starting units by using the translational vector and torsional rotations (Eyring Transformation equation).

(5.276)

By applying the atomic mass matrix M i , the dynamical matrix (DX = M −1/2 Bt FR BM −1/2 ) is given as

(5.277)

where −1/2 

Doi = M i

−1t −1 2 1t 1 0t 0 B 2t i−2 F Ro B i−2 + B i−1 F Ro B i−1 + B i F Ro B i + B i+1 F Ro B i+1

1 1t 2 1t 0 0t 1 + B 2t i−2 F Ri−2 B i−1 + B i−1 F Ri−2 B i−2 + B i−1 F Ri−1 B i + B i F Ri−1 B i−1 +  −1/2 −1 o ot +B −1t i+1 F Ri B i + B i F Ri B i+1 M i −1/2  −1t 1 1t 2 2t 2 B i+1 F Ro B 0i+1 + B 0t D+1 i F Ro B i + B i−1 F Ro B i−1 + B i−2 F Ri−2 B i−1 i = Mi −1t 1 0t 2 0t 0 1 + B 1t i−1 F Ri−1 B i + B i F Ri−1 B i−1 + B i F Ri B i+1 + B i+1 F Ri−1 B i  −1/2 −1t +B i+1 F Ri+1 B −1 i+2 M i+1 −1/2  0t 1 1t 2 0t 1 B i F Ro B 2i + B −1t Di+2 = M i i+1 F Ro B i+1 + B i−1 F Ri−1 B i + B i F Ri B i+1  −1/2 −1t 2 0 +B −1t i+1 F Ri B i + B i+1 F Ri+1 B i+2 M i+2   −1/2 −1/2 −1t 2 0t 2 1 D+3 B −1t i = Mi i+1 F Ro B i+1 + B i F Ri B i+1 + B i+1 F Ri+1 B i+2 M i+3  −1/2 −1/2  −1t B i+1 F Ri+1 B 2i+2 M i+4 D+4 i = Mi

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

565

Fig. 5.117 Comparison of the observed IR spectra of polyethylene with the calculated results. a at room temperature and b in the melt or at 160 °C. Reprinted from Ref. [68] with permission of Taylor & Francis, 1971

FRo is the quadratic force constants matrix of the interactions of atoms within the same chemical unit and FRi is for the interactions between the i-th and (i+1)-th chemical units. The g(ω) is calculated by using the thus-obtained D matrix.

566

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.118 Helical-handedness reversal motion of PTFE chains

Zerbi et al. calculated the g(ω) for the solid and molten states of polyethylene and compared with the IR spectra observed at room temperature and above the melting point, respectively [68]. Figure 5.117(a) is the case of room temperature. In addition to the normal mode bands of regular trans-zigzag chain conformation, the bands at around 1000 cm–1 are due to the TTTXTTT sequences, where X is such gauche conformation as G, GG, GTG, and GTTG. In the molten state (Fig. 5.117 (b)), the crystalline bands disappeared, and the gauche bands increased in intensity (or population). The observation of 1350–1365 cm−1 bands indicates the existence of short trans sequences in the melt. Another example is the IR spectra of PTFE above the phase transition point 19 °C [69]. The PTFE takes a (15/7) helical conformation above 19 °C. As the temperature is increased, the IR bands around 630 cm−1 become stronger, in particular the parallel component. At the same time, the bands at 300 cm−1 disappeared. The parallel band is observed at 778 cm−1 , which does not correspond to the helical band. The PTFE chains with the various conformations (2/1), (15/7), (10/3), and (4/1) were built and their g(ω)s were calculated. The appearance and disappearance of the characteristic peaks in g(ω) correspond well to the observed spectral change. The conclusion is that PTFE shows the conformational disordering from (15/7) helix at the low temperature to the mixture of helical and zigzag chains at the high temperature. This model is consistent with the X-ray diffraction data analysis about the diffuse scatterings. The right-handed and left-handed chain segments change along the helical chain axis, and the trans segments are existent on the way of exchange between these segments [70] (Fig. 5.118, see Sect. 1.15.1.4).

5.10.2 Band Intensity Not only the vibrational frequencies but also the intensities of the bands are important for the study of the structure. One method to predict the band intensity is the electrooptical theory [71–75]. As already mentioned, the infrared intensity of a band is given as follows using the transition dipole moment: I ∝ |∂ M/∂ Q k |2

(5.278)

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

567

M is the total dipole moment of the molecule and Qk is the k-th normal mode. M = jejμj

(5.279)

Here, ej is the unit vector of the j-th bond and μj is the magnitude of the bond dipole moment. The differentiation of M by the normal mode Qk gives     ∂ M/∂ Q k = j ∂ e j /∂ Q k μoj + j ej ∂μj /∂ Q k

(5.280)

Since the Qk -coordinate is related to the internal displacement coordinates Rj by R j = k L jk Q k

(5.281)

where L jk is the L vector component of the Rj related to the Qk mode.       ∂ ej /∂ Q k = n ∂ ej /∂Rn (∂Rn /∂ Q k ) = n ∂ ej /∂Rn L nk       ∂μj /∂ Q k = n ∂μj /∂Rn (∂Rn /∂ Q k ) = n ∂μj /∂Rn L nk Then,     ∂ M/∂ Q k = j n ∂ ej /∂Rn L nk μoj + j n eoj ∂μj /∂Rn L nk

(5.282)

  The derivative ∂ e j /∂Rn is obtained in the following way. The position vectors of the two atoms forming a bond vector rpq are expressed using X p and X q . The unit bond vector is epq .   r pq = X q − X p , epq = X q − X p /rpq

(5.283)

The differentiation of epq with respect to an internal coordinate Rn is  o   o  − epq ∂rpq /∂ Rn /rpq ∂ epq /∂ Rn = ∂ X q /∂ Rn − ∂ X p /∂ Rn /rpq

(5.284)

The terms ∂X p /∂Rn and ∂X q /∂Rn are the inverses of the s vectors related to the B matrix, which are now expressed as B−1 . Let us imagine a water molecule having the bond lengths r 12 (= R1 ) and r 23 (= R2 ) and the bond angle φ (= R3 ), where the atomic numbers are 1, 2 and 3. The above equation can be applied to these 3 internal coordinates R1 ~ R3 . X2 R1 X1

R3

R2 X3

568

5 Structure Analysis by Vibrational Spectroscopy

∂ e12 /∂ R1 = (∂ X 2 /∂ R1 − ∂ X 1 /∂ R1 )/R1◦ − e◦12 (∂ R1 /∂ R1 ) = (∂ X 2 /∂ R1 − ∂ X 1 /∂ R1 )/R1◦ − e◦12 /R1◦ ∴ ∂ e12 /∂ R1◦ = [(B −1 )21 − (B −1 )11 ]/R1◦ − e◦12 /R1◦ [∂ e12 /∂ R1 ]x = (1/R1◦ )[(B −1 )21 x − (B −1 )11x − e◦12x ] [∂ e12 /∂ R1 ] y = (1/R1◦ )[(B −1 )21 y − (B −1 )11y − e◦12y ] [∂ e12 /∂ R1 ]z = (1/R1◦ )[(B −1 )21 z − (B −1 )11z − e◦12z ] ∂ e12 /∂ R2 = (∂ X 2 /∂ R2 −∂ X 1 /∂ R2 )/R1◦ − e◦12 (∂ R1 /∂ R2 )/R1◦ = (∂ X 2 /∂ R2 )/R1◦ − 0 ∴ ∂ e12 /∂ R2 = [(B −1 )22 − (B −1 )12 ]/R1◦ − 0 [∂ e12 /∂ R2 ]x = (1/R1◦ )[(B −1 )22 x − (B −1 )12x − 0] [∂ e12 /∂ R2 ] y = (1/R1◦ )[(B −1 )22 y − (B −1 )12y − 0] [∂ e12 /∂ R2 ]z = (1/R1◦ )[(B −1 )22 z − (B −1 )12z − 0] ∂ e12 /∂ R3 = (∂ X 2 /∂ R3 − ∂ X 1 /∂ R3 )/R1◦ − e◦12 (∂ R1 /∂ R3 )/R1◦ = (∂ X 2 /∂ R3 − ∂ X 1 /∂ R3 )/R1◦ − 0 ∴ ∂ e12 /∂ R3 = [(B −1 )23 − (B −1 )13 ]/R1◦ − 0 [∂ e12 /∂ R3 ]x = (1/R1◦ )[(B −1 )23 x − (B −1 )13x − 0] [∂ e12 /∂ R3 ] y = (1/R1◦ )[(B −1 )23 y − (B −1 )13y − 0] [∂ e12 /∂ R3 ]z = (1/R1◦ )[(B −1 )23 z − (B −1 )13z − 0]

Similarly, by defining e23 = X 3 – X 2 , [∂ e23 /∂ R1 ]x = (1/R2◦ )[(B −1 )31x − (B −1 )21x − 0] [∂ e23 /∂ R1 ] y = (1/R2◦ )[(B −1 )31y − (B −1 )21y − 0] [∂ e23 /∂ R1 ]z = (1/R2◦ )[(B −1 )31z − (B =1 )21z − 0] [∂ e23 /∂ R2 ]x = (1/R2◦ )[(B −1 )32x − (B −1 )22x − e23x ] [∂ e23 /∂ R2 ] y = (1/R2◦ )[(B −1 )32y − (B −1 )22y − e23y ] [∂ e23 /∂ R2 ]z = (1/R2◦ )[(B −1 )32z − (B −1 )22z − e23z ] [∂ e23 /∂ R3 ]x = (1/R2◦ )[(B −1 )33x − (B −1 )23x − 0] [∂ e23 /∂ R3 ] y = (1/R2◦ )[(B −1 )33y − (B −1 )23y − 0] [∂ e23 /∂ R3 ]z = (1/R2◦ )[(B −1 )33z − (B −1 )23z − 0]

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

569

(5.285) Therefore, ∂ e/∂ R = R−1 [ · B −1 − E : 0]

(5.286)

Here, R−1 is the K × K diagonal matrix of the equilibrium bond lengths; K is 3x the number of bonds in the molecule. Δ is the K × n matrix of the difference between the derivatives ∂ X i /∂ R j and ∂ X k /∂ R j . n is 3x the number of atoms in the molecule. For the components of the matrix Δ, the atom of the end vector is −1 and the atom of the top vector is + 1. E is the K × K diagonal matrix of the equilibrium bond

570

5 Structure Analysis by Vibrational Spectroscopy

vectors e, and 0 is a K × (M–K) null matrix. B−1 is the inversed matrix of B matrix and is given as B−1 = M −1 Bt G−1 , as already mentioned.   In the actual calculations, the electro-optical parameters are ∂μ j /∂Rn and μ°j , which are empirically determined so that the observed intensity of the bands is reproduced well. Several examples of the electro-optical parameters are shown below for the CH2 group. μo (C − H) = 0.244 Debye ∂μ/∂ R

CH dipole change by CH stretching

− 0.906 Debye/Å

CC dipole change by CC stretching

0.010 Debye/Å

CH dipole change by HCH bending CH dipole change by HCC bending

± 0.027 Debye/rad 0.065 Debye/rad

The actually calculated IR spectrum of unoriented polyethylene crystal is compared with the observed spectrum (at a temperature 7 K) in Fig. 5.119 [76]. The force constants used are those shown in Sect. 5.8.6. The CH2 bending and rocking peaks were reproduced relatively well, but the CH2 wagging band is overwhelmingly higher compared with the actually observed band which is quite weak in spite of the mode with the transition dipole parallel to the chain axis. For Raman spectral intensity, the similar equation is derived as follows for the i-th Q mode [71–75].   2 Ii ∝ (v0 − vi )4 45 < α i >2 +13γ i

(5.286)

where < αi >= (1/3) u=1−3 (∂αuu /∂ Q i )   γ i 2 = (1/2) 3 uv=1−3 (∂αuv /∂ Q i )2 − 9 < αi >2 The matrix expression of ∂αuv /∂ Q i is given as below using the electrical optical parameters:       ˜ < α >/∂ R + eou eov t − (1/3)δuv I˜ ∂γ/∂ R + γo ∂ eou eov /∂ R L i ∂αuv /∂ Q i = δuv I(∂

(5.287)

where δuv is a Kronecker’s delta, K is a number of the bonds, I˜ is a Kdimensional vector of all unity entries, and the polarizability anisotropy between TO the longitudinal and transverse vibrations γok = αLO k − αk .

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

571

Fig. 5.119 Comparison between the observed and calculated IR spectra of polyethylene crystal; a PE-H and b PE-D [76]

 o o  o −1 −1 −1 o o ∂ eu ev /∂ R = E ov R−1 ΔB −1 Eu Ev : 0 u + E u R ΔB v − 2 R

(5.288)

R−1 ,  and 0 are the matrices defined already. E ou and E ov are the u and v components −1 of the K x K Eo matrix, respectively. B −1 u and B v are the u- and v-components of −1 the B matrix, respectively. The simulated Raman spectrum of the unoriented polyethylene crystal is compared with the observed spectrum, as shown in Fig. 5.120 [76].

5.10.3 Band Width and Molecular Motion When we compare the IR or Raman spectra between the liquid and solid states of the same compound, the band width is narrower for the solid state. This suggests the band width is related to the degree of molecular motion. In general, the band width is inversely proportional to the relaxation time τ: ν1/2 ∝ 1/τ; shorter τ means the higher activity of molecular motion. For example, Fig. 5.121 shows the temperature

572

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.120 Comparison between the observed and calculated Raman spectrum of polyethylene crystal [76]. The Raman spectrum was measured at 7 K

dependence of the IR spectra measured for atactic polystyrene of almost perfectly amorphous state. Passing through the glass transition temperature (T g ) the halfwidth of the amorphous bands increases remarkably, reflecting the increment of the micro-Brownian motion of the chains [77]. In the vicinity of T g, the band width became narrower, which might correspond to a critical phenomenon at the start of the molecular motion though the details are not clear. Figure 5.122a shows the Raman spectra of n-alkane in the solid and molten states [78–81]. The Raman band at 2883 cm−1 in the solid state is shifted to 2891 cm−1 in the melt. At the same time, this band shows the remarkable increase of the half-width in the transition from the solid to the molten state, as plotted in Fig. 5.122b. The band is assigned to the anti-symmetric CH2 stretching mode belonging to the symmetry species B3g . The symmetric stretching band of Ag symmetry species at 2850 cm−1 does not show such a very large change in the half-width. The difference originates from the effect of the rotational motion of the zigzag chain around the chain axis. In general, the Raman band profile is expressed as a Fourier transform of a correlation function of the polarizability tensor component αij in the following form.  Iij (ω) ∝

< αij (0)αij (t) > e−i(ω−ωo )t dt

(5.289)

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

573

(a)

(b)

(a)

Fig. 5.121 a IR spectra of atactic polystyrene in the heating process, and b the temperature dependence of the half width and peak position estimated for the 906 cm-1 band. Reprinted from reference [77] with permission of the American Chemical Society, 2002

where ω is an angular frequency and ωo is that of the vibrational mode. The reorientational motion of a zigzag chain around the chain axis is assumed to occur between several site directions (It must be noted that relatively short n-alkane molecule such as n-C20 H42 takes almost the zigzag form even in the molten state).

574

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.122 a b Raman spectra of n-C18 H38 in the solid (291 K) and in the melt (301 and 538 K) and the temperature dependence of the half-width of the bands of anti-symmetric (νa ) CH2 stretching mode, symmetric (νs ) CH2 stretching mode, and the CH2 twisting mode. c d Raman spectra of n-C20 H42 in the urea adduct in the solid and hexagonal phases and the temperature dependence of the half-width of the bands [78]. Reprinted from Ref. [78] with permission of the American institute of Physics, 1986

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

575

Using the transition probability K between the adjacent sites, the following equation is obtained finally: 2 −3K t e for the B3g mode < αij (0)αij (t) >∝ αxy 2 for the Ag mode (5.290) αzz  2  2 −3K t 2 αxx + αyy + αxx − αyy e for the Ag mode

The band profile is related to the term e−3Kt , which is contained for the B3g mode but is not for the Ag zz-component, reflecting on the difference in the half-width of the corresponding band (The Ag xx-component is almost constant since αxx ∼ αyy ).

5.10.4 Progression Bands 5.10.4.1

n-Alkanes

As shown in Fig. 5.123, the IR and Raman spectra of n-paraffins show many bands beyond the expectation from the relatively simple spectra of polyethylene [40–46, 82]. The band positions change with the length of n-paraffin molecule. They are called the progression bands. The band positions are predictable from the dispersion curves of polyethylene of planar-zigzag conformation. As already mentioned in Sect. 5.8.2 the end groups are assumed to be fixed (fixed ends) or free (free ends). Depending on the difference of the constraining condition of the end groups, the optically active phase angle between the CH2 units is changed as shown below. For the number of oscillators N, (fixed ends) δ = π k/(N + 1) (free ends) δ = π(k − 1)/N where k = 1, 2, …, N. The number of CH2 units contained is different depending on the vibrational mode of CH3 (CH2 )n−2 CH3 (the CH3 local modes are not coupled with CH2 group modes since the frequency region is different),

576

5 Structure Analysis by Vibrational Spectroscopy

For the CH2 stretching(ν), bending (δ), wagging (w), twisting (t), and rocking (r) modes For the CC skeletal stretching For the CCC skeletal bending () For the CCCC torsional mode (T)

N N N N

=n−2 =n−1 =n =n−4

Fig. 5.123 Infrared spectra of n-C23 H48 (upper) and n-C28 H58 (lower) in the solid state (-180 °C) to show the progression bands. Reprinted from Ref. [44] with permission of Elsevier, 1961

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

577

The simply coupled oscillators are drawn as shown in Fig. 5.124. The phase difference δ is obtained as indicated in this figure. The phase angle is defined by using a glide plane of PE chain (by Snyder [40–44]). (The phase difference is sometimes defined on the basis of C 2 s symmetry, by Tasumi [58].) Figure 5.125 is the dispersion curve of a CH2 rocking mode [r(CH2 )] of an isolated PE zigzag chain. The r(CH2 ) bands of CH3 (CH2 )6 CH3 (n-octane) may appear at the wavenumbers given by open circles; the phase difference δ = π/7, 2π/7, 3π/7, 4π/7, 5π/7, and 6π/7. This molecule has a point group C 2h . Among these vibrational modes, the modes of Au and Bu species are IR active, and Ag and Bg are Raman active. Some of the wagging modes are drawn as shown in Fig. 5.126. The arrows show the atomic displacements. The symmetric relation of these arrows corresponds to the symmetry species Bu and it is IR active. Since all the transition dipoles are parallel to the chain axis, the band is called the parallel band. Figure 5.127 shows the case of CH2 rocking

Fig. 5.124 Simply coupled oscillation modes for C6 molecule subjected to the fixed ends condition

Fig. 5.125 Dispersion curve of CH2 rocking mode [r(CH2 )] of an isolated PE zigzag chain

578

5 Structure Analysis by Vibrational Spectroscopy

 = 7

Bu (IR active, parallel) h C2 i

= /7

-1

-1

+1

h

i C2 = 6 /7

Ag (Raman) C2 i h +1

+1

+1

Fig. 5.126 The CH2 wagging mode [w(CH2 )] of n-C8 H18 under the fixed end condition

Au (IR) C2 i

 = 7

+

+

+

+1

+ +

+

+

-

-1

h -1

= /7

 = 6 7

+

+

+ Bg (Raman) C2 i h -1

-

-

+1

-1

-

Fig. 5.127 The CH2 rocking mode [r(CH2 )] of n-C8 H18 under the fixed end condition

mode. The symmetry and optical activity of a finite alkane chain are given in Table 5.14, where the z-axis // the chain axis, the x-axis is perpendicular to the z-axis and in the zigzag plane. For n-paraffin molecule with odd-number carbon atoms, the progression modes are drawn as shown in Fig. 5.128. One example is for the case of n-C7 H16 . The point group of this molecule is C 2v .

R(xy,yz,zx)

R(x 2 ,y2 ,z2 )

Activity

(n−2)/2

(n−2)/2

(n−2)/2

0

0

νs

δ

ω

t

r

(n−2)/2

0

Bending

Torsion

1

1

0

Antisymmetric (ν, δ, r)

Symmetric (ν, δ, r)

Torsion

CH3 modes

n/2

Stretching

CC skeletal modes

0

νas

1

0

1

(n−4)/2

0

0

(n−2)/2

(n−2)/2

0

0

0

(n−2)/2

Bg

Species

CH2 modes

n = even point group C 2h

Ag

n-Cn H2n+2

0

1

1

0

(n−2)/2

(n−2)/2

0

0

(n−2)/2

(n−2)/2

(n−2)/2

0

IR(z,x)

Bu

1

0

1

(n−2)/2

0

0

(n−2)/2

(n−2)/2

0

0

0

(n−2)/2

IR(y)

Au

0

1

1

0

(n−1)/2

(n−1)/2

0

0

(n−3)/2

(n−1)/2

(n−1)/2

0

R(x 2 ,y2 ,z2 ),IR(x)

A1

1

0

1

(n−3)/2

0

0

(n−3)/2

(n−1)/2

0

0

0

(n−3)/2

R(yz)

A2

n = odd point group C 2v

0

1

1

0

(n−3)/2

(n−1)/2

0

0

(n−1)/2

(n−3)/2

(n−3)/2

0

R(xz),IR(z)

B1

Table 5.14 Symmetry and number of progression bands of n-alkane. Modified from Refs. [40, 41] with permission of Elsevier, 1963

1

0

1

(n−3)/2

0

0

(n−1)/2

(n−3)/2

0

0

0

(n−1)/2

R(xy),IR(y)

B2

5.10 Analysis of Vibrational Spectra Characteristic of Polymers 579

580

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.128 The CH2 wagging and rocking modes of n-C 7 H16 under the fixed end condition

The analysis of the progression bands observed for many n-alkanes established the reasonableness of the dispersion curve of an isolated PE chain (Fig. 5.129).

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

581

Fig. 5.129 Dispersion curve of r(CH2 ) (π) and t(CH2 ) (0) mode of a planar-zigzag polyethylene chain. The phase relation is with respect to the c-glide plane. Reprinted from Ref. [40] with permission of Elsevier, 1963

5.10.4.2

Progression Bands of Polyethers, Nylons, Polyesters, and Oligomers

The progression bands are observed not only for n-alkanes but also for aliphatic polymers with the finite methylene segments: -[(CH2 )m X]n - where X is oxygen (polyether), amide (nylon), and so on. For a series of polyethers -[(CH2 )m O]n -, the oxygen atoms are assumed to be the fixed ends [83]. The treatment of the progression bands can be made similarly to the above-mentioned n-alkanes with CH3 end groups. For polyamides and polyesters, the situation is slightly different. Let us consider nylon mn with the chemical formula -[NH(CH2 )m NHCO(CH2 )n−2 CO]n - [84–87]. Figure 5.130 shows the temperature-dependent IR spectra observed for a model compound of nylon 1010, that is, N1010CC with the chemical formula CH3 (CH2 )9 – NHCO–(CH2 )8 –CONH-(CH2 )9 CH3 . The IR spectrum measured at room temperature is interpreted. The amide bands are picked up by referring to the literatures in which the vibrational modes of amide group are assigned (see Table 5.15). Many other bands detected in the wavenumber region of 700–1100 cm−1 are assigned to the progression bands of methylene segments. If we follow the idea derived for n-alkane molecules, the methylene moieties correspond to CH3 (CH2 )n CH3 and CH3 (CH2 )m−2 CH3 . The corresponding phase angles δ = kπ/(n + 1) and kπ/(m − 1), respectively. By using the dispersion curves obtained

582

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.130 Temperature dependence of IR spectra measured for a model compound N1010CC: a 400–1100 cm−1 , b 750–1190 cm−1 . Reprinted from Ref. [86] with permission of Elsevier, 2003

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

583

Table 5.15 Vibrational modes of amide unit Amide vibrational mode Wavenumber, cm−1

Vibrational mode

A, B

Overtone of Amide II, Fermi resonance of NH stretching mode

~3300, ~3100

I

~1700

C=O stretching

II

~1500

CN stretching (44%) + NH in-plane bending (63%)

III

~1300

CN stretching (5%) + NH in-plane bending (29%)

IV

~630

C=O in-plane bending

V

~650

NH out-of-plane bending

VI

~600

C=O out-of-plane bending

VII

~400, ~200

CN torsional mode

for a series of n-alkanes, the band positions corresponding to these phase angles can be predicted, which are compared with the observed band positions as shown in Fig. 5.131. Unfortunately, the thus-predicted band positions are not in good agreeCH2 rocking modes

C-C streching modes (a) Observed Spectra

(a) Observed Spectra Nylon 10/10

Nylon 10/10

N1010NN

N1010NN

N1010CC

N1010CC N10

N10 1050

1000

950

900

850

800

750

1150

1100

1050

1000

950

1100

1050

1000

950

1100

1050

1000

950

(a) end CH2 erased Nylon 10/10(10,8)

Nylon 10/10 (10,8) (10,9)

N1010NN

(10,9)

N1010CC

(10,8)

N1010CC

(10,8)

N10

(10,9)

N10

(10,9)

N1010NN

1050

1000

950

900

850

800

1150

750

(b) All CH2 considered Nylon 10/10 (10,10)

Nylon 10/10 (12,10) N1010NN

(12,10)

N1010NN

N1010CC

(11,10)

N1010CC

(10,10)

N10

(11,10)

N10

(10,10)

1050

1000

950

900

850

Wavenumber / cm-1

800

750

(10,10)

1150

Wavenumber / cm-1

Fig. 5.131 Comparison of the IR band positions observed for N1010CC with those predicted for the finite methylene segments: a CH2 end groups of the both sides are erased, and b all the CH2 units are included

584

5 Structure Analysis by Vibrational Spectroscopy

ment with the actually observed ones. It was found to be better to use the following equations for the good agreement between the observed and predicted band positions: δ = kπ/(n − 1) and kπ/(m − 3) which corresponds to the erase of CH2 units adjacent to the amide group from the original CH2 segments. In fact, a good correspondence is found as seen in Fig. 5.131 [84]. According to the NMR measurement, the CH2 units adjacent to the amide group are more active than the inner CH2 units because of the electro-drawing property of the amide group. The good agreement can be seen also for the various nylon mn samples. Figure 5.132 shows the IR spectra measured for the various model compounds of nylon. The bands are named 8P3C, for example. P indicates the rocking-twisting mode. “3” is the k value. “C” indicates the CH2 segment sandwiched by CO groups. The erase of 2CH2 units from (CH2 )8 segments of CO(CH2 )8 CO gives the (CH2 )6

Fig. 5.132 IR spectra measured for nylon 1010 and its model compounds

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

585

Fig. 5.133 Dispersion curves of CH2 rocking mode and CC stretching mode calculated for a planarzigzag polyethylene chain. The open circles are IR bands observed for the various nylon and nylon model compounds. Reprinted from Ref. [84] with permission of John Wiley & Sons, 2003

segment. By adding the CH3 groups at both the ends, we know this segment corresponds to n-C8 H18 molecule. The “8” indicates the thus-created alkane molecule. In other words, the observed band is assigned to the rocking mode of n-C8 H18 molecule with a phase angle δ = kπ/(n + 1) = 3π/7 (k = 3 and an effective number of CH2 oscillators n = 6). These bands are fitted well to the dispersion curves of the planar-zigzag polyethylene chain (Fig. 5.133). In Fig. 5.130b, we notice these bands decrease in intensity and become broader as the temperature is increased, and some new bands are detected. The bands detected at room temperature can be assigned to above-mentioned progression modes. The 10N and 8C indicate the bands originating from the NH(CH2 )10 NH and CO(CH2 )8 CO segments. The bands C9 and C7 correspond to the progression bands of n-alkanes (nC9 and n-C7 ) with shorter chain length than those of the original CH2 segments. That is to say, the zigzag chain segments at room temperature change to the shorter zigzag segments at a higher temperature by introducing gauche bonds. The band intensity is plotted against temperature. At around 80 °C the remarkable intensity change occurs, which is called the Brill transition temperature. In this way, the progression bands can be utilized for the detection of conformational change of trans-zigzag chain. Many aliphatic nylons exhibit the Brill transition at various temperatures depending on the methylene segmental lengths [88, 89]. Figure 5.134 shows the case of nylon 1010. The X-ray diffraction peaks 110/010 and 100 are separated at room temperature but they approach and merge into one peak above the Brill transition temperature of 140–180 °C, corresponding to the endothermic peak of the DSC thermogram. In parallel, the CH2 bands change their intensities: the trans bands of the original long methylene segments decrease in intensity and the bands of shorter CH2 segments start to appear. The amide V and VI bands observed in around 600 cm−1 region shift their positions only slightly in the Brill transition temperature region. The amide A band, which is sensitive to the NH…O hydrogen bond, shifts the position with increasing temperature but the shift is not very remarkable. This means

586

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.134 Temperature dependence of the various experimental data collected for nylon 1010 in the Brill transition. Reprinted from Ref. [85] with permission of Elsevier, 2003

the hydrogen bonds do not change very much during the Brill transition process. In this way, the triclinic crystal of nylon 1010 shows the Brill transition, in which the methylene segments exhibit the conformational disordering with the intermolecular hydrogen bonds kept almost unchanged. The averaged chain shape projected along the chain axis is almost cylindrical. The apparent merge of two X-ray peaks are due to such a structural change that the molecular chains approach each other to give apparently the pseudo-hexagonal packing structure, but the intermolecular hydrogen bonds constrain the free rotations of the molecular chains around the chain axis. In the higher temperature region immediately below the melting temperature, nylon 1010 and many other nylons show another phase transition, in which the hydrogen bonds are broken totally and the molecular chains rotate almost freely around the chain axis to show the hexagonal packing structure. The molecular chain conformation is disordered more drastically through the T-G exchange. The details will be described in a later section. The progression bands are observed also for aliphatic polyesters with long methylene segments -[(CH2 )m OCOCOO]n - and the regularly separated polymethylene chain by Br or Cl atoms at the constant periods -[(CH2 )m CHX]n - with X= Br or Cl [90]. For example, PE21Br is the polyester with m= 20 and X = Br. The Xray structure analysis revealed the crystal forms I and I’ take almost fully extended conformation with the all trans-zigzag CH2 segments (see Fig. 5.135). The polarized IR spectra are compared with the spectra of PE. The progression bands originated from the finite length of CH2 segment are detected for PE21Br and not for PE. In the

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

587

Fig. 5.135 Crystal structure and the polarized IR spectra of PE21Br. Red and green curves show the spectra with the electric vector of the incident IR beam perpendicular and parallel to the chain axis, respectively. Reprinted from Ref. [90] with permission of the American Chemical Society, 2014

latter case, only the CH2 vibrational modes with δ = 0 and π are optically active. The progression bands can be interpreted using the data of CH3 (CH2 )18 CH3 , indicating that the CH2 unit adjacent to the ester group must be erased from the coupled CH2 sequence, similarly to the case of aliphatic polyamides. The progression bands are not observed only for the methylene vibrational modes but also for a sequence of the other type of monomeric units. The typical examples are seen for poly(vinylidene fluoride) oligomers. Figure 5.136 shows the IR and Raman spectra measured at room temperature for a series of PVDF oligomers with the chemical formula CF(CF3 )2 (CH2 CF2 )n -I (n = 6 – 10) [91]. As will be described in Sect. 5.14.3.2, the molecular shape is essentially the same ¯ The strong main bands in the IR and Raman spectra as that of PVDF form II, TGTG. are almost common to all the members, while some weak bands are found to change their positions depending on the chain length n. These bands are considered to be the progression band components of CF2 bending mode by referring to the vibrational analysis of PVDF form II [37]. However, it is quite difficult to assign the phase angles φ predicted to these bands. The vibrational frequency-phase angle dispersion curves were calculated for PVDF form II chain using the GF-matrix method, which were

588

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.136 IR (left) and Raman spectra of a series of PVDF oligomers. Reprinted from Ref. [91] with permission of the American Chemical Society, 2002

used for the estimation of the most suitable phase angles for the observed bands. The results are shown in Fig. 5.137, where the band positions detected for all the oligomers are plotted. The fairly good agreements between the observed data points and the calculated dispersion curves suggest the reasonableness of the calculated dispersion curves.

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

589

Fig. 5.137 The fitting of the observed peak positions to the dispersion curves calculated for PVDF form II chain. Reprinted from Ref. [91] with permission of the American Chemical Society, 2002

5.10.5 LAM 5.10.5.1

LAM of n-Alkanes

As shown in Fig. 5.138, in the Raman spectra of n-alkane liquid and solid, the relatively strong bands are observed in the low-frequency region [63, 92–94]. The band position shifts to the higher frequency side as the chain length of n-alkane is shorter. This band is called the accordion mode or longitudinal acoustic mode (LAM). In the fully extended zigzag chain of finite length, the CH2 units displace along the chain axis with a slightly different phase angle, where the displacement magnitude increase as the CH2 unit separates its position from the center of the chain. The precise analysis clarified the characteristic behavior of this band. The peak position is given as a function of p = m/n (Fig. 5.139). n is the number of C atoms. m is the order of LAM mode.   v cm−1 = A × p + B × p 2 + C × p 3 + D × p 4 + E × p 5 + F × p 6

(5.291)

The coefficients A, B, C, D, E, and F are the constants which were determined so that the LAM band frequencies observed for a series of n-alkanes were reproduced well by this equation: A = 2495 ± 86 cm−1 , B = − (5.867 ± 2.855) × 103 cm−1 , C = (5.253 ± 3.537) × 104 cm−1 , D = − (3.485± 2.058) × 105 cm−1 , E = (7.329 ± 5.676) × 105 cm−1 , and F = − (4.724 ± 5.964) × 105 cm−1 . From the mechanical point of view, this band corresponds to the longitudinal acoustic mode starting from

590

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.138 Raman spectra of n-C36 H74 at 300 K. Reprinted from Ref. [92] with permission of the American Institute of Physics, 1967

the origin of the dispersion curve. In the low-frequency region, the acoustic mode is expressed as ) k ν˜ (L AM) = 2π Lc

E ρ

(5.292)

where k is the vibration order (k = 1, 3, 5, … for Raman active modes), L is the straight stem length, c is the speed of light in the medium, E is Young’s modulus of the zigzag chain, and ρ is the density. The initial slope in the plot of ν˜ versus k gives the E

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

591

Fig. 5.139 LAM frequency plotted against p = m/n. Reprinted from Ref. [92] with permission of the American Institute of Physics, 1967

592

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.140 a Two types of crystal structures (Orth I and Orth II). b The Raman spectra of orth II crystal form measured for n-C28 , n-C30 , and n-C36 paraffins. c Mechanical model to explain the LAM band splitting. Reprinted from Ref. [98] with permission of the American Institute of Physics, 1983

value of 358 GPa, which is too high compared with the value estimated by the other methods. The overestimation of E may come from the effect of end CH3 groups. The LAM frequency is affected by the mass of the end group and the interaction between the end groups of the adjacent layers [95, 96]. The most explicit observation of the intermolecular CH3 interaction is seen for the splitting of the LAM band [97, 98]. For example, these alkane molecules are crystallized in several different packing modes (Fig. 5.140). In the case of monoclinic form, one layer is repeated along the c-axis. On the other hand, the orthorhombic form II shows the double-layer structure with alternately opposite orientations. Since the inner structure of one layer is essentially the same between these two phases, the polytype phenomenon occurs between them. The LAM bands split into two bands in the orthorhombic form, while only one LAM band is observed for the monoclinic phase [98]. The simple equation is derived by assuming a series model of two springs which correspond to the Young’s modulus (E) of a lamella and the interlamellar interaction (the force constant = f ) connecting the adjacent lamellae. The vibrational frequency of LAM mode is expressed as below: vk = βπ k/L + 2β f /(π Ek)

and

vk = βπ k/L

(5.293)

where β = (E/ρ)1/2 /(2π c) and k = 1, 3,… The substitution of the observed LAM frequencies of the split bands, Young’s modulus E = 281 GPa and f = 2.92 × 1010 GPa/m. The theoretically calculated modulus is about 310 GPa.

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

593

Fig. 5.141 a LAM frequencies (cm−1 ) and b the derived lamellar thickness (▲) compared with that estimated by the SAXS data (●) observed for the polyethylene lamellar mats annealed at the various temperatures. In (a), the annealing temperature is A: room temperature, B: 79 °C, C: 99 °C, D: 112 °C, E: 120 °C, and F: 130 °C. Reprinted from Ref. [99] with permission of the American Institute of Physics, 1971

5.10.5.2

LAM of Polymers

The equation of LAM versus L can be utilized for the estimation of lamellae thickness. The lamellar structure of polyethylene grown from the melt crystallization shows the LAM band. In this case, the extended chain with the length equal to the lamellar thickness (stem) vibrates under the surface effect due to the folding part. By changing the crystallization temperature the lamellar thickness changes, which reflects on the long period of stacked lamellae as known from the SAXS measurement [99–101]. Figure 5.141 shows the lamellar thickness evaluated from the SAXS measurement and LAM measurement. They show good correspondence with each other, although the contribution of the folded part to the LAM frequency might be a serious problem (refer to Fig. 5.141). The LAM mode can be detected also for such helical chains as polytetrafluoroethylene oligomers [102], it-polypropylene [103], polyoxymethylene [103], and poly(ethylene oxide) [104, 105]. Equation 5.292 was applied to the observed LAM frequency, giving Young’s modulus along the chain axis: E= 206 GPa (PTFE), 37 or 88 GPa (it-PP), 189 GPa (POM), and 90 GPa (PEO). These values are as a whole larger compared with those derived by the X-ray diffraction method under tension [106, 107].

5.10.5.3

LAM of H-type Molecule

As one application of LAM concept, let us see the case of a complicated alkane molecule with the H character shape as shown in Fig. 5.142 [47]. The n value is

594

5 Structure Analysis by Vibrational Spectroscopy

(a)

CH (CH )n-

CH (CH )n-

HOCH-(CH ) -CHOH (CH )nCH

(CH )n-

(b)

CH (CH )n-

CH

(c)

(CH )n- (CH )n-

HOCH-(CH ) -C (CH )n-

CH

CH

OH

CH

CH

Fig. 5.142 a H-type molecule, b and c actually analyzed molecular shape. Reprinted from Ref. [47] with permission of the American Chemical Society, 2011

3∼18. They are named Cn HOH. The crystal structures of these H-shape molecules were analyzed with an X-ray diffraction method. The molecular shape looks like a giraffe. The IR spectra of these H-type molecules are expected to show the progression bands depending on the CH2 units. At the same time, the Raman LAM bands are observed which originate from the three legs (or necks) of alkane chain segments. However, these legs are connected with each other. Does such connection part affect the LAM bands of the extended chain parts? Figure 5.143 shows the actually observed LAM spectra. As the alkane leg is longer, the Raman band position shifts toward the lower frequency side, indicating they are LAM bands. When the spectra are compared with those of linear n-alkane molecules, it is found that the main LAM bands of H-molecule correspond to those of n-alkane with the carbon number of 2n + 1: for example, the peak positions of LAM bands of

(a)

(b)

(c)

Fig. 5.143 a LAM bands of n-C25 and the corresponding H-type molecule. b The LAM vibrational mode calculated for the H-type molecule, and c the Raman spectra observed for a series of H-type molecules. Reprinted from Ref. [47] with permission of the American Chemical Society, 2011

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

(a)

595

(b)

Fig. 5.144 a Frequency dispersion curves plotted against s, which are calculated for a chain consisting of simply coupled oscillators under the conditions of fixed ends, free ends, and free end and fixed end. b The same as (a) but plotted against the phase angle φ. Reprinted from Ref. [47] with permission of the American Chemical Society, 2011

C12HOH are almost the same as those of n-C25 H52 . This means the LAM mode of the long leg is not affected by the side chain connecting at the center position of the long leg. This is reasonable since the LAM is a purely longitudinal vibrational mode of long wavelength (see Fig. 5.143b). In this way, the LAM mode of the longest leg corresponds to the LAM mode of n-C2n + 1 alkane with the free ends. The other two shorter legs are predicted to vibrate with the condition of one fixed end and one free end. As already indicated in the previous Sect. 5.8.2, the vibrational frequency of a finite chain is affected by the condition of the end parts. The vibrational frequency ν changes with the vibrational order s (s = 1, 2, …, n). As shown in Fig. 5.144a, this s-dependence is different depending on the constraining conditions at the end parts. However, the dispersion curve expressed by the following phase angle φ is common to these three cases, Free ends: φ = π m/n Fixed ends: φ = π(m + 1)/(n + 1) One free end and one fixed end: φ = π(2m + 1)/(2n + 1)

(5.294)

where m = n – s and m= 0, 1, …, n − 1. For example, the 252 cm−1 Raman band of C7 HOH compound may have the phase angle φ of ca. 0.14π in the dispersion curve for the free end condition. However, one end is CH3 under no fixing and another end is CH group fixed to the other carbon atoms, i.e., the segment is subjected to the constraining condition of one free end and one fixed end. If the C atom fixed to the horizontal axis is ignored because of such a fixing condition, then the corresponding phase angle for n = 7 is

596

5 Structure Analysis by Vibrational Spectroscopy

φ/π = (2m +1)/(2n+1) = (2m +1)/15; for m = 0, φ/π = 0.067, and for m= 1, φ/π = 0.2. These φ values do not give any good fitting to the LAM dispersion curve under the free end condition. The atomic displacements of the central branch are almost parallel to the chain axis in the LAM mode (Fig. 5.145). An important point is that the atomic displacements occur over a whole chain segment including even the end OH group and almost no effect comes from the side CC bonds jointed to this central chain segment. This means that the LAM of the central chain segment CH3 -(CH2 )7 -OH can be approximated by the LAM of a straight alkane n-C9 with the free end condition. If it is so, the corresponding φ = π m/n = π/9(= 0.11π for m = 1), fitting well to the dispersion curve under the free ends. Similarly, the LAM band of C12 HOH molecule detected at about 182 cm−1 can be interpreted well by corresponding to the LAM of n-C14 subjected to the free end condition. In this way, all the alkane chain segments of the H-shape compounds can be interpreted on the basis of the LAM dispersion curve under the free end conditions with the terminal atoms (CH and OH groups) included as the members.

(a)

(b)

Fig. 5.145 (a) Comparison of the observed LAM frequencies of H-type molecules with the frequency dispersion curves calculated for a chain consisting of simply coupled oscillators under the various end conditions (b) The vibrational mode of C12 segmental part of C12 H. Reprinted from Ref. [47] with permission of the American Chemical Society, 2011

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

597

Fig. 5.146 IR spectra of a series of isotactic polystyrene composed of H and D monomeric units. Reprinted from Ref. [108] with permission of Wiley, 1968

5.10.6 Critical Sequence Length 5.10.6.1

Concept of Critical Sequence Length

In the solution or in the molten state, the molecular chain moves actively and changes its conformation with time. Some parts of the molecular chains may take short or long regular helical shapes. The IR and Raman spectra may detect these local segments. However, even when the spectrometer is so sensitive, all of these short and long helical segments cannot be necessarily detected. Some critical length might be existent for the detection. This concept is confirmed by such an observation of the IR spectra. Figure 5.146 shows the IR spectra measured for a series of isotactic polystyrene (PSt) copolymers with the various ratios of H and D monomeric units [108–110]. Different from the case of blend samples between pure PSt-H and D-PSt, the IR bands (e.g., 920, 1053 cm−1 ) of the H/D copolymers do not appear in proportion to the H/D content, but the bands intrinsic to the helical form appear for the copolymers with H/D content higher than some value. That is, the appearance of these bands needs to cross some critical sequential length of the monomer content. How can we know this “critical sequence length”? The band intensity reduced by the sample thickness, R(X) was evaluated and the ratio R(X)/R(1.0) was plotted against X or the molar fraction of the H monomeric unit. The result is shown in Fig. 5.147. Let us assume a 1D chain of regular conformation which consists of the random sequence of H and D units.

598

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.147 Plot of the reduced absorbance against the molar fraction H/D. The curves are calculated using Eq. 5.297. The behavior of the copolymer is different from that of the H/D blend sample as shown in the right-handed picture. Reprinted from Ref. [108] with permission of Wiley, 1968

The probability of an H unit is X. The probability Pn of taking the H sequence D(H)n D is given as Pn = (1 − X )2 X n

(5.295)

The length L of the segment distributes as L 1 , L 2 , L 3 , …, L m , ….. L ∞ at the different fractions. Among them, the fraction F(m) of L n segments longer than L m is calculated as follows.   ∞   ∞ n Pn / n=1 n Pn = X m−1 [m + (1 − m)X ] F(m) = n=m

(5.296)

The IR band intensity is proportional to F(m). Then, the ratio R(X)/R(1) is given as R(X )/R(1) = X F(m) = X m [m + (1 − m)X ]

(5.297)

Here, the R(X)/R(1) is not equal to F(m) but XF(m) since the total amount of H units in the copolymer is reduced by X compared with the homopolymer [R(1)]. This equation indicates that the plot of R(X)/R(1) against X is dependent on the critical sequence length m. The best m value can be estimated by finding the curve which can reproduce the observed plot of R(X)/R(1) as nicely as possible. In Fig. 5.147, the 920 and 1053 cm−1 bands give the best fit for m = 8–10. That is to say, these bands can be detected for the first time when the regular helical segment consisting

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

599

of about 10 H monomeric units exits in the system. In Fig. 5.147, the case of the H/D mixture is also shown, where the H band intensity changes linearly with H content. We have to be careful of the abovementioned data treatment: (i) the distribution of the H and D monomeric units in the copolymers should be perfectly random, (ii) the samples of the various H/D contents must be treated in the same way and they must have the same degree of crystallinity in the case of solid sample, (iii) the band intensity must be reduced by the accurate sample thickness, and (iv) the intensity is not the peak intensity but it should be the integrated intensity after the deconvolution of the overlapped bands. This data treatment and the evaluation of the critical sequence length are called the isotope dilution method and reported for quite limited kinds of polymers, including polyoxymethylene [111], isotactic polypropylene [108–110, 112], syndiotactic polystyrene [113], and vinylidene fluoride-trifluoroethylene copolymer [114]. In the last case, vinylidene fluoride (VDF) and trifluoroethylene (TrFE) monomeric units are used as basic units instead of the abovementioned H and D monomeric units.

5.10.6.2

Solvent-Induced Crystallization of Syndiotactic Polystyrene

syndiotactic Polystyrene (sPS) is melted and quenched into ice-water temperature, by which the amorphous sample is prepared. This sample is exposed to the atmosphere of toluene vapor and then the crystallization into the crystalline δ form starts to occur [115]. The δ form is the complex between sPS and toluene (or solvent) [116]. The time dependence of the IR and Raman spectra are measured in the solventinduced crystallization process [117, 118, 119]. Figure 5.148 shows the result of the IR spectral measurement. The IR band of toluene (465 cm−1 ) increases its intensity immediately after the supply of toluene. At almost the same time, the amorphous band at 568 cm−1 decreases in intensity and the IR bands characteristic of the regular helical form (549 cm−1 ) increase the intensity in parallel. After some time, the other bands of regular helical form (572 cm−1 ) start to increase. That is to say, the IR bands intrinsic to the regular helix appear at the different timing depending on the bands. The critical sequence length of the 549 cm−1 band is m = 7 ~ 12 and that of the 572 cm−1 band is m = 20 ~ 30. The later appearance of the 572 cm−1 band indicates the growth of the regular helical sequence. The amorphous band is observed to behave in such a way that the peak position shifts to the lower frequency side and the half-width increases in the early stage. As already mentioned, the half-width is related inversely to the relaxation time [118]. The increase of the half-width indicates the activation of the thermal motion of the amorphous chains by the supply of solvent vapor. Correspondingly, the peak position is also shifted toward the lower frequency side because of the weaker intermolecular interaction. At the same time, the intensity of the amorphous band decreases because of the increase of the crystalline part. By combining all these data, it may be possible to describe the process of the solventinduced crystallization of the amorphous sPS sample as follows [118, 119]: (i) stage A: as the solvent vapor is absorbed by the sample, the amorphous chain motion is

600

(a)

5 Structure Analysis by Vibrational Spectroscopy

(b)

(c)

Fig. 5.148 a Time dependence of IR spectra of sPS in the solvent-induced crystallization. b Plot of IR band intensity with time. c The schematic illustration of structural evolution process. Reprinted from Ref. [118] with permission of the American Chemical Society, 2002

activated (as known from the increase of the half width of the corresponding band) and the regularization to the helical form starts to occur. But the helical length is short, approximately 10 monomeric units at most or about three helical rotations of T2 G2 conformation. (ii) Stage B: the helical length becomes longer with time and the band of longer m value starts to be detected. The amorphous region is sandwiched by the crystalline regions and the amorphous motion is restricted as seen from the decrease of the half-width and the higher-frequency shift of the peak position of the amorphous band. The X-ray diffraction peaks (not shown here) start to be observed around here, indicating the formation of crystalline lattice of the longer helical segments. The thus-detected structural evolution is schematically illustrated in Fig. 5.148c.

5.10.6.3

Isothermal Crystallization of Isotactic Polypropylene

When a polymer substance is melted and cooled quickly to a predetermined temperature, crystallization starts to occur. By performing the time-resolved measurement of IR (or Raman) spectra during this process the structural evolution can be traced from the microscopic point of view. If the X-ray scattering can be measured at the same

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

(a)

601

(b)

A

B

C

(c)

Fig. 5.149 a Time-resolved IR spectral change measured in the melt-isothermal crystallization process of it-PP. b Time dependence of the various experimental data (IR, SAXS, and WAXD). c Illustration of structural evolution in the crystallization process. Reprinted from reference[112]with permission of the American Chemical Society, 2009

time, then the evolution of crystal lattice and the higher order structure consisting of crystalline and amorphous regions can be traced. For example, Fig. 5.149a shows the time dependence of FTIR spectra measured for isotactic polypropylene (it-PP) in the isothermal crystallization process from 200 °C to 130 °C [112]. The amorphous bands in 900–1100 cm−1 region are observed to decrease with time. At the same time, the IR bands intrinsic to the crystalline regular helices (841, 998, and 1220 cm−1 bands) started to appear and increased the intensity. When the integrated intensity of

602

5 Structure Analysis by Vibrational Spectroscopy

these bands is plotted against time, we can notice the appearance timing is different among the various bands. Such timing difference in the appearance of crystalline bands is considered to originate from the difference in the critical sequence length of these bands. The IR band at 998 cm−1 has m = 10, for the band at 841 cm−1 , m = 14 and for the band at 1220 cm−1 , m >> 15. The later appearance of the 1220 cm−1 band indicates the growth of the regular helical segments in the melt. By comparing these IR data with the SAXS data, the formation of domains consisting of the short regular helical segments in the melt is speculated. The average size of the domain is about 500 Å [(5/3)1/2 Rg ]. As known from the shortening of the correlation length ξ estimated from the SAXS data analysis (Debye-Buche plot), these domains have the correlation with the neighboring domains and approach each other to form the lamellar structure at the final stage of the early timing of the isothermal crystallization (as for the analysis of SAXS data, refer to Chap. 4).

5.10.7 Evaluation of Orientation of Polymer Crystals 5.10.7.1

Polarized Infrared Spectra

As described in Sect. 5.5.2, the polarized IR spectra are related to the angles between the transition dipole moments and the electric vector of the incident IR beam. By measuring the absorbances of the bands of the parallel and perpendicular polarization components, the degree of orientation of the corresponding units can be estimated [120, 121]. A polarized IR beam is incident to a polymer film (see Fig. 5.150). The orthogonal coordinate system is defined so that the incident plane of the beam is xy. The IR beam with the electric vector parallel to the z-axis is called the perpendicular IR beam. The IR beam with the electric vector in the xy-plane is the parallel beam. Using Fresnel’s law, we have the following equation: for the incident and refracted beams, ◦ ◦ cos(i), E y◦ = E // sin(i), E z◦ = E ⊥◦ E x◦ = E // ◦ ◦ E x = E // cos(r ), E y = E // sin(r ), E z = E ⊥◦

sin(i)/ sin(r ) = n Fig. 5.150 Optical path of incident and refracted IR beam

(5.298)

E//

y i z

E n r

x

5.10 Analysis of Vibrational Spectra Characteristic of Polymers Fig. 5.151 Geometry of chain axis and transition dipole moment M in the uniaxially-oriented polymer film

603

z E//

M

IR

a

E

Ma

θ

φ Mb

c

β Mc

b y

x

ϕ

Here, i is the angle between the incident direction and the y-axis, r is that of the refracted beam measured from the y-axis. n is the refractive index of the sample. The absorbance A is proportional to the square of the inner product between the electric vector E and the transition dipole moment M. A ∝ (M · E)2

(5.299)

For the parallel and perpendicular components, this equation is expressed below. A⊥ = Mz2 E z2

(5.300)

A// = Mx2 E x2 + M y2 E y2

(5.301)

Components E x and E z are expressed using the angle i and n.     E x2 = E o2 cos2 (r ) = E o2 1 − sin2 (r ) = E o2 1 − sin2 (i)/n 2 E y2 = E o2 sin2 (r ) = E o2 sin2 (i)/n 2

(5.302) (5.303)

The dichroic ratio D = A// /A⊥ is obtained as follows:     D = A// /A⊥ = Mx2 E x2 + M y2 E y2 / Mz2 E z2     = [Mx2 1 − sin 2 (i)/n 2 + M y2 sin 2 (i)/n 2 ]/ Mz2

(5.304)

From now, the incidence of IR beam is made at i = 0°. The refractive index n ~ 1. Using matrix T of the Eulerian angles, the components of the vector M x are expressed using the (a, b, c)-coordinate system fixed to the chain, as shown in Fig. 5.151. (Be careful here of such a point that vector M is fixed and the expression is changed from the abc-coordinate system to the xyz-coordinate system. If vector M itself is rotated, the transformation matrix is expressed as T −1 .)

604

5 Structure Analysis by Vibrational Spectroscopy

M

M it-PS

it-PP

s(CH

 

)

//

Fig. 5.152 Polarized infrared spectra of oriented polypropylene film. ⊥ and //: the electric vector of an incident IR beam is perpendicular and parallel to the oriented axis, respectively

⎛

⎞ Ma M x = T M = T ⎝ Mb ⎠ Mc

(5.305)

For the uniaxially oriented sample, using the distribution function f (θ) of the chain axis, the following equations are derived using the Eulerian angles. The z axis is the drawing direction of a polymer sample as illustrated in Fig. 5.151. The M x and M z components are expressed as below. Mx = M sin(β)(cos φ cos θ cos ϕ − sin φ sin ϕ) − M cos β cos ϕ sin θ Mz = M sin βsin θ cos ϕ + M cos βcos θ By expressing cosϕ = cϕ and so on, we have the following results. 

A x = A⊥ = M /4π 2

  = M 2 /4π 2

 0

2



π/2







θ=0−π/2 2π  2π 0

0

ϕ=0−2π

 φ=0−2π

Mx2 f (θ )dφdϕdθ

[(cφcθ cϕ − sφsϕ)sβ − cϕsθ cβ]2 f (θ )dφdϕdθ

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

'  2 = M /4 sβ 

2

π/2 



2 − 3sθ

2



 f (θ )dθ +

0

π/2

θ=0 2

605

2sθ 2 f (θ )dθ

= A y = (M 2 /4)[sβ 2 < 2 − 3sθ 2 > + < 2sθ >] 

A z = A// = M /4π 2

'    = M 2 /2 cβ 2

2





π/2



θ=0 π/2

θ=0

2π

ϕ=0



2π

φ=0

[sθ cϕsβ + cθ cβ]2 f (θ )dφdϕdθ

  2 − 3sθ 2 f (θ )dθ +



π/2s θ=0

( sθ 2 f (θ )dθ

= (M /2)[cβ < 2 − 3sθ > + < sθ >] 2

2

2

2

As a result,     D = A// /A⊥ = 2 cos2 (β) + S / sin2 (β) + S ,   S =< sθ 2 > / 1 − (3/2) < sθ 2 >

(5.306)

By modifying these equations,     (D − 1)/(D + 2) = 3 cos2 (β) − 1 /2 × 3 < cos2 (θ ) > −1 /2   = K × 3 < cos2 (θ ) > −1 /2 where   K = 3 cos2 (β) − 1 /2

(5.307)

Herman’s orientation function, which is equal to the second-order term of the Lagrange function= P2 (cos θ ) = 3 −1 /2 is obtained as   f = 3−1 /2 = (1/K )(D − 1)/(D + 2)

(5.308)

The value f changes depending on the orientation: for the unoriented sample, f = 0

⎫ ⎪ ⎬

for the perfect orientation, f = 1 ⎪ ⎭ for the in-plane orientation of the chains, f = −1/2

(5.309)

Another point to notice is that the dichroic ratio D = 1 for a certain condition even when the chain orientation is quite √ high. This is the case of K = 0. Using Eq. 5.307, K = 0 is when cos(β) = 1/ 3, or the angle between the chain axis and the transition dipole moment M, β = 54.7°. Good example to satisfy this condition is seen for the CH2 symmetric stretching mode in the (3/1) helical chain of (TG)3 , that is, the helices of isotactic polypropylene and isotactic polystyrene. As shown in

606

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.152, the transition dipole moment of CH2 symmetric stretching mode takes an angle of almost 54.7° from the chain axis. The situation is the same for the CH2 bending mode [δ(CH2 )].

5.10.7.2

Polarized Raman Spectra

The degree of crystal orientation is evaluated also by the polarized Raman data. The theoretical treatment is based on the work by Bower [122, 123, 124]. As shown in Fig. 5.153, the Cartesian coordinate system of the sample is O-x1 x2 x3 . An electric vector Eo of a laser beam incident to the sample has the direction cosines l = (l1 , l 2 , l3 ). The direction cosines of polarization vector P s of the scattered laser signal   are    l = (l 1 , l 2 , l 3 ). The transition polarizability tensor is expressed as α = αi j . As already mentioned, polarization P s is expressed as P s = α Eo

(5.310)

These vectors are expressed on the coordinate system O-x1 x2 x3 . P s  = l  Ps and Eo  = lEo and then, P s = l  αl E o

(5.311)

Since the scattering intensity is proportional to the (P  s )2 , we have the scattering intensity I s as  2 Is = Io i j li l j αi j

(5.312)

where I o is a constant depending on the incident laser light intensity and so on. The first indicates the summation of the scatterings from all the units contributing to the observed intensity. This summation is ignored from here in the derivation of the equations. As already mentioned, the orientation function is given as Fig. 5.153 Measurement of polarized Raman spectra

5.10 Analysis of Vibrational Spectra Characteristic of Polymers ∞ l l ω(ξ, φ, ϕ) = l=0 m=−l n=−l Mlmn Plmn (ξ ) exp(−imφ) exp(−inϕ)

607

(5.313)

The various scattering units are oriented in the various ways with the function ω(ξ, φ, ϕ), and then the scattering intensity is expressed as the average of the scatterings from individual units: 

2π

αi j α pq = No 

0 2π

= No 0

= 4π No 2

 i j pq Almn = 1/4π 2

 

2π



1

0

−1 2π  1

0

−1



ω(ξ, φ, ϕ)αi j α pq dξ dφdϕ ∞ l l l=0 m=−l n=−l Mlmn Plmn (ξ ) exp(−imφ)

exp(−inϕ)αi j α pq dξ dφdϕ i j pq

Mlmn Almn

lmn ∞  l   2π  2π  1  0

0

(5.314)

l 

−1 l=0 m=−l n=−l

Plmn (ξ ) exp(−imφ) exp(−inϕ)αi j α pq dξ dφdϕ

(5.315) ijpq

If αij (and αpq ) are known, the term Almn can be calculated. It must be noted that i j pq the nonzero values of Almn occur only for the even values of l, m, and n, and n ≤ 4; A000 , A200 , A400 , A202 , A402 , A404 under the group symmetry D2 . For example, for the uniaxially oriented sample: for α11 2 (ijpq = 1111),      A000 = (2)1/2 /15 3 α12 + α22 + α22 + 2(α1 α2 + α2 α3 + α3 α1 ) = A    A200 = 4 × (5/2)1/2 /210 3α12 + 3α22 − 6α32 + 2α1 α2 − α2 α3 − α3 α1 for α12 2 (ijpq = 1212),    A000 = (2)1/2 /15 α12 + α22 + α22 − (α1 α2 + α2 α3 + α3 α1 ) = D for α11 α22 (ijpq = 1122), A000 = A − 2D Here, α 1 , α 2 , and α 3 are the values of the polarizability tensor components along the principal axes.: αi j = aik a jk αk The coefficients aij are given as

(5.316)

608

5 Structure Analysis by Vibrational Spectroscopy

⎛

ai j

cθ cφcϕ − sφsϕ = ⎝ cθsφcϕ + cφsϕ −sθcϕ

−cθ cφsϕ − sφcϕ −cθsφsϕ + cφcϕ sφsϕ

⎞ sθcφ sθsφ ⎠ cθ

(5.317)

Just like the X-ray diffraction method, the experimental value of Raman band I s is obtained, then the moment M lmn is estimated. So the orientation function ω(ξ, φ, ϕ) can be derived. However, how can we know the components of the principal axis value αk ? Let us consider the case of uniaxially oriented sample, where the laboratory coordinate system is O-X1 X2 X3 and the sample coordinate system is O-x1 x2 x3 . The angle between X3 and x3 axes is β. The laser beam is incident along the X3 -axis and the scattered signal is measured along the X1 -axis (90°-scattering mode). The electric vector component of the incident beam directs along the X2 or X1 and that of the scattered beam along the X3 - or X2 -axis. The scattering intensity is given as  2 IX2 X1 = I21 = I0 αx2 x1 cos(β) − αx3 x1 sin(β)   = I0 αx22 x1 cos2 (β) − 2αx2 x1 αx3 x1 cos(β) sin(β) + αx23 x1 sin2 (β) Using cos2 (β) = [cos(2β) + 1]/2, etc., we have I21 = I0



     αx22 x1 + αx22 x1 − αx23 x1 /2 + cos(2β) αx3 x1 /2    − αx2 x1 αx3 x1 sin(2β) = I0 [I210 + I212 cos(2β))] (5.318)

Similarly,  2 I x3 x1 = I31 = Io αx2 x1 sin(β) + αx3 x1 cos(β) = Io [I310 + I312 cos(2β)] (5.319)  I x2 x2 =I22 = I0 αx2 x2 cos2 (β) − (1/2)αx2 x3 sin(2β) 2 − (1/2)αx3 x2 sin(2β) + αx3 x3 sin2 (β) =I0 [I220 + I222 cos(2β) + I224 cos(4β)]  I x3 x2 = I32 =I0 αx2 x2 sin(2β)/2 + αx3 x2 cos2 (β) 2 −αx2 x3 sin2 (β) − (1/2)αx3 x3 sin(2β) =I0 [I220 + I222 cos(2β) + I224 cos(4β)] where αx1 x2 = α12 , for example.   2 2 /2 + α13 I210 = I310 = α12

(5.320)

(5.321)

5.10 Analysis of Vibrational Spectra Characteristic of Polymers

609

  2 2 /2 I212 = −I312 = α12 − α13   2 2 2 I220 = 3 α22 + 2 α22 α33 + 3 α33 /8 + 4 α23   2 2 I222 = α22 − α33 /2   2 I224 = −I324 = (α22 − α33 )2 /8 − α23 /2   2 2 I320 = (α22 − α33 ) /8 + α23 /2 By measuring the scattering intensity I ij of the polarized Raman spectra, the components αi2j , etc. can be known. One example is shown for the uniaxially oriented poly(methyl methacrylate) i j pq [124]. As mentioned above, moments M lmn and Almn are limited to some values as below. .

αi j α pq = 4π 2 No

. l=0,2,4

i j pq

Ml00 Al00

(5.322)

is for all the units contributing to the scattering. The pair of ijpq is ijij or iijj. M 000 = 1/21/2 . M 200 is linearly related to the averaged value of tilting angle β, < cos2 (β) > . M 400 is linearly related to C 2 = < cos2 (β) > and C 4 = < cos4 (β) > . From Eqs. 5.318 to 5.321, we have the following equations, where t = 1 or 2 and α L = α3 − αt .     2 2 = I0 α22 = (1/8)No Io 8αt2 + 8αt α L + 3αt2 − 8αt α L + 6α L2 C2 + 3α L2 C4 I0 α11   2 I0 α33 = N0 I0 αt2 + 2αt α L C2 + α L2 C4 2 I0 α12 = (1/8)N0 I0 α L2 (1 − 2C2 + C4 ) 2 = I0 α 2I 3 = (1/2)N0 I0 α L2 (C2 − C4 ) I0 α23

    I0 α22 α33 = (1/2)N0 I0 2αt2 + αt α L + α L2 + α L αt C2 − α L2 C4 2 2 α11 α22 = α11 − 2 α12

α33 α11 = α22 α33 (5.323) These independent five equations (Eq. 5.323) give the unknown parameters (No Io )1/2 αt , (No Io )1/2 αL2 , (No Io )1/2 αt αL , C2 and C 4 . The polarized Raman spectra are measured by changing angle β shown in Fig. 5.153. The scattering intensity changes with the β angle as shown in Fig. 5.154. The finally obtained C 2 and C 4 values are shown in Table 5.16, where the result by NMR method is compared together. Depending on the sample, the Raman tensors (or their

610

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.154 Polar plots of the intensities I ij for 604 cm−1 band; a sample MF5 and b sample MF12. ◯● Experimental points, ––––least square fits. Reprinted from Ref. [124] with permission of Elsevier, 1974

Table 5.16 Reprinted from Ref. [124] with permission of Elsevier, 1974

Sample band/cm -1 M F5 AG M F12

 1 / 3

 2 / 3

C2

C4

486

-3.6

0.3

0.40

0.25

1732

- 3.4

-3.1

0.41

0.32

486

-3.0

0.7

0.43

0.32

1732

-1.5

-0.9

0.43

0.32

486

-2.5

0.4

0.53

0.39

1732

-1.5

-0.9

0.57

0.50

C 2 NMR

C 4 NMR

0.37

0.22

0.49

0.33

0.55

0.39

ratios) change depending on the samples (MF5, AG, and MF12), which might be due to the change in the conformation of molecular chains in the oriented samples.

5.11 Vibrational Anharmonicity

611

5.11 Vibrational Anharmonicity 5.11.1 Anharmonicity and Spectra In general, the IR and Raman spectra are treated in a good approximation under the assumption of harmonic vibrations. That is, the potential function of a normal mode Q is expressed in a simple form of V = (1/2)kQ2 with the harmonic force constant k. However, in the actual heating or stretching process, the sample causes the thermal expansion or compression of the unit cell and the shift of the band peak position. As long as we assume the harmonic vibration, it is impossible to interpret these phenomena reasonably. For example, let us consider the simple diatomic molecular vibration as shown in Fig. 5.155, the averaged bond length is always Ro even for the excited state with the parabolic potential function. Once the anharmonic term is introduced, the potential function changes as     V = Vo + (∂ V /∂ x)0 x + (1/2) ∂ 2 V /∂ x 2 0 (x)2 + (1/6) ∂ 3 V /∂ x 3 0 (x)3   + . . . ∼ (1/2) ∂ 2 V /∂ x 2 0 (x)2 for the harmonic mode   ∼ (1/2) ∂ 2 V /∂ x 2 0    +(1/3) ∂ 3 V /∂ x 3 0 (x) (x)2 for the anharmonic mode (5.324) The potential curve becomes asymmetric with respect to the center position and the averaged bond length becomes longer for the excited vibrational state. This results in the thermal expansion of the bond length. Similarly, the stretching of the diatomic molecule causes the peak shift by the change in bond length, which results in the change of force constant. The vibrational-mode calculation of a molecule by taking the anharmonic potential term into account is quite difficult, and the so-called quasiharmonic approximation is sometimes applied, where the higher order term is input into the harmonic force constant and the calculation is made as if it is harmonic. But, depending on the change of the bond length (or in general, the change of the

V

k ko

xo

x

k ~ ko + k’F

F

Fig. 5.155 Harmonic and anharmonic potential functions and the change of the force constant by the external force or temperature (F) effect

612

5 Structure Analysis by Vibrational Spectroscopy

internal coordinates), the force constant changes correspondingly. As shown already in Eq. 5.324, V = (1/2)k(x)2     k = ∂ 2 V /∂ x 2 0 + (1/3) ∂ 3 V /∂ x 3 0 (x) = k0 + k  x

(5.325)

k 0 and k  are a harmonic force constant and an anharmonic force constant, respectively. This treatment is useful for the prediction of the vibrational frequency shifts of polymer crystallites subjected to an external stress. In the above example of a diatomic molecule, the x is the change of bond length which is caused by an application of force. At the first stage of an infinitesimally small deformation, the force constant is k 0 , and so the x is expressed as x = k0 F under the tensile force F. At the same time, because of this geometrical change, the force constant k is changed to k = k0 + k  x = k0 + k  k0 F. This is repeated by changing the force F by a step F. As a result, the vibrational frequency is also changed in the form 1/2 of v ∝ k 1/2 = (k0 + k  x)1/2 ∼ k0 (1 + k  F/2). The concrete description will be made in the volume 2 of this book.

5.11.2 Fermi Resonance and Raman Spectra of Polyethylene Crystal The vibrational modes are independent of each other as long as the harmonic vibration is assumed. But the anharmonic vibration allows the coupling between the different normal modes. For example, the so-called overtone or combination modes are the typical examples. The overtone band, which appears at around the frequency position of 2νo for the normal mode frequency νo , is generated by the transition from the n = 0 state to the n = 2 state. In more general, if the vibrational frequencies of the two normal modes are νa and νb , then the combination band (including the overtone mode) may appear at the position of about νa + νb . This type of transition is originally forbidden as long as the vibration is harmonic. The symmetry species of these bands can be predicted from the character table of the molecule. Let us see the case of a water molecule with the point group C 2v . The character table is as follows. σ1 (yz)

σ2 (xz)

μ

1

1

1

z

E

C2

A1

1

α x 2 , y2 , z 2

A2

1

1

−1

−1

B1

1

−1

−1

1

x

xz

B2

1

−1

1

−1

y

yz

xy

5.11 Vibrational Anharmonicity

613

The overtone of the vibrational mode of B1 species should have the characters of A1 species since the direct product of the B1 ⊗ B1 gives the characters of A1 species. The overtone band can be detected in both the IR and Raman spectra in the present C 2v case. The symmetry species of the combination mode is predicted by performing the direct product of the characters of the corresponding two species. For example, the combination between B1 and B2 vibrational modes results in the mode belonging to the A2 , which is the IR inactive mode and can be detected as the Raman band with the polarizability component αxy . The combination mode between A1 and B2 modes gives the B2 combination band with the transition dipole vector parallel to the y-axis and the polarizability component αyz . The overtone and combination modes appear in the near IR frequency region. Since the near-IR spectra are quite complicated, the band assignment is made by referring to the predicted vibrational frequency or the sum of the vibrational frequencies of the 2 (or even 3) bands and by taking into consideration the polarization of the band. The overtone bands are useful for distinguishing the normal modes appearing at similar frequency positions. One good example is about the assignment of the OH stretching bands related to the hydrogen bonds in poly(vinyl alcohol) [125]. In the frequency region of 3300 cm−1 , the various OH stretching bands overlap to give the complicated profile. The slight difference in the frequency is enlarged in the overtone region since the frequency is almost two times higher. By referring to the polarization also, the OH bands at the different peak positions are related to the hydrogen bonds of different strengths and directions.

In the Raman spectra of the orthorhombic polyethylene crystal, the Raman band profiles in the frequency regions of 3000 and 1450 cm−1 are complicated beyond the expectation from the simple vibrational bands of CH2 stretching and bending modes, respectively [126, 127]. By referring to the dispersion curves of the orthorhombic polyethylene crystal, the anharmonic coupling of these normal modes with the overtones of the other normal modes is found to be the origin of these complicated profiles. This coupling is called the Fermi resonance [128–130]. The overtone of CH2 bending mode at 1450 cm−1 is expected to appear at around 2900 cm−1 , the frequency region of which is almost the same as that of the CH2 stretching mode. The coupling between these two modes (overtone and normal mode) causes the shift of the band positions. Let us check the essence of the Fermi resonance phenomenon. The wavefunction  is assumed to be a linear combination of the function of the normal mode n and that of the overtone o .  = an + bo

(5.326)

Hamiltonian Hˆ is expressed as Hˆ = Hˆ n + Hˆ o + Hˆ  where Hˆ n and Hˆ o are the Hamiltonians of the normal and overtone modes, respectively, and Hˆ  is the coupling interaction between n and o . The Schrödinger equation is given as

614

5 Structure Analysis by Vibrational Spectroscopy

Hˆ  = E or



 Hˆ n + Hˆ o + Hˆ  (an + bo ) = E(an + bo )

(5.327)

By applying the conjugated function * from the left side and integrating the equation, we have 

   ∗ an + b∗o Hˆ n + Hˆ o + Hˆ  (an + bo )dτ = E



  ∗ an + b∗o (an + bo )dτ

(5.328) By applying the variation principle for the energetically minimal condition (∂ E/∂a = 0 and ∂ E/∂b = 0), we have the following equations. In the calculation process, we assumed such an overlap integral as ∗n o dτ is zero. E n a + E  b = Ea where E n = have





∗n Hˆ n n dτ , E o = 

En E  E  Eo

and

E  a + E o b = Eb

∗o Hˆ o o dτ and E  =

     a E0 a = b 0 E b



(5.329)

∗n Hˆ  o dτ . Then, we

(5.330)

By solving this equation, we have the eigenvalues and eigenvectors (normalized) as follows:   2 1/2 /2 (i) E 1 = − δ 2 + 4 E  where = (E n + E o )/2 and δ = E o − E n 0 ' ( /   2 %  2 1/2  2 &2 1/2 / 4 E  − δ − [δ 2 + 4 E  ]1/2 a = δ − δ2 + 4 E  ( '   2 %   2 1/2 &2 1/2 2 b = 2E / 4 E − δ − [δ + 4 E ] 

(ii)

(5.331)

  2 1/2 E 2 = < E no > + δ 2 + 4 E  /2 %



a = δ + [δ + 4 E 2

  2 1/2 ]

( & '   %   2 1/2 &2 1/2  2 2 / 4 E + δ − [δ + 4 E ]

( '  2 %  2 &2 1/2 b = 2E / 4 E  + δ + [δ 2 + 4 E  ]1/2

(5.332)

If the coupling term E’ is zero, the eigenvalues are equal to E o and E n . The anharmonic effect causes the change of these energies as shown in Fig. 5.156. The magnitude of change or the energy difference between E 1 and E 2 is dependent on the closeness

5.11 Vibrational Anharmonicity

615

Fig. 5.156 Dependence of energy E on the energy difference δ(= E o − E n )

E

 E

CO2 overtone of δ (C=O) 667x2 = 1334 cm-1

νs(C=O) 1300 cm-1

δ (C=O) 667 cm-1

Fig. 5.157 Fermi resonance between the C=O symmetric stretching mode and the overtone of O =C=O bending mode

between the normal vibrational level (E n ) and the overtone level (E o ) δ = E o – E n . The Fermi resonance becomes stronger as the energy levels of these two vibrational states are closer. The relative intensity is also affected by such vibrational coupling. One typical example is the case of CO2 gas [129, 130]. The O=C=O bending normal mode is detected at 667 cm−1 . The overtone of this mode is expected to appear at 667 × 2 = 1334 cm−1 . This position locates near the Raman-active symmetric C=O stretching mode at 1300 cm−1 . As illustrated in Fig. 5.157, these modes are coupled together to give the shift of the vibrational frequencies as well as the intensity transfer. CO2 is trapped in a mineral. Depending on the pressure surrounding CO2 molecule, the degree of resonance is modified. This information can be used as a monitor of geobarometer for the minerals containing CO2 molecules. Now, let us consider the Fermi resonance in the Raman spectra of the orthorhombic polyethylene crystal [131–133]. In the 2900 cm-1 region, the normal modes are CH2 symmetric and antisymmetric vibrational modes, νs (CH2 ) and νas (CH2 ), respectively. These two normal modes belong to the symmetry species Ag and B3g , respectively, as learned from the factor group analysis of an isolated chain. We may predict the presence of two sharp Raman bands in the 3000 cm–1 region, which should correspond

616

5 Structure Analysis by Vibrational Spectroscopy

νs(CH2)

νas(CH2) The region resonated with 2δ (CH2)

Fig. 5.158 Raman spectrum of orthorhombic polyethylene, where the shadowed part is due to the Fermi resonance between the overtone of the CH2 bending mode and symmetric stretching mode. The CH2 bending modes are also coupled with the overtone of the CH2 rocking modes. Reprinted from Ref. [132] with permission of Elsevier, 1978

to these normal modes. However, in the actually-observed polarized Raman spectra, the broad band profile is detected as the Ag mode, excluding a sharp νas (CH2 ) band. This broad profile was interpreted to come from the Fermi resonance between the Ag νs (CH2 ) mode and the overtone of the CH2 bending mode [δ(CH2 )]. Besides, more importantly, all the points of the dispersion curve of the CH2 bending mode become the Ag species in the overtone mode. The coupling occurs between the νs (CH2 ) mode and the overtone of all the modes on the dispersion curve of the CH 2 bending mode. In addition, the band splitting due to the intermolecular interactions must be also taken into account. As indicated in Fig. 5.158, the 2 dispersion curves exist in the CH2 bending region, the overtones of which belong to the Ag species and resonate with the νs (CH2 ) mode in a wide frequency region. Besides, the overtone of CH2 rocking mode at around 700 cm-1 appears in the frequency region of δ(CH2 ) band. The Raman active δ(CH2 ) mode of the Ag species is coupled with the overtone of r(CH2 ) mode, and then the resultant band profile of the δ(CH2 ) mode becomes more

5.11 Vibrational Anharmonicity

617

complicated as a result of Fermi resonance with the 2r(CH2 ) mode. This complicated profile is coupled furthermore with the νs (CH2 ) mode, resulting in more complicated profile of the Raman band profile in the 2900 cm–1 region.

5.12 Circularly Polarized Spectra So far, we have focused on the IR and Raman spectra using the incident beams of linear polarization. These polarized spectra cannot distinguish the optically active molecules of the same chemical formula but with the opposite optical activity, such as D- and L-lactides, as illustrated below. By using the circularly polarized beams, we can distinguish these optically opposite species. These methods are called the circularly polarized vibrational spectroscopy [134].

L (S,S) lactide

D (R,R) lactide

5.12.1 Circularly Polarized Raman Spectra A circularly-polarized beam is incident on the sample (Figure 5.159). The electric field vector and magnetic induction vector of the incident light are E and B, respectively. The excited vibrational electric dipole (μ), magnetic dipole (m) and electric quadrupole (Q) are expressed in the following ways using the normal polarizability Fig. 5.159 Circularly polarized Raman scattering measurement

618

5 Structure Analysis by Vibrational Spectroscopy

tensor (α) and the higher-order polarizability tensors or the optical activity tensors G and A [135]. μ = α E + A∇ E + G B m = G∗ E Q = A∗ E

(5.333)

where ∇ E is the electric field gradient and * means the complex conjugate. It must be noted that the α, G, and A components are the so-called transition dipole and so on, which are equal to (∂α/∂ Q nv ), (∂G/∂ Q nv ), and (∂ A/∂ Q nv ), respectively for the normal mode vibration Qnv . For simplicity, they are expressed as given here. In the non-absorbable frequency region, according to the perturbation theory,





 αi j = αi◦j − iαi j ∝ r < n  |μi |r >< r μ j n > /(vrn − v) + n  μ j r >< r |μi |n > /(vrn  + v)]





 G i j = G ioj − i G i j ∝ r [< n  |μi |r >< r m j n > /(vrn − v)+ < n  m j r >< r |μi |n > /(vrn  + v)





  Ai jk = Ai◦jk − i Ai jk ∝ r < n  |μi |r >< r Q jk n > /(vrn − v)+ < n  Q jk r >< r |μi |n > /(vrn  + v)

(5.334) In this way, α, G and A are related to the interactions between μ and μ, μ and m, and μ and Q, respectively. The intensity of the scattered laser light is different between the right-circularly-polarized light and left-circularly-polarized light. (For the linearly polarized light, only the α components are considered.) The intensity ratio between these two components is given as follows: For the circular intensity difference (CID, ) with the component perpendicular to the scattering plane (ZY in Fig. 5.159), or the polarized Raman CID, ⊥ = (I Rz − I Lz )/(I Rz + I Lz )      = 2 7 i j αi j G  i j + i j αii G  j j + i j αi j l m εilm Alm j / 7 i j αi j αi j + i j αii αi j = [2 +7 + ]/[45 2 +7 2 ]

(5.335)

For the parallel  or the depolarized Raman CID, // = (I Rx − I L x )/(I Rx + I L x )      = 2 3 i j αi j G  i j − i j αii G j j − i j αi j l m εilm Alm j / 3 i j αi j αi j − i j αii α j j = 2(3 − )/(3 2 )

(5.336)

where εilm = 1 or −1 is depending on the even or odd substitution of XYZcoordinates. In these equations, = (1/3) j α j j = (1/3) j G j j = (1/2) i j (3αi j G i j − αii G j j )

5.12 Circularly-Polarized Spectra

619

(b)

(a)

b

a Fig. 5.160 a Crystal structure of NaClO3 [137]. b the depolarized 90°-scattering Raman spectra (lower) and the Raman optical-activity spectra (upper) measured for d- (—) and l- (- - -) NaClO3 single crystals with an excitation laser of 488 nm wavelength. Reprinted from Ref. [136] with permission of Wiley, 1995

= (3/2) i j αi j ( l m εilm Alm j ) 2 = (1/2) i j (3αi j αi j − αii α j j )

(5.337)

In the actual measurement, the linearly polarized laser beam is changed to the alternately circularly polarized lights using an optical modulator made of KDP crystal at a frequency of about 900 Hz. The scattered light components are collected by the detector alternately. The contribution due to the G and A effects is only 10–3 times of the normal α-component. One example of the circularly polarized Raman spectral data is seen for sodium chlorate (NaClO3 ) [136]. In the aqueous solution, the system is achiral. The crystallization to the cubic crystal P21 3 (#198, a = 6.7878Å) gives a 1:1 mixture of chiral d- and l-crystals. Almost pure crystals can be prepared under a strong stirring of the solution. Figure 5.160 shows the circularly polarized Raman spectra measured for the d- and l-single crystals, giving a difference in the Raman spectral profile between these two enantiomers.

5.12.2 Circularly Polarized IR Spectra A linearly-polarized infrared light with an electric field vector in x-y flat plane is incident along the z axis of the optically active substance (see Fig. 5.161). In case (a), the electric vector is parallel to the x axis (E x ). If the electric vector consists of

620

5 Structure Analysis by Vibrational Spectroscopy

(a) linearly-polarized

(b) linearly-polarized

(c) elliptically-polarized

(Ex)

(Ex and Ey in phase)

(Ex and Ey out of phase)

y

b x

z

(d) right-handedcircularly-polarized (Ex and Ey in 90 phase) o

a



(e) left-handedcircularly-polarized (Ex and Ey in -90o phase)

Fig. 5.161 Rotation of electric vector of the polarized light

the x– and y–components (E x and E y , respectively) and their timing is the same (or the phase is 0), the total electric vector is tilted by 45° from the x and y axes. Such linearly-polarized light is incident to an optically active substance, then the electric field vector is rotated more or less and the linearly-polarized light is changed to the light of an elliptical polarization, as shown in Fig. 5.161c. The rotation angle (or the orientation angle) ψ is defined as an angle between the long axis (a) and the x axis. Another important parameter is the length ratio between the long (a) and short (b) axes of the ellipsoid, a/b = θ, which is called the ellipticity. These phenomena (the orientation angle change and the ellipsoid shape) are caused by the difference of the absorbance A between the right-handed and left-handed circularly-polarized IR beams due to the effect of the interactions between the electric dipole (μ) and magnetic dipole (m) moments. That is, ellipticity θ ∝ A L − A R = A = circular polarization dichroic ratio (CD) (5.338) Using the extinction coefficient ε (Lambert-Beer’s law, A = εcd for the concentration c and thickness d), θ ∝ ε L (λ) − ε R (λ) = ε(λ)

(5.339)

5.12 Circularly-Polarized Spectra

621

where λ is the wavelength of the light. The intensity of the rotated light (R) is related to the interaction between μ and m vectors.  R = I m < i|μ| f >< f |m|i > ∝ (ε(λ)/λ)dλ (5.340) where i and f indicate, respectively, the initial and final states and Im means the imaginary term. On the other hand, the intensity due to the usual transition dipole is  D ∝ | | ∝ 2

(ε(λ)/λ)dλ

(5.341)

The anisotropy factor g is defined as g = ε/ε (or A/A) = 4R/D

(5.342)

which can be measured in the actual CD IR spectroscopic experiment. The block diagram of the measurement system is illustrated in Fig. 5.162. The linearly polarized IR beam after passing through the interferometer is incident to an IR PEM crystal (photoelastic modulator, λ/4 wavelength plate) to generate the r- (rCP) and l-circularly polarized (lCP) beams at an exchange frequency of 50 kHz. These beams are incident to the sample and their intensities are detected with an MCT (mercury-cadmium-telluride) detector. The rCP and lCP signals are detected alternately using a phase-sensitive lock-in amplifier controlled with a digital signal processing (DSP) system. The detected A is in an order of 10–3 ~ 10–4 compared with the noise level 10–5 . One example is shown in Fig. 5.163 [138]. Poly(lactic acid) has two enantiomers of D and L configurations. These two polymers are mixed together in chloroform solution at various ratios. In the amorphous region (case a), the ΔA changed almost

Fig. 5.162 Block diagram of vibrational circular dichroism measurement system. Reprinted from Ref. [138] with the permission of the American Chemical Society, 2017

622

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.163 IR circular dichroism ΔAbs and IR spectra measured for a a series of solution-cast PLLA/PDLA blend samples with the various L/D ratios and b a series of the annealed samples of (a). Reprinted from Ref. [138] with the permission of the American Chemical Society, 2017

linearly depending of the difference between D- and L-components, indicating the homogeneous distribution of PDLA and PLLA in the amorphous region. The annealing of these amorphous samples generates the crystalline region. The ΔA changes also depending on the blend ratio. It was proposed that PLLA and PDLA form the so-called stereocomplex (SC) in which these two enantiomers coexist in the crystallite only at 1:1 ratio. If this structure model is correct even for the asymmetric blend samples of PDLA and PLLA components, the crystalline bands of the SC should show always the zero ΔA value. In the actual spectra, the crystalline SC bands appeared strongly at around 900 cm−1 and 955 cm−1 in (b). The ΔA of these bands changes depending on the D/L ratio, indicating the PDLA and PLLA chains coexist in the SC at the various ratios, not only 1:1 ratio, just likely in the amorphous region.

5.13 Brillouin Scattering

623

5.13 Brillouin Scattering 5.13.1 Principle of Brillouin Scattering In the Raman scattering measurement, the observed bands are in the frequency region higher than 1 cm−1 approximately. In the lower frequency region, there are three peaks observed at around 0.1 – 1 cm−1 . They are called the Brillouin scatterings coming from the inelastic scattering of photon by the acoustic phonon modes (LA, TA1 and TA2) [139–141]. The related equations are the same as those of inelastic Raman and neutron scatterings (Fig. 5.164). E(scattered photon) = E(incident photon) ± E(acoustic phonon) (energy reservation) vs = vi ± vp where vp = vBrilouin shift

(5.343)

Wavevector (reservation of momentum) k = ks − ki

(5.344)

As a result, the relation between the acoustic phonon velocity and phonon frequency is expressed as follows. Since the wavelength of the acoustic phonon mode is 1/k = λp , the velocity vp is given as vp = vBrillouin shift λp

(5.345)

The wavelength λo of the incident light changes in the medium of the refractive index n as λi = λo /n. Equation 5.344 is equivalent to the X-ray diffraction case (the reciprocal lattice vector is equal to the phonon vector), indicating Bragg’s equation of reflection for the lattice spacing λp . By assuming that the wavelength of the incident laser light λi is the same as that of the scattered light λs since the Brillouin shift is quite small compared with νi and νs in Eq. 5.343, Fig. 5.164 Brillouin scattering. The incident photon is scattered by the acoustic phonon wave of the wavelength λ p

624

5 Structure Analysis by Vibrational Spectroscopy

v Brillouin shift

Rayleigh scattering Brillouin peak

θ = 70 o

θ = 30 o

Fig. 5.165 Brillouin scattering peak is shifted depending on the incident angle θ

k2 = (ks − ki )2 = ks 2 + ki 2 − 2ks ki cos(2θ ) ≈ 4ki2 sin2 (θ ).  2 Then 1/λ p ≈ 4(1/λi )2 sin2 (θ ) = 4(n/λo )2 sin2 (θ ). Finally 2λ p sin θ = λo /n

(5.346)

See Fig. 5.164 for showing this relation. The velocity of the acoustic phonon is given as follows using the Brillouin shift vBrillouin shift . vp = vBrillouin shift λ p = vBrillouin shift λ◦ /(2n sin θ )

(5.347)

This equation indicates that the Brillouin shift is dependent on the incident angle θ, as illustrated in Fig. 5.165. The velocity of the acoustic phonon vp is related to the elastic modulus c by ρv2p = c (ρ: density). Therefore, the frequency of the Brillouin scattering peak gives the modulus of the substance. For example, in the case of a cubic crystal, the three phonons propagating along the [100] direction are possible which have the atomic displacement directions [100] (LA), [010] (TA), and [001] (TA). They are related to the elastic constant components of c11 , c44 , and c44 , respectively. The first one is the longitudinal acoustic mode and the second and third ones are the transverse acoustic modes. In general, the equation related to the velocity and elastic constants is given by the following equation (Christoffel equation):   k j l ci jkl l j ll − ρv2p δik Uk = 0

(i, j, k, l = 1 ∼ 3)

(5.348)

where ρ is the density, li is the direction cosine of the acoustic wave. vp is the velocity, and U k is the atomic displacement. The δ ik is the Kronecker delta.

5.13 Brillouin Scattering

625

The Brillouin scattering intensity I Brillouin is expressed using the heat capacity of the substance. For liquid sample of heat capacities of constant pressure (C p ) and constant volume (C v ), using the fluctuation theory, we have   IRayleigh /(2IBrillouin ) = Cp − Cv /Cv

(5.349)

which is called the Landau-Placzek ratio [139, 140]. The Brillouin scattering is quite close to the Rayleigh scattering, and so the Fabry–Perot interferometer with the parallel etalon plates is used for the monochromator [142–144]. By increasing the optical path of the etalons, the resolution power becomes higher. In Fig. 5.166, the six-pass tandem Fabry–Perot interferometer is used for this purpose. The precise adjustment of the etalon plates is hard to do generally.

5.13.2 Analysis of Brillouin Spectra One example of the Brillouin scattering spectra measured for a single crystal of chain molecule is shown here [145]. The samples are the monoclinic and orthorhombic II crystals of stearic acid B form, the structures of which are shown in Fig. 5.167. The structure of one layer is the same between them but the layer stacking is different (polytype). The monoclinic type takes the stacking structure of the single layers, while the orthorhombic crystal takes the double-layer structure of the opposite direction. The Brillouin scattering spectra were measured by changing the angle between the k vector and the a-axis. The elastic constant tensor c is given as follows.

Fig. 5.166 Schematic illustration of Brillouin scattering system

626

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.167 Crystal structures of monoclinic and orthorhombic cells of stearic acid B form. Reprinted from Ref. [145] with permission of the American Chemical Society, 1991

For the monoclinic crystal, ⎛

c11 c12 c13 ⎜ c22 c23 ⎜ ⎜ c33 ⎜ c=⎜ ⎜ ⎜ ⎝ ∗

c14 c24 c34 c44

0 0 0 0 c55

⎞ 0 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ c56 ⎠ c66

(5.350)

Then Eq. 5.348 is given as follows in the case of the phonon propagation in the ab plane.



c11 l12 + c66l22 − ρv2 (c12 + c66 )l1l2 (c14 + c56 )l1l2





=0 c66l12 + c22 l22 − ρv2 c56l12 + c24 l22



2 2 2

c55 l1 + c44 l2 − ρv where l1 = sinφ and l 2 = cosφ for the angle φ measured from the b-axis.

(5.351)

5.13 Brillouin Scattering

627

Fig. 5.168 Brillouin spectra of stearic acid form B single crystal. Reprinted from Ref. [145] with permission of the American Chemical Society, 1991

For the orthorhombic form, ⎛

c11 c12 c13 ⎜ c22 c23 ⎜ ⎜ c33 ⎜ c=⎜ ⎜ ⎜ ⎝

0 0 0 c44

0 0 0 0 c55

⎞ 0 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎠ c66

(5.352)

628

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.169 Elastic eigenvalues of sound waves in the ab-plane of the monoclinic (Mon) and orthorhombic (Orth) forms of stearic acid B form. Reprinted from Ref. [145] with permission of the American Chemical Society, 1991

Then, the Christoffel equation for the phonon propagation in the ab plane is given below.



c11 l12 + c66l22 − ρv2 (c12 + c66 )l1l2

0



2 2 2 2 2

= 0 (5.353) c56 l1 + c24 l2 c66 l1 + c22 l2 − ρv



2 2 2

c55 l1 + c44 l2 − ρv The solution of these equations gives the angular dependence of ρv2 . The actually observed Brillouin spectra are shown in Fig. 5.168. The observed peak position gives the shift νBrillouin shift of the scattering peak from the Rayleigh scattering peak. The velocity v is calculated using Eq. 5.347, where the refractive index n = 1.544 was used as an approximation. The plot of ρv2 is shown in Fig. 5.169. The thus-obtained curves are fitted using Eqs. 5.351 and 5.353 to evaluate the elastic constant tensor components (in unit GPa). c11

c22

c33

c44

c55

c66

Mon

7.25

8.04

17.7

–

–

3.37

Ortho-II

7.40

8.94

31.5

7.10

2.31

3.39

c12

c13

c23

c14

c24

c34

c56

4.13

–

–

0

0

0

0

4.01

–

–

1.02

-2.31

10.06

0.48

5.13 Brillouin Scattering

629

Values c11 and c22 are similar to each other between these two crystal forms. The value of c33 is twice larger for the orthorhombic form than that of the monoclinic form. The interlayer interaction is different between them. In this way, the Brillouin scattering is related to the elastic constant. This method is often applied to the study of the structural phase transition of the ferroelectric crystal. The softening of the acoustic mode causes the change of the unit cell parameters and the atomic positions with the change of temperature. A detailed description of the soft mode will be made in a later section.

5.14 Practical Analysis of IR/Raman Spectra of Polymer Crystals 5.14.1 Polyethylene (H/D) Blends The polarized IR and Raman spectra of polyethylene are already shown in Figs. 5.54, 5.75 and 5.77. The factor group analysis was made for the single chain and the orthorhombic crystal. The band splitting originates from the intermolecular interactions between the corner and center chains in the unit cell. The band assignments were made in Table 5.13. The complicated Raman spectral profiles in the frequency regions of 3000 and 1450 cm−1 are due to the Fermi resonance between the normal mode and the overtone (720 cm−1 × 2 ~ 1450 cm−1 × 2 ~ 3000 cm−1 ). The spectra of n-alkane and related compounds were interpreted from the viewpoint of progression bands and LAM mode. All of these discussions were already made in the previous sections. In the present section, the effect of isotope on the vibrational spectra of polyethylene is discussed. One is a blend sample consisting of the H and D chain components and the other one is the H/D ethylene copolymers [146–149]. The blend samples of fully hydrogeneous high-density PE [–(CH2 CH2 )n –, hHDPE] and fully deuterated PE [–(CD2 CD2 )n –, d-HDPE] are not perfectly miscible in the crystalline region, although they are miscible in the molten state. Such a blend sample between the H and D species is used sometimes for the purpose to know the averaged size (Rg ) and trace of a polyethylene chain in the amorphous region by analyzing the SANS data. The D/H blend sample is mixed in the melt and cooled slowly to room temperature or crystallized at an isothermal crystallization temperature to obtain the “homogeneously” mixed H/D polymer components. However, the mixed sample is not necessarily perfectly homogeneously blended but it shows the partial phase separation between the H- and D-components. In fact, as seen in the DSC thermograms shown in Fig. 5.170a, the blend between h-HDPE and d-HDPE did not give a single peak but some shoulder was observed, indicating a partial separation. Most of the samples used for the SANS study were prepared by melt and quenching to the low temperature to keep the mixed state in the molten state. Such a quenched sample is in a sense a special sample and so the result cannot be necessarily applied to the normally crystallized sample. When a linear low-density polyethylene

630

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.170 The DSC thermogram in the cooling process from the melt measured for a the solution blended sample of h-HDPE and d-HDPE and their simply mixed sample, and b the blend sample of LLDPE and d-HDPE and the simply-attached sample of these two components. Reprinted from Ref. [146] with permission of the American Chemical Society, 1992

(LLDPE) sample with a low content of ethyl side groups (about 17 branches per 1000 skeletal carbon atoms) is blended together with d-HDPE sample, they were found to mix homogeneously even after the slow crystallization [146–150]. Figures 5.170b and 5.171 show their DSC data. The thermograms are observed to show only one peak in both the melting and crystallization processes for the blend sample between LLDPE and d-HDPE. Besides, the melting (and crystallization) single peak shifts systematically depending on the H/D content. Let us see the IR spectra of these LLDPE/d-HDPE blend samples with the various H/D contents. Figure 5.172 shows the IR spectra measured for the melt-slowly cooled samples. The IR spectra change also systematically with the H/D content [150]. The pure h-LLDPE sample shows the band splitting as already described. The pure d-HDPE sample also shows the splitting of the bands although the band position is shifted to the lower frequency side from 720 cm−1 to 520 cm−1 region because of the heavier hydrogen (D) atoms. The band splitting becomes smaller as the H (or D) content is diluted by the introduction of D (or H)-component. This phenomenon can be understood as the dilution of H…H (D…D) interactions by the invasion of D (H) chain stems into the H (D) chain stem matrix in the same crystallite. That is to say, the homogeneous co-crystallization phenomenon occurs for the blend samples of LLDPE and d-DHDPE pair. The IR band profile is separated into doublet and singlet components. The band splitting width ν and the relative content of the singlet component are shown in Fig. 5.173. As already mentioned, the band splitting occurs when the 2 chain stems are positioned along the 110 direction. Then, the 1D array of the statistically randomly mixed H and D stems along the 110 direction is considered for simplicity. These stems are

5.14 Practical Analysis of IR/Raman Spectra of Polymer Crystals

631

Fig. 5.171 The DSC thermograms in the cooling process from the melt measured for a series of solution-blended samples between LLDPE and d-HDPE with the different H/D molar ratios. Reprinted from Ref. [146] with permission of the American Chemical Society, 1992

Fig. 5.172 IR spectra measured for the blend samples of the various H/D ratios. Reprinted from Ref. [150] with permission of springer nature, 1999

632

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.173 a Band splitting width and b single band content plotted against the H-species fraction. Reprinted from Ref. [150] with permission of Springer Nature, 1999

vibrationally coupled due to the intermolecular interactions. The 1D array of these oscillators may be assumed using the simply coupled oscillator model, and the calculated frequency corresponds to the splitting width of the coupled stems ν. If the N stems or oscillators are arrayed along the 110 direction (Fig. 5.174), then the band splitting is expressed in the following way, where νo is the splitting width of the infinitely long array of the oscillators. v = v◦ cos[π/(N + 1)]

(5.354)

The averaged number of the stems of the same H (or D) species, N, is obtained for the H (D) content fraction X as follows. The H-H pairing probability of the sequence CD2 (CH2 )p (CD2 ) is (1-X)2 X p−1 and so the averaged sequential number N is N=

∞  p=0

p(1 − X ) X 2

p−1

∞  / (1 − X )2 X p−1 = 1/(1 − X )

(5.355)

p=0

Therefore, the band splitting width for the blend sample with the H content X is given as v = v◦ cos[π/(N + 1)] = v◦ cos[π(1 − X )/(2 − X )]

(5.356)

5.14 Practical Analysis of IR/Raman Spectra of Polymer Crystals

633

Fig. 5.174 Illustration of the linear array of the H stems along the 110 direction. Reprinted from Ref. [150] with permission of Springer Nature, 1999

The solid line in Fig. 5.173a shows the thus-calculated ν value [νo = 11.8 cm−1 , observed value for the H species], in good agreement with the observed one. On the other hand, the fraction of the singlet component of the observed IR band profile is calculated as follows. The DHD sequence gives the singlet band of the H species, the content of which is (1 – X)2 . The molar extinction coefficients of the singlet and doublet bands are, respectively, expressed as εsinglet and εdoublet . Using the Lambert-Beer law, the single content S is expressed as ⎡ S = εsinglet (1 − X )2 /⎣εsinglet (1 − X )2 + εdoublet   = εsinglet (1 − X )/ εsinglet (1 − X ) + εdoublet X

∞ 

⎤ (1 − X )2 X p−1 ⎦

p=2

(5.357)

The solid line in Fig. 5.173b is the calculated S value for εsinglet = 154.3 and εdoublet = 160.6 (arbitrary unit). Simply, if the same value is assigned to these two coefficients, then Eq. 5.357 changes to S = 1 – X. Figure 5.173 shows this linear relation also. Thus, we can say that the blend sample between LLDPE and DHDPE is a good system of cocrystallization between the D and H chain stems in the common crystallites. Besides, the H and D stems are randomly distributed in the crystal lattice. This conclusion can be extended to the discussion about the chain folding feature of polyethylene. The random positioning of the H and D stems suggests the random aggregation of the polyethylene chains and also the random reentry of the folded chains, as illustrated in Fig. 5.175, Refer to Sect. 4.5.3.

634

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.175 Illustration of crystal structure of a pure HDPE and b the blend sample between the H and D polymer species. c the chain folding models. The blend sample between LLDPE and d-HDPE species corresponds to the randomly reentry model . Reprinted from Ref. [150] with permission of Springer Nature, 1999

5.14.2 Structural Regularization of Polyethylene in Melt-Crystallization When n-alkane crystal is heated the orthorhombic crystal transforms to the hexagonal phase and then melts. The structure change can be monitored by measuring the IR and Raman spectra in addition to the X-ray diffraction as functions of temperature [151, 152]. A similar phase transition is observed also for polyethylene sample subjected to a tensile force [153]. In order to know the structure change in this phase transition, the measurement of the X-ray diffraction is useful for the check of the hexagonal phase [154]. The measurement of IR and Raman spectra can give us information on the conformational change in the phase transition [155]. Some descriptions were already made in Chaps. 1 and 4. The hexagonal phase is built up by the aggregation of the conformationally disordered chains with the trans-zigzag segments of 6 ~ 18 ¯ as detected in the IR and Raman CH2 units [156] and the gauche bonds of TGTG, spectral measurement.

5.14 Practical Analysis of IR/Raman Spectra of Polymer Crystals

635

Fig. 5.176 a IR spectral change observed in the isothermal crystallization process of PE from the melt. b The intensity change of the disordered trans and orthorhombic bands in the crystallization process, and c the illustration of the phase transition in the crystallization of PE. Reprinted from Ref. [157] with permission of Springer Nature, 1998

These information are useful for the study of the structural evolution in the isothermal crystallization process of PE from the melt. The highly resolved timedependent IR spectral change was measured in the temperature–jump isothermal crystallization process at T = 4 ◦ C [157]. Just before the appearance of the long trans-zigzag bands, the wagging (1360 cm−1 ) and rocking (720 cm−1 ) bands intrinsic to the gauche forms were observed to increase the intensity and then decreased with time (Fig. 5.176). This indicates the PE chains in the melt change to the locally regular but short trans-zigzag chain segments with the gauche bonds, which may be named the disordered trans state and exists in the hexagonal phase before the stabilization to the orthorhombic phase [158].

5.14.3 Vibrational Spectra of Poly(Vinylidene Fluoride) and its Copolymers We have already seen the IR spectra of PVDF in several sections. Here let us summarize the IR and Raman spectra of the various crystalline forms of PVDF [159, 160]. As already mentioned in the section of sample preparation method, PVDF crystallizes into various crystalline forms [161]. They are named the forms I (β), II (α), polar

636

5 Structure Analysis by Vibrational Spectroscopy

PVDF 1275

840

FormⅠ

510

Absorbance

1210 FormⅡ

765 795

610

530

1235 835 810

FormⅢ

1400

1200

1000

800

Wavenumber /cm

610

600

400

-1

Fig. 5.177 The IR spectra of PVDF forms I, II, cast III, and annealed III. Reprinted from Ref. [162] with permission of the American Chemical Society, 1981. The wavenumbers of the IR bands characteristic of each crystalline form are given in the right-side spectra

II or IIp or IV (δ, αp ), III (γ), and V (not confirmed). As already mentioned, form I consists of almost planar-zigzag chains of 2.55 Å period and these chains are packed so that the zigzag planes are parallel to the b-axis. This is a polar and ferroelectric ¯ conformation of the glide symmetry and is packed phase. Form II takes the TGTG in the cell with a point of symmetry or the non-polar crystal. Form III consists of the ¯ conformation and they are packed in anti-polar mode. The IR chains of TTTGTTTG spectra of these crystalline forms are shown in Fig. 5.177.

5.14.3.1

PVDF Form I

The form I takes the space group Cm2m–C 2v 14 . The factor group analysis is shown in Fig. 5.178 (refer to Sect. 5.7.4). Since the C-centered lattice is simplified to the primitive lattice containing one chain, as already shown in Fig. 5.96, this analysis is made for the latter structure. The planar-zigzag chain has a factor group C 2v . The polarized IR and Raman spectra of PVDF form I are shown in Figs. 5.179 and 5.180, respectively. The rolled (or the doubly oriented) sample was used for the polarized Raman spectral measurement from the various directions, where the Z-axis // the chain axis and the X-axis // the rolled plane. By comparing the IR spectra among the various crystalline forms and also the evaluation of critical sequential length based on the IR spectra of a series of VDF-TrFE copolymers, the band at 1273 cm−1 appears for the trans segment with CH2 CF2 units long than 4. The 840 cm−1 band appears when the trans-zigzag segment is longer than 3 monomeric units. The band assignment is listed in Table 5.17. The application of high voltage to the form I film (poling) causes the rotation of polar-zigzag chains so that the CF2 dipoles become parallel to the Y-axis. When the IR beam is incident to the film the A1 bands with the transition dipole parallel

5.14 Practical Analysis of IR/Raman Spectra of Polymer Crystals

637

Fig. 5.178 a Crystal structure of PVDF form I with the space group symmetries. The factor group analysis of form I crystal based on the primitive cell is shown in (b) Reprinted from Ref. [160] with permission of the American Chemical Society, 1985

Fig. 5.179 Polarized IR spectra of the oriented PVDF form I: a before poling and b after poling. The solid and broken lines show the spectra taken with the incident electric vector perpendicular and parallel to the drawing axis, respectively Reprinted from Ref. [160] with permission of the American Chemical Society, 1985

to the Y-axis decrease in intensity. In contrast, the B2 bands increase the intensity. Figure 5.179b shows the polarized IR spectra measured for the poled sample. The bands near 2900 cm−1 are assigned to the A1 and B2 for the symmetric (νs ) and antisymmetric (νa ) stretching modes, respectively. Comparison of the intensity change before and after poling indicates the above description clearly. In other words, the intensity ratio between these two bands indicates the degree of inversion of CF2 dipole moments of zigzag chains. The situation is seen also for the polarized Raman spectra of the poled sample (see Fig. 5.180). According to the factor group analysis, the B1

638

5 Structure Analysis by Vibrational Spectroscopy

(a)

(b)

rolled

X//b

c

Y//a

poled

X//a

c

Y//b

Fig. 5.180 Polarized Raman spectra of PVDF form I. a the rolled sample and b that after highvoltage poling treatment. The relation between the sample axes and crystal axes is changed: (before poling) X // b, Y // a. Z // c and (after poling) X // a, Y // b, Z // c. Reprinted from Ref. [160] with permission of the American Chemical Society, 1985

(1078 cm–1 ) and B2 (880 cm–1 ) modes have the transition polarizability components α’bc and α’ab , respectively. As seen in Fig. 5.180a, these bands were observed for (XZ) and (XY ), respectively. Therefore, before poling, the X-axis was parallel to the b-axis and the Y-axis to the a-axis. After poling the 1078 cm–1 band was detected clearly for (YZ) and the 880 cm–1 band for (XY ), and X // a and Y // b.

The polarized Raman spectra of form I are unique in such a point that the band position is shifted (LO-TO splitting) by changing the angle between the phonon vector (q = ki – ks ) and the polar b-axis: the LO mode for the b // q and the TO mode for the b ⊥ q. The band shift is a result of long-range dipole–dipole interactions. The A1 bands at around 1270 and 840 cm−1 regions show such an LO-TO splitting (Fig. 5.181). A little more detailed explanation of LO-TO splitting is made here. The LO is the longitudinal optical mode with the atomic vibrational direction parallel to the phonon propagation direction, while the TO is the transverse optical mode with the perpendicular relation between these two vectors. In the polar crystal, in general, the vibrational frequency is different between the LO and TO modes, which is called the LO-TO splitting. The equation of motion of an atom k with a mass mk is expressed

5.14 Practical Analysis of IR/Raman Spectra of Polymer Crystals

639

Table 5.17 Normal coordinates treatment of single chain of poly(vinylidene fluoride) form I. Reprinted from Ref. [160] with permission of the American Chemical Society, 1985 Species A1

A2 B1

B2

Wavenumber, cm−1

PED(%)c

Obsda

Calcdb

2980

2975

νs (CH2 )(99)

1428

1434

δ(CH2 )(81)

1273

1283

νs (CF2 )(40) − νs (CC)(22) + δ(CCC)(26)

840

844

νs (CF2 )(59) + νs (CC)(17)

508

513

δ(CF2 )(98)

980

980

t(CH2 )(100)

260

265

t(CF2 )(100)

1398

1408

w(CH2 )(58) − νa (CC)(34)

1071

1074

νa (CC)(53) + w(CH2 )(25) − w(CF2 )(22)

468

471

w(CF2 )(90)

3022

3024

νa (CH2 )(99)

1177

1177

νa (CF2 )(71) − r(CF2 )(18)

880

883

r(CH2 )(62) − νa (CF2 )(18) − r(CF2 )(19)

442

444

r(CF2 )(70) + r(CH2 )(24)

a Infrared

absorption spectral data (refer to Figs. 5.179 and 5.180) utilized model was of planar-zigzag conformation c Potential energy distribution. Symmetry coordinates: (ν ) antisymmetric stretching; (ν ) a s symmetric stretching; (δ) bending; (w) wagging; (t) twisting; (r) rocking. The sign + or – denotes the phase relation among the symmetry coordinates b The

as m k d 2 uk (r)/dt 2 = k  Rk,k  uk (r) − Q E(r)

(5.358)

P(r) = k Q k uk (r) + χ E(r)

(5.359)

  E(r) = −4πqo q ◦ · P(r)

(5.360)

where Rk,k is the force constant between the atoms k and k  , Q is the matrix representing the atomic charges, P(r) is the polarization, χ is the electric susceptibility, E(r) is the local electric field at the position r, and qo is the unit vector of a phonon q (qo = q/|q|) [163]. When the wavelength of a phonon is enough long compared with the unit cell size, the uk (r), P(r), and E(r) are expressed as uk (r) = u◦k exp[i(q · r − ωt)]

(5.361)

640

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.181 Frequency shift of the A1 Raman bands for the change of sample axis with respect to the phonon vector q. The angles shown are measured from the incident photon vector ki . Reprinted from Ref. [160] with permission of the American Chemical Society, 1985

P(r) = P exp[i(q · r − ωt)]

(5.362)

E(r) = E exp[i(q · r − ωt)]

(5.363)

where the ω is the vibrational frequency. By substituting these equations to Eq. 5.358, we have

 2   

 

Ω − ω2 δ j j  + 4π/ε∞ (q ◦ ) q ◦ · M j q ◦ · M j  = 0 j 1/2

M j = transition dipole = k Q k e j /m k

(5.364) (5.365)

ε∞ (q ◦ ) = dielectric constant at an infinitely high frequency = 1 + 4π χ   2   2 n i2 qi◦ (i = 1, 2, 3) (5.366) = εi∞ qi◦ =

5.14 Practical Analysis of IR/Raman Spectra of Polymer Crystals

641

(a)

(b)

(c)

Fig. 5.182 a the relation of the crystal axis and the phonon vector for the A and B modes of TGS, b a series of Raman spectra measured for the B modes by changing the angle θ, c the observed vibrational frequencies plotted against the angle θ for the bands of B species. The solid curves are the calculated results. Reprinted from Ref. [164] with permission of the IOP Publishing, 1987

ni is a refractive index. For the phonon vector qo perpendicular to the P(r), that is, in the case with only the short-range force R under E = 0, the following equation is given.

−1/2

m Rm−1/2 −



2

=0

(5.367)

where j (= ωTO )is the eigenvalue for this system. The eigenvector is ej in Eq. 5.365. In Eq. 5.364, the inner product (qo ·M j ) is dependent on the direction between the

642

5 Structure Analysis by Vibrational Spectroscopy

Table 5.18 Factor group analysis of TGS crystal with C 2 factor group E

C2s

μ

α

Internal modes

L(T)

L(L)

A species

+1

+1

b

a’a’, bb, cc, a’c

87

11

12

B species

+1

– 1

a’, c

a’b, bc

87

10

12

μ: transition dipole vector α: Raman polarizability tensor L(T), L(L): translational and librational lattice modes, respectively

qo and M j . For the vibration with qo ⊥ M j , that is, the TO mode, the eigenvalue = ωTO . The frequency gap between ωTO and ωLO is expressed approximately as   ω = ωLO − ωTO = 2π M 2 / ε∞ ωTO

(5.368)

The frequency gap is large for the larger transition dipole interaction M 2 , that is, for the vibrational mode with higher IR absorbance. The change of angle between qo and M will give the change of vibrational frequency. For example, Fig. 5.182b shows the Raman spectra (B modes) of a ferroelectric single crystal triglycine sulfate (TGS) [164]. The cylindrically cut single crystal was rotated around the b-axis or the c-axis to measure the Raman components of the symmetry species A or B, respectively (refer to Table 5.18). The angle θ is defined as the angle between the scattering vector q (= ki – ks ) and the b-axis (for A mode) or the c-axis (for the B mode) as shown in Fig. 5.182a. As the angle θ is changed, the Raman peaks are found to move the positions systematically. The wavenumber of the band peaks is plotted against the angle θ, as shown in Fig. 5.182c. Using Eq. 5.364, the vibrational frequency is expressed as a function of angle θ. In these calculations, the refractive index na’ = 1.583, nb = 1.483, and nc = 1.555 were used. The dielectric constant ε∞ (q) is a function of q vector and is expressed as below (Table 5.18). For the A mode, q = (sin θ, cos θ, 0)

  ε∞ (q) = εa∞ qa2 + εb∞ qb2 = q 2 n a2  sin2 θ + n 2b cos2 θ   ◦  2 + 4π M 2 cos2 θ − θ / n a2  sin2 θ + n 2b cos2 θ ω2A = ωTO

(5.369)

where θ o is the angle between M and the b-axis, which is 0° for the A mode. For the B mode, q = (sin θ, 0, cos θ ) ε∞ (q) = εa∞ qa2 + εc∞ qc2 ≈ q 2 n 2c  ◦ 2 ω2B = ωTO + 4π M 2 cos2 θ − θ /n 2c

(5.370)

5.14 Practical Analysis of IR/Raman Spectra of Polymer Crystals

643

The vibrational frequencies plotted in Fig. 5.182c, a case of the symmetry species B, are fitted well using these equations. A similar measurement was made for the PVDF form I sample subjected under the high electric field [160]. The results were already described in Fig. 5.181.

5.14.3.2

PVDF Form II

The factor group of a single chain of form II is C s since the glide plane passes the molecular chain. The factor group of the crystal is C 2h . As seen in Table 5.19, the mutual exclusion law is applied although the point of symmetry exists between the separated chains and the difference between the IR and Raman bands is not necessarily large. The representative gauche (TG) bands are at 800 ~ 700, 610, and 530 cm−1 , as shown in Fig. 5.183. By focusing the IR bands intrinsic to TT and TG conformations of forms I and II, respectively, the structural change in the thermally induced phase transition and/or the stress-induced transition between these two forms can be traced. For example, Fig. 5.184 shows the 2D image produced by the polarized IR spectra measured for the necked PVDF sample [165]. The necking phenomenon was observed in the central Table 5.19 Factor group analysis of PVDF form II crystal Species

Molecular modes

Lattice modesa

Ag

16

L(T a ), L(T c ) )0

Selection rules IR

Raman

–

Active

Bg

16

L(T b ), L(Rc

–

Active

Au

16

T b , L(Rc )π

⊥

–

Bu

16

T a, T c

// ⊥

–

translational mode, R: librational mode, 0 and π: phase difference between the two chains in the unit cell

a T:

Fig. 5.183 Polarized IR spectra and Raman spectra of the oriented PVDF form II. Reprinted from Ref. [159] with permission of the American Chemical Society, 1978

644

5 Structure Analysis by Vibrational Spectroscopy

A B CDE 0

150%

form I (1275cm-1 )

56 110 120 130 m

//

125%

125%

100%

form II (975cm-1

)

// 75%

Fig. 5.184 2D image derived from the polarized transmission IR spectra measured at the various points of the 125%-stretched PVDF form II film. Depending on the polarization of the incident IR beam, the images are different. The central part is the highly oriented form I and the other parts are the less-oriented form II. The red and green lines of the polarized spectra are for the electric vector of the IR beam perpendicular and parallel to the draw direction, respectively [165]

part of the sample. The microscopic 2D transmission IR spectral measurement was made at the various positions. The spectra at point E, near the original unoriented film, the IR profile is similar to that of poorly oriented form II with a small amount of form I. The spectra at positions A and B are those of perfectly transformed form I. Spectra C and D contain small amount of form II. The 2D images were produced by plotting the intensity of the bands at 1275 cm−1 (form I) and 975 cm−1 (form II). The transition from form II to I occurs quite sharply in the narrow region of the necking phenomenon. The oligomer samples are useful for the analysis of the IR and Raman spectra of the parent PVDF. A series of oligomers were extracted from the mixture using the super critical fluid CO2 chromatograph method and the individual samples were crystallized from the solution to get the single crystals. The X-ray structure analysis revealed the conformation and packing structure essentially the same as those of form II (see Fig. 5.185). These oligomers contain the finite sequences of CH2 CF2 units. Therefore, we can expect the progression bands. As already mentioned [91], the detailed analysis allowed the assignment of these progression bands on the basis of the vibrational frequency-phase angle dispersion curves calculated for the form II crystal (Sect. 5.10.4.2).

5.14 Practical Analysis of IR/Raman Spectra of Polymer Crystals PVDF form Ⅱ c = 4.6 2

7mer

Ⅱ PVDF formⅡ

7mer

645

cs = 4.6

a = 4.96

as = 4.98

3Å

Å

Å

Å bs = 9.72Å

b = 9.64Å

6mer

8mer

cs = 4.64 Å

6mer

b = 9.64Å

bs = 9.72Å

cs = 4.65

as = 4.92

as = 4.80

Å

Å

Å

bs = 9.76Å

bs = 9.76Å

bs = 9.76Å

bs = 9.76Å

Fig. 5.185 Crystal structures of PVDF oligomers in comparison with that of form II. The intermediate part of the chain is shown for comparison. (Left) along the a-axis, and (right) along the chain axis. Reprinted from Ref. [166] with permission of the American Chemical Society, 2002

Fig. 5.186 IR spectra of as-cast PVDF form III annealed at the various temperatures. Reprinted from Ref. [162] with permission of the American Chemical Society, 1981

5.14.3.3

PVDF Form III

PVDF film cast from the DMA (or DMSO) solution crystalizes into form III. This description is not necessarily correct. The IR spectrum measured for the as-cast film was compared with that of the annealed sample (Fig. 5.186). The relatively strong bands are detected at the positions of long trans bands (1273 and 840 cm−1 ) [162]. The annealed sample shows the decrease of these long trans bands and the increase of 610 cm−1 TG band. The bands at 810 cm−1 etc. are intrinsic to form III. The as-cast

646

5 Structure Analysis by Vibrational Spectroscopy

film consists of the disordered conformation parts of relatively long TTTT segments and the intrinsic TTTG segments. The heat treatment causes the transformation of TT ¯ to TG in the long TTTT segment and results in the regular sequence of TTTGTTTG of form III. 5.14.3.4

VDF Copolymers

VDF-TrFE copolymer shows the ferroelectric phase transition. The X-ray diffraction measurement revealed the repeating period along with the chain axis changes from 2.55 Å at room temperature to 2.30 Å in the high-temperature paraelectric phase, indicating the contraction of the chain conformation in the high-temperature region. The concrete conformational change can be investigated on the basis of the IR and Raman spectral measurements [167–169]. As shown in Figs. 5.187 and 5.188, the trans bands intrinsic to PVDF form I are detected at around 1275 and 840 cm−1 at room temperature. As the temperature approaches the transition point, these bands decreased in intensity and the TG bands started to appear at around 610 cm−1 . The intensity plot of these bands indicates the occurrence of the conformational exchange between TT and TG form. The 811 cm−1 band intrinsic to the TTTG segment of PVDF form III increased also the intensity. Therefore the molecular chains in the low-temperature ferroelectric phase take the trans-zigzag conformation, while the chains in the paraelectric phase exist in the statistically disordered form consisting ¯ and TG ¯ bonds. The T-G conformational change is clearly of the TTTG, TG, TTTG, detected for the copolymer with higher VDF content. The bulky TrFE parts are difficult to change the conformation, and the TG change occurs gradually without hysteresis in the heating and cooling processes (Fig. 5.188b). A more detailed study will be described in the Sect. 8.2.3.

Fig. 5.187 Temperature dependence of polarized IR spectra measured for VDF55% copolymer. Reprinted from Ref. [167] with permission of Elsevier, 1984

5.15 Simultaneous Measurement System

647

Fig. 5.188 a Temperature dependence of FT-Raman spectra measured for VDF55% copolymer. Reprinted from Ref. [168] with permission of the American Chemical Society, 1999. b Temperature dependence of intensity evaluated for the TT and TG Raman bands of VDF 52 and 73% copolymers

5.15 Simultaneous Measurement System Soft materials exhibit the various hierarchical structure consisting of the complicated aggregation of crystalline and amorphous regions. These higher order structures (including the chain conformation, chain packing mode in the crystalline region, the combination of the crystalline region with the amorphous region) changes sensitively depending on the slight variation of the external environment. Of course, such a complicated hierarchical structure must be seen from the various different points of view on the basis of the various analytical instruments. Therefore, it is ideal to collect these various data at the same time for the same sample, which is the so-called simultaneous measurement of the various instruments. The typical combination is the wide-angle and small-angle X-ray scattering methods. In addition, the combination of these methods with the vibrational spectroscopy or with the light scattering method is more useful.

5.15.1 WAXD/SAXS Simultaneous Measurement There are several ways to collect the wide-angle X-ray diffraction data and the smallangle scattering data together. The concrete description of the experimental systems was already made in Sects. 1.5.3 and 4.4.2 [170, 171].

648

5 Structure Analysis by Vibrational Spectroscopy

5.15.2 WAXD/SAXS/Raman Simultaneous Measurement In addition to the WAXD and SAXS simultaneous measurement, the combination of Raman spectral measurement is not very difficult. As shown in Fig. 5.189, a laser beam is incident on the sample, which is a target for the X-ray scattering measurement, and the backscattering component is collected using the same laser probe [172, 173]. The laser probe consists of the lens and optical fibers. The laser light is sent through the optical fiber and focused on the sample position using a lens, and the backscattered signals are collected with a bundle of optical fiber surrounding the lens. One application of this simultaneous measurement system of WAXD/SAXS/Raman data was made for the study of the ferroelectric phase transition of VDF52%-TrFE48% copolymer sample [174]. As shown in Fig. 5.190, the WAXD and Raman spectral data indicated the existence of crystal form I with the trans-zigzag conformation. Above the phase transition point the gauche bands were detected in the Raman spectra and the hexagonal packing structure of the contracted chains with the repeating period 2.30 Å was estimated from the WAXD pattern. This structural change was already explained in the section on vibrational spectroscopy (Sect. 5.14.3.4). The SAXS data measured at the same time indicated the change of the stacking structure of lamellae: the lamellae in the ferroelectric phase are tilted Fig. 5.189 Simultaneous measurement system of WAXD (CCD camera) and Raman spectra (CCD). Raman spectroscopy is of Chromex, Co. (USA)

(a)

5.15 Simultaneous Measurement System

(a)

649

(b)

PILATUS 300K (SAXS)

PILATUS 300K (SAXS+WAXD) Raman probe

Raman probe

Heating block

Heating block

Sample

Sample X-ray

X-ray

PILATUS 100K (WAXD)

VDF52%

(c)(a)

(b) 819cm-1 Raman Intensity (a.u.)

120℃ (HT)

Draw axis

2ndHeat 30℃ (CL) 1stCool 120℃ (HT) 1stHeat

855cm-1 1000

900

30℃ (LT)

800

Raman shift /cm-1

(c)

200/110(LT)

200/110(HT)

1st Heat

200/110(CL)

1st Cool

2nd Heat

30℃ (LT )

120℃ (HT)

30℃(CL)

120℃ (HT)

30℃ (LT )

120℃ (HT)

30℃(CL)

120℃ (HT)

(d)

Fig. 5.190 Simultaneous measurement system of a WAXD (Pilatus 100 k), SAXS (Pilatus 300 k), and Raman spectra (CCD) and b WAXD and SAXS (Pilatus300k) and Raman spectra (CCD). b Temperature dependence of WAXD, SAXS and Raman spectra measured for VDF52% sample. Reproduced from Ref. [174] with permission of the International Union of Crystallography, 2014

650

5 Structure Analysis by Vibrational Spectroscopy

from the draw axis, while they stood almost vertically above the phase transition temperature. This is caused by the thermally accelerated translational motion of the chains along the drawing axis.

5.15.3 WAXD/SAXS/IR Simultaneous Measurement The Raman spectra of polymer substance are difficult to obtain because of the overlap of strong fluorescence signals. In particular, the heat-treated sample and the sample measured in the heating process give overwhelmingly strong fluorescence, making the Raman spectral measurement actually impossible. The IR spectra are more useful in such a sense that they are not affected by the fluorescence. The reflection-type IR spectra, for example, ATR spectra, are easier to collect. But the detected band profile is not necessarily regular because of the anomalous dispersion effect of the refractive index, and the peak position and the relative intensity are not always the same as those measured in the transmission spectroscopic method. The ideal method is to use the transmission type. However, they need thin films to obtain the band intensity suitable for the quantitative analysis. These thin films are difficult for the X-ray diffraction measurement in the lab level. (Of course, the thick sample can be used for the transmission-IR measurement, which is useful for the X-ray scattering measurement, as long as the weak bands are used for the data analysis.) The combination of a synchrotron X-ray beam generated by an undulator with highly sensitive detector can solve this dilemma. Only one-second exposure gives enough strong WAXD and SAXS data of the sample film of only 1 μm thickness. The IR spectrometer suitable for the setting on the X-ray diffractometer must be small in size. A flexible optical fiber from the relatively large IR spectrometer might be possible, but the vibrational frequency range of the IR spectra detectable by the optical fiber is quite narrow and low signal-to-noise ratio. The usage of a miniature FTIR spectrometer fits this condition. The IR beam and X-ray beam must pass through the same position of the sample. One idea is to modify the optical path of the FTIR spectrometer so that the IR beam can pass the sample with a small tilting angle as illustrated in Fig. 5.191. The X-ray beam is incident horizontally to the sample and the IR beam passes the same position but with a small tilting angle (a few degrees), which does not affect the IR spectra [175]. One example using this system is described here [175]. The data are about the structural change induced in the oriented poly(tetramethylene terephthalate) (PTMT) film subjected to a tensile force. The sample was set to a remote-controlled stretching device installed on a sample stage of a synchrotron X-ray system, and the WAXD and SAXS data were collected in the usual way mentioned above. A miniature FTIR spectrometer was installed around the stretcher. The sample was stretched continuously at a constant stretching rate, during which the FTIR spectra were measured at every 7 s with the resolution power of 2 cm−1 and scanning time is 1. The undulator X-ray beam was incident on the sample and the WAXD and SAXS patterns were measured for 1 s exposure and 6 s rest, using a flat panel and an image-intensifier CCD, respectively. The thus-collected X-ray data are shown in Fig. 5.192. The detailed analysis results are described in Sect. 8.2.9.

5.15 Simultaneous Measurement System (a) mirror X-ray

651

Flat Panel (WAXD)

sample cell

FTIR mirror

to SAXS detector

(b) IR detector IR source FTIR

X-ray

mirror

mirror

sample cell

Fig. 5.191 Simultaneous measurement system of WAXD (flat panel), SAXS (CCD camera), and FTIR spectra. a Setting of a miniature FTIR spectrometer and b a snapshot of the system. Reprinted from Ref. [175] with permission of the American Chemical Society, 2014

Fig. 5.192 a Stretching device for the simultaneous measurement of PTMT sample. b 2D WAXD and SAXS patterns, and c the integrated data of (b). Reprinted from Ref. [175] with permission of the American Chemical Society, 2014

652

5 Structure Analysis by Vibrational Spectroscopy

Fig. 5.193 Temperature jump cell set on the simultaneous measurement system of WAXD, SAXS, and FTIR spectra [158]

The second example is about the temperature jumping experiment for the study of isothermal crystallization phenomenon. The temperature jump cell, the details of which were already described in Sect. 1.7.3.5, is set on the sample stage (Fig. 5.193) [158].

References 1. L. Pauling, E.B. Wilson, Introduction to Quantum Mechanics with Application to Chemistry (Dover Books on Physics, 1985) 2. G.M. Barrow, Introduciton to Molecular Spectroscopy (McGraw-Hill Book Co., New Yoor, 1962) 3. D.A. Long, The Raman Effect: A Unified Treatment of the Theory of Raman Scattering by Molecules (Wiley, New York, 2002) 4. J.M. Chalmers, P.R. Griffiths (eds.), Handbook of Vibrational Spectroscopy, vols. 1–5 (Wiley, New York, 2002) 5. N.B. Colthup, J. Opt. Soc. Am. 40, 397–400 (1950) 6. D. Lin-Vien, N.B. Colthup, W.G. Fateley, J.G. Grasselli, The Handbook of Infrared and Raman Characteristic Frequencies of Organic Molecules (Academic Press, London, 1991) 7. S. Takayama (ed.), Collection of IR absorption spectra of plastics, additives, and elastomers (in Japanese) (Spectrum Forum, 2010) 8. T. Sandner, C. Drabe, H. Schenk, A. Kenda, Miniaturized FTIR-spectrometer Based on Optical MEMS Translatory Actuator, in Proceedings of SPIE-The International Society for Optical Engineering 6466 (2007) 9. G. de Graaf, W. der Vlist, R.F. Wolffenbuttel, Design and Fabrication Steps for a MEMSbased Infrared Spectrometer using Evanescent Wave Sensing, Sens. ActuatorsA 142, 211–216 (2008) 10. D.B. Chase, Fourier Transform Raman Specroscopy. J. Am. Chem. Soc. 24, 7485–7488 (1986) 11. H. Tadokoro, K. Tatsuka, S. Murahashi, Infrared absorption spectra and structure of polyethylene terephthalate. J. Polym. Sci.59, 413–423 (1962) 12. K.H. Park, Y. Liang, S.H. Kim, H.S. Lee, New analysis method for three-dimensional orientation of PEN using polarized FTIR-ATR spectroscopy: multipeak reference method. Macromolecules 39, 1832–1839 (2006) 13. S.C. Park, Y. Liang, H.S. Lee, Quantitative analysis method for three-dimensional orientation of PTT by polarized FTIR-ATR spectroscopy, Macromolecules 37, 5607–5614 (2004)

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153. A.J. Pennings, A. Zwijnenburg, Longitudinal growth of polymer crystals from flowing solutions. VI. Melting behavior of continuous fibrillar polyethylene crystals. J. Polym. Sci. Polym. Phys. Ed. 17, 1011–1032 (1979) 154. S. Tsubakihara, M. Yasuniwa, Melting and crystallization of ultra-high molecular weight polyethylene with appearance of hexagonal phase II. Strain-induced crystallization of thermalcontracted sample. Polym. J. 28, 563–567 (1996) 155. K. Tashiro, S. Sasaki, M. Kobayashi, Structural investigation of orthorhombic-tohexagonal phase transition in polyethylene crystal: the experimental confirmation of the conformationally- disordered structure by x-ray diffraction and infrared/raman spectroscopic measurements. Macromolecules 29, 7460–7469 (1996) 156. Y. Cho, M. Kobayashi, H. Tadokoro, Polym. Prepr. Jpn. 30, 1842 (1986) 157. K. Tashiro, S. Sasaki, N. Gose, M. Kobayashi, Microscopically-viewed structural change of polyethylene during isothermal crystallization from the melt [1] Time-resolved FTIR spectral measurements. Polym. J. 6, 485–491 (1998) 158. K. Tashiro, H. Yamamoto, Structural evolution mechanism of crystalline polymers in the isothermal melt-crystallization process: a proposition based on simultaneous WAXD/SAXS/FTIR measurements. Polymers 11, 1316–1340 (2019) 159. M. Kobayashi, K. Tashiro, H. Tadokoro, Molecular vibrations of three crystal forms of poly(vinylidene fluoride), Macromolecules 8, 158–171 (1975) 160. K. Tashiro, Y. Itoh, M. Kobayashi, H. Tadokoro, Polarized Raman spectra and LO-TO Splitting of poly(vinylidene fluoride) crystal form I, Macromolecules 18, 2600-2606 (1985) 161. K. Tashiro, Crystal structure and phase transition of PVDF and related copolymers, ferroelectric polymers: chemistry, physics, and technology (H.S. Nalwa ed.) (Marcel Dekker Inc., 1995), pp.63–182 162. K.Tashiro, M.Kobayashi, H.Tadokoro, Vibrational spectra and disorder-order transition of poly(vinylidene fluoride) form III. Macromolecules 14 1757–1764 (1981) 163. S.M. Shapiro, J.D. Axe, Raman scattering from polar phonons. Phys. Rev. B6, 2420–2427 (1972) 164. K. Tashiro, N. Yagi, M. Kobayashi, T. Kawaguchi, Determination of the LO-TO splitting in the Raman spectra of ferroelectric triglycine sulfate (TGS). Jpn. J. Appl. Phys. 26, 699–706 (1987) 165. K. Tashiro, Time & Space dependence of sold-state structure of polymer materials on the basis of rapid-scan 2-dimensional infrared imaging technique. Polym. Prepr. Jpn. 56, 4095 (2007) 166. K. Tashiro, M. Hanesaka, Confirmation of crystal structure of poly(vinylidene fluoirde) through the detailed structure analysis of vinylidene fluoride oligomers separated by super critical fluid chromatography. Macromolecules 35, 714–721 (2002) 167. K.Tashiro, K.Takano, M.Kobayashi, Y.Chatani, H.Tadokoro, Structure and ferroelectric phase transition of vinylidene fluoride-trifluoroethylene copolymer: 2. VDF 55% copolymer. Polymer 25, 195–208 (1984) 168. K.Tashiro, R. Tanaka, M. Kobayashi, Detection of sharp DSC peak during the phase transition from the low-temperature phase to the cooled phase of vinylidene fluoride-trifluoroethylene copolymers. Macromolecules 32, 514–517 (1999) 169. K. Tashiro, M. Kobayashi, Vibrational spectroscopic study of the ferroelectric phase transition in vinylidene fluoride-trifluoroethylene copolymers: 1. Temperature dependence of the Raman spectra. Polymer 29, 426–436 (1988) 170. K. Tashiro, S. Sasaki, Structural changes in the ordering process of polymers as studied by an organized combination of the various measurement techniques. Progress Polym. Sci. 28, 451–519 (2003) 171. K. Tashiro, H. Masunaga, T. Kabe, Development of WAXD/SAXS simultaneous measurement system for the hierarchical structural study using a center-holed WAXD photon-counting detector. Polym. Prepr. Jpn. 68 1C23 (2019) 172. K. Tashiro, S. Kariyo, A. Nishimori, T. Fujii, S. Saragai, S. Nakamoto, T. Kawaguchi, A. Matsumoto, O. Rangsiman, Development of a simultaneous measurement system of X-ray

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diffraction and Raman spectra: application to structural study of crystalline-phase transitions of chain molecules. J. Polym. Sci. Polym. Phys. 40, 495–506 (2002) 173. K. Tashiro, H. Yamamoto, T. Yoshioka, T.H. Ninh, S. Shimada, T. Nakatani, H. Iwamoto, N. Ohta, H. Masunaga, Development of simultaneous measurement system of wide- and small-angle X-ray scattering and vibrational spectra for the static and dynamic analyses of hierarchical structures of polymers, vol. 69. (KOBUNSHI-RONBUNSHU (Special issue), 2012), pp. 213–227 174. R. Hirose, T. Yoshioka, H. Yamamoto, K.R. Reddy, D. Tahara, K. Hamada, K. Tashiro, Inhouse simultaneous collection of SAXS, WAXD and Raman scattering data from polymeric materials. J. Appl. Crystallogr. 47, 922–930 (2014) 175. K. Tashiro, H. Yamamoto, T. Yoshika, T.H. Ninh, M. Tasaki, S. Shimada, T. Nakatani, H. Iwamoto, N. Ohta, H. Masunaga, Hierarchical structural change in the stress-induced phase transition of poly(tetramethylene terephthalate) as studied by the simultaneous measurement of FTIR spectra and 2D synchrotron undulator WAXD/SAXS data, Macromolecules 47, 2052–2061 (2014)

Chapter 6

Computer Simulations

Abstract Computer simulation is now one of the most useful techniques for building up the energetically stable chain conformations and chain packing structures, for predicting the molecular motions in the aggregation structure of chains, for the calculations of the physical properties starting from the three-dimensional structures, for the reproduction of wide-angle and small-angle X-ray scattering patterns, and so on. The principles and concrete calculation procedures of molecular mechanics, molecular dynamics, Monte Carlo method, and quantum mechanics are learned using many examples. The importance of potential functions used in the energy calculations is also emphasized. Keywords Molecular mechanics · Molecular dynamics · Monte Carlo method · Quantum mechanics · Prediction of chain conformation · Packing structure

6.1 Significance of Computer Simulation The contribution of computer science to the development of structure analysis and property prediction of polymer materials has been increasing day by day. The remarkable development in the numerical calculation technique of the complicated mathematical equations (differential equations, integration, etc.) and the remarkable speed up and the increasing memory size of central performance unit (cpu) allow the concrete simulation of these information from the various levels of atoms, molecules, aggregation of molecules, meso-scale, and even macroscopic scale. As for the development of calculation speed, Moore law is famous, which is originally used for the development of transistor integration density and speed; the density increases about 4 times in every 3 years and the transistor speed increases twice in every 3 years [1]. This suggests that the cpu time of computer is accelerated by 2 × 4 times in 3 years. In one decade, the speed increases by 810/3 ∼ 1000 times in 10 years and 25.5 billion times in 50 years. The scale of computation is expressed in terms of flops or floating point operation per second. For example, 1 petaFlops (= 1015 Flops) means the 1015 times of computation in a second.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 K. Tashiro, Structural Science of Crystalline Polymers, https://doi.org/10.1007/978-981-15-9562-2_6

661

662

6 Computer Simulations 1T(tera) 10T 102

Number of atoms

100T 1P(peta)

103 ~ 104

10 P

100P

106 ~ 108

treatable by the first principle prediction of crystal structure prediction of chemical reaction prediction of protein structure

In these years, the cpu speed and memory size increase greatly even for personal computer. In particular, the technique of parallel computation accelerates the simulation speed remarkably, that is, the calculation of one program is made using a plural number of cpu. The construction of the so-called vector-type supercomputer with parallel construction of more than 1000 cpu allows us to perform the calculation of tremendously large structure model of a protein. One processor node (PN) consists of 8 cpu or 8 × 8 Gflops = 64 Gflops. If a super computer has 640 PN, for example, it corresponds to 8 cpu/PN × 640 PN = 5120 processors or 5120 × 8 Gflops ~ 40 Tflops. This supercomputer can perform the numerical simulation of ocean wave motion, typhoon route, and so on. The molecular-level calculation of physical phenomenon was started in 1950. The positions of rigid balls (Ar atoms) packed in a cell were traced by calculating the elastic collisions between the balls at the various temperatures. The packing structures of balls in the gas, liquid, and solid states were successfully reproduced, as shown in Fig. 6.1 [2]. Since then, various types of computer simulations have been developed for various kinds of molecules, crystals, amorphous phase, etc. In this chapter, the target is small organic molecules, polymer chains, and their aggregation states. The calculation methods may be classified into the following categories: molecular mechanics (MC), molecular dynamics (MD), Monte Carlo method (MC), and quantum mechanical method (QM).

(a)

(b)

(c)

Fig. 6.1 MD calculation of Ar atoms in a large cell. a gas, b liquid, and c solid. Reprinted from Ref. [2] with permission of the American Institute of Physics, 1959

6.2 Molecular Mechanics (MM)

663

6.2 Molecular Mechanics (MM) 6.2.1 Potential Functions The total energy of a molecule is calculated by summing up all the possible energy components listed in Table 6.1 [3, 4]. The molecular geometry is changed and the structure giving the lowest energy value is found out. This process is called molecular mechanics (MM). The lowest energy state can be found by performing the energy minimization, the principle of which is known in the textbook of a numerical calculation. (i)

Bond stretching energy

The harmonic potential is usually used (the first term), but the anharmonic term is also considered. Morse potential function is a more explicit function. The differentiation 2  of E(r ) = A 1−e−B(r −r o) gives the force F and fore constant K as shown in Fig. 6.2. Table 6.1 Various potential functions

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Fig. 6.2 The Morse potential, the corresponding force, and force constant plotted against the distance from the minimal value r min .

  F = −(∂ E/∂r ) = −2 ABe−B(r −r o) 1 − e−B(r −r o)

(6.1)

  K = ∂ 2 E/∂r 2 = −2 AB 2 e−B(r −r o) + 4 AB 2 e−2B(r −r o)

(6.2)

The graphs of E, F, and K are plotted against the distance r as shown in Fig. 6.2. The force constant itself changes with the change of the interatomic distance r, i.e., an anharmonic effect. (ii)

van der Waals interactions

Similar curves are for the van der Waals interactions (Fig. 6.3). Originally the van der Waals interaction comes from the interactions between the transient dipoles of the interacting nonbonded atoms. The electron cloud of an atom is deviated from the nucleus position when another atom approaches this atom. As a result, the dipoles are generated and they interact. However, this van der Waals interaction is different from the interactions between the permanent dipole moments. (iii)

Electrostatic interaction

This energy term decreases slowly with an increase of distance r or the long-range interaction. This point is different from the abovementioned van der Waals interaction, which is called the short-range interaction. The important point in the calculation of such long-range interactions is to check whether the energy sum reaches the equilibrium state or not, meaning the energy does not change anymore even when the r is changed furthermore.

6.2 Molecular Mechanics (MM)

(iv)

665

Torsional potential

The most important energy term in the prediction of a stable chain conformation is the torsional angles around the CC bonds. As seen in Table 6.1, the cosine terms are combined together but the main term is the third one around the C–C single bond (Fig. 6.4). (v)

Hydrogen bonds

Nylon, poly(vinyl alcohol), polyethyleneimine, and so on form the intra- or intermolecular hydrogen bonds of the type NH…O=C, OH…O, NH…N, etc. The hydrogen bond interaction is difficult to express using a simple function form since it consists of the summation of dipole-dipole interaction, charge transfer interaction, van der Waals interaction, and so on. The hydrogen bond potential energy is changed depending on the direction of the bond. (vi)

The cross terms

In addition to the diagonal energy terms shown in Table 6.1, the cross terms must be taken into account in the actual calculation. For example, one of the excellent sets of the potential energy functions is COMPASS which consists of the following many diagonal and cross terms [5, 6].

Fig. 6.3 Lennard-Jones-type van der Waals potential functionV (r ) (kJ/mol) = 10(1/r 12 − 1/r 6 )

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6 Computer Simulations

Fig. 6.4 Internal rotation (torsion) potential function. C cis, G gauche, S skew, and T trans. The curve is changed depending on the species of side atoms

Epot = [K 2 (r − ro )2 + K 3 (r − ro )3 + K 4 (r − ro )4 ] + [H2 (θ − θo )2 + H3 (θ − θo )3 + H4 (θ − θo )4 ]  +  V1 [1 − cos(τ − τ10 )]   +V2 1 − cos(2τ − 2τ20 ) + V3 [1 − cos(3τ − 3τ30 )]

 + K χ χ 2 +  Frr  (r − ro )(r  − ro ) +  Fθ θ  (θ − θo )(θ  − θo ) +  Fr θ (r − ro )(θ − θo ) +  Fr τ (r − ro )(V1 cos τ + V2 cos 2τ + V3 cos 3τ )

+  Fr  τ (r  − ro )(V1 cos τ + V2 cos 2τ + V3 cos 3τ ) +  Fθ τ (θ − θo )(V1 cos τ + V2 cos 2τ + V3 cos 3τ ) +  K τ θ θ  (θ − θo )(θ  − θo ) cos τ

The important point in the usage of these potential function terms is to find the most reliable parameters or the coefficients used in these functions. These parameters are determined suitably so that as many experimentally obtained structures and physical properties are reproduced as possible. Historically, the following sets were proposed as the best ones at those times. MM2, MM3, MM4 (Allinger, 1982 ∼ [8]) useful for hydrocarbons AMBER (Kollman, 1981 ∼ [9]) only the diagonal terms, no cross terms CHARM (Karplus, 1983 ∼ [10]) COMPASS (Sun [5, 6]) DREIDING, UNIVERSAL, CFF, DISCOVER [11, 12]. For example, Fig. 6.5 shows the comparison of the various physical parameters between the experimental and calculated values, where the COMPASS force field was used [5, 6]. The agreement is quite nice.

6.2 Molecular Mechanics (MM) (a) bond length

667 (b) vibrational frequencies

(c) compressibility(alkane)

Fig. 6.5 Comparison of the observed physical parameters with those calculated using COMPASS force field. a bond lengths, b vibrational frequencies, and c isothermal compressibility of n-hexane. Reprinted from Ref. [7] with permission of the American Chemical Society, 1998

6.2.2 Energy Calculation of Crystal Lattice The actual sample to measure the physical properties contains the Avogadro number of atoms. In the energy calculation it is, of course, impossible to calculate the energy for such an enormous system. The calculation must be made for a small system consisting of 102 –106 atoms though the size depends on the power of the computer machine. As a result, the surface effect of the system becomes serious. In order to avoid this effect, the periodic boundary condition is introduced, by which the surface does not exist anymore. As illustrated in Fig. 6.6, the fractional coordinate of an atom in the neighboring unit cell is expressed as follows using the coordinate of the basic cell. x(l, m, n) = x(0, 0, 0) + lea + meb + nec

(6.3)

where ei is the unit vector along the i-axis (i = a, b or c). It must be noted that the unit cell used in the simulation is not necessarily equal to that of the crystallographic unit cell. A large cell consisting of several tens or even hundreds of crystallographic unit cells might be used in the simulation. The large cell is important for the molecular dynamics calculation, as will be mentioned later. In any way, under the periodic boundary condition, the atoms going out of the unit cell take the same fractional coordinates as those of the original cell. The lattice energy is calculated by summing the interatomic interaction energies for all the atoms contained in the cell. For example, for the system containing 1000 atoms, the total number of the interatomic pairs is 500,000. For 10,000 atoms, the total pairs are 50,000,000. In order to shorten the total calculation time, the introduction of cut-off distance is needed, beyond which the energy values are almost constant (see Fig. 6.7). In this example, the cut-off distance is about 20 Å. The summation of electrostatic interaction energies over the wide crystal lattices is made by using the Ewald method. This energy is divided into two parts. One is

668

6 Computer Simulations x(l,m,n)

b

x(0,0,0)

a x (1)

x (2)

x (2) x (1)

Fig. 6.6 Atomic fractional coordinates in the unit cells. The atoms going out of the basic unit cell are put back to the original unit cell

(a)

(b) 15

5000 atoms

vdW energy (kcal/mol)

Number of nonbonded atomic pairs

106

10

5

0

6

8

10

12

14

Cuttoff distance

no

-30 -40 -50 -60 0

10

20 30 cut-off distance

40

Fig. 6.7 Effect of the cut-off distance on a number of atomic pairs and b van der Waals energy. Reprinted from the manual of biosym/discovery [7]

the part in which the interaction energy decreases steeply as known from the shape of the function (~1/r). This part is calculated by summing the interaction energies directly. The second is the slowly decreasing energy part. The direct summation up to the long distance takes a long time. By performing the Fourier transformation of this summation, the decreasing rate becomes faster, since the summation is made in the reciprocal lattice. This method is called the Ewald method.

6.2 Molecular Mechanics (MM)

669

In general, the lattice sum of the electrostatic interaction energies is expressed as E es = (1/2)

 i

= (1/2)

j

m Ai j / r i − r j − R L (L : lattice vetors)

L

 i

j

⎡

+ (1 − )F ⎣

m Ai j / r i − r j − R L

L

 i

j

⎤ m Ai j / r i − r j − R L ⎦

L

 = a convergence function = erfc(ηr ) = 1−erf(ηr )  ∞   = (2/π1/2 ) exp −s 2 ds

(6.4)

ηr

The parameter η can be chosen so that the conversion becomes faster. The first term converges quickly because  decreases rapidly. The second term is Fourier transform with respect to the reciprocal lattice, and the conversion becomes faster (A concrete equation is not given here). Let us see one example about the lattice sum. As illustrated in Fig. 6.8, the two atoms of plus and minus charges are arrayed alternately along the one-dimensional axis. The electrostatic interaction energy E es is expressed as below. E es =



(±)q 2 /ri

i

= −2q 2 [(1/R) − (1/2R) + (1/3R) − (1/4R) + · · · ]   = − 2q 2 /R [1 − (1/2) + (1/3) − (1/4) + · · · ]

(6.5)

Since ln(1 + x) = x − (1/2)x 2 + (1/3)x 3 − (1/4)x 4 + · · · , the term [1 − (1/2) + (1/3) − (1/4) + · · ·] = ln(1 + 1) = ln 2. Then, E es is obtained as     E es = − 2q 2 /R ln 2 = − q 2 /R M,

M = 2 ln 2

(6.6)

The constant M is called the Madelung constant. For the 3D crystal also the M value can be calculated in a similar way. For NaCl (face-centered cubic lattice) and CsCl (simple cubic lattice), the Madelung constant is given, respectively, as M(NaCl) = 1.747565 and M(NaCl) = 1.762675. For NaCl lattice, the number of pairs is counted as follows (see Fig. 6.9).

Fig. 6.8 1D lattice consisting of atoms with +q and -q charges

670

6 Computer Simulations

(a)

(b)

Fig. 6.9 Crystal structures of a NaCl and b CsCl

6 Cl- ions surrounding a Na+ ion at a distance R 12Na+ ions surrounding a Na+ ion at a distance 21/2 R 8Cl- ions surrounding a Na+ ion at a distance 31/2 R 6Na+ ions surrounding a Na+ ion at a distance 2R

The first neighborhoods: The second neighborhoods: The third neighborhoods: The forth neighborhoods: Then,

√ √ E es = −q 2 [6/R − 12/( 2R) + 8/( 3R) − 6/(2R) + · · · ] = −(q 2 /R) × 1.747565   = − q 2 /R M

(6.7)

6.2.3 Minimization Method The energy minimization is performed with various methods. The purpose is to find the approximate solution x satisfying the equation Ax = B

(6.8)

f (x) = (x · Ax)/2−B · x

(6.9)

as well as possible. A function

is introduced. To solve Eq. (6.8) is equivalent to minimize f (x) of Eq. (6.9). This is because the differentiation of Eq. (6.9) gives ∂ f (x)/∂ x = Ax−B

(6.10)

6.2 Molecular Mechanics (MM)

671

and the minimization condition is ∂f (x)/∂x = 0, which is equal to Eq. (6.8). If the answer x satisfies correctly Eq. (6.8), f (x) of Eq. (6.9) should be minimal. However, in the actual numerical calculation, the x does not necessarily satisfy Eq. (6.8). What we have to do is to obtain the x value giving the minimal value of f (x) as exactly and as quickly as possible. The simplest way is to find the steepest route of the first derivative ∂f (x)/∂x since the ideal condition of minimization is ∂ f (x)/∂ x = 0. An initial approximate answer of x, x(0) may be chosen as null; x(0) = 0. The difference (or residue) r (1) = B− Ax (0) is calculated. The vector p(1) going to the most plausible route to the answer is input. As a trial, p(1) is assumed to be equal to r(1) : p(1) = r (1) = B− Ax (0) = B. The next coefficient step answer is input as x (1) = x (0) + α(1) p(1) = α(1) p(1) . The   unknown α(1) is given so that the f (x) becomes minimal for the α(1) ; ∂ f x (1) /∂α(1) = 0. α(1) =  (1) (1)   (1)  p · r / p · A p(1) . As a result the first approximate answer is calculated as x (1) = x (0) + α(1) p(1) . Then, the residue r (2) = B − Ax (1) is calculated and p(2) = r(2) and x (2) = x (1) + α(2) p(2) . The process is repeated. The α (i) or p(i) corresponds to the steepest direction of the minimal f (x). This method is called the steepest descent method. The approach to the answer is rather rough and not necessarily rapid. Then some correction term is added to the vector: the vector p(2) = r (2) + β(1) p(1) where the coefficient β(1) is calculated so that the p(1) and p(2) are conjugate to each other; p(2) · A p(1) = 0 (The vectorsp(2) and Ap(1) are normal to each other). Then β(1) = − r (2) · A p(1) / p(1) · A p(1) . Using the thus-obtained p(2) , the next x(2) is     given as x (2) = x (1) +α(2) p(2) , where α(2) = p(2) · r (2) / p(2) · A p(2) . By repeating these processes, the final answer x(n) is obtained. The total process is summarized as follows. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

Put an initial answer x (0) = 0 Calculation of residue r (1) = B − Ax (0) Input the starting vector p(1) = r (1) = B − Ax (0) Calculation of x (1) = x (0) + [ p(1) · r (1) ]/[p(1) · A p(1) ] p(1) Calculation of residue r (2) = B − Ax (1) Calculation of the conjugate p(2) = r (2) − [r (2) · A p(1) ]/[ p(1) · A p(1) ] p(1) Calculation of x (2) = x (1) + α (2) p(2) , where α (2) = [ p(2) · r (2) ]/[ p(2) · A p(2) ] Repeat the processes (v) ~ (vii) [r (3) , p(3) , α (3) , x (3) . . .]

That is to say, the strict answer should be x (n) = x (0) + α(1) p(1) + α(2) p(2) + · · · · α(n) p(n) This is the combination of the steepest descent condition [α (i) ] and the conjugation condition [β (i) ], and it is called the conjugate gradient method (Fig. 6.10). The approach to the minimal point is more effective than the steepest descent method. The better approach to the minimal position is to use the Newton-Raphson method although the approaching rate is not very fast. As the first example, the multiple variable x is assumed to be a scalar x, for simplicity. A nonlinear equation f (x) = 0 is solved by developing the equation around an initial plausible answer x (1) (Fig. 6.11).

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6 Computer Simulations

Fig. 6.10 Minimization route of conjugate gradient method

x(1)

x(0) x(2)

Fig. 6.11 Newton-Raphson method

y=f(x)

y

x(0)

y=0 x(1)

    f (x) = f x (1) + [∂ f (x)/∂ x]x(1) x − x (1)    2 + (1/2) ∂ 2 f (x)/∂ x 2 x(1) x − x (1) + · · ·     ∼ = f x (1) + [∂ f (x)/∂ x]x(1) x − x (1)

x

(6.11)

If x = α is a correct answer, we should have f (α) = 0. Equation becomes approximately as     f (α) = 0 ∼ = f x (1) + [∂ f (x)/∂ x]x(1) α − x (1)

(6.12)

  α∼ = x (1) − f x (1) /[∂ f (x)/∂ x]x(1)

(6.13)

Therefore,

For the (k + 1)th approximate answer,   x (k+1) ∼ = x (k) − f x (k) /[∂ f (x)/∂ x]x(k)

(6.14)

  If the conversion condition f x (k+1) < δ is attained, where δ is an enough small value (~0.000001, for example), the best approximate answer is obtained. This Newton-Raphson method can be expanded to the n-dimensional equation. In the actual energy minimization, the f (x) is the energy of the whole system, and x is a set of the atomic coordinates or internal coordinates. The concrete numerical calculation programs are referred to in the mathematics textbooks.

6.2 Molecular Mechanics (MM)

673

6.2.4 Prediction of Chain Conformations This problem is mentioned already in the section of X-ray structure analysis. The regular helical conformation is generated using an equation of helix and the interatomic interaction energy is calculated by changing the skeletal torsional angles [13]. The local minimal energy points are extracted. If the molecular chain takes a glide plane symmetry, the torsional angles are changed with the conditions of …(τ1 τ2 )(−τ1 −τ2 )…, for example, which is opposite to the case of helical formation of …(τ1 τ2 )(τ1 τ2 )… where τ1 and τ2 are the torsional angles around the skeletal C–C bonds. Figure 6.12 shows the 2D conformational energy map calculated for an isolated it-PP chain. …-CH(CH3 )-CH2 -CH(CH3 )-CH2 -CH(CH3 )-… …

τ1 τ2

τ1

τ2 ….

The energy minima are found at the points A and B. The most stable conformation is A with (τ1 , τ2 ) = (G, T). This is a left-handed downward (Ld) helix. The conformation B with (τ1 , τ2 ) = (T, G) is curious. By changing the torsional angles of the conformation A from (G, T) to (T, -G), the helix A changes its helical handedness to the righthanded upward (Ru), where the shape projected along the chain axis is the same between Ld and Ru. It must be noted here that, if the configuration of an asymmetric Fig. 6.12 (a) 2D potential energy map for a pair of the skeletal torsional angles of it-PP helix. Reprinted from Reference [13] with permission of John Wiley & Sons, 1973. (b) The chain conformation of it-PP with the various conformations with the opposite carbon configurations.

_

(a)

B _

A

B A

B

A (b)

_

A-Ru

_

A-Ld

A’-Lu

A’-Rd

674

6 Computer Simulations

carbon atom CH(CH3 ) is changed from d to l, the situation is changed as follows (see Figure 6.12(b)). A

A’

_

The energetically stable conformation (A)

Ru

(τ1, τ2) = (T, -G)

(A)

Ld

(τ1, τ2) = (G, T)

The energetically stable conformation (A’)

Lu

(τ1, τ2) = (T, G)

Rd

(τ1, τ2) = (-G, T)

_

(A’)

The helical conformations A and A’ are in the relation of mirror symmetry. In the prediction of structure of poly(amino acid)s, the energy contour map named the Ramachandran map is often utilized (see Fig. 6.13) [14, 15]. The torsional angles around the amide group are φ and ψ. The energy calculation is made by changing the φ and ψ with the amide group fixed at the torsional angle of 180°. The twodimensional contour map is obtained by plotting the energy value as a function of φ and ψ. antiparallel sheets

parallel sheets left-handed helix

right-handed helix

Fig. 6.13 Ramachandran map. Modified from Ref. [15] with permission of Elsevier, 1968

6.2 Molecular Mechanics (MM)

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6.2.5 Prediction of Packing Structure The molecules with the energetically stable form are packed in a unit cell of an arbitrary size, and the total lattice energy is calculated by changing the atomic coordinates and the unit cell parameters. In some cases, the molecules are assumed to be a rigid body without any change of shape. The spatial relation of the molecules is generated by assuming a plausible space group symmetry. In the polymer crystal, the monomeric units are covalently linked together along the chain axis. This can be introduced so that the bond length is fixed to a constant value. The “Polymorph Predictor (Biovia)” is one example of the commercial programs to predict the packing structure of small molecules under the assumption of a suitable space group [16, 17]. The molecular shape is minimized at first. These molecules are packed in the cell and their positions and orientations are changed variously. The thus-obtained packing structures are compared and classified into several groups or clusters with similar energy values and packing modes. The representative model is picked up from these clusters and the energy minimization is performed. The thus-minimized structures are ranked in the order of energy. Figure 6.14 shows one example of the calculation [18]. The agreement between the X-ray analyzed structure and the energy-minimized model is almost perfect. This is a rare case. Usually, the probability of the good agreement is about 30% according to our experience.

Fig. 6.14 Crystal structure of poly-m-phenylene isophthalmide model compound. The red and black colors indicate the X-ray analyzed structure and the predicted structure, respectively. Reprinted from Ref. [18] with permission of the American Chemical Society, 2002

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The actual crystal structure is determined by the free energy, which is the summation of enthalpy and entropy terms. The latter term is due to the vibrational disorders of the atomic positions. The calculation is hard since the entropy term is calculated using the vibrational frequencies at the various reciprocal lattice points (the details were already mentioned in Sect. 5.9).

6.3 Molecular Dynamics (MD) 6.3.1 Principle of MD The time dependence of the coordinates of atoms contained in a large unit cell is calculated on the basis of a classical Newton’s equation of motion. Since the system size is too large to obtain the analytical solutions for such multi-dimensional equations, the numerical calculations are performed to solve the differential equations. The equations are expressed in the following ways (for the microcanonical ensemble system):   m i ∂ 2 r i (t)/∂t 2 = F i (t)

(6.15)

At the times t + t and t – t, we can develop the equations as below:   r i (t + t) = r i (t) + [∂ r i (t)/∂t] t + (1/2) ∂ 2 r i (t)/∂t 2 ( t)2 + · · ·

(6.16)

  r i (t − t) = r i (t) − [∂ r i (t)/∂t] t + (1/2) ∂ 2 r i (t)/∂t 2 ( t)2 + · · ·

(6.17)

Eqs. 6.16 + 6.17 gives r i (t + t) = 2r i (t) − r i (t − t) + (F i (t)/m i )( t)2 + · · ·

(6.18)

Eqs. 6.16 – 6.17 gives [∂ r i (t)/∂t] ≡ vi (t) = [r i (t + t) − r i (t − t)]/(2 t) + · · ·

(6.19)

for t = 0, r i ( t) = r i (0) + vi (0) t

(6.20)

These equations allow us to predict the change of atomic positions ri (t) with the passage of time. (i) (ii)

t = 0. Input ri (0) and vi (0). t = t, Eq. 6.20 gives the position ri ( t) and Eq. 6.19 gives the velocity vi ( t).

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(iii)

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t = 2 t. Using Eq. 6.18, ri (2 t) is calculated by giving Fi (t). The force Fi (t) is obtained by differentiating the total interaction energy at t = t with respect to ri (t). The concrete process is shown below. This method is called the leap-frog method.

******************* (Notes) Numerical solution of a differential equation There are many methods to solve the differentiation equation: for example, Euler’s method, leap-frog method, Runge-Kutta’s method, and so on. They are different in the accuracy of the solution. [dx(t)/dt] = f (x, t)

(i)

Plot the x(t) against time t by the Euler method.

Since x is a function of t, we have the following approximation: x(ti + t) ∼ = x(ti ) t =0 t = t t = 2 t ............... (ii)

[dx(t)/dt] = f (x, t)

+ [dx(t)/dt]t=ti t = x(ti ) + f [x(ti ), ti ] t x(0) = xo x( t) = xo + f (xo , t) t x(2 t) = x( t) + f [x( t), t)] t

Plot the x(t) against time t by the leap-frog method.

  x(ti + t) ∼ = x(ti ) + [dx(t)/dt]t=ti t + (1/2)d2 x(t)/dt 2 t=ti ( t)2 + · · · x(ti − t) ∼ = x(ti ) − [dx(t)/dt]t=ti t + (1/2) d2 x(t)/d2 t=ti ( t)2 + · · · The subtraction of these two equations becomes [dx(t)/dt]t=ti = [x(ti + t) − x(ti − t)]/(2 t) + . . . or t1 = t, t2 = 2 t, t3 = 3 t, ......

x(ti + t) = x(ti − t) + 2[dx(t)/dt]t=ti t (Euler method) x(t1 ) = x( t) = x(0) + f [x(0), 0] t x(t2 ) = x(2 t) = x(0) + 2[dx(t)/dt]t=t1 t = xo + 2 f [x(t1 ), t1 ] t x(t3 ) = x(3 t) = x( t) + 2[dx(t)/dt]t=t2 t = xo + 2 f [x(t2 ), t2 ] t ...............

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(iii)

[dx(t)/dt] = f (x, t) Plot the x(t) against time t by the Runge-Kutta method. k1 = f [ti , x(ti )] t k2 = f [ti + t/2, x(ti ) + k1 /2] t k3 = f [ti + t/2, x(ti ) + k2 /2] t k4 = f [ti + t, x(ti ) + k3 ] t x(ti + t) = x(ti ) + (k1 + 2k2 + 2k3 + k4 )/6

(iv)

  2 d x(t)/dt 2 = f (x, t) by Euler method By inputting y(t) ≡ dx(t)/dt, the equation changes to [dy(t)/dt] = f (x, t). y(t + t) ∼ = y(t) + [dy(t)/dt] t = y(t) + f (x, t) t x(t + t) ∼ = x(t) + [dx(t)/dt] t = x(t) + y(t) t t= 0 x(0) = x0 , y(0) = y0 t = t y( t) = y(0) + f [x(0), 0] t = y0 + f [x0 , 0] t x( t) = x0 + y(0) t = x0 + y0 t t = 2 t y(2 t) = y( t) + f [x( t), t] t x(2 t) = x( t) + y( t) t ······ ·········

(i)

  [dx(t)/dt] = f (x, t) = a x(t)−x(t)2 (Logistic equation)

The time spacing t is given. For example, in the range of t = 0 ~ t end , the curves shown below are obtained depending on the time step t = t end /N = 1/N, where N is the total number of steps. For a = 1, the answer y = x(t) is converged to the smooth curve by increasing the total number of points N or for the smaller step t. For a = 3.9, the curve changed apparently randomly depending on the iteration (this curve is not random but complicated). Besides, depending on the initial value y0 = x(0), the total pattern of the curve changes remarkably, in particular, in the last stage of the calculation (Fig. 6.15). This phenomenon is called the Chaos phenomenon (Lorentz chaos or butterfly effect). [The latter name indicates the difficulty of the weather casting. A weak wind caused by a butterfly at the initial point (Beijin) causes finally a large storm at the end point (New York).]

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Fig. 6.15 Plot of Logistic equation for the different a values. For a = 3.9, the y curve shows the complicated pattern, which changes remarkably differently depending on the initial value of y0 (red and blue lines, y0 = 0.160000 and 0.160001, respectively)

(ii)

Simple oscillation d2 x(t)/dt 2 = −x(t) using (a) Euler method and (b) Leapfrog method

The analytical answer is y = x(t) = cos(t). The differentiation z = dy/dt = − sin(t) and y 2 + z 2 = 1. The Euler method is rougher than the leap-frog method (Fig. 6.16). ********************************

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Fig. 6.16 Plot of simple oscillation. Depending on the method, the accuracy of the result is different

6.3.2 Statistical Ensembles 6.3.2.1

Ergodic Theorem

The molecular dynamics calculation is to predict the time dependence of the coordinates and velocities of the atoms contained in a unit cell (MD cell), from which the various physical parameters A(t) are calculated as a function of time. The physical parameters in the equilibrium state are evaluated by averaging A(t) in a time range of the MD calculation. On the other hand, the macroscopically observed physical parameters are estimated by averaging the experimental values observed for the various samples or the various positions of the sample, which are called the ensembles. 

t

time average time = lim

t→∞

¨ ensemble average ensemble =

A(t)dt/(t − to )

(6.21)

A( p, q)ρ( p, q)d pdq

(6.22)

to

where p and q are the position vector and momentum of the particles, respectively. In the three-dimensional system of N particles, the total dimension for all the particles is 6 N. The p–q space of 6 N dimension is named the phase space ( space). The ρ(p, q) is the density distribution function in the p–q phase space. The physical parameter is averaged by taking the distribution weight into account, which is called the ensemble average. (For example, a periodically oscillating particle along the x-coordinate axis has the total energy or a Hamiltonian H = mvx2 /2 + kx 2 /2. By using a momentum qx = mvx , then H = 2qx2 /m + kx 2 /2. The solution of the equation of motion gives x = X cos(ωt) and qx = −ωX sin(ωt). Therefore, x 2 + (qx /ω)2 = X 2 . The relation between x and qx shows the ellipsoidal curve. That is to say, the phase space gives the ellipsoidal pattern.)

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In statistical physics, these two averages are assumed to be equal to each other. time = ensemble

(6.23)

That is, the time-averaged results obtained by the MD calculation for one particular ensemble are assumed to be equal to the average for all the possible states of the various ensembles. This assumption is called the Ergodic theorem. The MD methods are different depending on the statistical systems, where N is the number of particles, V is the volume, E is the energy, T is the temperature, and P is the pressure.

6.3.2.2

Microcanonical Ensemble (Constant NVE)

The basic equation of motion m i d2 r i /dt 2 = F i

(6.24)

The other expression using Hamilton canonical equations dr i /dt = pi /m i ;

pi : momentum

d pi /dt = F i An arbitrary microscopic state appears at the same probability. The total number of the microscopic states is W. The entropy S = k B ln(W ). The temperature T = 1/(∂ S/∂ E)V . Pressure P = −(∂ E/∂ V )s. In this ensemble, T is not controlled and this method is difficult to apply to the equilibrium state.

6.3.2.3

Canonical Ensemble (Constant NVT)

The probability of an arbitrary microscopic state is ρ(r ) = exp(−Er /k B T )/Z

(6.25)

  The partition function Z = e−(K ( p)/k B T ) d p × e−(U (r )/k B T ) dr = Z K Z U . Here, K and U are kinetic and potential energies, respectively. The Helmholtz free energy F = −k B T ln(Z ). The basic equation of motion is given as follows: (The Newton equation cannot be used since this equation reserves the total energy constant.) There are various methods to satisfy this condition. For example, in the case of velocity scaling method, the thermal energy and the total kinetic energy are related by a constant scale β. The η is the total degree of freedom.

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η(k B T /2) = β 2



m i vi2 /2



(6.26)

i

with The velocities vi of the particles are changing  time. At every step of calcu lation, the velocity is scaled by applying β vi = βvi and the temperature T is kept unchanged. The equations of motion are given as below (U is the potential energy): d pi /dt = −∂U/∂q i m i dq i /dt = β pi

(6.27)

This NVT condition is appropriate for the thermal motion of the isolated molecules without periodic boundary conditions. Even when the periodic boundary condition is applied, the NVT condition is used if the pressure effect is not very large.

6.3.2.4

Constant Pressure (Constant NPT)

The Gibbs free energy G = −k B T ln(Z )

(6.28)

The pressure P is kept constant. P=



mvi2 /(3V ) +



i

F i · r i /(3V ), dP/dt = 0

(6.29)

i

The equations of motion are as follows: dr i /dt = pi /m i + χ(r · p)r d pi /dt = F i − χ(r · p) pi − ξ(r · p) pi dV /dt = 3V χ(r · p)

(6.30)

χ is the Lagrange constant. ξ +χ=



   2 pi pi · F i / i

This ensemble is used for the equilibrium state calculation at constant T and P. For studying the effect of stress on the structure and molecular motion, this method is useful.

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6.3.3 Physical Parameters Obtained by MD The physical parameters obtained by the MD method are as follows:  (i) Temperature by the translational motion = < m i r˙i2 /2>/(3N k B /2) i

Temperature by the rigid body rotational motion =
/(3N k B /2)

where I i is a moment of inertia and ω is an angular frequency.    (ii) Pressure