Polymers, Part B: Crystal Structure and Morphology. Methods of Experimental Physics, 16B

Methods of Experimental Physics VOLUME 16

POLYMERS PART 6: Crystal Structure and Morphology

METHODS OF EXPERIMENTAL PHYSICS: L. Marton and C. Marton, Editors-in-Chief

Volume 16

Polymers PART B: Crystal Structure and Morphology

Edited by R. A. FAVA ARC0 Polymers, Inc. Monroeville, Pennsylvania

I980 ACADEMIC PRESS

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COPYRIGHT @ 1980, BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED I N ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

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United Kingdom Ediiion published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NWI

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Library of Congress Cataloging in Publication Data Main entry under title: Polymer physics. (Methods of experimental physics ;v. 16) Includes bibliographical references and index. CONTENTS: pt. A. Molecular structure and dynamics. pt. B. Crystal structure and morphology. 1. Polymers and polymerization. I. Fava, Ronald A. 11. Series. QD381 .P612 547’.84 79-26343 ISBN 0-12-475957-2

PRINTED IN THE UNITED STATES O F AMERICA

80818283

9 8 7 6 5 4 3 2 1

CONTENTS CONTRIBUTORS. .

. . . . . . . . . . . . FOREWORD . . . . . . . . . . . . . . . PREFACE . . . . . . . . . . . . . . . . . CONTENTS OF VOLUME 16, PARTSA A N D C . . . CONTRIBUTORS TO VOLUME16, PARTSA A N D C . VOLUMESI N SERIES. . . . . . . . . . . .

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6. X-Ray Diffraction

6.1. Unit Cell and Crystallinity by JOSEPHE. SPRUIELL A N D EDWARD S. CLARK

6.1 . l . 6.1.2. 6.1.3. 6.1.4. 6.1.5. 6.1.6. 6.1.7.

Introduction . . . . . . . . . . . . 1 2 Basic Crystallography . . . . . . . . . Diffraction Theory . . . . . . . . . . 25 Experimental Techniques . . . . . . . . 39 Crystal Structure Determination . . . . . . 68 Disorder in Crystalline Polymers . . . . . 98 Measurement of Crystallinity by X-Ray Diffraction . . . . . . . . . . 114

6.2. Crystallite Size and Lamellar Thickness by X-Ray Methods by JING-IWANGA N D IANR. HARRISON 6.2.1. Introduction . . . . . . . . . . . 6.2.2. Crystallite Size by Wide-Angle Techniques . 6.2.3. Lamellar Thickness Using Small-Angle X-Ray Scattering (SAXS) . . . . . . . . . 6.2.4. Summary . . . . . . . . . . . . V

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128 129

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153

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CONTENTS

7. Electron Microscopy by RICHARDG . VADIMSKY 7.1. Introduction

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7.2. Fundamentals . . . . . . 7.2.1. Particle-Wave Concept 7.2.2. Image Formation . . 7.2.3. Image Interpretation .

185

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186 186 187 190

7.3. Electron Optics . . . . . . 7.3.1. Lens Theory . . . . 7.3.2. The Ideal Lens . . . 7.3.3. Image-Degrading Factors

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195 196 197 199

7.4. The Instrument . . . . . . 7.4.1. The Illuminating System 7.4.2. The Specimen Holder . 7.4.3. The Objective Lens . . 7.4.4. The Projection System .

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206 206 208 208 211

7.5. Operational Considerations . . 7.5.1. Specimen Preparation . 7.5.2. Focusing . . . . . 7.5.3. Resolution . . . . . 7.5.4. Magnification Calibration

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212 212 217 220 221

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7.6. Other Microscopy Techniques . . . . . . . . 7.6.1. Small-Angle Electron Diffraction . . . . 7.6.2. Stereomicrography . . . . . . . . . 7.6.3. Topographical Contrast Imaging . . . . . 7.6.4. Scanning Electron Microscopy . . . . . 7.6.5. Scanning-Transmission Electron Microscopy 7.6.6. Low-Loss Electron Microscopy . . . . .

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221

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224 227 227

. 222 . 222 . 222

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CONTENTS

7.7. Applications . . . . . . . . 7.7.1. Single Molecules . . . . 7.7.2. Polymer Single Crystals . 7.7.3. Melt-Crystallized Polymers 7.7.4. Oriented Polymers . . .

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228 228 229 230 233

8 . Chemical Methods in Polymer Physics by G . N . PATEL 8.1. Disorder in Polymer Crystals and Chemical Methods

. 237

8.2. Solvent-Etching

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8.3. Plasma-Etching

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8.4. The Surface Modification Techniques . . . . . . . 8.4.1. Halogenation . . . . . . . . . . . . 8.4.2. Dehydrohalogenation of Poly(viny1idene Chloride) . . . . . . . . . . . . . 8.4.3. Methoxymethylation of Polyamides . . . . . 8.4.4. Acylation of Polystyrene . . . . . . . . 8.5. The Surface Degradation Techniques 8.5.1. Selective Degradation Reagents . . . 8 S.2. Nitric Acid Degradation . . . . . 8.5.3. Ozone Degradation . . . . . . . 8.5.4. Deduction of Crystal Morphology . . 8.5.5. Location of Unsaturation and Branches 8.5.6. A Test for Random Attack . . . . 8.5.7. Preparation of Low-Molecular-Weight Fractions for GPC . . . . . . . 8.5.8. Hydrolysis . . . . . . . . . . 8.5.9. Degradation by a Mixture of Potassium Permanganate and Sulfuric Acid . . . 8.5.10. Hydrazinolysis of Polyamides . . .

245 245 252 253 255

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255 256 257 . . . 261 . . . 266 . . . 267

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8.6. Irradiation and Selective Degradation . . . . . . . 278 8.6.1. Radiation-Induced Chemical Changes . . . . 278 8.6.2. Location of the Chemical Changes . . . . . 281

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CONTENTS

9 . Thermal Analysis of Polymers

by JAMESRUNTA N D IANR . HARRISON 9.1. Introduction .

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9.2. Instrumentation and Method 9.3. Theory

287

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9.4. Basic Factors Affecting the DTA/DSC Curve . . . . 293 9.4.1. Instrumental Factors . . . . . . . . . 293 9.4.2. Sample Factors . . . . . . . . . . . 293 9.4.3. Reference Material . . . . . . . . . . 294 9.5. Melting Behavior of Polymers . . . . . . . . . 294 9.5.1. Introduction . . . . . . . . . . . . 294 9.5.2. Dependence of Melting on Crystallization Conditions . . . . . . . . . . . . . 297 9.5.3. Dependence of Melting on Molecular Weight and Molecular-Weight Distribution . . . . . 298 9.5.4. Annealing . . . . . . . . . . . . . 299 9.5.5. Effect of Heating Rate on Polymer Melting . . 300 9.5.6. Multiple Melting Peaks . . . . . . . . . 303 9.5.7. Dried and Suspension Crystals . . . . . . 308 9.5.8. The “True” Melting Point . . . . . . . . 313 9.6. Quantitative Methods

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9.7. Other Applications . . . 9.7.1. Phase Changes . 9.7.2. Glass Transition . 9.7.3. Polymerization . 9.7.4. Identification . . 9.7.5. Polymer Reactions 9.7.6. Crystallization . . 9.8. Summary

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316

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327 327 327 331 334 335 336

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337

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CONTENTS

10. Nucleation and Crystallization by GAYLON S. Ross A N D LOISJ . FROLEN

10.1 Introduction . . . . . . 10.1 .1 . Aims and Objectives .

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339 339

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340 340 341

10.2 General Background on Semicrystalline Polymers 10.2.1. Requirements for Crystallization . . . 10.2.2. Historical Development . . . . . .

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10.3 Experimental Methods for Measuring Crystallization Rates . . . . . . . . . . . . 10.3.1. General Considerations . . . . . . . . 10.3.2. Methods Using a Thin-Film-Type Specimen . 10.3.3. Methods Using a Bulk-Type Specimen . . 10.3.4. Differential Thermal Analysis (DTA) as Applied to the Determination of Tm0. . . . 10.3.5. Crystallization from Solution . . . . . .

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352 352 354 369

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381

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385 385 396

AUTHORINDEX FOR PARTB

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399

PARTB

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411

10.4 Nucleation . . . . . . . . . 10.4.1. Homogeneous Nucleation . 10.4.2. Heterogeneous Nucleation .

SUBJECT INDEX

FOR

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CONTRIBUTORS Numbers in parentheses indicate the pages on which the authors’ contributions begin.

EDWARDS. CLARK,Polymer Engineering, University of Tennessee, Knoxville, Tennessee 37916 (1) LOISJ. FROLEN, National Measurement Laboratory, National Bureau of Standards, Washington, D.C. 20234 (339) IAN R. HARRISON, College of Earth and Mineral Sciences, The Pennsylvania State University, University Park, Pennsylvania 16802 (128, 287) G. N . PATEL,Corporate Research Center, Allied Chemical Corporation, Morristown, New Jersey 07960 (237) GAYLONS . Ross, National Measurement Laboratory, National Bureau of Standards, Washington, D.C. 20234 (339) JAMES RUNT,College of Earth and Mineral Sciences, The Pennsylvania State University, University Park, Pennsylvania, 16802 (287) JOSEPH E . SPRUIELL,Polymer Engineering, University of Tennessee, Knoxville, Tennessee 37916 (1) RICHARD G. VADIMSKY, Bell Telephone Laboratories, Murray Hill, New Jersey 07974 (185) JING-I WANG,College of Earth and Mineral Sciences, The Pennsylvania State University, University Park, Pennsylvania 16802 (128)

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FOREWORD The thoroughness and dedication of Ronald Fava in preparing these volumes may be verified by this work’s impressive scope and size. This is the first time Methods of Experimental Physics has utilized three volumes in the coverage of a subject area. The volumes, in part, indicate the future development of this publication. Solid state physics was covered in Volumes 6A and 6B (edited by K. Lark-Horovitz and Vivian A. Johnson) in 1959. Rather than attempt a new edition of these volumes in a field that has experienced such rapid growth, we planned entirely new volumes, such as Volume 11 (edited by R. V. Coleman), published in 1974. We now appreciate the fact that future coverage of this area will require more specialized volumes, and Polymer Physics exemplifies this trend. To the authors and the Editor of this work, our heartfelt thanks for a job well done. L. MARTON C. MARTON

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PREFACE A polymer must in many ways be treated as a separate state of matter on account of the unique properties of the long chain molecule. Therefore, although many of the experimental methods described in these three volumes may also be found in books on solid state and molecular physics, their application to polymers demands a special interpretation. The methods treated here range from classical, well-tried techniques such as X-ray diffraction and infrared spectroscopy to new and exciting applications such as those of small-angle neutron scattering and inelastic electron tunneling spectroscopy. It is convenient to present two types of chapters, those dealing with specific techniques and those in which all techniques applied in measuring specific polymer properties are collected. The presentation naturally divides into three parts: Part A describes ways of investigating the structure and dynamics of chain molecules, Part B more specificially deals with the crystallization of polymers and the structure and morphology of the crystals, while in Part C those techniques employed in the evaluation of mechanical and electrical properties are enumerated. It should be emphasized, however, that this is not a treatise on the properties of polymeric materials. The authors have introduced specific polymer properties only incidentally in order to illustrate a particular procedure being discussed. The reader is invited to search the Subject Index wherein such properties may be found listed under the polymer in question. I have endeavored to arrange chapters in a logical and coherent order so that these volumes might read like an opera rather than a medley of songs. The authors are to be commended for finishing their contributions in timely fashion to help achieve this end. I also wish to acknowledge with thanks the support of ARC0 Polymers, Inc. and the use of its facilities during the formative stages of the production. R. A. FAVA

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CONTENTS OF VOLUME 16, PARTS A AND C PART A: Molecular Structure and Dynamics

1. Introduction by R. A. FAVA

Historic Development Definitions Formation and Conformation The Solid State 1.5. Orientation 1.6. Impurities I . 1. 1.2. I .3. 1.4.

2. Polymer Molecular Weights by DOROTHY 3. POLLOCK. AND ROBERT F. KRATZ 2.1. 2.2. 2.3. 2.4. 2.5.

Definitions of Molecular Weight Intensive Properties of Polymers Fractionation Gel Permeation Chromatography Miscellaneous Methods

3. Spectroscopic Methods

3.1. Infrared and Raman Spectra of Polymers by R. G. SNYDER 3.2. Inelastic Electron Tunneling Spectroscopy by H. W. WHITE and T. WOLFRAM 3.3. Rayleigh-Brillouin Scattering in Polymers by G. D. PATTERSON 3.4. Inelastic Neutron Scattering Spectroscopy by C. V. BERNEY and SIDNEY YIP

4. High-Resolution Nuclear Magnetic Resonance Spectroscopy by J. R. LYERLA xvii

xviii

CONTENTS OF VOLUME

16,

PARTS A A N D

c

5. Probe and Label Techniques

5.1. Positron Annihilation by J. R. STEVENS 5.2. Fluorescence Probe Methods by L. LAWRENCE CHAPOY and DONALD B. D u P d 5.3. Paramagnetic Probe Techniques by PHILIPL. KUMLER 5.4. Small-Angle Neutron Scattering by J. S. KING AUTHOR INDEX-SUBJECT INDEXFOR PARTSA, B,

AND

C

PART C: Physical Properties

11. Viscoelastic and Steady-State Rheological Response by DONALD J. PLAZEK

1 1 .O. 11.1. 11.2. 11.3. 11.4. 1 1.5.

Introduction Linear Viscoelastic Behavior Steady-State Response Nonlinear Viscoelastic Behavior Pressure Effects on Viscoelastic Behavior Sample Handling

12. Further Mechanical Techniques

12.1. Ultrasonic Measurements by BRUCEHARTMANN 12.2. Static High-pressure Measurements on Polymers by R. W. WARFIELD 12.3. Stress-Strain Yield Testing of Solid Polymers by JOHN RUTHERFORD A N D NORMAN BROWN 13. Production and Measurement of Orientation by IANL. HAY

13.1. 13.2. 13.3. 13.4.

Introduction The Production of Orientation Description of Orientation Measurement of Orientation

CONTENTS OF VOLUME

16,

PARTS A A N D

c

xix

14. ESR Study of Polymer Fracture by TOSHIHIKO NAGAMURA

14.1. 14.2. 14.3. 14.4.

Introduction Basic Theory and Experimental Techniques Radical Formation by Mechanical Fracture of Polymers Radical Formation during Tensile Deformation and Fracture of Oriented Crystalline Polymers 14.5. Fracture in Elastomers 14.6. Molecular Mechanism of Deformation and Fracture of Polymers 14.7. Limitations of ESR Method and Comparison with Associated Studies

15. Methods of Studying Crazing

by NORMANBROWN 15.1. 15.2. 15.3. 15.4. 15.5.

Introduction Structure Initiation and Growth Environmental Effects in Liquids and Gases Relationship of Crazing to Macroscopic Mechanical Behavior

16. Polymeric Alloys by J . ROOVERS

16.1. 16.2. 16.3. 16.4. 16.5. 16.6.

Introduction Thermodynamics Direct Observation Scattering Techniques Glass Transition Temperature Measurements Conclusion

17. Permeation, Diffusion, and Sorption of Gases and Vapors by R. M. FELDER and G. S. HUVARD

17.1. 17.2. 17.3. 17.4.

Introduction Historical Perspective Phenomenology Categories of Experimental Methods

xx

CONTENTS OF VOLUME

16,

PARTS A A N D

c

17.5. 17.6. 17.7. 17.8.

Pressure Measurement and Temperature Control Sorption Methods Integral Permeation (Closed Receiving Volume) Methods Differential Permeation and Weighing Cup (Open Receiving Volume) Methods 17.9. Sources and Minimization of Errors

18. Electrical Methods

18.1. Dielectric Constant and Loss by RICHARDH. BOYD 18.2. Static Electricity by D. KEITHDAVIES 18.3. Electric Breakdown by B. R. VARLOW

INDEX FOR PARTC AUTHOR INDEX-SUBJECT

CONTRIBUTORS TO VOLUME 16, PARTS A AND C

Part A

C. V. BERNEY,Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 L. L. CHAPOY, Instituttet for Kemiindustri, The Technical University of Denmark, 2800 Lyngby, Denmark DONALDB. D u P R ~Department , of Chemistry, The University of Louisville, Kentucky 40208 R. A. FAVA,ARCO Polymers, Inc., Monroeville, Pennsylvania 15146

J. S. KING,Nuclear Engineering Department, University of Michigan, Ann Arbor, Michigan 48109 ROBERTF . KRATZ,ARCO Polymers, Inc., Product Development Section, Monaca, Pennsylvania 15061 P. L. KUMLER, Department of Chemistry, State University of New York, College of Fredonia, Fredonia, New York 14063

J. R. LYERLA, IBM Research Laboratories, San Jose, California 95193 G. D. PATTERSON, Bell Telephone Laboratories, Murray Hill, New Jersey 07974

D. J. POLLOCK, ARCO Polymers, Inc., Product Development Section, Monaca, Pennsylvania 15061

R. G. SNYDER,Department of Chemistry, University of California, Berkeley, California 94720 J. R. STEVENS,Department of Physics, University of Guelph, Guelph, Ontario, NIG 2 WI, Canada

H . W. WHITE,Department of Physics, University of Missouri, University Park, Columbia, Missouri 65201

T . WOLFRAM, Department of Physics, University of Missouri, University Park, Columbia, Missouri 65201

S . YIP, Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 xxi

xxii

CONTRIBUTORS TO VOLUME

16,

PARTS A AND C

Part C

RICHARDH. BOYD,Department of Materials Science and Engineering, University of Utah, Salt Lake City, Utah 84112 NORMAN BROWN,Department of Metallurgy and Materials Science, University of Pennsylvania, Philadelphia, Pennsylvania 19174

D. KEITHDAVIES,Electrical Research Association Limited, Leatherhead, Surrey, KT22 7SA, England R. M . FELDER,Depurtment of Chemical Engineering, North Carolina State University, Raleigh, North Carolina 27607 BRUCEHARTMANN, Naval Surface Weapons Center, White Oak, Silver Spring, Maryland 20910 IANL. HAY,Celanese Research Company, Summit Laboratory, Summit, New Jersey 07901 G. S. HUVARD,Department of Chemical Engineering, North Carolina State University, Raleigh, North Carolina 27607

TOSHIHIKO NAGAMURA,* Department of Mechanical and Industrial Engineering, College of Engineering, University of Utah, Salt Lake City, Utah 841 12 DONALDJ. PLAZEK, Department of Metallurgical and Materials Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261 J . ROOVERS,Division of Chemistry, National Research Council of Canada, Ottawa, Ontario, K I A OR9 Canada

JOHN L. RUTHERFORD, Kearfott Division, The Singer Company, Little Falls, New Jersey 07424 B. R. VARLOW,Electrical Engineering Laboratory, University of ManChester, Manchester MI3 9PL, England R. W. WARFIELD,Naval Surface Weapons Center, White Oak, Silver Spring, Maryland 20910

* Present address: Department of Organic Synthesis, Faculty of Engineering, Kyushu University, Higashi-ku, Fukuoka 812, Japan.

METHODS OF EXPERIMENTAL PHYSICS E ditors-in-Chief L. Marton C. Marton Volume 1. Classical Methods Edited by lmmanuel Estermann Volume 2. Electronic Methods. Second Edition (in two parts) Edited by E. Bleuler and R. 0. Haxby Volume 3. Molecular Physics. Second Edition (in two parts) Edited by Dudley Williams Volume 4. Atomic and Electron Physics-Part A: Atomic Sources and Detectors, Part B: Free Atoms Edited by Vernon W. Hughes and Howard L. Schultz Volume 5. Nuclear Physics (in two parts) Edited by Luke C. L. Yuan and Chien-Shiung Wu Volume 6. Solid State Physics (in two parts) Edited by K. Lark-Horovitz and Vivian A. Johnson Volume 7. Atomic and Electron Physics-Atomic In erac ions (in two Parts) Edited by Benjamin Bederson and Wade L. Fite Volume 8. Problems and Solutions for Students Edited by L. Marton and W. F. Hornyak Volume 9. Plasma Physics (in two parts) Edited by Hans R. Griem and Ralph H. Lovberg Volume 10. Physical Principles of Far-Infrared Radiation Edited by L. C. Robinson Volume 11. Solid State Physics Edited by R. V. Coleman xxiii

xxiv

METHODS OF EXPERIMENTAL PHYSICS

Volume 12. Astrophysics-Part A: Optical and Infrared Edited by N. Carleton Part 6: Radio Telescopes, Part C: Radio Observations Edited by M. L. Meeks Volume 13. Spectroscopy (in two parts) Edited by Dudley Williams Volume 14. Vacuum Physics and Technology Edited by G. L. Weissler and R. W. Carlson Volume 15. Quantum Electronics (in two parts) Edited by C. L. Tang Volume 16. Polymers (in three parts) Edited by R. A. Fava Volume 17. Accelerators in Atomic Physics (in preparation) Edited by P. Richard Volume 18. Fluid Dynamics (in preparation) Edited by R. J. Emrich

Methods of Expe rim en taI Physics VOLUME 16

POLYMERS PART €3: Crystal Structure and Morphology

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6. X-RAY DIFFRACTION 6.1 Unit Cell and Crystallinity

By Joseph E. Spruiell and Edward S. Clark 6.1.l.Introduction

X-ray diffraction has been a major tool for studying the structure of matter since the German physicist Max von Laue first suggested in 1912 that x-rays could be diffracted by crystals. Experiments carried out by two assistants proved conclusively that crystals have periodic structures and that x-rays exhibit a wave nature with a wavelength of the same order as the periodicities found in crystals. Within a few short years, the basic theory of x-ray diffraction was de~elopedl-~ and the technique was applied to determine the structure of an ever increasing number of crystals. The use of x-ray diffraction to determine the structure of crystals thus predates the rise of the macromolecular hypothesis championed by Staudinger in the 1920~.~-'It is interesting, and perhaps significant of the difficulty of understanding polymer structure, that the early applications of x-ray diffraction to determine the crystal structure and morphology of polymers were misinterpreted. The x-ray diffraction measurements showed that the unit cell of polymers was generally no larger than those found for low-molecular-weight compounds.**0 Since it was presumed that the entire molecule must lie within a single unit cell, it was argued that the molecules must also be small. This presumption proved to be fallacious, of course. Arguments still persist about the validity of a two-phase model to describe the morphology of semicrystalline polymers and especially conM. von Laue, Muench. Sitzungsber. p. 363 (1912); Ann. Phys. (Leipzig) [4] 41, 989 (1913); Enzykl. Math. Wiss. 24, 359 (1915). * W. H. Bragg and W. L. Bragg, f r o c . R . Soc. London 88, 428 (1913); 89, 246 (1913). C. G . Darwin, Philos. Mug. [5] 27, 325 and 675 (1914). P. P. Ewald, Phys. Z . 14, 465 (1913); Ann. Phys. (Leipzig) [4] 54, 519 and 577 (1917). H. Staudinger, Ber. Drsch. Chem. Ges. 53, 1073 (1920); 57, 1203 (1924); 59,3019 (1926). H. Staudinger and J. Fritschi, Helv. Chim. Acru 5, 785 (1922). H. Staudinger and M. Luthy, Helv. Chim Acta 8, 41 (1925). E. Ott, Phys. Z . 27, 174 (1926); Nafurwissenschafren 14, 320 (1926). E. A. Hausen and H. Mark,Kolloidchem. Beih. 22, 94 (1926). @

1 METHODS OF EXPERIMENTAL PHYSICS, VOL. 168

Copyright @ 1980 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-475957-2

2

6.

X-RAY DIFFRACTION

cerning the details of models incorporating chain folding. One of the limitations is that x-ray diffraction analyses may not always yield a unique interpretation of the data. It is imperative that the scientist or engineer utilizing x-ray diffraction be aware of its limitations as well as its strengths. Often, other techniques need to be used to complement the diffraction experiments. Nevertheless, x-ray diffraction is still the single most powerful technique for establishing the structure of matter. The present chapter is dedicated to a description of the diffraction methods used to establish the crystal structure and crystallinity of polymers. Methods used for low-molecular-weight materials, but that are not readily applied to polymers, are omitted. It was also necessary to omit some topics used infrequently for polymers for the sake of brevity. In general, the approach taken is to present the material at approximately the level that could be read and understood by a first year graduate student, and at a level that is useful to the average polymer scientist. Thus many highly specialized techniques used by advanced crystallographers are omitted. The approach is more that of a textbook than of a literature review, and it is hoped that this approach will make the chapter useful to those scientists who have no previous experience in x-ray diffraction or who have experience only with some other aspect of its use such as orientation measurement or line broadening (crystallite size and strain) studies. These latter topics are not covered here, but are the subjects of other chapters in this volume. The present chapter should also serve as an introduction to these topics. Throughout this chapter we have relied heavily upon the “International Tables for X-Ray Crystallography.”1o This work serves as a major reference work and handbook for crystallographers throughout the world, and the serious student of crystallography should become familiar with its contents. 6.1.2. Basic Crystallography 6.1.2.1. Crystal Systems, Space Lattices, and the Unit Cell. A

crystal may be defined as a portion of matter within which the atoms are arranged in a regular, repeated, three-dimensionally periodic pattern. A direct consequence of this regular atomic arrangement is that crystals exhibit anisotropic properties. However, both the properties and the atomic arrangements within a crystal exhibit symmetry. A crystal is classified in one of seven large subgroups, called crystal systems, depending lo N. F. M. Henry and K. Lonsdale, eds., “International Tables for X-Ray Cryptallography,” Vols. I, 11, and 111. Kynoch Press, Birmingham, England, 1952, 1959, and 1962, resp.

6.1. U N I T CELL A N D CRYSTALLINITY

3

z

FIG.1 . Generalized set of axes showing definitions of a , b, c , a,fi, y.

on the symmetry exhibited by its atomic arrangement. The choice of crystal system provides an appropriate set of axes by which to describe the repeat unit of the crystal. The seven crystal systems are listed in Table I in order of increasing symmetry. The minimum symmetry required for each system and the axis system appropriate for the crystal’s repeat unit are also given in Table I. The a, 6 , and c are unit distances along the three noncoplanar axes and a,p, and y are the angles between the axes as defined in Fig. 1. The symbol # in Table I should be read “not necessarily equal to.” Periodic arrangements of any motif, e.g., a molecule, are generated by placing the motif at points located such that each point has identical surroundings. Such infinite arrangements of points are called lattices. (An example is shown in Fig. 2.) Bravais” showed that there are only 14 ways of arranging points in space such that each has identical surroundings. These 14 arrangements are the so-called Bravais or space lattices; they describe possible types of periodicities that crystals can have. The unit or period of the space lattice that most simply describes the nature of the space lattice and that will generate the entire arrangement of points in space when repeated by translation in three dimensions is called the unit cell of the space lattice. The unit cells of the 14 Bravais lattices are shown in Fig. 3. Each unit cell is a parallelepiped and can be described in terms of a set of vectors a, b, c along the edges of the cell. These translation vectors connect one point to another in the space lattice. When applied to crystallography, these lattice vectors are equivalent to the axes that define the crystal systems (Table I). Of the 14 space lattices, three belong to the II

A. Bravais, J . Ec. Polyrech. (Paris) 19, 1 (1850).

TABLEI. The Crystal Systems and Their Axes System

Axes

Axial angles

Minimum symmetry

# B # Y # 90" p # 90" ff=p=y=5Q" (2 = B = y = 90" Q = B = 90",y = 120" ff = p = y # 90"

None One twofold rotation axis (or rotary inversion) Three perpendicular twofold rotation axes (or rotary inversions) One fourfold rotation axis (or rotary inversion) One sixfold rotation axis (or rotary inversion) One threefold rotation axis (or rotary inversion) Four threefold rotation axes

~~

Triclinic Monoclinic Orthorhombic Tetragonal Hexagonal Rhombohedral Cubic

a#b#c a#b#c a#b#c a, = a, # c a , = up (=a,) # c a, = a, = as a, = a, = a3

ff

a = y = 5Qo,

ff=B=?=w

6.1.

UNIT CELL AND CRYSTALLINITY

5

FIG.2. A space lattice of points with several alternative unit cells outlined. (H.P. Klug and L. E. Alexander, “X-Ray Diffraction Procedures,” Wiley, New York, 1954.)

Endcentered monoclinic

Simple monoclinic

Simple orthorhombic

Triclinic

Body-centered orthorhombic

End-centered orthorhombic

I’ace-centered orthorhombic

fqjflm fgfg Simple cubic

Body-centered cubic

Face-centered cubic

Simple tetragonal

Body-centered tetragonal The unit cells of the 14 Bravais lattices. Each indicated point has identical surFIG.3. roundings. (L. H. Van Vlack, “Elements of Materials Science and Engineering.” 3rd ed., Addison-Wesley, Reading, Mass., 1975.)

6

6.

X-RAY DIFFRACTION

cubic system, four belong to the orthorhombic system, and the other seven are distributed among the other five systems. Some of the space lattices contain only a single lattice point per unit cell (one-eighth at each of eight corners). These are referred to as simple or primitive unit cells and are given the lattice symbol P. Other cells contain additional lattice points in the center of the cell (body-centered, I ) , in the center of each face ( F ) ,or in the center of a pair of faces (C). It should be realized that it is always possible to select a primitive unit cell to describe any of the 14 space lattices. The centered representations are preferred because crystals having such space lattices will have greater symmetry than is obvious from the primitive unit cell. 6.1.2.2. indices of Lattice Planes and Directions. Crystallographic directions are named by treating them as vectors. A set of integers u , u , w are sought such that r = ua

+ ub + wc,

(6.1.1)

where r is a vector pointing in the desired crystallographic direction and the basis vectors a, b, c are the translation vectors of the lattice (unit cell edges). A restriction maintained on u, u, and w is that they are the smallest possible integers. When naming directions, the indices are enclosed in square brackets, i.e., [uvw]. Examples of crystallographic directions in an orthorhombic unit cell are shown in Fig. 4.

FIG.4. Directions in an orthorhombic unit cell.

6.1.

a

b

Y

X

7

UNIT CELL A N D CRYSTALLINITY

Y

X C

d FIG.5. Lattice plane indices.

Two similar notations are used for defining lattice planes. Both notations provide information about the orientation of the planes relative to the axes of the unit cell. One notation also portrays information about the spacing between members of a set of planes, while the other notation does not necessarily provide this information. The former we refer to simply as the lattice plane indices while the latter are called Miller indices . To obtain the lattice plane indices for a given set of planes we note that the origin of our x , y, z coordinate system can always be chosen so as to lie on one plane of theset. Then the intercepts of the next plane in the set are given by a / h , b / k , c / l (see Fig. 5a). Here a, b, and c are the magnitudes of the lattice translation vectors and h, k. a n d l are the lattice plane indices. Thus we need only determine the rehdve intercepts l / h , I l k , 1 / 1 of the plane on the coordinate axes and take reciprocals to obtain the indices. The indices h, k, I are enclosed in parentheses to denote that they are planes rather than directions (hkl). Examples of the naming of lattice planes are shown in Fig 5 .

8

6.

X-RAY DIFFRACTION

The interplanar specing between members of the named set is equal to the perpendicular distance from the origin of coordinates to the plane used to determine the relative intercepts. Clearly then, planes with indices (200) have half the spacing of planes with indices (100). Similarly, the (333) planes have 1/3 the spacing of ( 1 1 1 ) planes. It should also be obvious that the (200) planes contain the (100) planes as a subset. The Miller indices of a set of planes can be obtained in a manner quite similar to that described for lattice plane indices. The main difference between the two notations is that multiple indices such as (333) are not retained. Such indices would be reduced to the smallest set of integers with the same ratio; thus (333) becomes ( 1 1 1 ) in the Miller index notation. Although the information concerning the orientation of the planes in space is retained, the information concerning the interplanar spacings is lost. In many applications this is no great problem and Miller indices are frequently used. But for x-ray diffraction experiments, there are advantages in using the unreduced lattice plane indices, and we use this notation throughout the remainder of this review. Thus we develop the wellknown Bragg law as A = 2dhklsin 8, where hkl are lattice plane indices, rather than nA = 2dhfkt,, sin 8, where h'k'l' are Miller indices. This is feasible because the nth-order reflection from planes of spacing d occurs at the same diffraction angle 8 at which first-order diffraction from planes of spacing d2 = d, f n occurs. 6.1.2.3. Symmetry of Crystals. Crystals exhibit definite symmetry in the way their atoms are distributed in space, The symmetry elements or operators that are possible in crystals include inversion centers, rotation axes, rotary-inversion axes, mirror planes, screw axes, and glide planes. Examples of these operators are shown in Fig. 6. The existence of an inversion center (or center of symmetry) means that with respect to a point in the unit cell (say the origin) if there is an atom at x, y, z, there will be an identical atom at --x, - y , - 2 . The origin of the unit cell is usually chosen to be a center of symmetry if possible since this makes calculations of intensity of x-ray reflections much easier as will be shown later. A threefold rotation axis requires that for a.&atom at ( r , 4) cylindrical coordinates there will be an identical atom at ( r , 4 + 120")and at ( r , 4 + 240"). The angular displacement of 120"is called the throw of the rotation axis. For an n-fold axis the throw is given by a = 360"/n.

(6.1.2)

Because of the periodic arrangement of atoms in crystals, only I - , 2-, 3-, 4-, or 6-fold axes occur.

a

b

C

e

f

FIG.6. Examples of symmetry elements. (a) inversion center, (b) 3-fold rotation axis, (c) 4-fold rotary inversions axis, (d) mirror plane, (e) 31 screw axis, and ( f ) glide plane.

10

6.

X-RAY DJFFRACTION

Axes of rotary inversion are combinations of a rotation and an inversion as shown in Fig. 6. Note that the axis of rotary inversion is not equivalent to a rotation axis plus an inversion center but involves an - _ _axis - is operation that is a combination of these two. A rotary inversion distinguished from a rotation axis by an overbar. Again, 1-, 2-, 3-, 4-, and - f o l d rotary inversions are possible although only the tetrad is unique; the other rotary inversion axes are equivalent to other symmetry elements. For example,T is equivalent to a mirror plane and6 is equivalent to a 3-fold rotation axis plus a perpendicular mirror plane. The operation of a mirror plane is self-evident; the points generated by operation of the mirror plane are related to the initial points as the virtual image is related to the real object for a true mirror. Note that the operation of a mirror plane on an asymmetric motif produces an enantiomorphic (related as the left hand to the right) motif. The operation of an inversion center also produces an enantiomorphic motif. Screw axes involve a translation parallel to the axis as well as a rotation component. The screw operation is designated as ns, where n is the fold of the rotation component (with throw (Y = 360/n) and S is an integer related to the translation component t by t = -S

n c~

(6.1.3)

where c is the length of the unit cell edge to which the screw axis is parallel. For example, consider a 3, screw axis. For an atom at ( r , 4, z) there will be a corresponding atom at ( r , d, + 120°, z + 4c) and also at ( r , d, + 240°, z + 3c). If there is a 3, axis, this relationship must hold for every atom in the crystal with respect to the 3-fold screw axis location in the unit cell. Possible screw axes are 2-, 3-, 4-, and 6-fold with subscripts from unity to one less than the fold number, i.e., 4,, 4,, %; it should be noted that 4, and 4, are enantiomorphs (mirror images). Glide planes combine a translation with a reflection across a plane. The translation is always parallel to the plane and in a direction that is along the unit cell edge or a face or body diagonal of the unit cell (a lattice translation). When parallel to a unit cell edge it must have a length of either a/2, b/2 or 4 2 . Such glide planes are designated as a, b, or c-glide planes. A mirror plane parallel to the (001) plane of an orthorhombic unit cell would become an a-glide plane if the structure on one side of the plane were translated a/2 before being reflected. Translations of half a face or body diagonal, say a/2 + b/2, define an n-glide plane or diagonal glide. So-called diamond glides (d-glide planes) are also possible with translations of one-fourth of a face or body diagonal (in face-centered or body-centered lattices, respectively).

6.1.

U N I T CELL A N D CRYSTALLINITY

II

The symmetry elements described above can occur alone or in consistent groups. The group of symmetry elements that describes completely the symmetry of the atomic arrangement within a crystal is called the crystal’s space group. A space group is an array of symmetry elements distributed in three dimensions and must be consistent with one of the 14 Bravais lattices. It is convenient to think of the symmetry elements of the space group as being placed at a particular location and in a particular orientation in the unit cell of the space lattice. Of course, not all symmetry elements are consistent with every Bravais lattice. For example, a 6-fold rotation axis is not possible with a tetragonal lattice. It can be shownI2that there are precisely 230 crystallographic space groups distributed among the 14 Bravais lattice types. Thus there are only 230 different symmetries among all of the possible atomic arrangements in crystals. The symbols used to designate space groups consist of the lattice symbol followed by a list of appropriate symmetry elements. The symmetry elements given are the various axes (rotation, rotary inversion, screw) along certain specified directions in the unit cell and/or the symmetry planes that are normal to these directions. The particular directions in the unit cell differ from one crystal system to another (see International Tables for X-Ray Crystallography,’O Volume I, for detailshereinafter referred to only as International Tables), but generally correspond to low index directions. For monoclinic cells the symbols given correspond to the unique b axis. The space group P2/m thus represents a primitive monoclinic space lattice with a 2-fold axis parallel to the b axis and a mirror plane m perpendicular to this axis. The space group Cmc2, corresponds to a C-centered orthorhombic lattice with a mirror plane perpendicular to the a axis, a c glide plane perpendicular to the b axis, and a 2, screw axis parallel to the c axis. The complete list of space groups together with a detailed description of each is given in International Tableslo (Vol. I, pp. 73-346). Pictorial symbols for the symmetry elements used in these tables are shown in Table 11. A typical space group listing from the International Tables is shown in Fig. 7. The space group shown is Pnma as indicated by the symbol in the upper right hand corner. (The symbol 0;: is the Schoenflies symbol from an older system of space group notation.) The pictorial representation of the space group is shown as the figure on the right side. The unit cell is shown in a projection along the c axis. The origin of the unit cell is in the upper left corner of this figure; the h axis points to the right and the a axis downward. Using Table 11, various symmetry elements are readily identified. I*

M . J . Buerger, “Elementary Crystallography.” Wiley. New York, 1963.

6.

12

X-RAY DIFFRACTION

TABLEIIA. Symbols of Symmetry Planes” Graphical symbol

Symbol

Symmetry plane

rn

Reflection plane (mirror)

a, b

Axial glide plane

Normal to plane of projection

- -- -............

C

n

Diagonal glide -. -. plane (net)

d

“Diamond” glide plane

Parallel to plane of projection

’ None

-.-. -

-.+.-.-.-

- - .-. +.-

1

Nature of glide translation None (If the plane is at z = 4 this is shown by printing4 beside the symbol.) 4 2 along [ 1001 or b / 2 along c / 2 along L axis; or ( a + b + c ) / 2 along [ 1 111 on rhombohedra1 axes ( a + b ) / 2 or (6 + c ) / 2 or (c + a ) / 2 ;or (a + b + c)/2 (tetragonal and cubic). (a 2 b ) / 4 or ( b 2 c)/4 or (c h a ) / 4 ; or (a f b h c)/4 (tetragonal and cubic).“

0 In the “diamond” glide plank the glide translation is half of the resultant of the two possible axial glide translations. The arrows in the first diagram show the direction of the horizontal component of the translation when the z component is positive. In the second diagram the arrow shows the actual direction of the glide translation; there is always another diamond-glide reflection plane parallel to the first with a height difference off and with the arrow pointing along the other diagonal of the cell face.

The figure in the upper left of Fig. 7 shows the equivalent positions in the unit cell. These are the positions at which an object, say an atom, would be repeated by the symmetry group from a single arbitrary starting position. The coordinates of the equivalent positions are also given in the listing. For a general point there are eight equivalent positions. Consider the point represented by an empty open circle with a plus beside it in the upper left corner. If we let the coordinates of this position be x, y, z (positive), then the point at X, y, Z is generated from the initial point by the inversion center at the origin. The comma indicates that this position is enantiomorphous to the starting point. By successive application of the symmetry operations all of the equivalent positions listed can be generated. An initial point chosen in certain “special positions” will result in fewer equivalent positions. For example, an initial point chosen on a mirror will not be repeated by the mirror. Such special positions in the space

TABLEIIB. Symbols of Symmetry Axes

Symmetry axis

Graphical symbol

Nature of right-handed screw translation along the axis

Rotation monad

None

None

4

Rotation tetrad

None

1

Inversion monad

0

None

4,

Screw tetrads

4 4

2

Rotation diad

0

None

Symbol 1

-

(normal to paper)

+

Screw diad

-

2c/4

4

3c/4

4

Inversion tetrad

None

6

Rotation hexad

None

6,

Screw hexads

cJ2

(normal to paper) (parallel to paper)

Either n J2 or b / 2

ss

Normal to paper

3 31

Rotation triad Screw triads

32

3

Inversion triad

None

A4 A

Symmetry axis

Nature of right-handed screw translation along the axis

s, -

(parallel to paper)

2,

Symbol

Graphical symbol (normal to plane of paper)

4 3

2c/6

63

3c/6

4

4c/6 5c/6

2c/3

None

c/6

6

Inversion hexad

None

14

6.

Orthorhombic m m m

X-RAY DIFFRACTION

P 2rln 2Jm 2Ja

Pnma

NO. 62

D%

-0 f so

9

i*

1-0

01-

-0 *0

0.

Origin at Number of pmitions, WyckoR notation. and point symmetry

1

Co-ordinates of equivalent positions

Conditions limiting possible rekctions General:

8

d

I

4 t . r , $ - v , J - Z ; f , t + y , i ; d-x.i..t+~; i9,i; I-.\.,i+y,l+z; X,l-YSj t + x , y , l - z .

hk1: No conditions Qkl: k t l = 2 n

X,Y,Z;

hOI: No conditions hkO: h-2n M)o:

(h-2n)

OkO (k=Zlr)

001: (I-&) Special: as above, plus no extra conditions

I

hkl: h + l = 2 n ; k = 2 n

Symmetry of special projections

(001)pgm; 1 7 ’ 4 2 ,b’-b

(100) cmm; b’=b, c’=c

(0IO)pg.g; c’=c, a’-u

FIG.7. Typical page from space group tables.

group are also indicated. In the case of Pnma a special position on a mirror plane reduces the multiplicity of the equivalent positions to four. Similarly a special position on an inversion center also reduces the multiplicity of the equivalent positions to four. The “conditions limiting possible reflections” refers to the fact that the symmetry of a crystal causes certain extinctions in its x-ray reflections (zero intensity). Thus the presence of only certain types of reflections can be used to help determine the space group of a crystal. Such conditions are also listed in Fig. 7. We discuss this feature in more detail later. Let us consider now the space groups of two important polymerspolyethylene and polypropylene. Figure 8 shows the unit cell and space

6.1.

15

UNIT CELL AND CRYSTALLINITY

-k b

Q

a+

_ _ _ _ - _!--__

---I -------I - - -

Pnam FIG.8. Structure and space group of polyethylene.

group symmetry notation for some of the symmetry elements of polyethylene. We have used the traditional notation of defining the c axis as the chain axis of the unit cell. According to Bunn13 the space group for this axis designation is Pnam. It is important to note that this is identical to the space group Pnma in the International Tableslo (Vol. I, p. 346) with the b and c axes interchanged. This interchanging of axes in space group notation is occasionally required to conform to the convention of defining the chain axis as the c axis. (However, this convention is not followed if the chain axis is the unique axis of the monoclinic system; in this case the chain axis is the b axis, as in nylon 6.) In interpreting the specifications l3

C . W.Bunn. Trans. Faraday SOC. 35, 482 (1939).

16

6. X-RAY DIFFRACTION

for the space group of polyethylene presented in International Tables as Pnma (Fig. 7), the interchange of b and c also requires interchange of y and z plus k and 1. Thus all atoms are in special positions (c) x, y, 4; R, 7, 2 ; 3 - x, 3 + y , 2 ;3 + x, 3 - y, f . The conditions for nonextinction are Okl: k + 1 = 2n; h01: 1 = 2n. The fractions by the carbon atoms in the sketch represent the height of the CH, groups above the bottom plane (ab) of the unit cell. There are four 2-fold screw axes parallel to the c axis. Nine are shown in the figure, but only four are independent of the translational identity of the unit cell. The 2-fold screw axes in the corners are easy to envision. A “zig” becomes a “zag” by a 2-fold screw operation about the c axis. But these screw axes relate to the entire crystal; thus screw axes are also found half way along the a axis. The right angle above the upper right hand corner with 4 by it means there is a mirror plane, parallel to the plane of the paper, 4 along the c axis (z = 4). Thus for every atom at x, y, 4 - z, there will be an atom at x, y, 4 z. All atoms are in the special position (c) in Fig. 7 and so z = 4 and 2 and all atoms lie on mirror planes at z = 4, 3. The short-long dashed lines at a = 4 and 2 mean there are n-glide planes parallel to (100)at a = 4 and 3. Note that the corner repeat unit will reflect c/2. Thus the into the center repeat unit after a translation of b/2 space group Pnam defines the unit cell as primitive with an n-glide plane perpendicular to the a axis, an a-glide plane perpendicular to the b axis and a mirror plane perpendicular to the c axis. There are also other symmetry elements present generated by these three. Of special note is the center of symmetry at the origin. Note also that it is necessary to give the position of only one CH, group and all of the other groups in the entire crystal are generated by the unit cell translations and symmetry elements. The symmetry of the unit cell of polypropylene (isotactic) is shown in Fig. 9.14 This is an end-centered monoclinic unit cell with a lattice point at the center of the ab face as well as at each comer, thus the C designation. The figure is a projection of the unit cell along the c axis on a plane perpendicular to the c axis. The fractions refer to the height of the atoms above a plane normal to the c axis. Note that (001) is not parallel to the plane of the page. Since the unit cell is monoclinic, the a axis and (001) are inclined to the plane of the projection. Thus the atoms at the extreme left at (h)c and (&)c are identical in translation along the a axis. There is no center of symmetry. There is a pair of helical molecular segments (crystallographic repeat units) at each lattice point -otie left handed and one right handed. Each molecular segment contains three chemical re-

+

+

’‘ G. Natta and P. Corradini, Nuovo Cimrnro, Suppl. 15, 40 (1960).

6.1.

17

UNIT CELL A N D CRYSTALLINITY

7/12

1/12

38

cc FIG.9. The structure and space group of polypropylene. (a) projection on (001) of the structure assuming the Cc space group, (b) symmetry elements of space group C c . Note that the mofif associated with each lattice point is an enantiomorphic pair of helical units. (After Natta and Corradini.")

peat units. The dotted line at 6 = 0 (and 3) means there is a c-glide plane parallel to (010). The left-handed repeat unit at a lattice point reflects into a right-handed repeat unit after a translation of z = (+)c. The short-long dashed lines denote the n-glide planes parallel to (010) with translation operations of 6/2 c / 2 . This n glide is a result of the combination of the c glide and centered lattice. Before leaving the subject of symmetry, we need to consider the specification of the symmetry of the crystal planes and directions in relation to each other. This is also the externally apparent symmetry of faceted macroscopic crystals and the symmetry that will control the variations of the physical properties of the crystal as a function of crystallographic direction. The point group of the crystal specifies this symmetry. Just as the space group is a group of consistent symmetry elements in space, the point group is a group of consistent symmetry elements that intersect at a point. Only rotation axes, axes of rotary inversion, inversion centers, and mirror planes participate in point group symmetrycombinations with translations are excluded. There are just 32 crystallo-

+

18

6.

X-RAY DIFFRACTION

graphic point groups and these define the 32 crystal classes. The point group consistent with each space group is given in International Tables.lo The symbol mmm in the upper left corner of Fig. 7 specifies the point group of any crystal whose space group is Pnma. Another group called the “Laue group” is concerned with the different point symmetries that can be distinguished by x-ray diffraction. Only 11 such symmetries can be distinguished from the symmetry of the x-ray pattern. 6.1.2.4. Specification of Polymer Structure. The size and shape of the unit cell, the symmetry of the atomic arrangement, and the coordinates of each atom in the unit cell is a complete description of the structure of any crystal, be it polymer or low-molecular-weight material. Because of the long, covalently bonded structure of polymer molecules, an alternative description of the structure of the molecules is emphasized. The long-chain molecules are assumed to be packed together in some specific way to generate the three-dimensional crystal structure. Although the chains can pack together in many ways, the chain axes are normally assumed to lie parallel to each other. The structure of the molecule is described in terms of (a) its configuration and (b) its conformation. The configuration of a molecule is its chemical structure but without regard to the different spatial arrangements that are possible because of rotation about single bonds. This includes specification of the chemical repeat unit and the way the chemical repeat units are bonded together to form the polymer chain, e.g., headto-tail or head-to-head and tail-to-tail. Tacticity is also a feature of the chain configuration. The three best known tactic forms for vinyl polymers-isotactic, syndiotactic, and atactic-are illustrated in Fig. 10. The complete description and nomenclature used to describe a polymer chain’s configuration will not be given here; it can be found in several readily available ~ o u r c e s .It~ should ~ ~ ~ ~be clear that stereoregularity along the chain is a prerequisite to the formation of a well-developed crystal structure. Chains that do not possess a stereoregular configuration cannot crystallize in the strict sense of the word, although their chains may pack together to form some minimum energy structure. As a general rule, chain molecules will aggregate to form a crystal if they have a regular (periodic) shape. And in order to have a regular shape the molecules must have regular chemical configuration. In some molecules, such as polyethylene, a regular configuration is inherent in the monomer. In others, such as polypropylene, chemical regularity reP. J. Flory, “Principles of Polymer Chemistry.” Cornell Univ. Press, Ithaca, New York, 1953). l6 F. W. Billmeyer, Jr., ”Textbook of Polymer Science,” 2nd ed., pp. 141-154. Wiley (Interscience), New York, 1971.

6.1.

U N I T CELL A N D CRYSTALLINITY

19

FIG.10. Tactic isomers of vinyl polymers. (a) syndiotactic, (b) isotactic, and (c) atactic.

quires special polymerization techniques from the asymmetric monomer. The conformation of a polymer chain refers to the specific atomic arrangement or shape taken by the molecule without variation of the chain configuration. The conformation can be changed by rotation about single bonds. The conformations of most common interest in crystalline polymers are the fully extended planar zigzag and various helical conformations. The periodic structure of the molecule is characterized by a “repeat distance” and a “repeat unit.” The linear chain is treated as a one-dimensional lattice in which the repeat distance is the distance between points and the repeat unit is the motif that is repeated at each lattice point. The chain repeat unit is composed of an integral number of configurational (or chemical) repeat units. The repeat unit is illustrated for two common polymers in Fig. 1 1. The repeat distance is equivalent, by convention, to the L‘ dimension of the polymer’s three-dimensional unit cell. In some cases the chain repeat unit also corresponds to the motif repeated at the points of the three-dimensional lattice, but this is not always true. The latter motif will, in general, be some integral multiple of the chain repeat unit. For example, in polypropylene the motif associated with a lattice point is an enantiomorphic pair of chain repeat units-a left-handed helical unit and a right-handed unit as illustrated in Fig. 9.

20

6.

X-RAY DIFFRACTION

FIG.1 1 . Chain conformations and repeat units of two crystalline polymers (a) polytetrafluoroethylene, (b) polyethylene.

Because of the common occurrence of helical conformations, it is useful to have suitable nomenclature for their description. Two systems are in use. One system is equivalent to the notation used for screw axes while the second system is called the helical point net system. 17-20 As an illustration of the helical point net system consider the helical molecule shown in Fig. 12, which represents the helical backbone of the -C-0 -C-0units of polyoxymethylene. The atoms lie on a common helix with five turns per 9 -C -0 -units or 18 atoms. Since alternate atoms are equivalent, the motif associated with each helical net point is a -CH2-0unit. The hydrogen atoms on each carbon are in equivalent positions and may be defined as regularly spaced along helices of larger radius. In some instances it is useful to define helices of different radii to describe various components of the chain, e.g., one helix for the backbone atoms and another for substituent atoms or groups. A helix may be defined in terms of the pitch P of the helix, which is the distance parallel to the chain axis corresponding to one turn of the helix, and p, which is the distance parallel to the chain axis corresponding to the distance between successive equivalent points or motifs. The distance p will normally be a function of the chemical or configurational repeat unit. The number of equivalent points per turn of the helix is thus P / p . It is also clear that equivalent points - 41 - - configurational motifs , per turn p t per turn

(6.1.4)

R. E. Huges and J. L. Lauer, J . Chem. Phys. 30, 1165 (1959). E. S. Clark and L. T. Muus, Meet. Crystallog. Assoc., 1964 Abstract F-12 (1964). lo L. Nagai and M. Kobayashi, J . Chem. Phys. 36, 1268 (1961). 2o L. E. Alexander, “X-Ray Diffraction Methods in Polymer Science.” Wiley (Interscience), New York, 1969. IT

6.1.

U N I T CELL A N D CRYSTALLINITY

21

where u is the number of configurational motifs in the crystallographic identity period c, and r is the number of turns of the helix in the identity period c. Both u and t are integers for ideal crystals (but see later discussion of irrational helical conformations) and the ratio u / t provides a convenient expression for the helical conformation in the helical point net system. In the example of Fig. 12, the repeat unit contains nine CH20 units in a distance of 17.3 A. The pitch distance P is 3.46 A. The axial spacing between CH20 motif groups, p , is 1.92 A. P / p = u / t = 3.46/1.92 = 9/5 = 1.80. The extended chain structure of polyethylene shown in Fig. 1 lb can be equally well described as a 2/1 helix, where the configurational motif is a CH2group. The structure of the polypropylene molecule, Figure 9, contains three configurational motifs in each crystallographic repeat, which also corresponds to one turn of the helix. Each motif corresponds to a propylene residue and the molecule can be described as a 3/1 helix. The polytetrafluoroethylene molecule shown in Figure 1 l a contains 13 CF2groups and six turns of the helix in its crystallographic repeat until; thus it may be called a 13/6 helix. In the conventional screw axis notarion the throw of the screw in degrees is given by a = 360/u.

(6.1.5)

The translational component A 2 = y ( c / u ) = y p , where y is an integer between 1 and u. The helix is then expressed symbolically as u y . The

FIG.12. Illustration of the nomenclature used in the helical point net system to describe the helical conformation of polyoxymethylene.

22

6. X-RAY

DIFFRACTION

symbols for the two helices illustrated in Fig. 11 in the screw axis notation are 2, and 1311. The model of Fig. 12 is a 92 helix. The relation between the two notations (18) is given by yf = EL4

+ 1,

(6.1.6)

where ~isaninteger. Forexample 13/6 = 1311:11 x 6 = 136 + 1 where E = 5 ; also 9 / 5 = 92where 2 X 5 = 9~ + 1 and E = 1. In the discussions that follow, the helical point net system will be used exclusively. 6.1.2.5. The Reciprocal Lattice. A useful concept that simplifies many crystallographic calculations as well as the mathematics of diffraction is the definition of a “reciprocal lattice.” This concept has its origins in the mathematical definition of reciprocal vector sets. Let a, b, and c be the translation vectors that describe the size and shape of the unit cell of a particular space lattice. A set of vectors a*, b*, c* exists such that a * a* = 1, b . a* = 0, c * a* = 0,

a - b* = 0, b . b* = 1, c * b* = 0,

a * c* b . c* c - c*

= =

0,

=

1.

0,

(6.1.7)

The vectors a*, b*, and c* are said to be reciprocal to a, b, and c. Just as we can generate a space lattice with translation vectors a, b, and c, another space lattice can be generated with translation vectors a*, b*, c*. This latter lattice is said to be the reciprocal lattice of the former one. Using the conditions of Eq. (6.1.7), it is readily shown that a*

=

b x c a * b x c’

b*

=

c x a a *b x c’

c*

=

a x b (6.1.8) a - b x c’

The scalar triple product a b x c is equal to the volume of the unit cell of the original space lattice. a* is perpendicular to the plane of b and c, b* is perpendicular to the plane of c and a, and c* is perpendicular to the plane of a and b. Equations (6.1.8) provide a straightforward way to evaluate a*, b*, and c* from any given values of a, b, and c. Figure 13 shows two examples of the relationship between the unit cell of a given space lattice and the unit cell of its reciprocal lattice. Note that in lattices with orthogonal axes, the reciprocal lattice translation vectors a*, b*, and c* are parallel to a, b, and c, respectively; however, this is not generally the case if a, b, and c are not orthogonal. The magnitudes of the vectors a*, b*, and c* are simply the reciprocals of the magnitudes of a, b, and c in orthogonal cases, but this simple relation does not generally hold when a, b, and c are not orthogonal. It is readily observed from Eq. (6.1.7) that the reciprocal relationship between the two lattices is symmetric; that is, the reciprocal of the recip-

6.1.

I

FIG.

23

U N I T CELL A N D CRYSTALLINITY

*

13. Reciprocal lattice unit cells for (a) orthorhombic lattice, (b) hexagonal lattice.

rocal lattice is the original space lattice. Consequently,

a =

b* X c* a* b* x c*'

b =

c* x a* a* b* X c*'

c = a*a*b* 'x* c*

.

(6.1.9)

Thus if either lattice is known, the other is readily calculated. The application of Eqs. (6.1.8)and (6.1.9) to the various crystal systems give the results shown in Table III'O (Vol. I , p. 13). In Table 111, V is the volume of the original unit cell, V* is the volume of its reciprocal, and K is an arbitrary scale constant. Let r* = ha* + kb* lc* be a vector from the origin to any given point in the reciprocal lattice. Here h , k, and I are the integers of the lattice plane indices. An important property of the reciprocal lattice is that r* is perpendicular to the planes (hkl) in the lattice whose translation vectors are a, b, and c. Furthermore, the magnitude of r*, lr*l, is equal to the reciprocal of the interplanar spacing of the (hkl) planes:

+

r* I (hkl)

and

dhrl = l/lr*l.

(6.1.10)

TABLE111. Reciprocal Lattice Relationships for Crystal Systemsa Triclinic

a* =

K b c sin a , b* - Kca sin p * - K a b sin y , c V V

where V = abc(1

+ 2 cos a cos p cos y -

cos* a- cos2 p- cos* y)ln

= 2 abc{sin s. sin (s - a).sin (s 2s=a+p+y

p). sin (s - y)}l'*; V*

-

Orthorhombic

Tetragonal

(I*

= b* = -K c* = -K a' c

Cubic

Hexagonal

Rhombohedra1 cos a* = cos p* = cos y* From International Tables,Io Vol. I.

V

cos y cos a- cos p , cos y* = cos a cos p- cos y sin y sin a sin a sin p K K K C* = -, a* = p* = w, y* = 180" - y 1st setting: a* = - b* = a sin y b sin y' c K K K 2nd setting; a* =-a sin p b* = F =* =-, c sin p = y* = 900, p * = 1 w - p cos a* =

Monoclinic

=

=

cost a- cos a sin* a

-

_ -

COS

(1

a

+ cos a)

6.1.

U N I T CELL A N D CRYSTALLINITY

25

P

FIG. 14. Correspondence between crystal planes and the reciprocal lattice points.

As a result of these relationships it is helpful to think of the reciprocal lattice as a representation of the crystal lattice in which the (hkl) planes of the crystal are each represented by a lattice point of the reciprocal lattice. Each reciprocal lattice point hkl is located on a line through the origin perpendicular to the corresponding planes (hkl) of the crystal and at a distance from the origin that is the reciprocal of the crystal plane spacing (see Fig. 14). Equation (6.1.10) provides a convenient relationship from which to derive the interplanar spacing formulas for each crystal system in terms of the lattice plane indices and unit cell parameters. The resulting expressions are given in Table IV. 6.1.3. Diffraction Theory

We now treat the nature of the diffraction of x rays from crystals, and the relationship of the diffraction pattern to the structure of matter. In order to develop this subject, we consider first the scattering of radiation from elemental charged particles (i.e., electrons and protons) and then consider how the resultant scattering from spatial distributions of these particles can be derived. Throughout the discussion we assume that the x-ray beam incident on our sample is monochromatic unless otherwise specified. We also treat only the kinematical theory. For our purposes this simply means that we ignore the possibility that a wave scattered from one particle might be rescattered by another particle. Finally, in adding waves scattered by different particles together we use the Fraunhofer approximation. This assumes that the point of observation of the resultant wave is so distant from the scattering source that the wavelets to be added from each scattering center can be considered parallel. 6.1.3.1. Scattering by Electrons. Electrons scatter radiation in two ways: either (1) coherently, or (2) incoherently. From the standpoint of diffraction theory the coherent scattering is much more important be-

TABLEIV. Spacing Formulas for the Crystal Systems I _ -

Triclinic

(pkI

(I

+ 2 cos a cos p cos y

2kI +(cos p cos y bc

1

- COS' a -

COS*

p - COS*

y)

uc

Monoclinic, a = y = 90" Orthorhombic, a = p Tetragonal, u = b, a

=

=

Rhombohedral, u = b

y = 90"

p =y

= c,

=

90"

a =p

=

y

1 _ - (hz + k Z + I z ) sinz a + 2(hk uz(l

d%ki

Hexagonal, a = b , a = /3 = W, y = 120" Cubic, a = b = c, a

= /3

= y = 90"

1 _ - h*+kJ+P &kl

a*

1

21h - cos a) + -(COS y cos a - cos p)

+ kl + f h ) ( c o s Za - c o s a) + 2 c0s3 a - 3 c o s z a)

6.1. UNIT CELL

AND

CRYSTALLINITY

27

Y

I.

FZ

wave-

FIG. 15. Schematic illustration of scattering from a classical electron.

cause the coherently scattered waves can interact and produce the interference effects, which we refer to as diffraction. According to classical electromagnetic theory21 x rays can be represented by a transverse wave consisting of perpendicular electric and magnetic fields. The intensity of the x-ray beam is proportional to the square of the amplitude of the wave representing the electric field. When the electric field interacts with an electron a secondary wave is set up by the acceleration of the electron. This secondary wave is the wave “scattered” from the electron. Thornson22used classical scattering theory to show that the coherent intensity scattered by a single electron is given by

I,

=

I,

~

e4 (1 m2c4R2

+ c;?

29,

(6.1.11)

where e is the charge on electron, m the mass of the electron, c the velocity of light, R the distance between the electron and point of observation (counter), and 28 the scattering angle as illustrated in Fig. 15. Notice that Eq. (6. I . 11) holds as well for other charged particles as for electrons provided the correct charge and mass are used. Thus the intensity scattered by a proton is negligible compared to the intensity scattered from an electron due to the inverse square dependence on the mass of the particle; therefore we can neglect the presence of protons in treating the phenomenon of x-ray scattering by atoms. Likewise, the neutrons having no charge can be ignored. The trigonometric factor in parentheses in Eq. (6.1.11) is called the polarization factor and has the form given only if the incident beam is un21 A . H. Compton and S. K . Allison, “X-Rays in Theory and Experiment,” 2nd ed. Van Nostrand-Reinhold, Princeton, New Jersey, 1935. J . J . Thomson, “Conduction of Electricity Through Gases,” 2nd e d . , p. 321 (see also, Compton and Allison,z’ p. 117).

28

6.

X-RAY DIFFRACTION

polarized. This is the usual situation when the beam of x rays comes directly from the x-ray tube. However, if the beam is monochromated by diffraction from a crystal, it will be partially polarized when it strikes the sample. 6.1.3.2. Scattering by an Atom. An atom with atomic number Z contains Z electrons. If all of these Z electrons were located at the same point in space, the scattered wavelets from the Z electrons would be in phase with each other and the intensity scattered from the Z electrons would be equivalent to the intensity scattered by a point charge of mass Zm and charge Ze. For this case, Eq. (6.1.11) readily gives The wavelengths used in x-ray diffraction are of the same order of magnitude as the atomic diameters. Hence the waves scattered from electrons at different locations in the atom are not in phase. The net result is that the intensity scattered by an atom will generally be less than predicted by the above expression. In order to envision this, consider adding the waves scattered from two volume elements in the electron charge cloud of the atom as shown schematically in Fig. 16. Since the intensity is proportional to the square of the amplitude of the resultant wave, we need not write out the time-dependent form of the waves but only the amplitudes and phase relations. The complex amplitude of the resultant wave is (6.1.12)

where E, is the amplitude scattered by a single classical electron, p the electron density in the volume element dV, and 4 the difference in phase between the two waves. Let Soand S be unit vectors in the direction of the incident beam and scattering direction, respectively, A the wavelength of the radiation, and r a vector that gives the relative positions of the volume

FIG.16. Illustration of the phase difference between the wavelets scattered from different parts of an atom due to the spatial distribution of electrons.

6.1. U N I T

CELL AND CRYSTALLINITY

elements. The phase difference

4

29

4 is given by 2T A

2.rr

= - (path difference) = - (S - So) r.

A

(6.1.13)

The atomic scatteringfactorfis defined as the ratio of the amplitude coherently scattered from an atom to the amplitude scattered from a single classical electron. Thus (6.1.14)

Therefore, the intensity coherently scattered by an atom Z, is given by I, = f V e .

(6.1.15)

The vector (S - So)/A is called the diffraction vector. It is convenient to write (S - &)/A = s and this shortened notation is used henceforth. The magnitude of the diffraction vector is (6.1.16)

where 8 is half the angle between the incident beam and the scattering direction (see Figs. 15 and 16). Therefore,fis a function of (sin @/A, and it is generally tabulated as a function of this parameter. In order to determine f from Eq. (6.1.14), the electron density distribution function must be known. This is normally obtained through quantum-mechanical techniques (approximate or otherwise). A typical source of tabulated values is given in International Tableslo (Vol. 11, p. 201). Figure 17 gives scattering factors for several elements of importance to polymer crystallography. Equation (6.1.14) indicates that as the scattering angle approaches zero, f approaches Z as would be expected based on Eq. (6.1.11). In general, it is found that fdecreases rapidly as (sin @/A increases as shown in Fig. 17. As already mentioned, the quantum-mechanical treatment of scattering shows that there is also a contribution of the intensity scattered from an atom whose wavelength is modified; this is the Compton modified scattering. Since the wavelength of this radiation is slightly longer than the incident wavelength and varies with scattering angle, there is no definite phase relationship between waves scattered by different electrons and the intensities scattered from each electron simply add. The Compton scattering thus occurs as a weak diffuse scattering that gradually increases with scattering angle. It is occasionally necessary to compute this contribution in order to correct experimental data for the modified scattering.

6. X-RAY

30

o

0.2

0.4

DIFFRACTION

0.6

sine 7 * (k‘ ”1

0.8

1.0

FIG.17. Atomic scattering factors for fluorine, oxygen, nitrogen, carbon, and hydrogen. (Source: “International Tables for X-Ray Crystallography,” Vol. 111, p. 202.)

The interested reader should consult Compton and Allisonz1 or an advanced text such as that of Warren.z3 Before leaving the subject of the scattering from an atom, it should be pointed out that a more rigorous t ~ e a t m e n t ~shows ~ * ~that ~ * an ~ ~additional effect on the atomic scattering factor occurs if the wavelength of the radiation being scattered is near an absorption edge of the scattering atom. In this case, the corrected atomic scattering factor is given by

f=fo

+ Aft + i A f “ ,

(6.1.17)

wheref, is the uncorrected scattering factor found in tables, and Af’ and Af” are real and imaginary parts of the dispersion correction. These correction terms are also available for commonly used wavelengths (see International Tables,lo Vol. 111, p. 213). 6.1.3.3. Scattering by an Assemblage of Atoms. In the same manner that the waves scattered from different volume elements in an atom were summed, the waves scattered from different atoms in an assembly of N atoms may be summed to obtain the resultant complex amplitude A: N- 1

A

a, exp(2ris r,),

=

(6.1.18)

n=O

B. E. Warren, ”X-Ray Diffraction.” Addison-Wesley, Reading, Massachusetts, 1969. R . W. James, “The Optical Principles of the Diffraction of X-Rays.” Cornell Univ. Press, Ithaca, New York, 1965. p6 W. H. Zachariasen, “Theory of X-Ray Diffraction in Crystals.” Wiley, New York, 1945. Is

6.1.

UNIT CELL A N D CRYSTALLINITY

31

where u, is the amplitude scattered from the nth atom. This atom is located, relative to the zeroth atom, which is at the origin, at the tip of the position vector r,; see Fig. 18. Since the intensity is given by

I = IAI2 = AA*

(6.1.19) (6.1.20)

=

I,

xx m

exp[2mis * (r, - r,)].

fJn

(6.1.21)

n

Equation (6.1.21) is a general expression of the kinematical theory; it is applicable to solids, liquids, and gases. 6.1.3.4. Scattering from Identical Atoms Placed at the Points of a Simple Space Lattice. Let us use Eqs. (6.1.18) and (6.1.21) to calculate the intensity scattered from N identical atoms; each atom is located at a lattice point of a simple (noncentered) space lattice. We may consider the crystal as a small parallelpiped containing N 1 atoms (and unit cells) along the a lattice translation, N z along b, and N , along c. Relative to an origin at the corner of the crystal, the location of the nth atom is given by

r, = n,a

+ nzb + n3c,

(6.1.22)

where the n are integers whose values range from zero to N , - 1, N z - 1, N3 - 1, respectively. We may also write s = hla*

+ hzb* + h3c*,

(6.1.23)

where the base vectors are the reciprocal lattice vectors. The h can be thought of as coordinates, not necessarily integers, in reciprocal space that define the location of the tip of the vector. Substituting Eqs. (6.1.22)

w

0 -* S

FIG. 18. Scattering from an array of atoms.

32

6. X-RAY

DIFFRACTION

and (6.1.23) into 18 and simplifying, NI -1

A =a

2

Nz-1

exp(2rinlh,) a?-0

nl =O

N3-1

exp(2rin2h2)

exp(2rin3h3), (6.1.24)

)u=O

where N1, N 2 , and N 3 are the number of lattice points along the three respective directions. Each of the sums of Eq. (6.1.24) is the sum of a geometric progression, and hence each can be evaluated in closed form. Performing this sum and then multiplying A by its complex conjugate, we obtain

Equation (6.1.25) expresses the intensity scattered from the crystal in terms of coordinates h l , h 2 , h, in reciprocal space. This function is shown schematically in one dimension in Fig. 19. The peaks occurring at integral values of h, have values that are proportional to N,2 while their breadth is proportional to l/N1. Thus as N 1 ,the number of lattice points, becomes larger, the sharp peaks become taller and narrower. This is the basis for understanding small-particle broadening of x-ray diffraction peaks. For reasonably large values of N the intensity is appreciable only when hl , h2 , and h3 are approximately integral. This condition is equivalent to saying that the tip of the vector s = (S - &)/A touches a reciprocal lattice point. When this occurs, reinforcement of the waves scattered from the different atoms occurs in the direction of the unit vector S, and we say that a djffructed beam has been formed in this direction. According to this criterion, a diffracted beam occurs when h, = h, h2 = k , h3 = I , so that (6.1.26) Taking the absolute value of each of these vectors, substituting from Eq. (6.1.16) and recalling that d h k l = l/lrtkl1, gives 2 sin e l k

=

l/dhkl

or

A = 2dhkl sin 8

(Bragg’s law).

(6.1.27)

Thus, the condition embodied in Eq. (6.1.26) is equivalent to satisfying the well-known Bragg law for the (hkl) planes. Note, however, that in addition to satisfying Eq. (6.1.27) the diffraction condition, Eq. (6.1.26), requires that the incident and diffracted beams make equal angles with the so-called reflecting planes (hkl) and that the incident beam, the diffracted beam, and the normal to the (hkl) planes must be coplanar. The Laue equations may also be obtained from Eq. (6.1.26). Taking the scalar product of (S - So)/A with a, b, and c, respectively, yields

25

20

15 N

.fn \

5

E

10

N

E ._

m

5

0

h

h N = 100

8C N

0 1

g

6C

N

c ._ v) \

5

C c .-

40

N

In 20

b.L

1 h

FIG. 19. The function (sinz a N h ) / s i n z r h for (a) N

=

5 , (b) N = 20, (c) N

=

100.

6.

34

X-RAY DIFFRACTION

+ kb* + Ic*)

0. a = (ha*

A

or (S - So) a = hA,

(S

-

So) b = kA,

a,

-

(S - So) c = 1A.

(6.1.28)

Examination of the derivation of Eq. (6.1.26) makes it clear that the three Laue equations, (6.1.28), must be simultaneously satisfied for diffraction. The simplifications of the solution of the Laue equations (6.1.28) by the use of Bragg’s law [Eq. (6.1.27)] was a major milestone in the development of the mathematics of diffraction. 6.1.3.5. The Ewald Sphere. The conditions for diffraction embodied in Eq. (6.1.26) may be understood by the useful geometrical construction of the Ewald ~ p h e r e This . ~ sphere of rejection is illustrated in Fig. 20. The Ewald sphere is a sphere of radius l/A passing through the origin of reciprocal space. The incident beam direction is along a diameter of the Ewald sphere. The vector So/h in this direction is also a radius of the sphere. The vector (S - So)/A is a chord extending from the origin of reciprocal space to a point on the sphere as shown in Fig. 20. According to Eq. (6.1.26), this vector corresponds to a reciprocal lattice vector rZkl when the conditions for diffraction from the (hkl) planes are satisfied. In addition, this means that the conditions for diffraction are satisfied whenever a reciprocal lattice point touches the Ewald sphere; when this occurs, a diffracted beam is emitted in the direction S/A corresponding to the direction from the center of the sphere through the point on the sphere

4

I

4 D i f 1 ril c t e d

FIG.20. Sphere of reflection in a reciprocal lattice.

6.1.

U N I T CELL A N D CRYSTALLINITY

,520

J

35

DIFFRACTED

,’BEAM DIRECTION

FIG.21. Illustration of Ewald sphere construction for polyethylene.

touched by the reciprocal lattice point. The Ewald sphere, while perhaps seemingly complex at first consideration, has proved to be a valuable tool for interpreting geometrical relationships between the reciprocal lattice of a crystal and the locations of the diffraction spots on an x-ray pattern from the crystal. The use of the Ewald sphere and reciprocal lattice is usually one of the most difficult of the techniques to be learned by the crystallographer as well as one of the most important, particularly in polymer crystallography, in which the reciprocal space may consist not only of sharp spots but also diffuse streaks. A simple illustration of its use is shown in Fig. 21. 6.1.3.6. Scattering by Complex Crystal Structures. Suppose that in each unit cell of the space lattice there are Q atoms. These atoms are not necessarily identical, but the 4 t h atom has atomic scattering factor f,. The position of the 4th atom relative to the origin of the unit cell in which it lies is r,. Thus a vector from the origin of the space lattice to the 4th atom in unit cell n is

%,

=

r,

+ r,

= n,a

+ n2b + ngc + r,.

(6.1.29)

36

6. X-RAY DIFFRACTION

In this case the equation for the amplitude from the crystal, analogous to Eq. (6.1.24),is 4

A

=

C a, exp(2nis

r,)

x exp[2ni(h,nl

+ h2n2 + h3n3)].

(6.1.30)

After multiplying A by its complex conjugate, the intensity may be written in the form

where 4 Fhlh2h3

=

fq

exp(2Tis *

(6.1.32)

rq).

Q

Because the sine terms in Eq. (6.1.31)have appreciable values only when h, , h2, and h3 are integers, we generally need to evaluate Fh,hlh3 for the case h, = h, h2 = k , h3 = 1. Here hkl are integers and are the indices of the planes from which diffraction is occurring. We may determine F h k l more explicitly if we let r,

=

x,a

+ y,b + z,c,

(6.1.33)

where x,, y,, z, are coordinates of the 4th atom in the unit cell relative to the translation vectors a, b, c. Therefore, using Eq. (6.1.23), 4

f, exp[2ni(hxq +

Fhkl =

hy,

+ Iz,)].

(6.1.34)

P

In this form F h k l is known as the crystal structure factor or structure amplitude. This function can be expressed in a more convenient form for computation by incorporating into Eq. (6.1.34)the identity e2n*s= cos 2nx + i sin 2nx. The value of F h k l * F&.l = lFhk1l2 needed to compute the intensity of the hkl reflection is then given by lFhkl12

=

[< f, P

+

COS

[:

2n(hxq

fi sin 2n(hx,

ky, -I- 12,)

l2

+ ky, + Iz,)]

2

.

(6.1.35)

Notice that if the unit cell contains a center of symmetry, that is, if for every atom at x,, y,, z, there is an identical atom at -xq, -y,, - z q , the sine terms cancel in pairs and thus only the cosine terms need to be considered.

6.1.

37

U N I T CELL A N D CRYSTALLINITY

Also note that of the terms in Eq. (6.1.31) Fhkl alone contains information about atomic locations in the unit cell. Considering this fact together with the discussion following Eq. (6.1.23, we can see that the locations of diffracted beams in space, i.e., the spots on a diffraction pattern, are determined by the size and shape of the unit cell of the crystal’s space lattice while the intensity of each diffracted beam is affected by the location of the atoms in the unit cell. As a specific example to show the dependence of reflection intensity on the atomic locations in the unit cell, consider the relative intensities of various reflections from a crystal with a body centered lattice, e.g., poly-ortho-methylstyrene.2s In the case of centered lattices an atom in an arbitrary starting position is repeated by the lattice-centering translations. Thus we can write Eq. (6.1.34) as a product of two terms. The first term represents the sum over the atoms in the motif at a given lattice point, and the second term represents the centering translations required to fill the unit cell with the remaining atoms. For a body-centered lattice, we may take the origin 000 at the initial lattice site. The centered site is then at t 4 3 and we may write

[ 1 + exp [ 2 a i (i+ 4

+)I}.

(6.1.36)

The sum over m now runs over all the atoms that compose the motif at each lattice point. The second term represents the fact that there are two lattice sites in the unit cell. Notice that if h + k + I = 2n 1, where n is any integer, then Fhkl = 0 irrespective of the number or location of the atoms in the motif. This “extinction,” which produces zero intensity whenever h + k + I = 2n + 1, is caused by the relative positions of the atoms produced by the body-centering translations. Reflections of this type will not appear on the x-ray diffraction pattern, but if h + k + I = 2n then

+

which shows that the intensity of those reflections which are not extinquished due to the lattice translations are still functions of the relative positions of the atoms in the motif repeated at each lattice point. The fact that reflections of the type h + k + I = 2n + 1 are missing from the diffraction pattern is readily used to identify the existence of body-centering

*‘

P. Corradini and P. Ganis, Nuovo Cimento, Suppl. 15, 96 (1960).

6. X-RAY

38

DIFFRACTION

in a crystal's lattice. This idea can readily be extended to other centered lattice types; each has a specific set of extinctions that identifies it. As we shall see in a later section, systematic extinctions can be used to identify other symmetry operators in a crystal's space group. This can often ultimately lead to the identification of the space group itself. 6.1.3.6. The Effect of Thermal Motion. So far we have considered a crystal as a collection of atoms located at fixed points in each unit cell. Actually, the atoms undergo thermal vibration about their mean positions. At any instant a given atom may be displaced slightly from its mean position. The effect of thermal motion on the scattered intensity can be analyzed by including the displacement a,, in the equation defining the position of each atom and then averaging the resulting expression, analogous to Eq. (6.1.30), over time. The results of such an analysis2324 show that there are two basic effects of thermal motion.? First, the intensities of the Bragg diffraction peaks are reduced by a factor that depends on the diffraction angle and on the root mean square displacement of the atoms in a direction normal to the diffracting planes. The second effect is that the intensity lost from the Bragg peaks shows up as diffusely scattered radiation. The latter is of little significance in crystal structure determination and is not discussed further. The former effect must be taken into account. The analysis referred to above shows that this may be done by including a factor e-MQin the expression for the structure amplitude. Hence

Fhkl =

X f q c M exp[2ri(hxq q + kyq + k,)],

(6.1.38)

Q

where sin2 8 Mq = BqA'

(6.1.39) *

Here the coefficient Bq depends on the mean square displacement of the qth atom of the unit cell in a direction normal to the diffracting planes. It may therefore be affected by the symmetry and bonding in the unit cell and by the temperature. The net effect of Eqs. (6.1.38)and (6.1.39) is that the Bragg peaks are more greatly reduced in intensity as the temperature and scattering angle increase. For a more detailed discussion of the effects of thermal motion see Warren23and James.24 6.1.3.7. Description of the Crystal in Terms of Electron DensityFourier Series Representation. Although we have considered the crystal to be made up of discrete atoms, we could equally well consider it t See also Section 6.1.6, especially Section 6.1.6.2.

6.1. U N I T C E L L A N D CRYSTALLINITY

39

to be represented by a continuous electron density function. The electron density is a triply periodic function with period equal to the unit cell of the crystal. Continuous, periodic functions can be represented mathematically by Fourier series and we may write p(x, Y , 2 ) =

h

2 c C,,, k

l

where the sums run from - M to

exp [2ni ( h X a

+ a.

+ k by + /z)], c

(6.1.40)

Fourier coefficients are given by

The integration in Eq. (16.1.41) is over the volume of the unit cell. In terms of a continuous electron density function, the structure factor Fhkl can be written as

Equation (6.1.42) is readily understood by reference to Eqs. (6.1.14) and (6.1.34). Comparing Eqs. (6.1.41) and (6.1.42), we see that 1 chkl

=

Fhkl?

(6.1.43)

and

Thus the structure factors are the coefficients of a Fourier series for the electron density function. If we can evaluate the complex quantities F h k l , we can immediately construct an electron density distribution map using the above expression. It should be noted, however, that Eq. (6.1.31) relates the intensity of a diffraction peak to the modulus of the structure factor. The modulus of Fhkl is thus all that can be determined from an experimental measurement of the intensity, i.e., the phase of Fhkl cannot be directly evaluated by experiment. The evaluation of the phases of the Fhrl is thus one of the major problems in crystallography. 6.1.4. Experimental Techniques 6.1.4.1. Instrumentation. 6.1.4.1.1. X-RAY SOURCES. X rays are generated when electrons emitted from a filament at high temperature impinge on a metal anode and are decelerated by interaction with the atoms in this target material. With sufficient accelerating voltage these electrons will cause displacement of orbital electrons in the atoms of the

6.

40

X-RAY DIFFRACTION

anode metal resulting in production of both the characteristic wavelengths useful in diffraction as well as a continuum. This is an inefficient process and produces a large amount of thermal energy sufficient to melt the anode metal unless the anode is cooled. This, plus the fact that x rays cannot be focused by a lens assembly, places significant restrictions on the design of x-ray tubes and the intensity of x rays that can be directed to useful purposes in a particular diffraction instrument. The design of x-ray tubes has changed relatively little in the past 60 years (although fortunately major improvements have been made in generation and stabilization of high voltage).

high-voltage transformer 0

-

transformer

1

autotransformer

t

F:c.22. (a) Schematic illustration of X-ray tube design, and (b) illustrative wiring diagram for self-rectifying tube. (B. D. Cullity, "Elements of X-Ray Diffraction," 2nd ed.. Addison-Wesley, Reading, Mass., 1978.)

6.1.

U N I T CELL A N D CRYSTALLINITY

41

The basic design and wiring diagram of an x-ray tube is illustrated in Fig. 22. A helically wound filament of tungsten is enclosed in a vacuum envelope. A small voltage is applied to the filament, thereby heating it to emit thermal electrons, which are then accelerated to strike the anode by a high voltage between the filament and anode, typically 40 kV for a copper anode. Cooling of the anode is provided by circulation of liquid coolant (water or oil) through the interior of the anode. X rays are emitted in all directions from the planar surface of the anode but, for practical purposes, only a small portion exits through thin-walled windows of the tube to be used in a diffraction experiment. In the general-purpose x-ray tube, the filament has a helically wound conformation, with the result that electrons are emitted from a retangular area of the anode on the order of 2 x 10 mm. An important consideration of the filament geometry is that a “line source” of x rays exits through a window parallel to the long dimension of this rectangle and a “spot source” exits through a window perpendicular to this direction. Thus the line source is preferred for instruments employing slits and the spot source is used for pinhole systems. As a first approximation the power emitted is comparable for these two directions: the shape of the beam is different. Another aspect of the tube geometry, evident from Fig. 22, is that intensity approaches zero in a direction parallel to the plane of the anode and it is necessary to take the beam off at a small angle to this plane, on the order of 6“. An unfortunate feature of the tube design is that a large portion of the x-ray intensity produced must be wasted. Since focusing of the beam is not generally practical, collimation of the x-ray beam is provided by selection of a narrow geometrical band by a slit o r pinhole assembly. The usable intensity produced is generally much lower than would be desired for optimum use of the diffraction instruments. For this reason several types of x-ray tubes have been developed to increase the intensity of the useable x-ray beam. The principle restriction in the permanently sealed general-purpose tube is the limit on the number (tube current) of electrons striking the anode imposed by the melting point of the metal. One design whereby additional heat may be dissipated is through a rotating-anode device. The anode is a rapidly rotating cylinder, the surface of which,provides a constantly changing target. With this method of reducing the local electron beam heating a much higher tube current and usable x-ray intensity is possible. However, this system is much more expensive than a general-purpose sealed-tube unit and maintenance problems may be significant. These tubes are designed to be disassembled for simple replacement of components and require vacuum pumps. A different tube design for increasing usable intensity is the microfocus

42

6.

X-RAY DIFFRACTION

tube for pinhole use. The area of x-ray emission from the anode is greatly reduced by focusing electrons from the filament to a small anode area. With less total heat to dissipate due to the small area, beam intensity or “brightness” from this anode area may be increased. Units providing high intensity have been manufactured with easily disassembled tubes for simple replacement of filaments that burn out and anodes that tend to become pitted from overheating. For reasons of economy perhaps 90% of all polymer wide-angle x-ray diffraction investigation is performed with general-purpose sealed tubes. By far the most commonly used anode metal is copper. This is primarily due to copper’s high thermal conductivity (for heat disipation), the fact that the characteristic K a wavelength normally used is not highly absorbed by air paths, and that the K a wavelength lies sufficiently far from the absorption edges of the elements in most polymers to avoid appreciable fluorescent background emissions. Since the atomic numbers of most elements in polymers are low, this latter effect is normally not serious for any of the common anode materials. In general the wavelength chosen will also depend on th‘e number of reflections and their disposition desired on the x-ray pattern. According to Bragg’s law, short wavelength radiation will produce more reflections in the available 28 range (0 to 180°), but a given pair of reflections will be closer together and hence more difficult to resolve. Fortunately, copper K a also represents a good compromise in this regard. 6.1.4.1.2. X-RAY DETECTION SYSTEMS.The basic experimental arrangement for obtaining x-ray scattering data from a polymer specimen consists of the collimated x-ray beam, the specimen, and an x-ray detection device to record the scattered radiation. The two detection systems used are (1) photographic film and (2) electronic counters. Each plays an essential role in polymer crystallography and employs instrumentation of specific design. 6 .I .4. I .2. I . Photographic Recording. The oldest, simplest, and least expensive means of recording the scattered radiation is photographic film. It is still one of the techniques most used in polymer applications since a large amount of semiquantitative data is displayed at one time. No film is manufactured specifically for x-ray diffraction applications. The film most used is the type designed for medical use called “no-screen medical x-ray film.” This film is coated on both sides (for increased sensitivity) with an emulsion particularly sensitive to x-ray wavelengths. This type of x-ray film is manufactured for medical use today and film quality may be highly variable, a factor that must be considered in quantitative applications. For rapid recording of x-ray data (with a corresponding loss of resolution), an “instant” film is manufactured by the Polaroid Corpora-

6.1.

U N I T CELL A N D CRYSTALLINITY

43

tion, which must be used with a special plane film camera employing an intensifying screen. Such a system is very useful for sample alignment and rapid survey purposes. As discussed elsewhere in this review, the primary use of photographic methods is the identification of polymer species, the determination of phases present, the determination of lattice constants, semiquantitative estimation of crystallinity and orientation, and other uses that do not require precision intensity measurements. With care, photographic film may also be used for determination of diffraction intensity values. An approximately linear relationship exists between density and exposure. In a processed film, the density D is expressed as D =

loglO(Ii",ident/It,,n,,itte~),

(6.1.45)

where I is the intensity of the light beam in a photometer. In practice, many problems are found in measuring intensity accurately from flms including the coarse graininess of x-ray film, variation in quality of film, and background fog. However, it should be realized that for many purposes, even crystal structure determination, high accuracy of data is not always essential and photographic intensity data have been widely used. However, for structure refinement, that is, determination of the accurate positions of atoms in the unit cell, special photographic techniques or electronic counting methods must be used. A discussion of intensity measurements may be found in the International Tableslo (Vol. 111, p. 133). 6 . I .4.1.2.2. Electronic Counters. Two basic types of counting systems for soft (copper) x-rays are in general use: ionization detectors and solid-state detectors. Each type has particular advantages, but for general-purpose use the solid-state detector is usually preferred for reasons of economics. A discussion of several detector systems may be found in the International Tables.lo One form of solid-state detectors known as a scintillation counter is illustrated in Fig. 23a and consists of a fluorescent crystal and a photomultiplier tube. The scintillation crystal (usually a doped NaI crystal) is sealed in a cavity with metal foil (beryllium and aluminum) on one side to allow penetration of the scattered x-ray beam from the diffractometer while rejecting visible light. The back of the crystal is sealed with glass to allow the fluorescent light photons to enter the photomultiplier tube. An x-ray photon penetrates the metal foil window at X (Fig. 23a), and enters the scintillation crystal (SC), producing a pulse of visible light from excitation of the atoms of the crystal. This visible light pulse is subsequently transmitted to the photomultiplier tube to produce an electrical voltage pulse of sufficient magnitude to be processed by the associated electronic circuitry. Since the amplitude of the voltage pulse from the scintillation counter is

44

6.

C"",,',

X-RAY DIFFRACTION

1

X

(C)

FIG. 23. Schematic diagrams of (a) scintillation detector, (b) Geiger counter detector tube, (c) proportional counter detector tube. (Source: "International Tables for X-Ray Crystallography," Vol. 111, p. 145.)

directly related to the wavelength of the impinging x-ray photon, it is possible to feed this signal to a pulse height analyzer to provide a degree of monochromatization through selection of pulses of amplitude corresponding to the desired wavelength (e.g., copper K a radiation) as discussed in Section 6.1.4.1.3. Unfortunately, the energy or wavelength resolution of the scintillation counter is too poor to provide fully monochromatic radiation by this means. Thus, two aspects are important in deriving the value of the scattered intensity-the wavelength selection related to the amplitude of the voltage from the photomultiplier and the number of photons and corresponding number of voltage pulses. The scintillation counter has an advantage over ionization counters in longevity and reliability. It is to be expected that in the course of operation or alignment procedures, the highly intense main x-ray beam will impinge on the detector. The scintillation counter is resistant to damage from the

6.1.

U N I T CELL A N D CRYSTALLINITY

45

intense radiation, whereas ionization detectors may be permanently damaged due to decomposition of the essential quenching gases. The life of the scintillation counter is indefinite as long as the seal to the hydroscopic crystal remains intact and voltage requirements of the photomultiplier tube are not exceeded. Another advantage is the uniform and nearly 100% efficiency of response of the scintillation detector to a wide range of wavelengths commonly used for diffraction. A limitation of the scintillation counter has been the background noise level of the photomultiplier. With proper selection of these tubes, this is a minor problem in most polymer applications. The proportional counter inherently can be expected to give a slightly better signal to noise ratio if this aspect is critical. Another type of solid-state counter, the lithium-drifted silicon semiconductor detector, possesses outstanding wavelength resolution. For this reason it has found application in fluorescence analysis devices where it can be used to identify the wavelengths present in the radiation impinging on it without the need for dispersion by a crystal. Because of the need to operate these detectors at the temperature of liquid nitrogen they have not gained wide acceptance for diffraction measurement. The ionization detectors may be divided into three types: Geiger counter, proportional counter, and position-sensitive proportional counter. These are illustrated schematically in Fig. 23b,c. The original design for the ionization detector is the Geiger counter. The x-ray quantum enters a gas-filled tube through a thin wall. A high voltage is maintained between a tungsten anode (A) and the cathode tube wall (C). The photon excites the gas within the tube, usually argon, to cause a Townsend avalanche of electrons and a resulting finite electrical current between cathode and anode. While this avalanche occurs, the Geiger counter cannot react to a second photon. Therefore, the Geiger counter is highly limited to the number of photons that can be detected in a unit time, the limit of linearity being only several hundreds of counts per second. Since linearity is desirable to many thousands of counts per second, the Geiger counter is of limited application to diffractometry. However, it is the most sensitive of detectors for weak radiation and is inexpensive. Thus, it plays a critical role in the x-ray laboratory as a means for evaluation of radiation safety protection. There is no known “safe” level of x-radiation and both portable Geiger counters and fixed continuous operation monitors are important adjuncts to all x-ray procedures, whether counter or photographic. The limitation of linear counting rate may be raised and provision for electronic monochromatization may be provided by modification of the Geiger counter in the form of the proportional counter. This counter, which may be of the end- or side-window type (Fig. 23b,c), is operated at a lower voltage than the Geiger counter. A photon entering the counter

46

6. X-RAY DIFFRACTION

tube creates a much smaller avalanche of electrons and the “dead time” for detecting a subsequent electron is much shorter. Furthermore, the amount of current that flows from the cathode to the anode for each photon event is proportional to the energy (wavelength) of the photon. Thus, electronic monochromatization of pulse height analysis is possible and resolution is much better than for the scintillation counter. The counting rate can be increased to high levels (tens of thousands of counts per second) through the addition of a small amount of halogen gas to quench the avalanche. However, as mentioned previously, the effectiveness of this quenching gas is diminished.with exposure of the tube to the intense main x-ray beam. The proportional counter produces a much lower electric current than the Geiger counter and linear electronic amplification of high quality must be provided. The proportional counter has low background noise and is suitable for applications requiring very long counting times as in small-angle x-ray scattering. For wide-angle uses, the scintillation counter is generally preferred for reasons stated above. A modification of the proportional counter approaching commercial development deserves a brief description’. In the side-window proportional counter illustrated in Fig. 23c, the avalanche of electrons from an entering photon is confined to the region of the tungsten wire anode A in the vicinity of the initial ionization event. With the addition of specially designed ~ircuitry,~’ timed electrical pulses along the anode combined with the Townsend avalanches may be combined to yield the number of pulses detected at a particular position on the anode wire. With a window along the length of the anode and scanning circuitry, a plot of intensity vs. position may be obtained. Thus the need for scanning with a counter on a point-by-point basis with a goniometer is made unnecessary and the time for recording x-ray data is greatly reduced. This technique may be extended to use of wire anode grids providing scattering data in two dimensions comparable to a photographic fiber pattern. Present limitations of a lack of linear resolution and high cost are expected to be reduced and the position-sensitive proportional counter can be expected to gain increased utility in the polymer x-ray laboratory in the 1980s. 6.1.4.1.3. MONOCHROMATIZATION OF X-RAYBEAM.Most investigations on the structure and orientation of polymer specimens require a single x-ray wavelength, a monochromatic source. However, by virtue of the physics of x-ray generation, the main beam is polychromatic and consists of two types of radiation superimposed as illustrated in Fig. 24. These are (1) the continuous radiation for which the spectral distribution

’’

C. J . Borkowski and M. K . Kopp, Rev. Sci. Instrum. 39, 1515 (1968); IEEE Truns. Nucl. Sci. 17, 340 (1970).

6.1 UNIT CELL A N D CRYSTALLINITY

iu

47

LO‘

>

t ul

z w

f

White

radiation

T

T

-1-

2.0

1.6

0.4

0.8

WAVELENGTH ( A )

0.4

1.2

WAVELENGTH (A)

0.8

1.2

1.6

2.0

WAVELENGTH ( A )

FIG.24. Schematic illustration of the intensity distribution from a Cu target x-ray tube (a) unfiltered, (b) filtered through nickel foil, (c) filtered through a balanced cobalt filter.

is a function of the voltage accelerating the electrons to the tube anode, and (2) a characteristic radiation that is a sequence of monochromatic lines with wavelengths depending solely on the choice of anode metal (copper). It is necessary to devise a method of selecting only a single of the characteristic spectral peaks and eliminating or minimizing the other spectral peaks and the continuum. This necessity of monochromatization is particularly important for polymers, as will be discussed later. The simplest solution for partial monochromatization is to make use of the “absorption edge” effectlo (Vol. 111, pp. 73-79). For example, if a copper anode is used, a nickel foil placed in the beam path at some point between the source and the detecting system will selectively absorb the K P line reducing its intensity greatly with much less diminution of the K a doublet. The reduction in intensity of the K P relative to the K a line is a function of the thickness of the nickel film. A typical choice is a

48

6.

X-RAY DIFFRACTION

thickness of 0.015 mm nickel foil, which will give a 99.6% reduction in intensity of the KP line and a 50% reduction in the K a doublet. Specific details on absorption foil material for different anodes and their optimum thicknesses may be found in International Tablesio (Vol. 111, pp. 73-79). The most desirable location for the nickel filter is directly in front of the detecting system, since the filter will then serve to reduce the intensity of the radiation scattered by air in the beam path. This is easily accomplished in electronic recording systems where only the receiving slit need be covered by nickel foil. In a photographic system, the entire film surface must be covered by foil. This is feasible but awkward; usually the nickel foil is placed between the x-ray tube and pinhole collimating system. The advantage of the metal (nickel) foil filter is its simplicity and negligible cost. The principal disadvantage, particularly significant for polymer work, is the presence of the low-wavelength continuous (white) radiation in the filtered beam (see Fig. 24b). The elimination of the white radiation when using photographic films requires a monochromator, usually a single crystal with one face cut parallel to a major set of crystal planes. A suitable single wavelength, say copper Ka, is diffracted from these planes when the monochromator crystal is adjusted to the proper angle 8 in satisfaction of Bragg’s law. While the monochromator has the great advantage of producing “clean” x-ray photographs, the disadvantages of loss of beam intensity (longer exposure time), adjustment problems, polarization effects, etc., reduce its practicality. Consequently, crystal monochromated x-ray photographs ’are usually confined to special cases. A discussion of monochromator techniques may be found in the International Tableslo (Vol. 111, p. 79). In practice, the experimenter learns to recognize and to discount the effects of white radiation in x-ray photographs. The principal effect of the white spectrum is easily seen in a highly oriented fiber pattern such as Fig. 25. Note the streak along a locus between the beam impingement location (beam stop) and a spot of high intensity. This streak corresponds to the broad spectral white peak in Fig. 24b. In unoriented specimens this effect is less evident, although of equal significance. It is important to realize that the presence of white radiation grossly distorts the amorphous portion of the x-ray scattering pattern and any accurate method for estimation of crystallinity requires monochromatic radiation. Thus, electronic recording devices (scintillation counter) are generally used for analyses requiring resolution of crystalline and amorphous contributions in the data. A number of simple, practical approaches for obtaining effective monochromatization have been devised for electronic counting systems. These are generally available as accessories from the x-ray instrument manufacturers. Probably the most satisfactory device is the curved

6.1.

U N I T CELL A N D CRYSTALLINITY

49

FIG.25. Flat plate x-ray pattern from polypropylene fiber. Note radial streaking due to "white" radiation that passed the nickel filter.

crystal monochromator placed between the receiving slit and detector. The commercial devices are simple to align and have proved highly satisfactory. The monochromator can easily resolve (remove) the K P and white radiation from the K a and no filter foil is needed. However, the Ka,, K a 2 doublet cannot be resolved with this simple attachment; this is usually an insignificant disadvantage. A disadvantage of the monochromator is that, at a particular setting, say for K a detection, certain other, much shorter wavelengths in the continuous spectrum can also be reflected into the counter ( K a / 2 , K a / 3 , etc.). This problem is easily eliminated by modern pulse height discriminator circuits. While the use of a monochromator reduces the intensity of the beam reaching the counter, this loss is offset by the improvement in signal to noise ratio. Although the curved crystal monochromator is the most satisfactory method of obtaining monochromatic data from polymers, alternative devices are available and are satisfactory. The simplest and least expensive of these is the Ross balanced-filter technique. Two foil filters with absorption edges slightly above and slightly below the K a wavelength are used sequentially to record intensity at a fixed diffraction angle. With

50

6. X-RAY DIFFRACTION

proper selection of filter thickness, the two results can be subtracted to yield an effective monochromatic datum as indicated by the difference between Figs. 24b and 24c for the combination of nickel-cobalt filters for copper Ka. This method is mechanically cumbersome, slow, and has largely been replaced by the more convenient pulse height discrimination technique, which is much more practicable and nearly as effective. As noted earlier, a useful characteristic of proportional and scintillation counters is the proportionality between the energy of the x-ray quantum striking the detector and the amplitude of the electrical pulse in the output of the detector. Thus, there is an essentially linear relationship between the pulse height amplitude and the wavelength of the impinging radiation. Using a pulse height analyzer circuit, pulses corresponding to the K a quantum energy are transmitted to the recording device and pulses from other wavelengths are rejected. Unfortunately, the level of discrimination is limited by the resolution of the detectors. Either detector combined with a pulse height analyzer reduces the white radiation to an acceptable level; a foil filter is needed to reduce the K p contribution. As mentioned above, the addition of a monochromator at the detector gives a substantially better signal to noise ratio and improved monochromatization. This combination is advisable for precision measurements such as crystallinity determination and line profile analysis. If quantitative measurements are to be made with electronic counting circuitry, it is important that the intensity of the beam from the tube remain constant. A general-purpose sealed tube with a modern stabilized generator is satisfactory, but large variations in the temperature of the anode cooling water must be avoided since thermal expansion of the anode can change the position of the beam origin. Rotating-anode sources may be particularly troublesome in this respect and on-line monitoring of the emergent beam may be required. 6.1.4.1.4. X-RAYCAMERAS. For photographic recording of data, three basic camera types are of primary interest in polymer crystallography: plane film, cylindrical film, and the Debye-Schemer powder camera. Each type serves a special purpose. The most commonly used camera is the plane-film or flat-plate camera (often mistermed a Laue camera). It can be used for either oriented or unoriented polycrystalline samples. Flat-plate patterns from oriented fibers are often simply called “fiber patterns.” The great virtue of the flat-plate camera is simplicity, speed, and minimum cost. The basic elements are a pinhole system for collimating the x-ray beam, a mechanism for holding the specimen, which is placed in the collimated beam, and a plane film to record the exiting scattered beam; the main beam is intercepted by a beam stop. A wide variety of commercial types is available.

6.1.

UNIT CELL A N D CRYSTALLINITY

51

One particular type designed by StattonZ8and marketed by the W. H. Warhus Company is shown in Fig. 26a. This camera has the added advantage that it can also serve as a small-angle camera. The angular range recorded is determined by the distance of the film cassette from the sample, which is placed at the exit end of the collimator. Of particular importance t o polymer crystallography in selecting camera is the material used to prevent room light from exposing the film. Black paper is commonly used but since long exposures (up to several hours) may be required, the film cover must have minimum stopping power for x rays and must not be of nonuniform texture, which can cause a “texture” shadow on the film. Therefore, black paper of selected smoothness and quality must be used. Carbon-filled polyester film has also proved satisfactory. Alternatively, a camera of light-tight design can be selected, which requires no film cover, although the akwardness of camera loading in a darkroom must be considered. For a random polycrystalline (or powder) sample, diffraction from all of the crystals in the sample having ( h k l ) planes making the proper Bragg angle with the incident beam produces a cone of radiation with semiapex angle 28, which intersects the film of a flat plate camera along a circle called a Debye ring (see Fig. 27). Different circles represent diffraction from different ( h k l ) planes in the sample. The plane-film record can quickly reveal semiquantitative aspects such as the existence or absence of crystallinity as in Fig. 28 or molecular orientation as in Fig. 29. Note in Fig. 28 that two camera designs were used: one with a beam stop mechanically supported in the beam, the other a preferred design with a lead beam stop glued to the paper film cover. A beamstop can be made very simply by cutting a disk from a lead sheet with a cork borer, covering the circle with fluorescent material and locating the position for gluing in dim room light while holding the beam stop in the center of the main beam with long forceps. (Be sure to observe proper x-ray exposure protection for the hands, face, and body in any alignment operation!) Exumple. Figure 28a is a plane-film pattern of spherulitic (unoriented) polyoxymethylene recorded with nickel-filtered copper K a radiation (1.54 A) with a specimen-to-flm distance of 40 mm. The Bragg angle for a diffraction ring can be calculated from the equation

e

=

it tan-’(2~/2D).

W. 0. Statton. private communications

(6.1.46)

FIG.26. (a) Flat plate camera designed by Statton (courtesy of W. H. Warhus Co.). (b) Debye-Scherrer powder camera during exposure and open (courtesy of Philips Electronic Instruments, Inc.).

6.1.

UNIT CELL A N D CRYSTALLINITY

53

-FI L t4

DEBYE R I N G

Fic. 27. Diffraction from random polycrystalline (or powder) samples. (a)Condition that must hold for each crystal diffracting from the ( h k l )planes. (b) Resulting cone of diffracted rays produced from diffraction by random polycrystalline sample.

The d spacing for the diffracting planes is given by d=

A

2 sin[$ tan-'(2x/2D)] '

(6.1.47)

where A is the wavelength, 2x the diameter of the diffraction ring, and D the sample-to-film distance. The indices of the diffracting rings from this hexagonal unit cell are known and so from the measurement of the ring diameter 2x, the unit cell constants can be calculated using the appropriate equation from Table IV. The measurements from Fig. 28a are given in Table V. The lattice constants from these rough measurements compare favorably with the accepted values of u = 4.47 A, c = 17.2 A. Figure 29 shows the diffraction pattern of an oriented fiber of polyoxymethylene recorded in the same flat-plate camera. Because of the orientation of the unit cells, the continuous rings of the unoriented specimen (Fig. 28a) become arcs (or "spots" if the degree of orientation is high). These arcs are arrayed along layer lines that have the form of hyperbolas

54

6.

X-RAY DIFFRACTION

FIG.28. Flat plate patterns of (a) crystalline, unoriented polyoxymethylene and (b) noncrystalline polymethylmethacrylate.

6.1.

U N I T CELL A N D CRYSTALLINITY

55

FIG.29. Flat plate pattern of oriented polyoxymethylene fiber. Fiber axis vertical (Ni filtered, Cu radiation).

on a flat-plate film. As illustrated in Fig. 30, the zero layer is termed the equator. The locus of the center of the layer lines (which is parallel to the fiber axis) is called the meridian and spot positions along nonzero layer lines are in the quadrants. TABLEV n

Index

2x (rnrn)

d

U

100 105 110 115

34 55 68 89

3.86 2.60 2.23 1.89

4.46

Measurements from Fig. 28a.

c

17.6 4.46 17.8

56

6.

X-RAY DIFFRACTION

FLAT PLATE F I L M Brogg reflectton spot jappeors in four quodronfsl

FIBER A X I S

-C Y Ll NDRlC A t F I L M

L

FIG.30. Illustration of fiber pattern nomenclature and measurement of "2x" and "2y."

It is often preferable for geometrical reasons to record data using a cylindrical-fdm camera. The film is coaxial with the fiber specimen, and as illustrated in Figs. 30 and 31 the layer lines appear as straight lines. In the fiber texture, the crystals of the microcrystalline fiber tend to have a specific crystallographic direction, usually the chain direction in polymers, aligned with the fiber axis. Because of the cylindrical symmetry of the fiber, crystal directions perpendicular to the chain axis tend to be distributed at random about the fiber axis. Thus the diffraction pattern of a highly oriented fiber is equivalent to that from a single crystal rotated rapidly about the chain axis direction. This provides a basis for understanding why the reflections all lie on horizontal layer lines. The layer lines are caused by the, fact that all diffracted beams are confined to the generators of cones that are coaxial with the fiber axis as shown in Fig. 32. This restriction is imposed by the necessity that for diffraction, the Laue equations must hold for the crystals in the sample. Since the chain direction is assumed along the fiber axis in each crystal, the Laue equation corresponding to the crystallographic c axis (chain direction) is the same for all crystals and gives the equation for the generators of the cones along which all diffracted beams lie, i.e., Ih = c * (S - So).

(6.1.48)

6.1.

UNIT CELL A N D CRYSTALLINITY

57

FIG.31. Cylindrical film pattern of polyoxymethylene fiber. Fiber axis is vertical (Ni filtered, Cu radiation).

This reduces to lh

=

c cos y

(6.1.49)

because the incident beam is normal to the fiber axis and hence to c. The layer lines are numbered according to the value of 1 in Eq. (6.1.49). When 1 = 0, the cone reduces to a plane that intersects the film along the equator of the pattern. If the fiber axis corresponds to the crystallographic c axis, then all diffraction spots on the equator have indices of the type hkO. Those on the first layer above the equator have indices of type h k l , etc. From the geometry of Fig. 32 it is clear that cot y = ylrF,

(6.1.50)

where YF is the radius of the cylindrical film and y the height above the equator of the film at which a given layer line is located and is directly obtained from measurements on the film (Fig. 30). The repeat distance along the c axis is given by

(6.1.51)

58

6.

X-RAY DIFFRACTION

FIG.32. Formation of layer lines from cones of diffracted rays.

Indexing is relatively straightforward when the size and shape of the unit cell are known, but when they are not it can be a tedious trial and error task. Successful indexing in such a case provides the unit cell size and shape, i.e., the lattice constants.

Example. The pattern from a filament of polyoxymethylene shown in Fig. 31 shows several layer lines with weak reflections that were not observed on the flat plate pattern of Fig. 29 due to insufficient exposure time. The cylindrical camera has a diameter of 57.3 mm. The intense upper layer line has 1 = 5 at a 2y value of 29 mm. The calculated repeat distance is c = 17.1 A. A series of layer line spacings may be measured and averaged for a more accurate determination of the repeat distance. In cases where the spots are spread into arcs, the 2y measurement is made from the arc centers. Accuracy of lattice constants from the cylindrical film fiber pattern is limited; however, these data are very useful for indexing the reflections prior to making more accurate d-spacing measurements with another camera with greater precision such as with the Debye-Scherrer powder camera. Complete indexing of the fiber pattern requires the determination of the h and k values for each spot on the pattern. On a given layer line the h and k values are controlled by the other two Laue equations, which must be simultaneously satisfied for diffraction [see Eq. (6.1.28) and discussion pertaining thereto]. However, the most straightforward method of indexing is by use of the reciprocal lattice and Ewald sphere construction. The procedure for indexing is discussed more fully in Section 6.1.5.2.

6.1.

U N I T CELL A N D CRYSTALLINITY

59

The pattern of unoriented polyoxymethylene recorded with a cylindrical camera is shown in Fig. 33. This gives a comparison between cylindrical film pattern and the flat film photograph of Fig. 28a. By virtue of the cylindrical geometry, many more reflections may be measured. The most accurate measurements are made along the equator, the locus normal to the fiber/film axis. The measured 2x equatorial values may be converted into 28 values by the equation 28

=

180"(2~)/~~,

(6.1.52)

where D is the camera diameter. These 28 values, indexed with the use of a fiber pattern (Fig. 28 or 30), may be converted to d values for accurate determination of lattice constants. For convenience in obtaining accurate data from unoriented specimens, a standard Debye-Scherrer powder camera (Fig. 26b) should be used. The most widely used powder camera has a diameter of 114.6 mm and records the equatorial data on a band of 35 mm wide film. By special methods of mounting the film in the cylindrical powder camera such as placing the cut ends of the film around 90" 28 (the Straumanis mounting), systematic errors can be minimized. Cameras of this type are readily available commercially. Systematic errors such as that due to film shrinkage, sample offcentering, camera radius inaccuracies, and absorption must be eliminated

FIG.33. Cylindrical film pattern of unoriented polyoxymethylene.

60

6. X-RAY

DIFFRACTION

from the final results. Methods of extrapolation for obtaining the best values of lattice constants are described in the International Tableslo(Vol. 11, p. 225). In addition to the three common types of cameras discussed above, several other types are of occasional but important use. If it is necessary to index the diffraction spots from polymer whose unit cell is completely unknown, it is very helpful to have a specimen with some tendency toward preferred orientation in three dimensions. This provides information in addition to that which can be obtained from the cylindrically averaged fiber sample, i.e., information more nearly equivalent to that from a single crystal. Specimens with a tendency toward threedimensional orientation may sometimes be prepared by rolling a cylindrically symmetrical specimen (oriented filament) along its axial direction. Such samples are usually examined with a Weissenberg camera, a complex mechanical device in which the specimen is rotated about its axis in synchronization with an axial translation of the cylindrical film. The operation of this camera is time consuming but straightforward. Details on use of the Weissenberg camera may be found in the International Tableslo (Vol. 11, p. 185). In theory, another type of moving-film camera, the Buerger precession camera, may also be used but in practice, polymer patterns become “blurred” unless the preferred orientation is extremely high and this method is recommended for specimens that are true single crystals. 6.1.4.1.5. COUNTER DIFFRACTOMETERS. The counter diffractometer offers an alternative to cameras with photographic film for recording x-ray scattering data with an inherent advantage of recording data quantitatively and often with a much shorter exposure time. Although it might appear that the counter diffractometer should be the primary means for collection of x-ray data from polymers, in practice it is a complementary tool. The intensity scattered from polymers is inherently weak and collection of data on a point-by-point basis may be required, with the net time for data collection being as long as for photography. A greater limitation is the lack of resolution of weak peaks at an intensity useful in structural analysis. Aspects of fiber patterns such as the diffuse streaks in layer lines important to the understanding of lattice disorder are not easily collected with present counter systems. On the other hand, counter techniques are the primary methods for measurement of crystallinity and quantitative orientation functions, the latter by means of pole figure analysis. Since the time for data collection at a point is short and data must be compared quantitatively on a point-by-point basis, the x-ray generator must produce an x-ray beam of constant intensity, a feature unnec-

6.1.

U N I T CELL AND CRYSTALLINITY

61

essary in photographic recording. This requires highly stabilized excitation voltage and tube current. 6.1.4.1.5.1. Dijfractometer Geometry. The basic components of the diffractometer are a stabilized x-ray source, generally a sealed-beam tube with copper target, a means for holding and positioning the sample, a detection device (counter) with associated electronic circuitry, and a means for positioning the counter so as to receive the diffracted beams. Positioning of the sample and counter are carried out by a goniometer. The goniometer contains a mechanical arm supporting the detector, which can be moved to different positions for recording of the scattered radiation. The angle 28 between the transmitted x-ray beam and the counter position can be read out directly. Resolution is provided by a slit (or pinhole) system: a divergence slit at the beam exiting from the tube and a receiving slit before the detector. Additional slit systems such as Soller slits are often provided for better combinations of intensity and resolution of the beam. A discussion of some commonly used slit systems and their effects on resolution may be found in the International Tableslo (Vol. 11, pp. 220-225). A newer type of goniometer device approaching commercial distribution employs a fixed counter termed a position-sensitive proportional This type of device is similar to an ordinary film camera with the film replaced by the position-sensitive counter. In this device a collimated beam impinges on the sample and the scattered beam is recorded on the surface of a one- or two-dimensional counter device. By means of special timed circuit pulses, the level of intensity vs. position can be obtained. Resolution is provided by the design of the complex counter-circuitry combination. Diffractometers in commercial production that are in general use in polymer studies are of two types: the powder diffractometer and the single-crystal orienter. Both of these types are of general-purpose use for all crystalline materials and have limitations when applied to polymers. These limitations arise frequently from the low intensity of the x rays scattered by polymers. 6 .I .4.1.5.2. Powder Dijfractometer. The most widely used counter technique for obtaining maximum intensity of x-ray scattering by a specimen is a combination of slit systems with parafocusing geometry. This is illustrated in Fig. 34. The source of the x-ray beam is a so-called “line source” on the surface of the copper tube target. The primary x-ray beam is divergent, with the angular spread being limited by the divergence *@

C. J. Borkowski and M. K. Kopp, IEEE Trans. Nucl. Sci. 19, 161 (1972). R . W. Hendricks, J . Appl. Crystallogr. 11, I5 (1978).

62

6.

X-RAY DIFFRACTION

FIG.34. (a) Photograph of powder diffractometer and (b) optical principle of the powder diffractometer (courtesy of Philips Electronic Instruments, Inc.).

slit near the tube window. The divergent beam impinges on a large area of the sample (about 1 cm2). Since different portions of the beam impinge upon the sample at slightly different angles, the diffracted rays also travel along slightly different directions. The design is based on a focusing principle that causes the diffracted rays to converge to an approximate focus at the receiving slit placed before the counter. Perfect focusing occurs only if the sample surface conforms to the curvature of the so-called focusing circle. Normally, this curvature is not maintained and a flat sample is used instead. The defocusing caused by the flat sample is kept small by limiting the beam divergence to 1-4". As shown in Fig. 34, the receiving slit position corresponds to the Bragg angle 20. This geometry also lends itself readily to the use of a monochromator between the receiving slit and the detector. The parafocusing powder diffractometer has found wide use in many types of crystallographic investigations. A significant limitation in its use for polymers is the requirement that the sample be unoriented since even a small degree of preferred orientation in a polymer sample can greatly affect the recorded intensities. As can be seen from observation of the highly oriented fiber patterns, an unoriented sample will have a large degree of overlap of peaks from different layer lines. Identification of the individual reflections and resolution from each

6.1.

U N I T CELL A N D CRYSTALLINITY

63

other may present a formidable task in the diffractometer scan from an unoriented sample. One specialized use of the powder diffractometer may be of value to the analysis of fiber patterns. If many fibers are mounted in parallel and are placed in the sample holder so that the fiber axes lie parallel to the diffractometer axis, the resulting diffractometer pattern is an accurate record of the equatorial reflections. This simple use of the powder diffractometer may be useful in study of polymorphism. It must be noted that a random diffraction scan of the fiber specimen so mounted cannot be obtained by rotating the specimen in its plane! In principle, the powder diffractometer can be used in the transmission mode for many polymers due to the low absorption of the x-ray beam. In this case, the sample holder is modified for transmission with the plane of the sample rotated 90"to its position in the reflection geometry. In this position the plane of the sample bisects the angle between the incident beam and the diffracted beam. However, the benefits of parafocusing are not operative in the transmission mode. Resolution of diffracted peaks will be poor because of large peak widths. Improvement of resolution can be obtained with the use of narrow slits but the intensity of the beam will be decreased greatly. 6 .I .4. I S . 3 . The Four-Circle Diffractometer or Single-Crystal Orienter. The single-crystal orienter is a versatile type of goniometer designed for collecting intensity data from single crystals and is widely used for this purpose. Various devices are available commercially and employ an incident beam collimated by a pinhole system with a corresponding pinhole-receiving collimator placed before the detector. The sample is a small (1 mm or less) single crystal mounted on a goniometer capable of rotating the sample about three different axes. Thus any reflection plane may be brought into position for diffraction of the impinging beam into the detector. An illustration of this type of system is shown in Fig. 35. Unfortunately, single crystals of polymers large enough for use are not

FIG. 35. Principle of the four-circle diffractometer (single crystal orienter).

64

6.

X-RAY DIFFRACTION

available. However, this system has been used with success for measurement of pole figures in polymers. In this mode, the detector system is fixed at the Bragg angle to receive reflections from a single diffracting plane. By measuring the relative intensity diffracted by the sample as a function of its angular position on the goniometer, the relative orientation of the reflection planes within the sample can be evaluated. With a series of pole figures for different reflections, the relative orientation of unit cells within the sample can be determined. Although in principle the single-crystal orienter can be used to collect intensity data from fibers for use in structure determination, this application is difficult in practice. In the single-crystal orienter as designed, resolution is provided by the specimen. The small single-crystal specimen is completely enveloped by the impinging beam and only a small portion is scattered into the receiving pinhole. In polymers, the specimen must be much larger, on the order of the dimensions of the collimated beam (-2 mm) for there to be sufficient usable diffracted intensity. Thus, resolution of peaks is poor (large line breadth). Resolution may be improved through use of smaller pinholes but usually with unsatisfactory loss of diffraction intensity. Therefore, the use of the single-crystal orienter is confined primarily to pole figure analysis from the few most intense reflections from the sample. 6.1.4.1J . 4 . Automation. Despite the practical limitations in the application of counter techniques to polymer structure determination, continuing improvements in instrument design, x-ray sources, improved methods of sample preparation, etc., promise to make counter techniques gain wider use. The principal limitation, as mentioned, is the low intensity of the diffracted beam. This problem can be overcome to a large degree by collecting data with the counter fixed for an extended period of time at each point. While many hours may be required for collection of data, this method lends itself readily to automation and both powder diffractometers and single-crystal orienters used for polymer analysis should be provided with one of the numerous microprocessors now available for instrument control. An additional advantage to the use of microprocessors is that diffraction data are recorded in a form suitable for input to a computer. 6.1.4.2. Correction and Analysis of Intensity Data. 6.1.4.2.1. PRELIMINARY CONSIDERATIONS-BACKGROUND SUBTRACTION. The intensity data measured directly by counter diffractometer or microphotometer techniques generally contain several sources of error. In addition to the random errors associated with counting statistics, these include (1) extraneous scatter due to x-ray scattering from the air or other nonsample material in the beam path, (2) scattering due to unwanted wavelengths in

6.1.

UNIT CELL A N D CRYSTALLINITY

65

the incident beam (lack of monochromatic radiation), (3) fluorescent radiation emitted from the sample (usually not important unless an absorption edge for atoms in the sample occurs at slightly longer wavelength than the radiation used), (4)variation of intensity caused by absorption in the sample or changes in the number of atoms irradiated with scattering angle, and ( 5 ) other factors associated with specific techniques (these are considered below). Further, the interpretation of diffraction data is always based on the coherent radiation; thus, the Compton modified scattering must also be removed before the data are used for computation of structure factors or other interpretation. It is possible to calculate the Compton scattering and subtract it from the data provided the intensities are measured in absolute units. This requires tedious experimental and computational procedures and is usually not considered worthwhile in view of the problems of rigorously eliminating or correcting for the sources of extraneous scattering and fluorescence. In practice it is often satisfactory to assume that these contributions are all slowly varying functions of scattering angle, and that they contribute to a background scattering on which the coherent crystalline diffraction peaks are superimposed. With this assumption, these extraneous conditions can be eliminated by using a linear background interpolated under the diffraction peaks, which is then subtracted from the total scattering curve. This approach is quite satisfactory for highly crystalline samples with crystallite sizes large enough to produce sharp crystalline diffraction peaks. For semicrystalline polymers allowance must also be made for the scattering contribution from the amorphous fraction. While this can sometimes be treated as a part of the background scattering, difficulties arise when crystallinity is low, the crystallite sizes are so small that appreciable peak broadening occurs, or the sample is paracrystalline. In these latter cases it may be necessary to use the most rigorous methods of eliminating or correcting the measured scattering curve. For a more complete discussion of such corrections the reader is referred to Warren23 and James.24 6.1.4.2.2. EXPERIMENTAL EVALUATION OF STRUCTURE FACTORSTHEINTEGRATEDINTENSITY. It would appear that the modulus of the structure factor, IFhkl(,could be determined directly from E q . (6.1.31) and an experimental determination of the corrected intensity I at the Bragg peak. Actually, this is not readily accomplished because of the dependence of the peak intensity on the perfection of the crystal and on the details of the experimental techniques used to record the Bragg peak. It was pointed out previously that the height and breadth of the Bragg peaks depend on the crystal size. Most real crystals contain substructure or slight misorientations between different regions (mosaic structure), which

66

6.

X-RAY DIFFRACTION

also affects the peak breadth and height. The peak shape will also be affected by the presence of inhomogeneous elastic strain fields such as those that surround dislocations; these strains in effect change the local interplanar spacings. In polymers severe lattice displacements leading to paracrystallinity can occur. The tacit assumption is made in deriving Eq. (6.1.31) that the incident beam contains negligible divergence; actually this is seldom the case. Because of these considerations it is necessary to obtain experimental values of IFhkll from the integrated intensity rather than the intensity at the peak of the Bragg reflection. The integrated intensity is the total energy diffracted into the Bragg peak. It is independent of crystal shape, mosaic structure, and strain, and is proportional to the volume of the sample that is irradiated. Although the integrated intensity still depends on the details of the experimental technique, the dependence is predictable and has been worked out for the common techniques. Consider the case of a small crystal that is rotated about an axis perpendicular to a slightly divergent incident x-ray beam. The hkl reflecting planes lie parallel to the axis of rotation and are turned through the Bragg angle 8 for reflection at a fixed angular velocity O . During this rotation all parts of the crystal make the Bragg angle with each ray of the incident beam. A counter is set to receive any radiation scattered in a direction corresponding approximately to the precise Bragg angle. The total energy diffracted into the counter as the crystal is rotated is given by Ehkl

= $1 dt d A ,

(6.1.53)

where t is time, A the area of the receiving slit in front of the counter, and I the intensity given by Eq. (6.1.31). This expression can be integ ~ a t e dto ~ give ~,~~ (6.1.54) H e r e v is the volume of the crystal and V the unit cell volume. Since I, is given by Eq. (6.1.1 I), we can rewrite Eq. (6.1.54) as

The terms preceding lFhk1I2 in Eq. (6.1.55) consist of universal and experimental constants; they can be combined into a single proportionality constant K. The final 8 dependent factor following IFhkl12 is known as the Lorentz polarization factor for a single crystal in an unpolarized x-ray beam. This factor is often given the symbol L , .

6.1.

UNIT CELL A N D CRYSTALLINITY

67

Since E is an experimentally measurable quantity, values of IFhkll can be obtained from Eq. (6.1.55). Although Eq. (6.1.55) gives the relationship between ( F h k l I and Ehklin absolute units, it is generally not necessary to make measurements of Ehkl in absolute units. For most studies of crystal structure, relative integrated intensities for various hkl reflections are quite sufficient. Thus, it is usually unnecessary to evaluate the proportionality constant K. Equation (6.155)holds for a crystal small enough that its absorption of the incident and diffracted x-ray beams can be neglected. It is not always convenient to work with such small samples, consequently it is in general necessary to correct for the effect of absorption. This correction will depend on the size and shape of the crystal (or sample) and on its linear absorption coefficient for the wavelength being used. Since polymers generally contain rather low atomic number elements, absorption is not as great a problem for them as it is for higher atomic number materials. Absorption corrections for special cases that are often encountered in studies of polymers have been described by AlexandeP and in a variety of other sources (see International Tables,lo Vol. 11, pp. 291-315; Warrenz3;James24). One important result is that for an effectively infinite thickness sample, examined in a parafocusing powder diffractometer, the absorption correction is independent of the diffraction angle 8. For some diffraction techniques more than one set of crystallographically equivalent planes may contribute to the recorded integrated intensity. This is more common for techniques that use polycrystalline samples such as the powder technique and fiber techniques. Such techniques are quite important in studies of polymers because it is impossible to obtain single crystals of polymers of sufficient size to study by x-ray diffraction. Whenever more than one set of planes contribute to the recorded intensity, it is necessary to account for this fact when computing lF),kll values from measured integrated intensities. This is done by including a “multiplicity factor” in the expression for the integrated intensity. For the powder technique all planes having the same spacing diffract to the same cone of diffracted rays (see earlier discussion of powder technique) and hence contribute to the measured integrated intensity. For a random powder the multiplicity factor becomes a function only of the crystal symmetry. For a cubic material, for example, there are three sets of planes of the form {loo}, namely, (loo), (OlO), and (001). We may also 31 L. E. Alexander, “X-Ray Diffraction Methods in Polymer Science,” pp. 69-81. Wiley, New York. 1969.

6.

68

X-RAY DIFFRACTION

count the parallel planes C‘TOO), (OiO), and (001). - For (1 11) type planes, however, there are (1 I l ) , ( I l i ) , (IF), ( l i l ) , (1 1 I ) ( I 1 l), e l i ) , i.e., four sets of planes with identical spacings. Thus for a random powder the probability that the { 1 1 1) planes will be correctly oriented for reflection is 4/3 that of the { 100) planes, and the intensity of the 1 1 1 reflection will be 4/3 that of the 100 reflection, other things being equal. The multiplicity factor m for the 100 reflection is 6 and that of the I 1 1 reflection is 8. The multiplicity factor for the 100 reflection on a powder pattern of a tetragonal crystal is only 4. This is due to the fact that the (001) and (OOi) planes do not have the same spacing in the tetragonal system as the (loo), (TOO), (OlO), and (010) planes. Multiplicity factors that apply for different crystal systems and common techniques are available in the International Tableslo (Vol. I, p. 31). We may write, in general, that the relutive integrated intensity ELkl is given by an expression of the form Ehi

=

K’mlFhkl12A(0)Lp,

(6.1.56)

where K ’ is a proportionality constant, m the multiplicity factor, Fhkl the structure factor modified for the effects of thermal motion, A ( @ the absorption factor, L the Lorentz factor, and p the polarization factor., As was already discussed, all of the terms in Eq. (6.1.56)except Fhkldepend on the diffraction technique. The appropriate Lorentz factors for two of the common techniques are given in Table VI. Note that when absorption corrections are negligible, the square of the modulus of the structure factor for each peak (aside from a proportionality or normalization constant) is given by E;tkl divided by the product mLp.

6.1.5.Crystal Structure Determination 6.15 . 1 General Description. Classically, crystal structure determination proceeds in several steps. First, a diffraction pattern (or a series of diffraction patterns) is obtained using an appropriate technique. OrdinarTABLEVI. Lorentz and Polarization Factors

Technique Powder, filtered radiation Powder, crystal monochromated radiation (incident beam) Fiber pattern and rotating single crystal a

Polarization factop

i(i + cosz 28) 1 + COSz 28, COS2 28 I

+ cosz 28,

(as above)

0, is the diffraction angle from the monochromator crystal.

Lorentz factor (sin2 8 cos

8)-1

cos e)-, (sinz 28

-

e)-1’2

6.1.

U N I T CELL A N D CRYSTALLINITY

69

ily, these patterns will be obtained on film; their major purpose is to define the location in space of the diffracted beams. Single-crystal photographs such as those obtained from the rotating- and oscillating-crystal techniques, the Weissenberg technique, or the precession technique are most suitable for this purpose. The Bragg angle 8 can be determined for each diffraction spot and the d spacings computed from Bragg’s law. Interpretation proceeds to the indexing of the diffraction patterns and the determination of the size and shape of the crystal’s unit cell. This normally involves determining the coordinates of the diffracted spots in reciprocal space. Determination of the reciprocal lattice coordinates of each diffraction spot is equivalent to assigning indices to each reflection and determining the size and shape of the reciprocal lattice unit cell. The calculation of the size and shape of the crystal’s space lattice then follow directly from Eqs. (6.19) (or Table 111). The results obtained may at this point be considered tentative, approximate, and subject to change. This procedure can generally be carried through in an unequivocal way when using single crystals and the more powerful techniques such as the, Weissenberg and precession cameras, but this is not always possible for other techniques. Polymers present difficulties in this regard because the available single crystals are too small to use in such x-ray diffraction experiments. Consequently, it is necessary to use polycrystalline samples. As noted in Section 6.1.4.1.4,the samples may have a random distribution of crystal orientations or they may exhibit a high degree of preferred orientation. Diffraction patterns from randomly oriented crystals (powder patterns) are quite useful for many purposes, but they are very difficult to index if the crystal system has a symmetry lower than that of the hexagonal or tetragonal system. Since many polymer crystals have lower symmetry than this, it is usually necessary to use highly oriented samples, such as drawn fibers. Because of its importance in determining the size and shape of the unit cell, the procedure for indexing fiber patterns is described in detail below. Once the size and shape of the unit cell have been established, the density is measured and the number of chemical formula units per unit cell is calculated. The relationship needed is D

=

nM/AV,

(6.1.57)

where D is measured density, n the number of chemical formula units, M the molecular weight per formula unit, A Avagadro’s number, and V the volume of the unit cell. The diffraction patterns are then examined for systematic absences (extinctions). The goal is to establish as much about the symmetry of the crystal as possible; ultimately the crystal’s space group is desired. The

70

6.

X-RAY DIFFRACTION

procedures used to establish the space group are considered in more detail in a subsequent section. The task of determining the locations of the atoms in the unit cell begins at this point. The number of atoms of each type is known. If the space group is also known, then some information about possible locations of the various atoms in the cell can often be obtained by inspection of the equivalent positions of the space group. For example, consider the case of alum, KAI(S04)2.12H20,whose unit cell is cubic with four formula weights per unit cell. The space group of alum can be determined uniquely from the extinctions and is found to be Pa3. This space group contains 24 equivalent points in the general positions, and has special positions with a multiplicity of 8 (xxx, etc.) and two sets of special positions with a multiplicity of 4. These are 4(&)at 4 3 3; 3 0 0; 0 3 0 ; 0 0 3 and 4(a) at 000; 0 3 3 ; 3 0 3 ;3 3 0. Obviously the four potassium atoms and four aluminum atoms in each unit cell cannot occupy general positions whose multiplicity is 24. They must in fact occupy positions with a multiplicity of four; thus one group, the potassium atoms, occupies 4(a) while the other group the aluminum atoms, occupies 4(b). Furthermore, the eight sulfur atoms must occupy positions with a multiplicity of 8 and hence they have only one undetermined coordinate x. In this rather atypical case a great deal can be learned from the knowledge of the space group. Additional information such as atomic radii, bond distances, and bond angles are used to try to establish a trial structure model by locating all the atoms in the unit cell in a manner that is consistent with all of the known constraints. The modulus of the structure factor lFhkll is experimentally determined by making intensity measurements on each observable reflection. This can be done from film patterns, but is most accurately done using counter-diffractometer techniques. Details of this procedure were described in Section 6.1.4.2. Structure factors are calculated for the trial structure model and compared with the experimentally measured values. The calculation is readily carried out using computer techniques. Simplified formulations of the structure factor expression for each space group are presented in the International Tableslo (Vol. I, pp. 353-525). Improvements in the model can proceed by trial and error or by more sophisticated refinement techniques. In either case the agreement between the model and the actual crystal structure is judged by comparing the calculated and experimental values of IFhkl[. The agreement is generally expressed in terms of a reliability factor R given by

(6.I .58)

6.1. U N I T CELL A N D CRYSTALLINITY

71

The trial structure is considered to be partially correct for values of R less than 0.45.32 Such models are considered worthy of additional refinement. Models are usually assumed to be substantially correct if R is less than 0.2. 6.1 5 2 . Indexing Fiber Patterns. The determination of the size and shape of the unit cell of a polymer is often carried out by indexing either cylindrical or flat-plate patterns of highly oriented fibers. This procedure is based on the relationship between the position of the diffraction spots on the x-ray pattern and the reciprocal lattice of the crystal as embodied in the Ewald sphere construction previously described. As discussed in Section 6.1.4.1.4, the fiber pattern has reflections arranged on layer lines. The I index of each reflection is determined by the layer line on which the reflection lies and the c (chain repeat) crystallographic axis is readily determined from the layer line spacing and Eq. (6.1.51). The values of h and k for each reflection are best determined from the reciprocal lattice and Ewald sphere construction. Consider what happens in reciprocal space when a single crystal is rotated about its c axis. Reference to Fig. 36 shows that if the rotation axis is a major zone axis, e.g., the c axis, the reciprocal lattice for the crystal will consist of sheets or layers of lattice points lying in planes perpendicular to the rotation axis. As the crystal is rotated its reciprocal lattice rotates with it and several of the reciprocal lattice points will eventually touch the reflection sphere establishing the proper conditions for diffraction. Because of the nature and symmetry of the technique, it is more convenient to determine the cylindrical coordinates of reciprocal lattice points from the location of a given reflection on the x-ray pattern. The approROTAT ION A X I S EWALD

RECIPROCAL

X X BEAM

FIG. 36. Illustration showing that rotation of a crystal and hence its reciprocal lattice causes contact of reciprocal lattice points with the Ewald sphere, thus producing diffraction. 32

M . J . Buerger, "Contemporary Crystallography," p. 214. McGraw-Hill, New York,

1970.

72

6. X-RAY

DIFFRACTION ali on Ii rot or fiber axis

I FIG.37. Cylindrical coordinates of the point hkl in reciprocal space.

priate cylindrical coordinates are shown in Fig. 37. The coordinate {/A measures the height of the reciprocal lattice point above the zero level while ,$/A measures the radial distance to the reciprocal lattice point normal to the fiber axis or axis of rotati0n.i The angle I) between a reference direction such as the x-ray beam and the radius ,$ is indeterminate in a rotating crystal or fiber pattern, since the reciprocal lattice points always take on all possible values of I). This fact makes indexing of fiber patterns difficult and somewhat tenuous. It is clear from Fig. 36 that all reciprocal lattice points on the same layer have the same value of 5. Thus 5 is related to the c repeat distance and the 1index; it does not give any additional information about the h and k indices. The value of ,f alone contains additional information about h and k. It is easily shown that c = /A/.

h

*,... . . .

10

0

5th layer

.

,,~ 2 ' " ~

06

0

02

0.4

o6

RkSl

FIG.5 1. Comparison of observed intensities and cylindrically averaged intensities calculated for the models of LiquorP, Bunn and Holmes3, and Allegra ef U I . ~The vertical bars show the observed relative intensities of the reflections. [T. Tanaka, Y. Chatani, and H. Tadokoro, J . Polym. Sci., Polym. Phys. Ed. 12, 515-531 (19741.1

for the 8/5 helix of Liquori,so the 8 / 5 helix of Bunn and Holmes,s2 and the 8/3 helix of Allegra et ~ r l . ~In ' these calculations an isotropic thermal parameter B = 8 A2was used to modify the atomic scattering factors, and Bessel functions with In( 6 12 were included. The results of these calculations are shown in Fig. 5 1 . The vertical bars show the relative intensities of the observed reflections corrected for Lorentz and polarization factors. Based on these results, Tanaka et al. considered the 8/3 model ~ 'be the more likely, and this result was said to be born of Allegra et ~ r l . to out by more detailed structure factor calculations including intermolecular interference effects. The latter calculations were carried out using the following equation:

(6.1.75)

6.

96

X-RAY DIFFRACTION

rb 4I

. . i..

/

, .

+

@

i-

.. ,.

1

I

4

FIG. 52. Molecular arrangement of polyisobutylene in the unit cell of the space group P2,2,2,. [T. Tanaka, Y. Chatani, and H . Tadokoro, J . Polym. Sci., Polym. Phys. Ed. 12, 515-531 (1974).]

In applying Eq. (6.1.75) the sum over p runs over the two molecules in the PIB unit cell and the pth helical chain in the unit cell is displaced from the origin to the point (x,/a, y,/b, z , / c ) and is rotated about the chain axis by the angle 4,. Some of these parameters for the two PIB molecules can be fixed from the symmetry of the structure. Tanaka et a/.= note that the chain axes must coincide with twofold screw axes of the space group because the space group P2,2,2, has four equivalent general positions. Further, in order to satisfy the space group the two molecules must have the same conformation, but an antiparallel packing arrangement. These conditions restrict the locations of the molecules to ( x , / a , y d b , zl/c = 0.25, 0.00, z,/c), (b, = (b and (x2/a, y 2 / b , z2/c = 0.75, 0.50, - z , / c ) , (b2 = - (b as shown schematically in Fig. 52. Tanaka et al. evaluated the unknown parameters zl/c and (b by first noting that the intensity of reflections on the eighth layer line are almost independent of (b. The intensities of these reflections were thus calculated for a range of values of zl/c (from 0.00 to 0.125) corresponding to the pitch of one monomeric unit. The calculated values compared more favorably with the o b e r v e d experimental intensities when zl/c was either 0.05 o r 0.11. Calculations for all layer lines were then made with 4 varying from 0 to 180" while maintaining zl/c at 0.05 o r 0.11. This gave two models for which zl/c = 0.05 and (b = 30" o r zl/c = 0.11 and (b = 100". The respective reliability factors for these two models were 0.28 and 0.33. These values indicated reasonable trial structures had been determined, but further refinement was needed. Refinement of the structure was carried out in stages by least-squares methods and the usual crystal structure factor based on fractional atomic coordinates. It was necessary to remove the previously assumed restricbe maintained. This was tion that the exact 8/3 helix of Allegra et consistent with the crystal symmetry which only requires the twofold screw axis symmetry for the polymer chain. Because of the limited

6.1.

U N I T C E L L A N D CRYSTALLINITY

.r

c>

9 5"

-

55'

-

51"

97

-

*+ N

Y

3'

9

c-3

3"

' 130"

- 167

P

4

0

- 4 7

FIG.53. Molecular structure of polyisobutylene. Bond lengths, bond angles, and torsional angles are shown. [T. Tanaka, Y. Chatani, and H. Tadokor0.J. f o l y m . Sci., f o l y m . f h y s . Ed. 12, 515-531 (1974).]

amount and quality of the intensity data, it was deemed necessary to use a constrained least-squares analysis originally proposed by Arnott and Wonacott.s3 This amounted to fixing certain well-established bond lengths and angles at fixed values in order to reduce the total number of unknown variables. Application of these methods to the two trial structures showed that the first modelcould ultimately be refined to give a very satisfactory R value of 0.13, while the second model was substantially poorer. The latter model was thus discarded in favor of the former. The final molecular conformation achieved is shown in detail in Fig. 53; the crystal structure is shown in Fig. 54 and the atomic coordinates of the 16 carbon atoms in a crystallographic asymmetric unit are given in Table XIII. In the molecular structure of PIB the average carbon to carbon bond length is 1.54 A and the average C-CH,-C and C-CM,-C bond angles are 128 and 1lo", respectively. The latter angle is close to the tetrahedral angle of 109.5", but the former bond angle is opened consider-

83

S . Arnott and A. J . Wonacott, Polymer 7, 157 (1966).

98

6.

X-RAY DIFFRACTION

FIG.54. Crystal structure of polyisobutylene. [T. Tanaka, Y. Chantani, and H. Tadokoro, J . Polym. Sci., Polym. Phys. Ed. 12, 515-531 ( 1 9 7 4 ~ 1

ably, probably as a result of steric hindrance between adjacent methyl groups. The bond rotation angles along the chain consist alternately of nearly gauche ( - 47 to - 62") and nearly trans ( - 160 to - 167") conformations. 6.1.6 Disorder in Crystalline Polymers

In previous sections we alluded to the fact that polymeric materials may exhibit numerous imperfections in the ordered arrangement of their molecules, but we did not deal with the nature of these imperfections nor the effects they have on the diffraction pattern. We deal briefly with this subject in the following sections. 6.1.6.1. The Nature of Disorder in Polymers. It is common knowledge that some polymers do not crystallize, but remain amorphous under

6.1. TABLEXIII.

U N I T CELL A N D CRYSTALLINITY

99

Atomic Coordinates and Termal Parameters of PIB

~

0.3607 0.3441 0.3010 0.5727 0.1646 0.2164 0.04% 0.0283 0.3677 0.3166 0.5757 0.3550 0.2892 0.2884 0.0967 0.4554

0.0422 0.0242 0.1639 0.0256 0.0184 0.0259 -0.0926 0.1180 -0.0422 -0.0357 0.0008 -0.1652 0.0694 0.0378 0.1270 0.1538

0.0473 0.1293 0.0277 0.0248 0.1769 0.2561 0.1619 0.1577 0.2991 0.3814 0.2858 0.2779 0.4280 0.5076 0.4057 0.4153

5.5 6.5 8.7 7.4 6.0 9.4 9.3 7.4 4.2 8.1 6.9 6.1 6.4 6.4 7.9 7.1

all conditions. In general these polymers lack the stereoregularity along the polymer chain to allow any semblance of three-dimensional periodicity to exist. The order that exists in such a polymer, unlike that in crystalline materials, is short range and similar to that existing in other amorphous substances such as liquids. This order results from the fact that over short distances from any given origin the locations of the atoms or atomic groups in the structure are not totally independent of each other, but are affected by the nature of near-neighbor intramolecular bonding and the intermolecular packing. Polymers with sufficient molecular stereoregularity to allow crystallization may also be amorphous if given insufficient time to crystallize in the temperature range between their glass transition and melting point. Although the degree and nature of order that exists in amorphous polymers are of interest, we largely limit further discussion to samples exhibiting some degree of crystal-like struct ure . The measured density of a polymer sample that has been observed to “crystallize” always lies well below the theoretical density computed from the unit cell volume and the mass of its contents based on a crystal structure analysis. This difference in density is often much greater than is observed in the case of most low-molecular-weight crystalline substances, and it is commonly interpreted to mean that polymers are rarely, if ever, fully crystalline. We often speak of the “degree of crystallinity” as a numerical measure of the order existing in the sample. This concept of t h e “degree of crystallinity” is, in general, not well defined. A popular,

100

6.

X-RAY DIFFRACTION

plausible (but not unique) interpretation of the density defect is a twophase model consisting of reasonably perfect crystals embedded in an uniform amorphous matrix. On this basis the “degree of crystallinity” would best be defined as the weight fraction of crystalline phase in the sample. The problem arises because the structure of real polymers is not so simple as this model. The density and order of the amorphous phase may vary due to molecular orientation, and the crystals, which are extremely small, many themselves contain numerous defects. For some polymers the crystal lattice may be so highly distorted that a structure that may be considered intermediate between a true crystal and an amorphous structure is produced. Such structures have been called paracrystalline.s4 In some cases the morphology of a given sample may be better interpreted as a single paracrystalline phase than in terms of the twophase crystal-amorphous model. The problem of disorder in so-called semicrystalline polymers thus divides naturally into two parts. One part is concerned with the nature of the imperfections in the crystalline or paracrystalline regions, while the other part is concerned with the relative amount of such material in comparison to the amorphous phase. Let us now consider the nature of crystal distortions, with emphasis on the major disorders that may occur in polymer crystals. In general, the distortions that occur in crystal lattices are classified as distortions of the first kind or as distortions of the second kind. The basic difference between these two classes of distortions is that the long-range order of the crystal lattice is retained when only distortions of the first kind are present, but the long-range order is destroyed by distortions of the second kind. Both types of distortion introduce local displacements of atoms, groups of atoms, or molecules from their positions in an ideal lattice. For distortions of the first kind the magnitude of the displacement is independent of the location in the crystal and long-range order is preserved (Fig. 55b). The opposite is true of distortions of the second type; the displacements appear to increase with distance from any chosen origin. Eventually the displacements are greater than the lattice spacings and there is no correlation with the lattice over large distances (Fig. 55c). The most obvious example of a distortion of the first kind that exists in all crystals is thermal motion. The atoms or groups of atoms vibrate about their average positions so that, at any instant, the atoms do not form a perfectly periodic arrangement, but any given atom or group of atoms is never more than a fraction of a lattice spacing from its location in the ideal lattice. The long-range order of the average lattice is thus maintained (see Fig. 55b). Other distortions of the first kind are produced by R. Hosemann, Z . Phys. 128. 465 (1950); Polymer 3, 349 (1962).

6.1.

UNIT CELL A N D CRYSTALLINITY

101

a

C

d

FIG.55. Schematic illustration of (a) perfect (undistorted) lattice, (b) lattice distortions of the first kind, (c) lattice distortions of the second kind, and (d) amorphous structure (no lattice). Note that in (b) the average lattice is given by the points of intersection of the lines, while the actual location of the atomic groups is indicated by the filled circles. The large dashed circles surrounding each of the average lattice sites indicates that the atomic groups always lie within such a circle. This is not the case in ( c ) and (d). In (c) the displacements get larger with distance from some chosen origin (0).

such localized point defects as vacancies, interstitials, or solute atoms or molecules. I n a polymer the occasional incorporation of a second type of mer into the polymer chain could give rise to such effects, as could the incorporation of chain ends. Mixed crystals or solid solutions also exhibit distortions of the first kind. It is to material with distortions of the second kind (Fig. 5%) that the term paracrystalline is applied.84 Such distortions can be thought of as being generated when there is a definite preferred way of packing nearneighbor molecules in the structure, but tLis packing arrangement is imperfect. In the case of polymers there are a number of packing disorders that can be readily visualized. Figure 56 illustrates some of the disorders that can produce paracrystallinity in polymers. Because of the relatively weak bonding between chains and the flexibility of the chains, slight tilting or twisting of the chains relative to each other is possible, as illustrated in (a). In (b) relative rotations of the chains about their axes are illustrated. Such a defect would presumably occur more readily for molecules that tend to take on molecular conformations that have near cylindrical shape. The molecules may be parallel but neighbors are shifted slightly in an irregular manner along the chain axis as in (c) or their intermolecular spacings may vary while maintaining registry along the chain

102

6.

X-RAY DIFFRACTION

d

e

f

FIG.56. Several packing disorders which might occur in polymers. (a) chain tilting or twisting, (b) relative chain rotation (end view), (c)longitudinal chain displacement, (d) variation of intermolecular spacings (e) imperfect parallel chain arrangement, and (f) imperfect antiparallel chain arrangement.

direction as in (d). The latter state is sometimes referred to as smectic while a combination of (b), (c), and (d) in which the molecules are parallel but there is little or no further order is referred to as nematic. If the chains possess a directional character, then distortions might occur because of the lack of either perfectly parallel (e) or perfectly antiparallel (f) packing. Finally, in the case of molecules with helical conformations, the helix may be irrational and never exhibit a precise repeat along the chain (not shown). As already implied these various types of disorders may occur in combination with each other and with various disorders of the first kind. 6.1.6.2. The Effect of Distortions of the First Kind on the Diffraction Pattern. Distortions of the first kind can all be analyzed by methods analogous to those used in the analysis of thermal motion. Qualitatively similar results are obtained in every case. The intensities of the sharp crystalline Bragg peaks are reduced, without broadening, and this intensity is redistributed in reciprocal space as diffuse scattering. The reduction in the intensity of the Bragg peaks increases with distance from the origin in reciprocal space. The underlying reasons for the qualitative features described in the preceding paragraph can be understood from the following analysis, which follows closely that presented by other a ~ t h o r s ~ especially ~ - ~ ~ ~ ~ ~ ~ G ~ i n i e r . ~ Referring ' to Eqs. (6.1.21) and (6.1.31) we write for a crystal whose unit cells contain only distortions of the first kind

I(s)

=

I,

2m 2n F ,

exp[2ris

(rm -

rJ],

(6.1.76)

6.1.

U N I T CELL A N D CRYSTALLINITY

103

where F , is the structure factor for the mrh unit cell in the crystal. Since we assume that the crystal is distorted, the structure factor varies from one unit cell to the next and cannot be removed from the double sum as in deriving Eq. (6.1.31). Note that the sums run over all the unit cells in the crystal and the vectors r, and r, locate the points of the average lattice. Consider the nature of the Fm in Eq. (6.1.76). We can write 4

Fm(s) =

f,m

exp[hi(s

Rqm)],

(6.1.77)

Q

wheref,, is the atomic scattering factor for the qth atom in the mth unit cell and Rqm is the vector from the mth lattice point to the qth atom in the mth unit cell. It is clear that two types of disorder of the first kind can be distinguished. In one case the qth atom in every unit cell is the same, but its location is displaced by a vector 6, from its position in the ideal crystal, so that =

%m

Rq +

Sqm

(6.1.78)

and (6.1.79) m

i.e., the averaged lattice constant is unaffected. In the second type of disorder the 4th atom in one unit cell is not identical to the qth atom in another cell, but the atom positions are not displaced from their location in the ideal crystal. This is referred to as substitutional disorder. In like manner the averaged unit cell constant is unaffected. In order to derive an expression for the intensity lost from the Bragg peaks and distributed into the diffuse background, let us now return to Eq. (6.1.76)and group the terms in the double sum into pairs separated by the same vector

rm

-

r,,

=

r,.

(6.1.80)

We obtain

where the overbar indicates that the average over many unit cells is to be as R . Hosemann and S. N . Bagchi, “Direct Analysis of Diffraction by Matter,” pp. 239-246 and 654ff. North-Holland Publ., Amsterdam, 1962. 88 A. J . C. Wilson, “X-Ray Optics.” Methuen, London, 1949. 67 A . Guinier, “X-Ray Diffraction in Crystals, Imperfect Crystals, and Amorphous Bodies,” pp. 154ff. Freeman, San Francisco, California, 1963.

6. X-RAY

104

DIFFRACTION

taken. The sum over m represents a sum over all the lattice points for which the point m + p also lies in the crystal. This is true because F;,, will be zero and, consequently, so will Fm F;,, when one node lies outside the crystal. The number of terms in the sum over m in Eq. (6.1.81) is thus somewhat less than the number of unit cells N, and is smaller the greater the value of p. Let us set Fm=F+Pm,

(6.1.82)

whereF is the average structure factor for a unit cell in the disordered lattice, i.e., (6.1.83) and

Pm represents the deviation from that

average value. Then

(F + P m ) (F* + ~ k + p ) _ _ = FE* + FPk+p + F*Pm + P m P ; + p .

FmFk+p =

-

(6.1.84)

~

(6.1.85)

Clearly,

Substituting this back into Eq. (1.6.81) and rewriting, we find ~(s= ) I, C,

(c,PI') exp(2ris

r,)

~m

(6.1.88) The first term in Eq. (6.1.88) is identically equal to the intensity diffracted from an undistorted crystal in which the structure factor for each unit cell is equal to the average structure factor of the unit cells of the distorted crystal. The second sum represents a diffuse scattering since the parameters +p are assumed to rapidly drop to zero as p increases. This is equivalent to saying that the correlations between the disorder in one cell and that in another cell are only short range. It is convenient to divide out the term for which p = 0 in the second part of Eq. (6.1.88). Note that

- -

PI2+ 40, 40 = (F,12 - PI'.

FmFk = IFmI2=

(6.1.89)

6.1.

U N I T CELL A N D CRYSTALLINITY

105

Finally, for a crystal containing disorder of the first kind,

-

2 4,

+ I,

I ( S ) = IeN(JFm12-

exp(2.rris rP)

p= 1

xx

+ 1~/3l~ exp[z.rris . (r, m

- rn)].

(6.1.90)

n

Hence, the first term is clearly a slowly varying function in reciprocal space and is a component of diffuse scattering, which is present in all disordered samples. The second term also represents diffuse scatter and depends on the correlations between these distortions for unit cells separatedby the vector - _r, . If there are no correlations, even for neighboring cells pmp;T(+p = pmpm+P = 0 by Eq. (6.1.86). This means that the 4pare zero and the second term in Eq. (6.1.90)then contributes nothing to the intensity distribution. The final term represents, again, the sharp crystalline diffraction peaks. As a simple specific example of the application of Eq. (6.1.90),consider thermal motion. For simplicity we consider the case for a crystal containing only one atom per unit cell. Then the structure factor of the mth unit cell is given by (6.1.91)

F m = fexp[2.rri(s Sm)].

We must average this value over time in the case of thermal motion. Expanding the exponential in a series and taking the average exp[2ri(s

S,)]

=

1

+ 2.rri(s - 6,)

-

+ --

2d(s*

*

. (6.1.92)

Here the sum of the averages is equal to the average of the sum. The second term on the right is zero because the vector 6, has an average value of zero. For small displacements higher-order terms than the quadratic term can be neglected and exp[2.rri(s S,)] = 1 - 27?(s

= 1

- 27r2 s2 pm,(6.1.93)

where urnis the projection of 6, on s. The structure factor can be written

-

Fm = f ( l

-

-

2.rr2s2~,2)= fexp(-MI,

=

8 ~ ~ u sin2(8)/A2. ,*

(6.1.94)

where

M

-

(6.1.95)

The factor c Mis known as the Debye LWaller temperature factor, which is frequently expressed as exp[-B sin2(8)/A2],where B is an empirical factor representing the degree of disorder of the first kind. 6.1.6.3. Distortions of the Second Kind-Paracrystallinity. As noted previously, paracrystalline distortions destroy the long-range order

106

6.

X-RAY DIFFRACTION

of the crystal lattice. These distortions therefore cause very marked effects on the x-ray diffraction pattern. A fundamental difference from disorder of the first kind is that disorder of the second kind causes broadening of the Bragg reflections in addition to attenuation of peak intensity with increasing angle. A somewhat different statistical approach must be taken in order to analyze distortions of the second kind, since there is no longer an average lattice. The analysis of these effects is due largely to Hosemann and c o - w ~ r k e r s . ~ ~ ~ ~ ~ ~ Let us write the intensity scattered from the paracrystal as follows:

Z(s)

=

ZeN(F- p)+ ZeNiq2Z(S).

(6.1.96)

The first term on the right of this equation represents diffuse scattering and it is present for the same reason that it appears in Eq. (6.1.90), namely, every unit cell of the paracrystal has a different structure factor because of the relative atomic displacements. We can use an average structure factor provided we include this term. The second term on the right then represents the intensity distribution in reciprocal space for a paracrystal in which each cell has the structure factor corresponding to this average. Our main concern here is the nature of the function Z ( s ) , which may be called the paracrystalline lattice factor. Z ( s ) plays the same role for the paracrystal that the Laue interference function plays for an ideal crystal; compare Eq. (6.1.96) to Eqs. (6.1.25) and (6.1.31). In order to evaluate Z ( s ) it is necessary to have a statistical description of the distribution of the points of the paracrystalline lattice. If we let z(r) represent the distribution function for the paracrystalline lattice points, then z(r) dV is the probability of finding a lattice point within the volume element dV at the end of the vector r. The problem of evaluating z(r) for the general three-dimensional paracrystalline lattice has never been fully solved, although Hosemannse has treated a special three-dimensional case, which he calls the “ideal paracrystal.” In this treatment he assumes independent distribution functions for nearest neighbors in each of three directions. The solution is thus essentially a superposition of three one-dimensional cases. The main features of the treatment of the effect of the paracrystalline distortions on the diffraction pattern can be understood by considering a simple one-dimensional case. The discussion presented here draws heavily upon the treatment given by G~inier.~~ Let the lattice points in the x direction be located at positions 0, 1, R. Hosemann and S. N . Bagchi, “Direct Analysis of Diffraction by Matter,” pp. 302ff. North-Holland Publ., Amsterdam, 1962. A . Guinier, ”X-Ray Diffraction in Crystals, Imperfect Crystals, and Amorphous Bodies,” pp. 295ff. Freeman, San Francisco, California, 1963.

’@

6.1.

U N I T CELL A N D CRYSTALLINITY

107

FIG.57. One-dimensional paracrystalline lattice and the distribution function h , ( x ) for the distances between nearest neighbors.

2, . . . , n , as illustrated in Fig. 57. The distance between any two nearest neighbors is assumed to vary about some average value a according to a distribution function h,(x). This function may, but does not necessarily, have a Gaussian shape. The probability that the distance x1 between any two nearest neighbors lies between x and x + dx is h,(x) dx. The overall distribution function z(x) for the entire one-dimensional paracrystalline lattice is composed of a sum of terms as follows: m

Z(X) =

S(X)

+C n= 1

hn(x) +

m

2 hn(-x).

(6.1.97)

n=1

The sums run over all possible neighbors of a given point taken as origin. Thus h,(x) represents the distribution function for second nearest neighbors, h 3 ( x ) the distribution function for third nearest neighbors, etc. The term S(x) is a Dirac delta function and represents the unit probability that a lattice point exists at x = 0 if we choose our origin at a lattice point. We can evaluate the h n ( x ) from h l ( x ) as follows. Consider the case for second nearest neighbors, h2(x). The distance x2 between second nearest neighbors is the sum of two adjacent first-neighbor distances, x1 and x;. If we let x1 = y and xi = x - y (see Fig. 57), then the probability that x1 = y is hl( y ) and the probability that xi = x - y is h,(x - y). The probability that both these events occur simultaneously is the product of their individual probabilities. Since we are interested in the probability of a certain x whatever the value of y, we must integrate over all possible values of y. The total probability is thus

M x )=

lom

hl(y)hl(x - y) d y .

(6.1.98)

This integral is the convolution of h l ( x ) by itself and may be denoted h,(x)*h,(x). By a similar argument h3(X)

= h,(x)*h,(x) = h,(x)*h,(x)*h,(x),

(6.1.99)

6.

I08

X-RAY DIFFRACTION

1

0

6.00

o=5A

A = 0.2

I

- 400 Y

N

2 00

0 00 000

2000

4 0 0 0 6000

8000

I0000

X

1,

z.::lb7, 1.20

0 = 5 i A = 0.7

N

0.00 0.00

20.00 40.00 60.00 80.00 10000 X

080

0 = 5 i

-

A.10

X

060 -

040020 -

FIG.58. Distribution function z(x) for one-dimensional paracrystalline lattice computed fromEq. (6.1.97)andaGaussianh,(x)givenby Eq. (6.1.105)forthecasethatthemeannearest neighbor distance a = 5A for three different values of the parameter A.

h,(x) = h,(x)*h,(s)* . . . * h , ( x ) . \

n times

'

(6.1.100)

J

The overall distribution function z(x) for the lattice points as given by substitution of Eq. (6.1.100) into Eq. (6.1.97) is illustrated schematically for an arbitrary h,(x) in Fig. 58.

6.1.

U N I T CELL A N D CRYSTALLINITY

I09

The paracrystalline lattice function Z ( s ) is the Fourier transform of z(x). This transform can be evaluated using the well-known relation that the transform of a convolution product is the product of the transforms of the terms in the convolution. Applying this relation gives a

Z ( s ) = transform z(x) = 1

+2

[ H ( s ) l m+

m=l

[H*(s)lm, (6.1.101) m=l

where H ( s ) is the Fourier transform of h,(x), i.e.,

H ( s ) = J h , ( x ) exp(2risx) dx.

(6.1.102)

If we let H ( s ) = IH(s))exp(i+), then m

Z(s) = 1

+2 2

IH(s)lmcos m 4 ,

(6.1.103)

m=1

which can be rewrittense as (6.1.104) Consider the evaluation of Z(s) for the special case that h,(x) is Gaussian of width ( 2 ~ ) lA, / ~i.e., (6.1.105) where

Jh1(x) dx/h(a) = ( 2 ~ ) A. "~

(6.1.106)

Now H ( s ) = exp(2risa)Jh(x) exp[2ris(x - a)] dx =

[

1 1 e x p ( 2 r i s a ) ~- exp - 5 (x A

(6.1.107)

a)2]

exp[2ris(x - a)] dx (6.1.108) =

exp( - 2bAs2) exp(2risa).

(6.1.109)

Note that IH(s)l = exp(-2r2A2s2),

(6.1.110)

4 = 2rsa.

(6.1.111)

Substituting these quantities into Eq. (6.1.104) gives the function Z(s). This function contains a series of maxima, as shown schematically in Fig. 59, which occur near the integral values of h = sa. But as s or h increases, the peaks become broader and broader. The integral breadth p

110

6. X-RAY

DIFFRACTION

I

40.00

30.00

N 20.00

10.00

L

0.00 0.00

0.20

0.40

0.60

6.00 -

-

0.80

1.00

1.20

1.00

1.20

a=5A

A.0.7

I 4.00 -

v

N

2.00

-

0.0 0 0.00

020

0.40

0.60

0.80

S FIG.59. The paracrystalline lattice functions, Z(s), for the one-dimensional paracrystalline lattices whose distribution functions, z(x), were shown in Fig. 58. Compare to the Laue interference function (Fig. 19) for a finite perfect crystal.

of the hth-order diffraction maxima is given approximately bys51seJ0 1

P = - [2a I 'O

- exp(-2~2gZhZ)],

(6.1.112)

M. Kakudo and N . Kasai, "X-Ray Diffraction by Polymers," Kodansha Ltd., Tokyo,

1972.

6.1. U N I T CELL AND CRYSTALLINITY

111

or for small g, /3

1 a

- .rr2g2h2,

(6.1.1 13)

where g = A / u is a relative measure of the magnitude of the paracrystalline distortions. In three-dimensional paracrystals, the nature of the distortions can be considerably more complicated than those illustrated by the above onedimensional model. In general, there exists a matrix of A or g values corresponding to the crystallographic anisotropy. The basic effects on the diffraction pattern are, nevertheless, similar in that increased broadening of the diffraction peaks with increasing values of A and s are expected. The values of g for several polymers have been discussed by Hosemann." The results are of order g = 0.02. The morphological view of semicrystalline polymers espoused by Hosemann (74) is that polymers consist of microparacrystalline domains connected by a threedimensional network of tie molecules. 6.1.6.4. General Equations for the Intensity Scattered from Disordered Crystals. In the preceding discussions the effects of crystal size were largely ignored. It was assumed that the crystals or paracrystals were large enough so that their size had little effect on the diffracted intensity. We include here a brief description of crystal size effects for the sake of generalizing the results given. A detailed discussion of crystal size effects and the measurement of crystal sizes in polymer samples is treated in Chapter 6.2. The intensity scattered from any assemblage of atoms is equal to the square of the modulus of the resultant amplitude scattered from the group of atoms. In terms of the general theory of diffraction this amplitude can be written as the Fourier transform of the electron density:

In Eq. (6.1.114) there is a contribution to A ( s ) only when p(r) is nonzero, i.e., within the finite boundaries of the scattering object. In order to isolate the effects due to crystal size, it is convenient to define a shape function d r ) such that d r ) = 1 everywhere within the scattering object and is zero for all r outside the object. The actual electron density function for the object can thus be written as the product p(r) = p d r )

*

dr),

(6.1.115)

where p,(r) is the electron density for an infinite size object having other'1

R. Hosemann, Mukromol.

Chrm., Suppl. 1, 559 (1975).

6.

I12

X-RAY DIFFRACTION

wise the same structure as the actual finite object. Thus A ( S )=

J 0

-

p,(r)a(r) exp(- 2ris r) d ~. ,

(6.1.116)

The Fourier transform of a product of two functions is the convolution of the transform of each of the two functions. If we let S(s) =

and A,(S)

=

lom lom

(6.1.11 7)

a(r) exp( - 2ris * r) dV,, p,(r) exp(-2ris

r) d ~ , ,

(6.1.118)

then A ( s ) = A,(s)

* S(s).

(6.1.119)

A further elaboration of this a p p r o a ~ h ~results ~ , ' ~ in the following expressions for the scattered intensity. For crystals containing only distortions of the first kind

Z(S) = ZeN(F- D21FI2)+ 7 Ie (FpD2L2 * lS(s)I2, (6.1.120) where N is the number of unit cells, D is called the distortion factor, V is the volume of the unit cell, F the structure factor, L the Laue interference function for an infinite crystal, and S ( s ) the Fourier transform of the shape factor. For the case that the distortions are due only to thermal motion, then D becomes the Debye-Waller factor D = exp ( - sin2 B ~e ) .

(6.1.95)

The equivalent exprqssion for paracrystals containing distortions of both the first and second kinds is

z -

Z(S) = ZeN(F- PlF'12)+ $ IFI2PZ(s) *

lS(s)I2.

(6.1.121)

Here Z(s) is the paracrystalline lattice factor. The effect of finite crystal size is to broaden all the Bragg reflections of crystals by the same amount as measured in reciprocal space (i.e., as a function of s). According to Eq. (6.1.121) a similar broadening is superimposed on the already broadened and attenuated scattering from a paracrystal. 6.1.6.5. Disorder in Polymers Analyzed in Terms of Helical Transforms. Long-chain molecules by their linear nature and highly aniso-

'* M. Kakudo and N . Kasai, "X-Ray Diffraction by Polymers," pp. 134ff. Kodansha Ltd., Tokyo, 1972.

6.1.

U N I T CELL A N D CRYSTALLINITY

I13

tropic bonding can undergo certain disordering on their crystal lattice of types that are not found in isotropic crystals. Examples include rotational and translational displacements of the chains with respect to their axes. The effects of these types of disorder on the x-ray diffraction pattern have been analyzed independently by Clark and MU US^^ and by Arn ~ t t . ' ~Their analyses are based on the helical transform theory developed by Cochran et discussed previously. The atomic arrangement of a linear polymer molecule may be defined in terms of atoms regularly spaced along helices o f t turns in u motifs (atoms). When the chains are in parallel array, the diffraction pattern will fall on layer lines in agreement with the selection rule 1 = tn

+ um,

where I is the layer line number, n the order of Bessel function J , controlling the intensity in the layer line, and m any integer. Thus each layer line I is associated with a Bessel function of order n. When the parallel array of molecules conforms to a three-dimensional lattice, the layer lines will consist of sharp Bragg reflections. When there is rotational or translational disorder of the chains about their axes, the intensity of the Bragg reflections is reduced and a continuous transform (streak) occurs along the layer line. This reduction of intensity of reflection is related to the disorder type by specific functions of the parameters 1 and In1 of the selection rule. For example, small angular displacements cause the intensity of a Bragg reflection to be reduced by the approximate factor (6.1.122) w h e r e 2 is the mean square angular displacement of a chain with respect to its neighbors. Thus, the intensity reduction is related solely to In1 and is independent of 1. For small longitudinal displacements along the chain axis of mean square v a l u e 2 , the intensity reduction of a spot is approximated by Z/Z, exp(-PF). (6.1.123)

-

In this case, the intensity reduction is a function of the layer line number only and is independent of Inl. Note that the selection rule requires both 1 and n to be zero on the equator. Therefore, angular and rotational displacements have little effect on the equatorial reflections. (However, if these disorders are accompanied by displacements of the chain axes themselves, the equatorial pattern will be altered; in this case, appropriate expressions for disorder of the first or second kind apply.) Large angular displacements at random of chains about their axes reduce the diffraction pattern to continuum (streak) layer lines except on "

s. Amott, Trans. A m . C r y s t d o g r . Assoc.

9, 31 (1973).

114

6.

X-RAY DIFFRACTION

those for which n = 0, Thus only the equator and those upper layer lines for which the selection rule gives n = 0 will have sharp spots. These upper lines will correspond to those for which 5 = l / p , where p is the separation of the helical motif units measured along the chain axis ( u l t = PIP). Large transitional displacements allow sharp reflections only on the equator. Additional discussion on the effect of these types of disorder including screw displacements on fiber patterns is given by Clark and M ~ u s . ~ ~ Clark and MU US'^ have used this analysis to interpret the unusual diffraction effects observed in the fiber patterns of polytetrafluoroethylene. This polymer exhibits phase transformations at 19 and 30°C. Below 19"C, the fiber pattern exhibits numerous Bragg reflections on all observable layer lines (Fig. 60a) and corresponds to a helical structure having six turns in 13 CF2 units. As shown in Fig. 60, diffraction patterns taken at temperatures above 19°C exhibit considerable reduction of the Bragg intensities (and corresponding increase in the continuum streak) on all layer lines except the equator. The conformation of the molecule has seven turns in 15 CF2 units, thus giving a selection rule 1 = 7n

+ 15m.

The fiber pattern at 25"C, Fig. 60b, has sharp spots only on those layer lines whose intensity is controlled by low-order Bessel functions (In1 = 0, 1, 2, 3). Those layers whose intensity is controlled by higher-order Bessel functions exhibit only a continuum streak. These observations are consistent with small angular rotations of molecules about their long axes in accord with Eq. (6.1.122). This equation predicts from the selection rule that the layer 1 = 15 for which n = 0 should show sharp reflections. This was proved to be correct using tilted fiber patterns. Above 3WC, Bragg reflections disappear on all layer lines except those for which n = 0. This is interpreted in terms of random rotational displacement with no translational disorder. An interpretation of these disorders in terms of a dynamic model has been given by Clark.7s 6.1.7. Measurement of Crystallinity by X-Ray Diffraction 6.1.7.1. Theoretical Basis for Crystallinity Measurements. Let us now consider the problem of determining the relative amount of crystalline or paracrystalline material in a polymer sample. E. S . Clark and L. T. Muus, Z . Kristullogr.. Kristallgeom., Krisiallphys., Kristallchem. 117, 1 I9 (1962). E. S . Clark, J . Mncromol. Sci.. Phys. 1(4), 795 (1967).

FIG.60. Fiber patterns of polytetrafluoroethylene (a) 15"C, (b) 25"C, (c) 35°C.

6.

116

X-RAY DIFFRACTION

All valid methods of measuring the “degree of crystallinity” by x-ray diffraction techniques are based on the fact that the total coherent scattering from N atoms is the same, independent of their state of aggregation. This is a direct consequence of the law of conservation of energy. Mathematically we can express this physical principle in terms of an integral of the scattering over all of reciprocal space, i.e.,

J Z(s) dV*

(6.1.124)

const.

=

Z is equivalent to Eq. (6.1.25) and, as already mentioned, is proportional to the number of atoms irradiated. Let us assume that there is no preferred orientation in the sample or else the sample is mechanically randomized by cutting into small particles or rotating about some axis. Under these conditions the measured scattering from the sample at any value of s is equivalent to the spherical average value in reciprocal space. Thus

JZ(s) dV* = 47r

lorn

(6.1.125)

s2Z(s) ds,

where s is the magnitude of s and corresponds to the radial position in reciprocal space. Let us further assume that the sample consists of two distinct phases. It is not necessary at this point to require that one phase be ideally crystalline and the other be ideally amorphous. In fact the two phases might in principle be two different crystalline phases, neither of which is ideal. Nevertheless, we refer to the intensity scattered from one phase as Zc and that from the other as I , . The usual connotation in a crystallinity measurement would be that one phase is “crystalline” the other “amorphous.” For this two-phase sample we can write 41r

lorn

s2Z(s) ds = 47r

s2Zc(s) ds

+ 41r

lorn

s2Z,(s) ds.

(6.1.126)

The integral on the left-hand side of Eq. (6.1.126) is proportional to the total number of atoms irradiated, NT . The first integral on the right-hand side of Eq. (6.1.126) is proportional to the number of atoms that are in the crystalline phase Nc , and the second integral on the right is proportional to the number of atoms in the amorphous phase N a . Thus rrn

Nc

Nc + Na

= -Nc -

NT

J -

s2Zc(s) ds

lorn

(6.1.127)

0

s2Z(s) ds

*

For a homopolymer, the chemical composition of the crystalline and

6.1.

UNIT CELL A N D CRYSTALLINITY

1 I7

amorphous phases is the same and (6.1.128) where X, is the weight fraction of the polymer in the crystalline phase. Finally,

(6.1.129) An equation analogous to Eq. (6.1.129) can obviously be written for the amorphous fraction X, . Equation (6.1.129) or its equivalent is the basis for most x-ray techniques for determining the degree of crystallinity of polymers. The actual application of Eq. (6.1.129) is subject to many difficulties, which often result in approximations. To begin with, due allowance should be made for any experimental factors that affect the shape of the scattering curve as a function of s such as variation in absorption, the number of atoms irradiated, or time spent counting at a given s value (Lorentz factor). Further, the intensities Z, and Z are the coherently scattered intensities. The experimentally measured scattering should be corrected by subtracting the incoherent scattering; but this requires measurement in absolute intensity units, a difficulty not often considered worth the effort. Other extraneous sources of background such as air scattering and lack of monochromatic radiation, should be avoided or be removed by a correction procedure. Measurement of the scattered intensity over the full range of s values as required for application of Eq. (6.1.129)is impossible; in practice the scattering curve is only measured over some finite range of s values. It is assumed that coherent scattering occurring outside this range is insignificant. The value of Z, refers to all scattering from crystalline regions of the sample, including diffuse components such as thermal diffuse scattering and other types of disorder scattering arising from distortions and imperfections in the crystalline regions. This fact makes it very difficult to accurately separate the crystalline contribution from the amorphous contribution to the scattering curve. This is one of the most troublesome problems of all. In the following paragraphs a few selected techniques in the literature are reviewed with emphasis on the approximations used and the degree to which the authors treat the major difficulties described above. 6.1.7.2. Approximate Methods for X-Ray Crystallinity: The Crystallinity Index. It is obvious from even a cursory inspection of the physics of

118

6. X-RAY

DIFFRACTION

diffraction from polymers that calculation of an absolute crystallinity value from x-ray data is a formidable task. Nevertheless, the concept of crystallinity as a gross physical property is valuable, such as in the correlation between crystallinity and mechanical properties or in certain aspects of polymer chemistry such as tacticity. For many of these purposes, an absolute crystallinity is not needed. It is often adequate to calculate a reproducable value from the x-ray data, which will compare, on an arbitrary basis, the crystallinity of different samples of the same polymer. Such a relative evaluation is termed a “crystallinity index.” Two simple methods have been developed-one in which the x-ray data are resolved precisely, but somewhat arbitrarily into “crystalline” and “amorphous” peaks, and the other an empirical method for comparing observed data with data for crystalline and amorphous “standards.” 6.1.7.3. Sample Preparation and Data Collection. Both of these methods require uniformity of sample preparation and data acquisition. For quantitative data, the standard reflection diffractometer with circuitry for pulse height analysis and, if available, a monochromator is preferred. Data obtained by densitometry from a photographic film (powder pattern) may be used but it must be recognized that nonmonochromatic radiation, even with a (Ni) filter, will cause large distortions of the data and corresponding index values. All diffractometer scans should be made with the same geometry and the slit system should be selected so that the specimen surface is no smaller than the x-ray beam at any recorded angle. The area irradiated is easily seen by placing a fluorescent screen in the sample holder. (Observe safety precautions.) The diffractometer data is unaffected by sample thickness if it is greater than a minimum (6.1.130)

where p m is the mass absorption coefficient, p’ the macroscopic sample density, and 8 the maximum Bragg angle recorded. In the example to be shown for molded polyoxymethylene, p m = 8.98 cm2/g, p’ = 1.4 g/cm3, 8 = 13”, and rmln = 0.6 mm. One of the requirements for a precise crystallinity index is that the sample be unoriented. Correction of the data for orientation cannot readily be made. However, since orientation of the crystalline fraction is often accompanied by orientation of the amorphous fraction, the effects of orientation do not negate this simplified approach completely, although it does introduce a serious source of error. 6.1.7.4. Determination of the Crystallinity Index. 6.1.7.4.1. PEAK RESOLUTIONMETHOD. This method is based on the approach of

’’

A. Klug and L. E. Alexander, “X-ray Diffraction Procedures,” p. 252. Wiley, New York, 1954.

6.1.

1 I9

U N I T CELL A N D CRYSTALLINITY

Hermans and Weidinger” for cellulose and is applicable to polymers having a limited number of highly intense diffraction peaks such as polyethylene and polyoxymethylene. A scan is made over a limited preselected range to include the intense peaks and the underlying amorphous peak as in Fig. 61. An arbitrary but precise method is defined to resolve the “crystalline peaks,” the “amorphous peak” and the “background.” The following assumptions are made: (1) The total scattering from the sample is divided between crystalline peaks from the “crystallites” and amorphous peaks from the remaining “amorphous regions.” (2) The total scattering from the sample is that included in the resolved “crystalline” and “amorphous” regions. (3) The relative areas of the “crystalline” peaks and the “amorphous” peak are respectively proportional to the number of electrons (and thus mass) in the “crystallite” and the “amorphous” regions.

With these assumptions the crystallinity index X can be calculated from the resolved peak areas: -- 1 x, = A(Cr)A(Cr) + KA(Am) 1 + KR’

(6.1.131)

where R is the ratio of amorphous to crystalline peak areas and K is a constant. For comparative purposes, K may be set to unity. If it is desired for the crystallinity index to have a value approximating the absolute ”

P. H. Hermans and A . Weidinger, Makromol. Chem. 44-46, 24 (1961); 50,98 (1961).

Intensity

15

20

25’

28

FIG.61. Resolution of the diffractometer scan of polyoxymethylene into crystalline and amorphous portions.

120

6. X-RAY

DIFFRACTION

crystallinity, the value of K must be determined from another accurate measurement of crystallinity for the same sample. This measurement may be made by an x-ray technique such as that of Ruland7* to be described later, but excellent results have been obtained using accurate estimates of specific volumes for the amorphous regions and the crystallites (unit cell density). Using carefully prepared, void-free specimens the specific volume can be determined with a density gradient tube. The absolute crystallinity Xabsis assumed to be related to the specific volumes of the crystallites and amorphous regions by Xabs =

Vam

- vx - Vcr

( x 100).

(6.1.132)

Note that the idealized relationship is linear between crystallinity and specific volume rather than the density. Thus a useful value for K may be determined from combinations of Eqs. (6.1.132) and (6.1.131). Ideally K should be constant over the entire range but an averaged value usually is used. An example of the peak resolution method is shown in Fig. 61 for polyoxymethylene. A diffractometer scan is made with copper radiation from 5 to 26" (28) and resolved into one crystalline peak (100 peak) and one amorphous peak. The center of the amorphous peak is defined as exactly 2" (28) below the top of the 100 peak and a vertical line is constructed. The crystalline peak is resolved precisely from the rest of the scan by constructing a line intersecting the top of the amorphous peak and tangent to the scan in the 28 region 25-26'. The amorphous peak is resolved with precision (if not high accuracy) by constructing a background line parallel to the 28 axis and coincident with the scan at 28 = 5-6". The area of the amorphous peak is defined as twice the area of the portion on the low-angle side from the defined center. Greater precision can be obtained for planimetry by performing the scanning operation in two sections with different (but exactly known) multiplication factors on the chart recorder. Similar computations also can be made on a computer from digital diffractometer data. Using crystalline and amorphous specific volumes of 0.622 and 0.825 cm3/g the K value for this method of resolution of the polyoxymethylene scan is 0.56: X =

1

+ 0.56R ( X 100).

(6.1.133)

This method for polyoxymethylene is adapted from that of Hammer et ~ 1 . ' A~ more elaborate method for determination of the proportionality

'* lo

W. Ruland, A r m Crystallogr. 14, 1180 (1961); Polymer 5, 89 (1964). C. Hammer, J. Koch, and J . F. Whitney, J . A p p l . Polym. Sci. 1, 170 (1959).

6.1.

U N I T CELL A N D CRYSTALLINITY

121

factor K may be found in a study of isotactic polystyrene by Challa er U ~ . ~ A convenient method for resolution of polyethylene data has been devised by Matthews er af.81but their data included a substantial contribution from air scatter and white radiation, which must be discounted. 6.1.7.4.2. DIFFERENTIAL INTENSITY MEASUREMENT. A second, simple method for using the x-ray scan to obtain crystallinity indices for comparing samples of the same polymer is applicable to polymers having complex scans with several strong reflections such as poly(ethy1ene terephthalate). Resolution into crystalline and amorphous regions may not be feasible. Therefore, two “reference scans” are prepared between precisely set limits in 28-one for a sample of very high crystallinity and one for a sample of very low crystallinity. The unknown sample is also scanned between the selected limits. The three scans are normalized to the same total intensity between the 28 limits. An example of these three scans for PET is shown in Fig. 62 from work of Statton.82 At appropriate increments of 28, the differences between the normalized intensity values are determined: (Z, - 1,) and (Z, - la). An “integral” crystallinity index is calculated from (6.1.134) G . Challa, P. H. Hermans, and A. Weidinger, Makromol. Chem. 56, 169 (1962). J . L. Matthews, H. S . Peiser, and R. B. Richards, Acta Crystallogr. 2, 85 (1949). W. D. Statton, J . Appl. Polym. Sci. 7, 803 (1963).

INTENSITY

I TI I I

I.

-I.

I,

-I.

A

\\

/\UNKNOWN CRYSTALLINE

STANDARD

DIFFRACTION

ANGLE

FIG.62. Diffractometer data for three poly(ethy1ene terephthalate) samples illustrating the differential intensity measurements needed for calculation of a crystallinity index. [W. 0. Statton, J . Appl. Polym. Sci. 7, 803 (1963).]

O

I22

6.

X-RAY DIFFRACTION

with the summations over a series of 28 values between the set limits. A small computer is required and automatic data collection is advisable. Alternatively, the data may be treated to obtain a “correlation” crystallinity index from a linear regression analysis of the expression

(I,

-

Z,)

=

X(Z,

-

I,)

+ B.

(6.1.135)

A plot is made from (I, - Z,) values vs. (Z, - I,) values at a series of 28 increments. By linear regression, the slope is the crystallinity index. Although more calculations are required for the “correlation” index, it has been found to be preferred to the “integral index.” It should be noted that this method, which is based on reference scans, defines the reference samples as X = 0 and 1, i.e., 0 and 100% crystallinity. 6.1.7.5.Ruland’s Method. R ~ l a n d has ’ ~ given a significantly improved method for determining x-ray crystallinity, which is based on fundamental principles. Noting that it is easier to experimentally measure the intensity concentrated within the sharp Bragg peaks rather then the total crystalline scattering (including diffuse components), Ruland assumes that Eq. (6.1.129) can be written as

{lsp

s 2 L ( s )dsl

X, =

Is:

F).

&so, s p , D,

(6.1.136)

s2Z(s) ds

Here Zcr(s)is the coherent intensity concentrated in the sharp Bragg peaks. The integration limits so and sp must be finite in practice and are chosen so that

(6.1.137) w h e r e 7 is a mean square atomic scattering factor for the polymer, given by

f.= w , f ; 2 / z N , .

(6.1.138)

N1 is the number of atoms of type i in the empirical formula. The right-hand side of Eq. (6.1.137) is the total scattering in the range so to s,, if the atoms scatter independently. The nature of K can be understood by noting thatt

t In his original paper, Ruland uses D instead of Dzin Eq. (6.I . 139). This, of course, is of no consequence.

6.1.

U N I T CELL AND CRYSTALLINITY

123

(6.1.139) where D is the disorder function and N the number of atoms in the unit cell of volume V. Equation (6.1.139) reduces to Eq. (6.1.136) if K

=

1"szp

ds/ / " s ~ 0 2 ds.

so

(6.140)

80

The major problem in the application of Ruland's method is the evaluation of K. For a given upper limit of integration s, the only unknown in Eq. (6.1.136) is D, the disorder function. For samples containing only disorders of the first kind D2 = exp(-ks2).

(6.1.141)

For samples containing disorders of the second kind (paracrystallinity) Ruland notes that, according to the theory of Hosemann,64the loss of intensity in the diffraction peaks is given by an expression of the form (6.1.142) But Ruland argues that Eq. (6.1.141)gives sufficiently accurate values of D for the evaluation of K , even in cases where there is significant paracrystalline disorder. In such a case the parameter k is assumed to be given by a sum

k =

kT

+ k1 + k2,

(6.1.143)

where kT accounts for thermal motion, k, for other disorders of the first kind, and k2 for paracrystalline disorder. It is therefore possible to compute D and hence K for a wide range of conceivable values of k and for various s, values. This gives a nomogram such as that shown in Fig. 63. Since the actual value of k is not generally known in advance but may be assumed constant for a given sample, Ruland suggests that Eq. (6.1.136) can be solved by seeking a constant value of k that gives constant crystallinity X , for a series of values of s,. In general, solution must proceed by trial and error. Figure 63 is used to find appropriate K values for arbitrarily chosen trial values of k. The application of Ruland's method thus proceeds as follows. First the experimental data are obtained over a wide range of scattering angles for each sample. Great care is necessary to avoid, eliminate, or reduce a variety of experimental problems such as lack of monochromatic radia-

6. X-RAY

124

DIFFRACTION

7-

6-

5-

%

4-

3-

2-

1-

L . . 0.2 0.4

.

~

0.6

I

I

0.8

,

,

1 .o

1.2 SP

FIG.63. Nomogram for K values as function of k and s p . The chemical composition is assumed to be (CH,),and so = 0.1. [W. Ruland, Acra Crysrallogr. 14, 1180 (1961).]

tion or absorption. The 8 scale is converted to the s scale (s = 2 sin 8/A) and the intensity data are converted to absolute units. Corrections are made for Compton scattering and any remaining systematic experimental errors (e.g., air scatter, absorption) and the resulting coherent scattering curve is plotted as s2Z(s) vs. s as shown in Fig. 64. Several ranges of integration are chosen with a constant lower limit so, but different upper limits s,. The crystalline peaks are separated from the remaining scattering by drawing a smooth curve from tail to tail following the general slope of the continuous scattering. According to R ~ l a n d ’this ~ restricts the designation “crystalline” to ordered regions larger than 20-30 A and containing paracrystalline imperfections not exceeding rms deviations in the nearest-neighbor distances of about 10%. Equation (6.1.136) is then solved by trial to simultaneously obtain values of X, and k (or D ) as discussed in the preceding paragraph. Some typical results for polypropylene are given in Table XIV and for nylon 6 in Table XV. In the case of polypropylene Ruland noted that the disorder function was not affected by sample history while the crystallinity varied substantially as shown in Table XIV. This led to the conclusion that most of the disorder in the crystalline regions of polypropylene

6.1.

I25

U N I T CELL A N D CRYSTALLINITY

FIG. 64. Plot of s21(s) versus s for a polypropylene sample. [W. Ruland, A m Crysrallogr. 14, I180 (1961).]

was associated with thermal motion, i.e., there was little paracrystallinity in the samples examined. Both nylon 6 and nylon 7 exhibited marked variations of k as well as crystallinity with sample history, suggesting that lattice disorder above and beyond thermal motion occurs. Based on an analysis of diffraction from a single-phase paracrystalline substance Ruland was able t o arrive at a criterion for discerning whether a given sample should be considered a single-phase paracrystalline material TABLEXIV. Weight Fraction of Crystalline Phase X , in Polypropylene Samples as a Function of k and spa

k=O

k = 4

k=O

k = 4

Sample 3 k=O k = 4

k=O

k=4

0.1-0.3 0.1-0.6 0.1-0.9 0.1-1.25

0.270 0.159 0.105 0.067

0.329 0.294 0.305 0.315

0.353 0.222 0.145 0.095

0.431 0.411 0.421 0.447

0.546 0.333 0.220 0.145

0.120 0.078 0.044 0.029

0.146 0.144 0.128 0.136

Mean X ,

Sample 1

Sample 2

Interval (so-sp)

0.31

0.43

0.666 0.616 0.638 0.682 0.65

Sample 4

0.14

Data of R ~ l a n d . ' ~Sample 1: Melted, quenched in water at room temperature. Sample 2: Same as 1, heated 1 hour at 105°C. Sample 3: Same as 1, heated 30 minutes at 160°C. Sample 4: High atactic content.

TABLEXV. Weight Fraction of Crystalline Phase X, in Nylon 6 samples as a Function of k ans spo Sample 3 Interval (sO-sp)

Sample 4

Sample 6

Sample 7

Sample 8

Sample 9

Sample 10

k=O

k=3.0

k=O

k=4.2

k=O

k=3.0

k=O

k=5.6

k=O

k=3.9

k=O

k=4.4

k=O

k=3.7

0.10-0.40

0.260

0.10-0.65 0.10-0.95 0.10-1.25

0.175

0.338 0.306 0.327 0.330

0.169 0.102 0.072 0.050

0.242 0.216 0.238 0.245

0.216 0.139

0.281 0.242 0.265 0.273

0.214 0.119 0.076 0.054

0.340 0.308 0.322 0.341

0.232 0.143 0.101 0.070

0.324 0.289 0.314 0.318

0.253 0.143 0.101 0.072

0.367 0.312 0.346 0.367

0.241 0.154

0.331 0.302 0.317 0.329

Mean X, a

0.129 0.091

0.33

0.105

0.075

0.24

Data from Ruland," which see for sample histories.

0.27

0.33

0.31

0.35

0.106

0.076

0.32

6.1.

U N I T CELL A N D CRYSTALLINITY

I27

or a two-phase crystalline (or paracrystalline) plus amorphous mixture. Paracrystallinity effects both the integral breadth of the diffraction lines [see Eq. (6.1.112)] and the calculated values of crystallinity. By comparing these two effects Ruland was able to show that a two-phase structure must exist if the degree of crystallinity satisfies an inequality that may be expressede3as Xc < (1 - 2dp)Al - dp),

(6.1.144)

where d is the interplanar spacing corresponding to a given reflection and

p the integral breadth of the peak corrected for instrumental broadening.

S . Kavesh and J . M.Schultz, J . Po/ytn. S r i . . Pnrf A-2 8, 243 (1970).

6.2 Crystallite Size and Lamellar Thickness by X-Ray Methods

By Jing-l Wang and Ian R. Harrison 6.2.1. Introduction

In the study of crystalline polymers, two sizes have been reported that are of interest to polymer scientists: microparacrystal or mosaic block size and lamella thickness. Let us briefly review material contained earlier in this volume. Polymer chains crystallize in the form of lamellae (platelet-like objects). Their thickness is of the order of 50-500 A and their lateral dimensions usually lie in the range of 1-25 pm. Within these platelets the polymer chains are arranged to be approximately perpendicular to the top and bottom surfaces. Further, since the individual chains are many times longer than the crystal thickness, they are envisioned as folding back on themselves and reentering the crystal core. The nature of the fold, the number of units involved, and their organization is still a matter for debate. However, the concept of chain folding implies that lamellae are two-phase systems, at least in the simpler models. There exists a top and bottom (fold) surface of a more or less disordered structure (liquidlike or amorphous) and sandwiched between is a crystalline core. This model naturally implies that a single lamella (or single crystal) is not 100% crystalline. Turning to the crystalline core of the lamella, evidence in several forms suggests that this core is far from being a perfectly regular crystalline lattice. Rather, the lattice is considered as paracrystalline containing defects of the second kind (see Chapter 6.1). On the basis of this model we would then envision the central core to be a platelet whose lateral dimensions are several microns in length. However, using X-ray techniques it can be shown that the lateral dimensions of diffracting units in the central core are not several microns but rather on the order of hundreds of angstroms depending on the heat treatment of the sample. The central core is therefore considered to be composed of paracrystalline material, which is further separated into blocks. These blocks (microparacrystals, mosaic, crystallites) are separated by families of kink jogs, which effectively create screw dislocation between crystallites. The blocks are slightly misaligned so that crystallographic planes in one block are not in register I28 METHODS OF EXPERIMENTAL PHYSICS. VOL. 168

Copyright 0 1980 by Academic Press. Inc All rights of reproduction in any form reserved

ISBN 0-12-415957-2

6.2.

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

129

with the same index planes in adjacent blocks. As a result the individual blocks act effectively as independent diffracting units. It is the objective of this chapter to examine the ways in which lamellar thickness (fold period or long period) and crystallite size can be measured using X-ray techniques. Instrumentation used in wide-angle X-ray diffraction has been previously described, those used for small-angle X-ray work will be dealt with in this chapter. A number of corrections used in X-ray studies will be considered in more detail. Finally, results on a variety of polymers will be reported and an attempt made to examine the implications of alternative explanations. For a more detailed review of material presented here readers are referred to the original works of referenced authors. We would also suggest the following authors whose texts the reader may wish to peruse: Alexander,’ Klug and Alexander,2 and Kakudo and Kasaia3 6.2.2. Crystallite Size by Wide-Angle Techniques 6.2.2.1. Introduction. An ideal crystal irradiated at the appropriate angle by a parallel and monochromatic beam of X rays gives a sharp spike (intensity vs. 28) in accordance with Bragg’s law. In practice the diffracted X-ray beams from real crystals are broadened. In addition to the crystallite size effect, which will be our primary concern, there are other sources of broadening in any experimental system. These may include lattice distortions due to thermal vibration, microstrain, and the paracrystalline nature of the lattice. There are also “instrumental factors” such as misalignment, vertical divergence, lack of monochromaticity, and the absorption of X rays by the sample. In this section we first consider the effects of finite size on the diffracted beam, assuming that no other sources of broadening are present. 6.2.2.2. Broadening Due Solely to Crystallite Size. 6.2.2.2.1. THE SCHERRER EQUATION (DERIVATION). Scattered intensity by an ideal infinitely large crystal can be observed only when all diffracted beams are in phase. For the above ideal system the diffracted intensity for a given set of planes should be a spike at O0, the Bragg angle for those planes. For crystals of finite size, destructive interference does not take place for rays that are slightly “off” the Bragg angle. As a result, line broadening occurs around the diffraction spike. Broadening is often measured as the L. E. Alexander, “X-ray Diffraction Methods in Polymer Science.” Wiley, New York, 1969. H. P. Klug and L. E. Alexander, “X-Ray Diffraction Procedures of Polycrystalline and Amorphous Materials,” 2nd ed. Wiley, New York, 1974. M. Kakudo and N. Kasai, “X-Ray Diffraction by Polymers.” Am. Elsevier, New York, 1972.

6. X-RAY DIFFRACTION

130

breadth of the diffraction peak at the half-height of the peak. A crude relationship between crystallite size and peak breadth is given in Fig. 1 for various diffraction angle^.^ A more exact relationship between the breadth of the diffraction peak (p) and crystal size (3) is given by the Scherrer equation:

-

Dhkl =

KA/P

(6.2.1)

COS 60,

where Z h k l is the “crystallite size,” A the wavelength of the x-ray beam, K the Scherrer shape factor, and eo the Bragg angle for the particular reflecting planes (hkl). The Scherrer equation was initially derived with the following assumptions: (a) the X-ray beam is monochromatic and parallel, and (b) no X rays are absorbed by the sample. At the Bragg angle O0, the path difference between diffracted X rays from two adjacent reflection planes is given by the distance ABC (A/) in Fig. 2, where ABC = A1 = 2d sin

eo = nA

(6.2.2)

for discrete diffraction (constructive interference of X rays), where d is the interplanar spacing. If the incident beam strikes the reflection planes with a small angular deviation t from the Bragg angle, then the path difference arising from two neighboring planes is given by

Al’

=

2d sin(8,

+ t).

(6.2.3)

As t is very small, Eq. (6.2.3) can be approximated by

eo + 2dt cos e, = nA + 2dt cos 8,. (6.2.4) The phase difference 24 due to diffraction from two neighboring planes Al‘

=

2d sin

is then 2 ( p = -2- ~ “ ’ - 2nn + 47rdt cos A A

eo ’

the effective phase difference being 24 =

47rdt cos 60 A ’

(6.2.5)

Now if A is the amplitude diffracted by a single lattice plane then, assuming no absorption of X rays by the crystals, the amplitude is the same for each diffraction plane, For crystals with N layers of reflection planes, the resulting amplitude is5

‘R. C. Rau, A d v . X-Ray Anal. 5, 104 (1962). A. Guinier, “X-ray Diffraction in Crystals, Imperfect Crystals and Amorphous Bodies.” Freeman, San Francisco, California, 1963.

I

I 0.15

I

I 0.20

I

I 0.25

I

I 0.30

I

I 0.35

I

I 0.40

I32

6. X-RAY

DIFFRACTION

FIG.2. A set of planes with separation d . Two in-phase incident rays have a path length difference given by ABC after reflection.

AN = A sin N+/sin

4.

(6.2.6)

The scattered intensity I is defined as

I = AA*,

(6.2.7)

where A* is the complex conjugate of A. The scattered intensity at diffraction angle 28 for N layers of reflection planes is then Z(28) = A2 sin2 N4/sin2 4.

(6.2.8)

For Bragg diffraction at 8,, all the diffracted beams from N planes are in phase so that the scattered intensity is z(2e0) =

~

2

~

2

.

(6.2.9)

Substituting (6.2.9) into (6.2.Q then Z(28) = Z(28,) sin2(Nt$)/N2 sin2+.

(6.2.10)

For small values of 4, Eq. (6.2.10) becomes Z(28) = 428,) sin2(N+)/N2$2.

(6.2.11)

Equation (6.2.11) can then be solved for the condition that r(2e) = I(2e0)/2, which is satisfied when N 4 = 0.444~.

(6.2.12)

By combining Eqs. (6.2.12) and (6.2.5) we can find that value oft, namely which gives an intensity of half that at 8,: tl12= 0.222A/Nd cos 8,.

The width at half-height of the diffraction peak

(6.2.13)

(pllZ)is defined as

6.2.

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

pl12= 4f112

=

0.89h/Nd cos 8.,

133

(6.2.14)

Using Eq. (6.2.14) it can be seen that the Scherrer shape factor is 0.89 and crystallite size (5)is equal to Nd. This is effectively the perpendicular distance through a set of N parallel planes that are d apart. 6.2.2.2.2. THE SCHERRERSHAPECONSTANT (K). In the original derivation of the Scherrer equation it was assumed that the crystal was cubic and that the line profile was approximated by a Gaussian The Scherrer shape constant K obtained was 0.94. The value of K is in fact a function of crystal shape, indices of diffraction plane, definition of line breadth (p), and definition of average crystallite size 5. For exbut integral breadth ample, if one uses not half-width of the peak (pl12) (PI) defined as

a = swe) d(2e)/z(2eo),

(6.2.15)

then

p,

=

A/@ cos eo),

(6.2.16)

i.e., K from Eq. (6.2.1)or (6.2.14)is unity. The same definition of 5 is still used, namely, perpendicular distance through a set of planes. Strictly speaking, D derived using Eq. (6.2.16) represents an apparent crystallite size that is a volume average of the crystallite dimension 5 normal to the diffraction planes being used. If “true particle size” is defined as the cubic root of the particle volume, i.e., p = V113, then the Scherrer equation becomes more complicated. Wilson7has shown that if line broadening is solely due to small crystallite size, then the true particle size ( p ) can be related to variance (W20) using

where (2e2 - 2e1) is the angular range over which one evaluates the variance W20, W2, is defined as

(6.2.18) and (2e) is the centroid of the line profile. The Scherrer shape constants KW and L a W , for various crystal shapes with low-index reflection, are given in Table I.8 B. E. Warren, “X-Ray Diffraction.” Addition-Wesley, Reading, Massachusetts, 1969. A . J . C. Wilson, Proc. Phys. Soc., Lundon 80, 284 (1962). A. J . C. Wilson, “Mathematical Theory of X-Ray Powder Diffractometry.” Phillips Technical Library, New York, 1962.

134

6. X-RAY

DIFFRACTION

TABLEI. Small-Particle Size" Cube

Tetrahedron

Octahedron

hkl

KW

Law

KW

LaW

KW

Law

100 110 111 210 211 22 1 310 311 320 32 1 410 322 41 1 33 1 42 1 332 430 43 1 510 511

1 1.4142 1.7321 1.3416 1.6330 1.6667 1.2649 1.4076 1.3868 1.6036 1.2127 1.6977 1.4142 1.6059 1.5275 1.7056 1.4000 1S689 1.1767 1.3472

0 1 2 0.8 1.6667 1.7778 0.6 1.2727 0.9231 1.5714 0.4706 1.8824 1 1.5790 1.3333 1.9091 0.%00 1.4615 0.3846 0.8148

2.0801 1.4708 1.8014 1.8605 1.6984 1.7334 1.9733 1.8815 1.7307 1.6678 2.0180 1.7657 1.9611 1.6702 1.8156 1.7739 1.6641 1.6318 2.0397 2.0016

2.8845 1.4423 2.1634 2.3076 1.9230 2.0031 2.5961 2.3601 1.9970 1.8543 2.7148 2.0785 2.5640 1.8597 2.1977 2.0978 1.8461 1.7751 2.7736 2.6708

1.6510 1.1647 1.4298 1.4767 1.3480 1.3758 1.5662 1.4933 1.3737 1.3237 1.6017 1.4015 1.5565 1.3256 1.4411 1.4079 1.3208 1.2951 1.6189 1.5886

1.8171 0 0.9086 1.0903 0.6057 0.7067 1.4537 1.1564 0.6989 0.5192 1.6033 0.8017 1.4133 0.5260 0.9518 0.8260 0.5088 0.4193 1.6773 1.5479

a The Scherrer constant KW = p / z , appropriate for use with the variance of the line profile and the taper parameter Law. They are tabulated for three regular crystal shapes; for a sphere they have the constant values 1.2090 asnd zero, respectively.

In practice, the second term (taper parameter) in Eq. (6.2.17) is normally negligible. Therefore, by rearranging one obtains (6.2.19)

where

The advantage of using the variance method is that variances are additive and independent of the shape of diffraction lines. If, instead of the variance term pw , the integral breadth PI is used, then the true particle size p can be evaluated by the relations p = K,

AlpI cos e,.

(6.2.20)

The Schemer shape constants K, for crystals of various shapes (cube,

6.2.

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

135

tetrahedron, octahedron, and spheres) have been derived and calculated by Wilson* and are given in Table 11. Note that 5 as given by Eq. (6.2.14) or (6.2.16) is an apparent size. However, widely different apparent sizes derived from different diffraction planes can give additional information regarding the overall shape of the diffracting unit. From the above brief description it can be seen that within the Scherrer equation the value of the shape factor K is dependent on (a) the definition of line breadth: pllZ,PI, pw;(b) the definition of crystallite size: Dhkl,p ; (c) the crystallite shape, known or assumed: (d) the indices of the diffraction plane. In general, it is true to say that the majority of crystallite size work reported for polymers has used half-width PI,* or integral breadth &. Remember also that so far we have assumed that the only source of broadening present is that due to crystallite size. In real systems other sources of broadening must be removed before the above equations can be applied. 6.2.2.3. Removal of Instrumental Broadening. 6.2.2.3.1. INTRODUCTION. A number of different approaches have been used in order to TABLE11" Reflection

Tetrahedron

Cube

Octahedron

100 110 111 210 211 22 1

1.3867 0.9806 1.2009 1.2403 1.1323 1.1556 1.3156 1.2543 1.1538 1.119 1.3453 1.1772 1.3074 1.1135 1.2104 1.1826 1.1094 1.0878 I ,3597

1.oooO

1.1006 1.0376 1.1438 1.1075 1.1061 1.1185 1.1138 1.1211 1.0902 1.0955 1.1123 1.1304 1.1207 1 .W63 1.1133 1.1334 1.0786 1.0835 1.1101

3 10 311 320 32 I 410 322 41 1 33 I 42 1 332 430 43 1 510

1.0607

1.1547 1.0733 1.1527 1.1429 1.0672 1.1359 I .0698 1.1394 1.0583 1.1556 1.1174 1.1262 1.1324 1.1513 1.0667 1.1240 1.0506

a The Scherrer constant K, = p/D, appropriate for use with integral breadths; for a sphere it has the constant value 1.0747.

136

6.

X-RAY DIFFRACTION

separate out broadening due solely to crystallite size. It should be remembered that there are two additional sources of broadening; instrumental factors and sample factors other than crystallize size. Typically the approach has been to remove instrumental broadening and then to analyze the resultant peak(s) for other factors. Three different methods have been used to remove instrumental broadening. Probably the most direct is to identify all possible sources of broadening, e.g., slit width, sample geometry, and to multiply intensities by appropriate factors that will be a function of angle. This approach yields the complete peak profile. The second method is to assume that the experimental diffraction peak profile h(x) is the convolution of the pure line profile of the sample f ( x ) and the instrumental function g(x): h(x) = S f ( Y ) g ( x - Y ) d ~ .

(6.2.21)

The instrumental function g(x) is found by running a standard sample that is presumed to be free of any crystallite size-broadening effects. There are additional restrictions on this standard as will be shown later. Appropriate techniques enable one to deconvolute the experimental curve and obtainf(x), the pure line profile. Finally the quickest, simplest, and most widely used technique involves making assumptions about peak shape. If one can assume that h(x) and g(x) are, for example, Gaussian then one can readily determine the half-width off(x). Note that this latter method does not give the complete profile of the peak, only the half-width. 6.2.2.3.2. INSTRUMENTAL FACTORS. Let us consider those instrumental factors that could play a role in line broadening. 6.2.2.3.2.1. Wavelength Distribution. In the absence of pure monochromatic radiation there will be a range of angles over which the Bragg diffraction conditions are satisfied. For example, if the Bragg equation is written A = 2d sin 8, then A(28) = 2 tan 8(AA/A).

(6.2.22)

For any distribution of wavelengths represented by AA we shall have a range of angles A(28) over which Bragg diffraction will occur. Experimental X-ray diffraction patterns are generally broadened by the presence of, for example, copper Kcq, K a z doublets in the incident beam. Filters are normally used to remove KP and single-crystal monochromators can be used to remove Ka2. However, the latter technique has not been widely used because of the reduction in intensity. (Recently introduced graphite monochrometers only reduce intensity to approximately 30% of the original value.) The separation angle of the K a l , K a z doublet increases with increasing diffraction angle as shown in Fig. 3.

6.2.

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

I

-".

4-

i I i , I'

Z

0

5

1

i I i ;

0,

4 L

137

3i

W

l

0 28 DEGREES

FIG.3. The angular separation o f the K a doublet as a function of diffraction angle (28) for some common radiations.

For routine work the correction curve by Jonesg (Fig. 4) is quick and easy to use to allow for K a doublet broadening. In Fig. 4, d is the angular separation B, and b,, the observed line breadths, and B and b the corrected line breadths for the sample and standard, respectively. Rachinger's iteration method', to separate the KCXdoublet becomes suitable if computers are used. Recently Gangulee" and Kirdon and De Angeles12evaluated and separated the KCXdoublet by a more rigorous Fourier analysis. 6.2.2.3.2.2. Slit Width. Broadening due to slits can be divided into that due to finite source slit width and finite receiving slit width. Broadening due to the finite source slit width is given by13 (6.2.23) where x = 28 - 200, ws = (h/R)(57.3)tan et, A is the width of source slit, R the sample to source and sample to diffractometer distance, and et the take-off angle (3-6"). F. W. Jones, Proc. R. Soc. London, Ser. A 166, 16 (1938). W. A. Rachinger, J . Sci. Insfrum. 25, 254 (1948). A. Gangulee, J . Appl. Crysrallogr. 3, 272 (1970). ** A. Kirdon and R. J. De Angelis, Acta Crystallogr., Sect. A 27, 596 (1971). l3 L. E. Alexander, J . Appl. Phys. 25, 153 (1954).

@

6 . X-RAY DIFFRACTION

138

\' 0.7'1 0.65

I

I

I

I

l , . , l , l

0.3

0.4

0.5

d/b,

or

I

,

0.8

0.9

0.60 0

0.1 0.2

0.6

0.7

I

.o

d/B,

FIG.4. Curves for correcting line breadths for K a doublet broadening. d is the angular separation for the doublet at the appropriate diffraction angle (Fig. 3). Bo and 6, are the observed line breadths and B and b the corrected line breadths for standard and sample, respectively. As seen later, various assumptions can be made for the line shapes of the standard and sample. A, Integral breadths of back-reflections; B, half-breadth of Cauchy-Gaussian; C, half-breadth of Gaussian-Gaussian; D, half-breadth of Cauchy- Cauchy.

The broadening due to receiving slit width is expressed as13 (6.2.24) x = 28 - 2d0, and w, is the angle subtented by the receiving slit of the

goniometer . Broadening due to slit widths is effectively negligible when the diffractometer is properly aligned. 6.2.2.3.2.3. Vertical Divergence. If the axial dimensions of the source and receiving slits are small in comparison to that of the specimen, then broadening due to vertical divergence is8*14 gb

= (2x cot

el-"',

0 >x

> tm,

(6.2.25)

where t , = 62 cot e/(4)(57.3),

with 6' the vertical divergent of the X-ray beam with Soller slits. The distortion due to vertical divergence is greatest at very small and high diffraction angles. I'

J. N. Eastbrook, Br. J . Appl. Phys. 3, 349 (1952).

6.2.

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

139

6.2.2.3.2.4. Sumple Geometry. A flat sample is often used in the X-ray diffractometer. Since the flat specimen is tangential to the focus circle, it causes the diffracted beams to be displaced from the ideal focal point. The line-broadening function due to the use of the flat sample rather than a curved sample can be represented by the angular divergence of the diffractometer's equatorial divergent slit. Changes of the profile are significant at very small angles. At moderate values of beam divergence and Bragg angles larger than lo", line broadening due to flat sample surfaces is negligibly small. 6.2.2.3.2.5. Absorption Broadening. For samples of high absorption coefficient, diffraction takes place mainly in the surface layers of the specimen. Broadening and displacement of the diffraction line with this type of sample is negligible. If the absorption coefficient of the sample is low, as is the case with most polymers, then diffraction can take place inside the sample and cause an asymmetric broadening and shift of the peak positions. The effects of a low absorption coefficient on the diffraction lines have been treated and discussed by A l e ~ a n d e r , 'Langford ~ and Wilson,15 and Keating and Warren.lS The line-broadening function of low-absorption-coefficient materials is g, = exp(K,x),

(6.2.26)

where x = 28 - 28,, K, = 4p,R/114.6 sin 28, p, is the absorption coefficient, and R is the sample to source and sample to diffractometer distance. 6.2.2.3.2.6. Misulignment. Misalignment arises from missetting the zero position and missetting the 2 : 1 ratio gears and is primarily due to mechanical imperfections in the instrument. The empirical function for misalignment is13 (6.2.27) where t = 28 - 28,, k6 = 2/w, and w is half-maximum breadth of the source. 6.2.2.3.2.7. Polarization and Lorentz Effect. The polarization factor P is used to correct for polarization of the X-ray beam by the crystal and is described by the equation P

=

1/2(1

+ C O S ~28).

(6.2.28)

The Lorentz factor L allows for differences in time and geometrical Is l8

J . 1. Langford and A. J . C. Wilson, J . Sci. Instrum. 39, 581 (1962). D. T. Keating and B. E. Warren, Rev. Sci. Instrum. 25, 519 (1952).

6.

140

X-RAY DIFFRACTION

opportunity, which different reciprocal lattice points have to intersect the Ewald sphere. For randomly oriented crystalline powders irradiated by monochromatic X-ray beams,

L

=

l/sin 28 sin 8.

(6.2.29)

For oriented samples, for example, a fiber whose axis has preferred orientation along some specified crystal axis, the Lorentz factor is1'

L(hkl) = l/sin2 e cos 8 sin & k l J ,

(6.2.30)

where &&is the angle between the normal to the reflection plane and the fiber axis z. The Lorentz and polarization factors are often combined as a single geometrical correction factor. For the small angular range of a single diffraction peak any change in broadening due to the Lorentz polarization factor is negligible. 6.2.2.3.2.8. Thermal Vibration and Temperature Effect. Atoms in a crystal are not at rest, but undergo thermal vibration about their equilibrium positions. The displacement due to thermal vibration causes the diffracted beam to be smeared and decreases the scattered intensity with increasing diffraction angle. The effects of thermal vibration on the peak intensity profile are given by the Debye- Waller temperature factor:

D

=

exp[ - 2B(T) sin2(e)/A2],

(6.2.31)

where B(T) is the temperature factor, related to the mean square displacements of atoms in the crystal, and increases with increasing temperature. At higher temperatures and higher diffraction angles, the scattered intensity is very weak as shown in Fig. 5 . Although thermal vibration causes the intensity of the diffraction peaks to decrease and become more diffuse, it does not increase line broadening, as indicated by Debye,18 Waller,l0and G ~ i n i e r .If~ there is no coupling between the thermal vibrations of the single atoms in a real crystal, a more complete treatment of the effect of temperature suggests that superimposed on the weakened crystal diffraction peaks there will be broad maxima. These broad maxima are the result of so-called temperature-diffuse scattering. We are not aware of any papers in the polymer literature that account for these effects.

P. M.DeWolff, J . Polym. Sci. 60, 534 (1962). P. P. Debye, Verh. Drsch. Phys. Chem. [N.S . ] 15, 678 and 857 (1913). I@ I. Waller, Z . Phys. 17,398 (1923).

6.2.

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

141

100

80

c

60

>

t vl z W

$ -

40

20

0

ze DEGREES FIG.5. A plot of intensity vs. 28 for a sample with a mean crystallite size of 300 A. The intensity maximum is set at 30" and the curve shape shown for various values of B ( T ) , the temperature factor [Eq. (6.2.31)]. Curve A ( B = 0) is normalized at 100, curve B ( B = 2) and curve C (B = 5) show a decrease in intensity but no change in half-width.

6.2.2.3.3. DECONVOLUTION PROCEDURES. It should be apparent from the above descriptions of the various factors that the corrections involved are quite straightforward. Perhaps it should be noted that this is true for any individual correction. However, carrying out all corrections on a number of peaks in a diffraction scan would be quite tedious if computers were not available. Assuming that access to computers is becoming standard, then the deconvolution procedures mentioned earlier appear to be a more realistic approach. It will be recalled that the observed diffraction peak profile h(x) is the convolution of the pure line profile of a sample f ( x ) and the instrumental (weight) function g(x) given by the relation h(x) = SflvMx - Y ) dY.

(6.2.32)

142

6.

X-RAY DIFFRACTION

Further, g(x) was to be determined by running a standard that showed no line broadening due to crystallite size. It should also be apparent from the review of instrumental factors that there will be other restraints on the standard. The standard should have the same absorption coefficient as the sample. If this is not the case, then an additional source of broadening may be present. Naturally the geometry of standard and sample should be identical. Due to the presence of K a doublets, the standard should have a diffraction peak in the same angular region as the sample. If this is not so, then doublet broadening in the sample may be under- or overcompensated. Provided that the instrumental factors are the same in both standard and sample runs, the deconvolution procedures will remove all instrumental effects to yield the pure line profile. However, it should be remembered that as crystallite size increases, line broadening decreases. That is, for large crystallites the controlling factor determining line breadth will be the instrument. It follows that the experimental set up be carefully examined with a view to obtaining the minimum possible line width, consistent with sufficient intensity. Two basic methods of deconvolution have been described in the literature. The first is commonly known as Stokes’ methodz0and represents a deconvolution procedure using Fourier transformations. This has been widely applied to metals but has found only limited application in the polymer area. Recently Kirdon and De Angeliszl modified this technique with a least-squares analysis to reduce the errors and the fluctuations of the Fourier coefficients. However, the compution times are approximately an order of magnitude greater than for the direct Fourier transform method as reported by Kirdon and Cohen.zz An iterative folding method has recently been updated by E r g ~ n .In~ ~ general, this is a simpler and far less costly procedure in terms of computer time than is Stokes’ method. We are not aware that any applications of this technique to polymer systems have been reported in the literature. At the present time, in our laboratory, we are using this approach, although our data have not yet been submitted for publication. 6.2.2.3.4. APPROXIMATION METHODS. By far the largest amount of data in the literature reports crystalline sizes using a variety of approximations. The approximations arise from assumptions that are made regarding the shape of the diffraction peak. For example, one may asA. R. Stokes, Proc. Phys. Soc.. London, Ser. A 61, 382 (1948). “S. M. D. Symposium on Computer-Aided Engineering’’ (G. M. L. Gladwell, ed.), p. 285. Univ. of Waterloo Press, Waterloo, Ontario, zo

** A. Kirdon and R. J . De Angelis, in

1971. ps

A. Kirdon and J. B. Cohen, J . Appl. Crystallogr. 6, 8 (1973). S. Ergun, J . Appl. Crysiallogr. 1, 19 (1968).

6.2.

CRYSTALLITE SIZE AND LAMELLAR THICKNESS

143

sume that all intensity profiles are Cauchy functions. Then the halfwidths or integral breadths are additive: p=B-b,

(6.2.33)

where p is the line broadening due to small crystallite size only, B the experimentally observed line broadening, and b the instrumental broadening, measured using crystals of effectively infinite dimensions. Using the above definitions of p, B, and b and assuming Gaussian peak shapes, then p2

= B2 - b2.

(6.2.34)

The above treatment assumes that the standard crystals used gave peaks in the same angular region as the sample. If this is not the case then an additional correction for K a doublet broadening (Jones’ method) would be required. It is also possible, and in some cases more realistic, to use combinations of Gaussian and Cauchy curves. A third alternative is to determine the variance of the curve [Eq. (6.2. lS)]. The advantage of using this method is that variances are additive and independent of the shapes of the line profiles:

w, = w, - w,.

(6.2.35)

However, the variance obtained depends on the choice of the background level. If the tails of the profile slowly approach the background, then the range over which the variance is evaluated should be large. It is apparent that truncation errors can lead to erroneous values for variance. If the tail of the profile varies as the inverse square of the range cr, then the variance can be evaluated by a variance-range function of the form24.25 -

wi = K,V + w,,

cr = 2e2 - 28, = ~ ( 2 e ) ,

(6.2.36)

where W, is the variance of the profile calculated by using a particular truncation range u and Ki the true variance of the profile determined from the slope of a W, vs. v plot. W , is a term related to the taper parameter mentioned earlier. 6.2.2.4. Separation of Size and Distortion Broadening. 6.2.2.4.1. INTRODUCTION. There are at least two other major sources of broadening that may be present in polymer systems. These are broadening due to lattice distortions as a result of microstain and broadening arising from the paracrystalline nature of the lattice. It follows that, if one removes instrumental broadening by using a standard and deconvo24 J . !. Langford and A. J. C. Wilson, in “Crystallography and Crystal Perfection” ( G . N . Ramachandran, ed.), p. 207. Academic Press, New York, 1963. 25 A. J. C. Wilson, Nature (London) 193, 568 (1962).

6.

144

X-RAY DIFFRACTION

luting, this standard has further restraints imposed upon it. The standard must have either the same strain and/or paracrystalline lattice or it must be essentially free of strain and/or paracrystallinity. The latter case is the more usual and as a result one must allow for the broadening effect arising from these two possible sources. Before considering the treatment of the two possible sources of broadening, we wish to remind the reader of the diffraction space variable s: so = l/dhkl

or

s =

2 sin 8/A.

(6.2.37)

Up to this point the half-width pl12or integral breadth p, has been expressed in radians. In subsequent equations breadth will be expressed in s units (ps); (6.2.38) 6.2.2.4.I . 1. Paracrystal Broadening. The concept of paracrystallinity was proposed by Hosemann26to describe distortions in the crystal (see Chapter 6.1, this volume). The paracrystal posses short-range order but not long-range order. These distortions are distortions of the second kind, and depend on the order of diffraction line. The integral breadth due to paracrystallinity (pSp)is given by2'

psp= ( 1 / 2 d ) [ l - exp( - 2rgP2rn2)].

(6.2.39)

When the value of 2r2gP2rn2 is sufficiently small, Eq. (6.2.39)reduces to

psp = (1/2d)(27r2gp2rn2),

(6.2.40)

where g, is the degree of statistical fluctuation of the paracrystalline distortion relative to the separation distance of the adjacent lattice cell (Adld),rn is the order of reflection, and d is the interplanar spacing for the first-order reflection. As will be shown later, g, can be determined and for many polymers has' values of approximately 2%.28 6.2.2.4.1.2. Strain Broadening. The X-ray diffraction lines of coldwork metals are broadened. This broadening is due to lattice distortions in the crystals produced by microstrain. The integral breadth due to microstrain (&') is given by20

ps'

=

( 1 ld)[(2d112( gt">112rnl,

(6.2.41)

z8 R. Hosemann and S. N. Bagchi, "Direct Analysis of Diffraction by Matter." North-Holland Pub]., Amsterdam, 1%2. e7 R. Bonart, R. Hosemann, and R. L. McCullough, Polymer 4, 199 (1963). R. Hosemann and W. Wilke, Fuserforsch. Texrilrech. 15, 521 (1964). C. N. J. Wagner and E. N . Aqua, Adv. X-Ray Anal. 1, 46 (1964).

6.2.

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

145

where (g?) represents the mean square lattice distortion over the crystallite size L . A simpler, although more vaguely defined, value for the “upper limit of lattice distortions” has been given by Stokes and Wilson30 as e (Ad/a?hki, (6.2.42) 2 :

where the relationship between e and (8:) is e

2 :

1.25 (g,2)112.

(6.2.43)

Using the Wilson definition of strain31the integral breadth due to microstrain becomes (&”) (pSw) = 2 e m / d . (6.2.44) From the above definitions it follows that once the pure diffraction profile has been obtained (instrument broadening removed) it may be broadened by factors other than crystallite size. Methods must therefore be developed to remove distortion-broadening effects. Note also that line broadening produced by small crystallite size is independent of the diffraction order. In contrast, broadening due to distortions from strain or the paracrystalline nature of the lattice is a function of the order of the reflection. This then is the basis for separating out size and distortion parameters. 4.2.2.4.2. SEPARATION ASSUMING O N E SOURCE OF DISTORTION. 4.2.2.4.2.1. Assuming Line Shape. A number of simplified methods are available if only one source of broadening is present, i.e., paracrystalline or strain, in addition to size broadening. Furthermore, one can make an assumption regarding the shapes of the contributing line profiles provided at least two orders of reflection are present. 1. Cauchy function approximation. For strain broadening

ps =-

1

+

(2T)’” ( g , z ) 1 / 2

m.

(6.2.45)

do

Dhkl

For paracrystalline broadening,

ps =- 1 Dhkl

+ dgp2m2 do

.

(6.2.46)

2. Gaussian function approximation. For strain broadening (6.2.47) 30

A. R. Stokes and A. J. Wilson, Proc. Phys. Soc., London 56, 174 (1944). A. J. C. Wilson, “X-ray Optics.” Methuen, London, 1949.

146

6. X-RAY

DIFFRACTION

For paracrystalline broadening (6.2.48)

ps represents the integral breadth in s units after removal of instrument broadening, the other parameters have been previously defined. It is apparent that appropriate plots of ps or p: vs. m ,m2, or m4 will yield an intercept (at m = 0) that corresponds to l / i r h k l or 1 / B h k F . It is also apparent that the slopes of the lines are related to the distortion parameters g, or ( g : ) . Slightly more complex relationships are available with no assumptions made regarding line shape. 4.2.2.4.2.2. Variance Method. Wilson32has shown that for lattice distortions due to small local strain, the strain variance w d is given by wd

=

4 tan2 eo ( e w 2 ) ,

(6.2.49)

where e = Ad/d, d is the interplanar spacing, and (ew2)the variance of the lattice strain distribution. The variance due to particle size and local strain is described by the r e l a t i ~ n ~ ~ . ~ ~

w -- 2 d E cos eo + 4 tan2 eo (ew2>

(6.2.50)

28

or

Refer to Eq. (6.2.17) for a definition of W,, and note that the second term, the taper parameter, has been neglected. Particle size is then obtained from a plot of

and from the slope of the line one may determine the distortion parameter. Kulshreshtha et ~ 1 showed . ~ that ~ the variance range function due to distortions produced by a paracrystalline lattice can be expressed as W(s) = Kos d s )

+ EO,,

(6.2.52)

s* A. J. C. Wilson, Proc. Phys. Soc., London 81, 41 (1963). C. N. J. Wagner, in “Local Atomic Arrangements Studied by X-ray Diffraction” ( J . B.

Cohen and J. E. Hillard, eds.), AIME Conf., Vol. 36, p. 271. Gordon &Breach, New York, 1966. 34 A. K. Kulshreshtha, N. R. Kothari, and N . E. Dweltz, J . Appl. Crysrallogr. 4, 116 ( 197 1 ).

6.2.

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

147

where

W(s) =

w28-

cos2 8 0 h2 '

and do is the d spacing of the first reflection. In order to determine crystallite size, Km must first be determined for each reflection from a plot of values of K , are then W ( s )as a function of u (s); see Eq. (6.2.52). These plotted vs. m2 to yield an intercept of 1/27r2 D . 6.2.2.4.2.3. Fourier Analysis Method. Briefly, using a Fourier transform (Stokes' method), the Fourier coefficients A , and B, for a pure diffraction profile can be obtained directly. The Fourier cosine coefficients A , are further assumed to be composed of two terms; one due to particle size (A,Ps) and one due to lattice distortions (A:) due to strain, such that6.35.36 A,

=

AnpsA:.

(6.2.53)

For small strains the Fourier cosine coefficients can be expressed by the relation In A,(m) = In Anps - 27r2m2n2( e 2 ) ,

(6.2.54)

where m is the order of reflection and n a variable integer representing the number of unit cells in a direction perpendicular to the reflection planes. For small values of m and n a plot of In A,(m) vs. m2 should be linear. So it is possible by extrapolation of the above plots to m2 = 0 to obtain In Anpsas a function of assumed values of n. One now sets Anpsat n = 0 to be unity and normalizes the other values o f A n p S .Then for small values of n , AnPs(n)can be approximated by AnPs(n)= 1 - n / N ,

(6.2.55)

where N multiplied by the d spacing gives 0, the mean crystallite size perpendicular to the diffracting planes. N is obtained by plotting AnPs(n) vs. n . At higher values of n the plot is no longer linear and N is obtained by extrapolating the linear portion of the curve obtained at low values of n until it intersects the n axis at AnPs(n)= 0. While the above treatment is valid only for strain distortions a similar treatment has been proposed for the paracrystalline l a t t i ~ e . ~ ~ , ~ ' 35

36

B. E. Warren and B. V. Averback, J . Appl. Phys. 21, 585 (1950). B . E. Warren, f r o g . Mer. Phys. 8, 147 (1959).

6 . X-RAY DIFFRACTION

148

At this point it should be noted that the sizes obtained by Fourier transform techniques are not the same as those obtained from, for example, a measure of integral breadth. It is probably a natural assumption that the complex manipulation of the Fourier method will give a more accurate measure of crystallite size. However, it can be shown that the two methods are complementary. The integral breadth method gives a “weight-average” (size-average) dimension and the Fourier technique a “number-average” d i m e n ~ i o n . ~ The ~ crystallite sizes obtained by the integral breadth method should always be greater than or equal to those obtained by the Fourier technique. Note also that given a weight and a number average, if one is prepared to make an assumption about the nature of the crystallite size distribution, i.e., Gaussian, then one can completely characterize the distribution with a mean and standard deviation. We are not aware that this has ever been reported in the literature. 6.2.2.4.3. SEPARATION WITH Two SOURCES OF DISTORTION. Up to this point it has been assumed that only one source of distortion is present in the sample in addition to size broadening. In order to test if, for example, the paracrystalline model is the sole contributor to distortion in the polymer systems, one requires at least three orders of diffraction. However, due to the low crystal symmetry of most polymers, three orders of reflection of sufficient intensity and definition are rarely observed. However, KajP recently measured the crystallite size of highly oriented nylon 6 using integral breadth. He found that it is valid to assume that, even in a stressed sample, distortions are purely paracrystalline in origin. Hosemann,26*27 who is largely responsible for the paracrystalline lattice theory, had previously demonstrated with his c o - w o r k e r ~that ~ ~ polyethylene ~~*~~ reflections are consistent with a paracrystalline lattice. In contrast, the work of Buchanan and MilleF on isotactic polystyrene emphasizes the need for a careful evaluation of the full line profile. Using a variety of approximations the above authors concluded that with their samples they could not unambiguously distinguish between paracrystalline and microstrain distortions. However, they did state that a better fit of the data was obtained assuming microstrain distortions. It has also been assumed that only one size (range) of crystallites is present. Hosemann and WilkeZ8have assumed that (hkO) reflections are composed of two overlapping Gaussian profiles in a highly drawn polyethD. R. Buchanan, R. L. McCullough, and R. L. Miller, Acra Crystallogr. 20,922 (1966). K . Kaji, Makromol. Chem. 175, 311 (1974). W. Wilke, W. Vogel, and R. Hosemann, Kolloid-Z. Z . Polym. 237, 317 (1970). 40 A. Schonfeld, W. Wilke, G . Hohne, and R. Hosemann, Kolloid-Z. Z . Polym. 250, 110 37 38

(1972).

D. R. Buchanan and R. C. Miller, J . Appl. Phys. 37, 4003 (1966).

6.2.

CRYSTALLITE

SIZE A N D LAMELLAR THICKNESS

149

ylene sample. These two Gaussians were resolved and gave rise to two distinct crystallite sizes, 83 and 175 A. These values could be interpreted as the diameters of a single ultrafibril and a cluster of four fibrils. There is good agreement between these values and those reported by the same authors using small-angle X-ray scattering. One point that is not quite clear from the above data arises from the values of the paracrystalline fluctuation parameter g,. For the ultrafibrils this is stated to be 3.2% and for the fourfold clusters it is 2.2%. It is hard for us to rationalize how the fourfold clusters could be less dissordered than the single ultrafibril. 6.2.2.5. Experimental Results. Following are a series of examples taken from the literature that demonstrate the differences obtained using different approximations and different instrumental corrections. In Table I11 we see the effects of a variety of instrument-broadening corrections on a set of (hkO) reflections with h = k . (These same data were further corrected using different distortion corrections in Table IV.) The main feature to note here is that in all cases the apparent crystallite size decreases as the order of the reflection increases. From our previous discussion it is apparent that a source of broadening is present in addition to the crystallite size effect. Further, the effect of certain distortions on first-order reflections is quite small. For example (Table IV, column b, number 5 ) , the paracrystalline-corrected hkO reflections of Gaussian instrument-corrected profiles yield an average crystdlite size of 156 A. The (1 10) reflection instrument corrected (Gaussian) itself gives 153 A. The data shown above serve to illustrate the relative values typically obtained using a variety of approximations. Differences between instrumental broadening techniques result in sizes that are at most approximately 8% different from each other. However, for any chosen instrument-broadening method, the various assumptions regarding the naTABLE111. Corrections for Instrumental Broadeninp

110 220 330

142 118

153 124

106

110

228 171 144

164 143 119

169 145 122

Data of Buchanan and MilleP for isotactic polystyrene (hM)) reflections. Crystailite size E (A). * Uncorrected (observed). Corrected by integral breadth method-Gaussian function. * Corrected by integral breadth method-Cauchy function. Using integral breadths direct from Fourier analysis (Strokes' method). Using integral breadths from Fourier coefficients p = l/Z,"An.

150

6. X-RAY DIFFRACTION TABLEIV. Corrections for Lattice Distortions' b

d

C

Distortion Corrections 1. Fourier transform 2. Cauchy 3. Gaussian

4. Cauchy-Gaussian

130 208 172 176

192 160 176

5 . Paracrystal

162

156

Warren- Averbach method = ps + P O

p

p'

B

= ps' + = ps +

PO'

&'/a

'Data of Buchanan and Miller" for isotactic polystyrene (hM)) reflections Crystdite Size E (A). * p, experimental broadening; ps, size broadening; PO, distortion broadening. Instrumental broadening corrected by Fourier analysis. Instrumental broadening corrected by Gaussian function. ture of the distortion and the peak shape may produce as much as a 20% difference in crystallize size. This latter observation does not include the Fourier method. As previously mentioned this yields a number-average size, which in this case is much lower (20-35%) than the weight-average size derived from integral breadth methods. Similar information is contained in Table V. The similarity between the sizes obtained from the first-order reflection observed (020) and the paracrystalline corrected data (row g) should be quite apparent. The implication is that if all we require is a size then that obtained from a low-order reflection is well within experimental error. This of course presumes that the only additional source of broadening is that due to the paracrystalline lattice and that the lattice distortion parameters are within the normal range for polymers, i.e., less than 4%. The data in Table VI illustrate a feature that is often overlooked in wide-angle work. It has previously been stated that most polymers crystallize as lamellar structures with their chain axis (c axis) perpendicular to the top and bottom fold surface. It follows that (001) planes lie parallel to the top and bottom surface of the lamella. Further, crystallite sizes obtained from reflections off these planes will give a measure of the thickness of the crystalline core. This follows quite naturally, since the size obtained from diffraction peaks is a dimension perpendicular to the planes giving rise to those peaks. From a knowledge of the crystal core thickness 70.2 8, (Table VI, d) and a separate measure of the fold period or total thickness of the lamella 89.1 8, (Table VI,e), it follows that one may readily compute the thickness of the amorphous fold surface LA = 18.98,.

6.2.

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

151

TABLEV. Corrections for Instrumental Broadening of Tufcel (Polynosic Viscose)” OM

b

C

d

e

f

R

020 040 080 h

98 96 51 -

110

182 139 62 183

123 I09 53 125

137 124 55 148

111

*

102 52 112

Data of Kulshreshtha er crystallite size (A). Observed. Gaussian approximation. Cauchy approximation. Cauchy-Gaussian. Jones’ method. Variance method. Corrected for paracrystalline lattice distortions.

A similar approach has been used on polyethylene fibers by S t a t t ~ n ~ ~ and on solution-crystallized polyethylene mats by Kobayashi and KelleP3 Thielke and B i l l m e ~ e r . ~ ~ Notice that this sort of calculation is dependent upon the assumption that the c axis is perpendicular to the fold surface. Along the same lines several authors have questioned the validity of the mosaic block model as a necessary one for as-formed crystals. For example, using polyoxymethylene single crystals it has been reported that extremely large (or nonexistent) blocks are seen.45 However, this occurs only when the TABLEVI“

y-002 y-004 Y-M

67.5 64.4 58.5

67.5 64.5 58.5

z:::]

70.2

89.1

18.9

5Y.2

* Data of Kaji3*; nylon 6 (y form), (stressed): crystallite s i z e 3 (A).

* Uncorrected.

Corrected for the doublet. Corrected for instrumental broadening by Gaussian approximation. Corrected for paracrystalline distortion by Cauchy approximation. Long spacing (fold period). 42 W. 0. Statton, in “Newer Methods of Polymer Characterization” (B. Ke, ed.), Chapter 6. Wiley (Interscience), New York, 1964. Y. Kobayashi and A. Keller, J . Muter. Sci. 9, 2056 (1974). 4rl H. G. Thielke and F. W. BiUmeyer, J . Polym. Sci., Part A-2 2, 2947 (1964). 4a I. R. Harrison and J. Runt, J . Polym. Sci.. Polym. Phys. Ed. 14, 317 (1976).

6.

152

X-RAY DIFFRACTION

crystals are prevented from drying down and kept in their as-formed state. As soon as the crystals are removed from suspension one finds mosaic blocks of the usual size, 150-200 A. Similar arguments have been presented for polyethylene crystals. In this case chain inclination can lead to an upper limit on observed crystallite size without invoking mosaic blocks. In addition, electron diffraction customarily gives large (several thousand angstroms) size blocks, whereas X-ray line broadening typically yields sizes of several hundred angstrom. Discrepancies between the two techniques may also be explained in terms of chain i n c l i n a t i ~ n .These ~ ~ observations are not in conflict with the basic paracrystalline lattice theory, nor do they question the existence of mosaic blocks in dried-down samples. They do, however, question the existence of such units in crystals that have never been dried down. Needless to say, with the limited amount of data available for suspension crystals, the controversy still continues in the literature. Returning to the determination of crystal core thickness, using 001 reflections, a novel approach has recently been reported by Windle.47 On a well-characterized carefully prepared sample of polyethylene single crystals it is possible to observe not only the 002 peak, but also subsidiary maxima. One can calculate both the theoretical 002 profile and the correlation function for various models that describe the electron density distribution on traveling perpendicularly through a crystal. On this basis it is then possible to match calculated and experimental data. Fitting the data is complicated by the relatively low intensities of the subsidiary maxima 1. R. Harrison, A. Keller, D. M. Sadler, and E. L. Thomas, Polymer 17, 736 (1976). A. H. Windle, J . Muter. Sci. 10, 252 (1975).

t

= 32i

+ 5%

-30% I

I

I

-1

4

D = 85i t5%

:-

I

I I

,

I

OVERALL THICKNESS 1171 + 5 % -12%

FIG.6. Trapezoidal model for the electron density distribution through a lamella (Tsvankin type). Measured fold period by SAXS was 1 1 1 A. Wide-angle data from the 002 peak and its subsidiaries give an overall thickness of 117 A and a total transition zone thickness of 64 A.

6.2.

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

153

(1.7% of main 002 peak). In addition, the 002 peak and its subsidiaries sit on a diffuse halo. However, using three different matching techniques the author concludes that the data are best fit by the trapezium model shown in Fig. 6. It should be noted that in each case a trapezium model was assumed. Briefly, the trapezium model states that on traveling from the center of the crystal toward the fold surface, one encounters a gradual decrease in the crystal-amorphous ratio, which extends over quite a large distance, in this case, 64 out of 117 A. This model is reconciled with a discontinuous transition between crystal and amorphous regions, by assuming that there is an increasing frequency of amorphous regions as one approaches the fold surface (buried-fold 6.2.3. Lamellar Thickness Using Small-Angle X-Ray Scattering (SAXS) 6.2.3.1. Introduction. In general, SAXS systems have an extremely simple geometry, in contrast to many wide-angle units, i.e., precession or Weisenberg cameras. However, because SAXS by definition implies collecting data close to the main beam, the system requirements are much more stringent than those normally encountered in wide-angle diffractometry. In SAXS we could generally require (a) a highly collimated narrow primary beam with small divergence, (b) no parasitic scattering, i.e., no scattering in the absence of a sample, and (c) a monochromatic primary beam or a detector that can separate out a “single” frequency. The methods used to accomplish these requirements usually reduce the primary beam or detected beam intensity. This can lead to extremely long exposure times. In part, this can be offset by the use of special high-intensity tubes or detectors, which accumulate all scattered data at one time, i.e., position-sensitive detectors. A schematic diagram of the SAXS system is shown in Fig. 7. With polymeric systems two types of SAXS phenomena are used. Regular periodic arrays of lamellae produce “discrete scattering.” That is, a plot of scattered intensity vs. scattering angle has a maximum or distinct shoulder in the (typically) 0-2” range.42 If the scattering units are randomly arranged then “diffuse scattering” is produced. Intensity decreases monotonically as scattering angle increases. Both types of data can be used to obtain lamella thickness or fold period. 6.2.3.2. Experimental Techniques. In many SAXS systems the various components are interrelated. However, for the sake of clarity we shall try to discuss the components independently. Is A KeUer, E. Martuscelli, D. J . Priest, and Y. Udagawa, J . Poly. Sci., Part A-2 9, 1807 (1971).

154

6. X-RAY

DIFFRACTION

MONOCHROMATOR

FIG.7. A schematic diagram of a typical SAXS system.

6.2.3.2.1. DETECTION OF SCATTERED RADIATION. Detectors consist of two basic types; film and counter. (a) Film 1. This will detect the scattered intensity at all angles simultaneously. As such, it is the simplest position-sensitive detector. Because the film detects intensity at all angles simultaneously, there are no stringent requirements on the stability of the source. We shall see that this is not true for counter systems. 2. On film one can rarely detect intensity ranges greater than 100: 1. Experimentally, in SAXS one often encounters intensities that range from 1000: 1. This problem can be overcome by using different exposure times for different areas on the film or by using several thicknesses of film, the films acting as attenuators for the scattered radiation. 3. The main beam cannot be allowed to impinge on the film. Not only does it produce an intense spot on the film but there is a “halo” effect. That is, the film is also blackened around the point at which the main beam impinges. As a result diffraction near the main beam may be obscured. Hence one would normally have a “beam stop” to eliminate the main beam. This can present an alignment problem in some cameras. 4. Because a “reasonable” amount of intensity is required to darken a film, one would normally anticipate longer exposure times than with counters. In general, film is essential for pinhole collimated systems, and pinhole collimation is essential for samples that have unknown orientation. More sophisticated position-sensitive detectors may make the requirement for film obsolete. Film is usually used to determine relative changes in spacing in situations where the Bragg equation can be applied. That is to say, it can be used for certain discrete diffraction systems; however, it is normally less desirable for diffuse scattering where one wishes to determine additional sample parameters.

6.2.

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

155

(b) Counter systems can be subdivided into four general types. 1. Geiger-Muller, a highly stable rugged counter that is essentially linear up to -400 counts per second (cps). Above this rate the counter is nonlinear, but can be calibrated up to rates of about 1500cps. As a result, this is an excellent counter for weakly scattering systems. The pulses produced by a G-M counter are of the same amplitude regardless of the energies (wavelengths) of the incident X-ray photons. 2. Proportional counters are linear up to 500 cps and can be used up to 10,000 cps with calibration. Their name is derived from the fact that the amplitude of the pulse that the counter produces is proportional to the energy of the incident photon. It should also be noted that the efficiency of these counters (G-M and proportional) is to some extent a function of wavelength. This implies that to some degree they are both acting as monochromators. 3. Scintillation counters have approximately the same characteristics as proportional counters, with the exception that the efficiency of these counters is not a function of wavelength. Further, their efficiency is close to 100% compared to 60-80% for the G-M and proportional counters. These features make the scintillation counter an ideal choice for SAXS systems. 4. Position-sensitive detector systems, based on flow proportional counters, have been developed ~ e c e n t l y . ~These ~ * ~ detectors ~ consist of an anode wire with an active length of some 40-80 mm. The pulse induced by the radiation is detected at both ends of the wire. The difference in rise time of the pulse at each end of the wire is related to the position of the event. Hence for any one event two signals can be generated. One of them is proportional to the energy of the pulse, the second gives the position along the wire where the pulse was initiated. The resolution of such devices is such that with infinitely narrow entrance slits the units can separate events occurring roughly 50 pm apart along the wire. Needless to say some sort of multichannel analyzer or computer is essential to store and display the data. Note that when counters are used, the recorded data will be sensitive to any fluctuations in main beam intensity. This arises simply from the fact that data are taken at different positions at different times by scanning the counter through the appropriate angle. With position-sensitive detectors based on proportional counters, the situation is a little more complex. If one specifies a fixed number of counts per “position,” then the counter C . J . Borkowski and M. K. Kopp, IEEE Trans. Nucl. Sci. 17(3), 340 (1970). so Y. Dupont, A . Gabriel, M . Chabre, T. Gulik-Krezywicki, and E. Schechter, Nature

(London) 238, 33 1 (1972).

156

6.

X-RAY DIFFRACTION

system is sensitive to main beam fluctuations, since it will take different amounts of time to accumulate the same number of counts at various positions. In contrast, if one specifies a fixed time per “position,” then different numbers of counts will be accumulated. The error, which is inversely proportional to the square of the number of counts, will be different for different positions. 6.2.3.2.2. MONOCHROMATION OF RADIATION. Monochromators are essential in SAXS since no X-ray source produces radiation of a single wavelength. Rather a distribution of wavelengths is produced and it is necessary to select a particular one. If this is not done, then in discrete diffraction one obtains peaks that are broadened due to the range of wavelengths present. In diffuse diffraction no meaningful data can be obtained because of the wavelength “smearing” of the diffracted intensity. Monochromatization can be achieved through the use of filters, single-crystal monochromators, or pulse height analyzers. Details of these techniques have been covered in Chapter 6.1 in this volume. 6.2.3.2.3. COLLIMATION SYSTEMS. Collimation of the main beam can be achieved in a variety of ways. Perhaps the simplest method is through the use of pinholes or slits. X-Ray sources usually produce either a lineor a point-collimated beam. It is the job of the collimation system to define the main beam shape, and at the same time to reduce parasitic scattering resulting from the main beam impinging on the defining slits. The position of a series of slits is shown schematically in Fig. 8. The size of slits 1 and 2 and the distance of slit 2 from the source define the divergence of the beam. As collimation increases intensity decreases. Note that since the main beam impinges on slit 2 the edges act as scatterers. A third set of slits is placed to cut out the parasitic scattering from slit 2 but so as not to touch the main beam. The sample is placed directly behind the third slits. The detector pivots around the

2

3

FIG.8. A schematic for the arrangement of slits or pinholes in a typical SAXS collimation system. Note the third set of slits should not touch the main beam.

6.2.

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

I57

Beam Defining Blocks

Ll

t

Specimen

stop

FIG.9. The Kratky slit system. Parasitic scattering is essentially removed from one side of the main beam by the blocklike slits.

sample position and will have a fourth set of slits in front of it. A similar arrangement holds for pinhole collimation with the additional feature that the detector is usually a film and a beam stop is incorporated. Note that in Fig. 8 some parasitic scattering will be produced as parasitic scatter from slit 2 hits slit 3. While this is relatively weak, the Kratky system has been developed to remove all parasitic scattering due to slit edges, at least on one side of the beam.51 This is achieved with blocklike slits (see Fig. 9),52which are arranged so that one side of the main beam effectively traverses a pair of long slits. Both of the above geometries lead to a major decrease in beam intensity. One way around the reduction in intensity is to use some sort of curved surface to focus the beam to a point or line. The surface can be glass, gold-plated glass or resin, or a monochromator crystal that is correctly cleaved and then bent to the required curvature. Either one or two such surfaces may be required (see Fig. There is some parasitic scatter from surface inhomogeneties . A more sophisticated system of curved surfaces is the toroidal mirror (Elliot) system, Fig. 11. This geometry accepts a greater angular range of the main beam, with a resulting higher beam intensity at the There will still, of course, be some losses on reflection, but these are generally not as great as at monochromator crystal surfaces. 6.2.3.2.4. RADIATIONSOURCES. The X-ray source in most SAXS systems is of one of two basic types. Either a “microfocus” tube or a rotating anode. These modifications of the customary X-ray sources are designed to increase main beam intensity in order to accommodate the de0. Kratky, in “Small Angle X-Ray Scattering” (H. Brumberger, ed.), p. 63. Gordon & Breach, New York, 1967. P. J . Harget, Norelco Rep. 18, 25 (1971).

158

6.

X-RAY DIFFRACTION

Quartz Cr y s t a l

Defining

””

Guard

/

FIG.10. A single curved surface, effectively focusing the main beam to a line.

crease in intensity caused by collimation, monochromation, and elimination of parasitic scattering. A microfocus tube manufactured by Siemens is specifically designed for Kratky geometry. The line focus is 7 x 0.25 mm2 and the tube itself is rated at 1400 W,i.e., it can be operated at 40 kV, 35 mA. In comparison, a standard tube may be 10 x 0.075 mm2 and 750 W, approximately 40 kV, 18 mA. “Rotating anode” type systems are available commercially in various wattage ratings. The largest unit can be run at approximately 60 kV, 1000 mA; smaller units run at 60 kV, 100 mA. If one is using the SAXS system for film work, or for readily defined discrete peaks using a counter, then source stability is not a problem. However, with diffuse scattering or with poorly resolved discrete peaks

Specimen

FIG.11. point.

I

A toroidal mirror arrangement (Elliot), which will focus the main beam to a

6.2.

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

159

or if detailed line shape analysis is performed, then stability becomes a major concern. Thermal stability usually requires a minimum 2 hour warm-up period, preferably overnight, to avoid shifting of the focal spot. Voltage stabilization is inherent in the machine, as most commercial generators are adequate in this regard. Coolant temperature for the tube head can be responsible for a 10% change in intensity if the coolant temperature varies by 3”C.51 This usually implies the use of a closed-loop cooling system. Room temperature fluctuations of 1°C can lead to intensity changes of roughly 1%. There also exist elegant and somewhat costly systems that monitor the main beam intensity while the sample is being run. In this way any fluctuations in main beam intensity can be observed and corrected. 6.2.3.2.5. PARASITIC SCATTERING. Parasitic scattering from slits, pinholes, and curved surfaces has been noted in the collimator section. Parasitic scattering is any scattering that is present in the absence of a sample. As noted, the primary causes of such scattering are the edges of the collimator systems or the nature of the reflection surfaces. However, there are a number of additional sources of such scattering within the system: 1. Air in the direct beam. This effect can be eliminated by removing the air by enclosing the main beam and evacuating the systemJ3 o r by replacing air with helium. 2. Windows for X rays, which are essential if one wants to enclose the main beam, are additional sources of scattering. They will also attentuate the main beam. The usual windows are made of cellulose acetate, Mylar, beryllium, o r mica. 3. Sample containers are necessary if one wishes to study solutions or suspensions. These are usually thin-walled glass tubes, but may also be polymeric, beryllium, o r mica. Once again these will act as both scatterers and attenuators of the main beam. Before commenting on the various corrections for parasitic scattering, it is necessary to first consider the interaction of X rays with the sample. When a beam of X rays hits the sample two things happen. First, the beam will be diffracted at some characteristic angle or over an angular range and, second, the beam will be absorbed (attenuated) by the sample. The thicker the sample the more diffraction; however, the thicker the sample the more the beam will be absorbed. Obviously if one is interested in obtaining maximum diffracted beam intensity there will be an op53

R. W. Hendricks, J .

Appl. Crystallogr. 3, 348 (1970).

160

6. X-RAY

DIFFRACTION

timum sample thickness. This can be shown to occur when Z/Z, = l/e, where Z and Z, are the transmitted and incident beam intensities, respectively.“ So maximum diffracted intensity occurs when Z ZE 0.37Z0,where Z is measured at zero angle. There are a number of methods of correcting for parasitic scattering: 1. Completely remove the sample and measure the scattered intensity as a function of angle. Remember that the sample would normally attenuate this parasitic scattering. Therefore the measured intensity with no sample present must be corrected by a sample transmission factor. 2. The sample is removed from its customary position and placed between source and collimator (attenuating position). In this position the detector will see no scattering from the sample, but the main beam will be attenuated by the sample. As a result the transmission correction mentioned above is carried out automatically. 3. The sample is replaced by an essentially nonscattering system such as a metal foil. Naturally the foil should have the same transmission as the original sample.

Note that an implicit assumption of all of the above corrections is that parasitic scattering is directly proportional to the main beam intensity. This may not always be the case. In addition, remember that the samples used are not necessarily homogeneous, but may have large voids or not be of uniform thickness. As a result, extreme care should be taken to ensure that the beam passes through the same part of the sample regardless of whether the sample is in the scattering or attenuating position. This is probably the largest single source of error in background corrections. 6.2.3.2.6. SAMPLES. Many different forms of samples can be run on SAXS systems. Films, fibers, plates, or rods are easily and directly mounted in the main beam. Small-diameter rods often prove difficult to use with slit-collimated systems if it is necessary to run with the rod and beam parallel. Because the beam is of finite width it will actually “see” a section of the rod, but this section should be centered on a diameter of the rod. If this is not the case, the beam can be reflected off the rod and give rise to spurious diffraction data. Certain powder samples can be compressed into disks or pellets and then run as a solid sample. This assumes that the parameter of interest is not affected by the compaction procedure. If it proves impossible to run powders this way, then they can be run in tubes or cells as one would normally run solutions or suspensions of particles in liquids. In this case one B. D. Cullity, “Elements of X-Ray Diffraction.” Addison-Wesley, Reading, Massachusetts. 1956.

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161

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

must allow for the scattering inherent in all types of containers. That is, a blank must be run t a determine the background scattering, which is then subtracted from the sample-plus-container scattering. In the case of solutions or suspensions, the background run should consist of the container plus the liquid used in order to allow for the inherent liquid diffuse scattering . 6.2.3.3. Experimental Observations (Discrete Diffraction). 6.2.3.3.1. SINGLE CRYSTALS. If polymer single crystals are allowed to settle out of suspension, they form an oriented mat. A cross section through a dry mat would reveal a periodic structure formed by the stacking of the two-phase lamella (Fig. 12). Bragg has demonstrated that for period fluctuations in electron density the following equation holds: nA = 2d sin

e0,

(6.2.2)

where A is the wavelength of the scattered radiation (see Fig. 2). This equation is customarily applied to planes of atoms in a crystal, where d is the periodicity of the planes. For example, dllo for polyethylene is approximately 4.1 A and, with copper radiation (A = 1.54 A), the scattering angle 28 22". In the case of the crystal mats there is an additional periodicity of, for example, 100 A. This implies that 28 should be approximately 0.8". In practice, properly oriented mats of single crystals can give rise to five or more diffraction peaks, which represent higher orders of the Bragg reflections, i.e., n = 1, . . . , 5 in the above equation (Fig. 13). J

CRYSTAL CORE

A SINGLE CRYSTAL

PERIODIC REPEAT DISTANCE EQUAL TO THICKNESS OF A SINGLE CRYSTAL

FIG.12. An idealized arrangement of crystal cores and fold surfaces within a mat of lamellae. The periodic repeat is equivalent to the crystal core thickness plus two fold surface thicknesses.

162

6. X-RAY DIFFRACTION

I

I

1

2 28 DEGREES

FIG.13. A plot of log intensity vs. 20 for a mat of polyethylene crystals grown isothermally at 90°C from xylene. Three orders of reflection are clearly shown.

The d spacing obtained from such mats is often referred to as the fold period of the lamella. This repeat distance represents the perpendicular distance through one crystal core and two fold surfaces, i.e., the thickness of a single lamella. Two additional points need to be made. With single-crystal mats the fold period obtained from the first peak is in

6.2.

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

I63

excellent agreement with the thicknesses obtained for isolated lamellae using electron microscopy .42*55 Furthermore, good agreement with microscopy can be obtained without performing any of the corrections, which will be discussed later. Note that as previously stated any additional peaks represent higher-order reflections. It has, however, been noted that the position or indeed the presence of discrete diffraction peaks is affected by the manner in which crystal mats are made.56 6.2.3.3.2. MELT-CRYSTALLIZED SAMPLES. At this point we should consider some of the experimental results for melt-crystallized polymers. Here the predominant morphology is spherulitic and the lamellas that compose the arms of the spherulite will be randomly oriented with respect to the main beam. One should still observe diffraction, however, since within the spherulite there will be small regions where lamellae are stacked together and correctly oriented with respect to the main beam. This would be analogous to a powder pattern in wide-angle diffraction. Several points arise from experimentally reported SAXS data on melt-crystallized polymers, particularly rapidly cooled or quenched samples: 1. In comparison to single-crystal mats, melt-crystallized polymers generally show only a poorly resolved broad maximum or shoulder superimposed on a diffuse scattering background. 2. When two or more maxima are observed, the peak positions are generally not simple integer multiples of each other. That is, the peaks do not appear to be higher order reflections of the same periodic structure. 3. Observations by electron microscopy have not shown good agreement with sizes inferred from SAXS. The SAXS size is usually greater than that seen using microscopy. The above effects have been discussed in detail by Geil,57Kavesh and Schultz,JBand Burmester and Gei1.59 There are, however, a number of rather obvious points that should be made to help explain these observations. First, melt-crystallized samples generally have thicker lamellae than solution-crystallized polymers. In the crystallization of polymers it is generally accepted that lamellar thickness is inversely proportional to supercooling. The degree of supercooling is given by the difference between equilibrium melting point and crystallization temperature (for melt-crystallized), or the difference between dissolution temperature and A. Keller and A. O'Connor, Nurure (London) 180, 1289 (1957). S. Mitsuhashi and A. Keller, Polymer 2, 109 (1961). 's P. H. Geil, J . Polym. Sci.. Purr C 13, 149 (1966). S. Kavesh and J. M. Schultz, J . Poly. Sci., Purr A-2 9, 85 (1971). A. Burmester and P. H. Geil, Adv. Poly. Sci. Eng., Proc. Symp., 1972 p. 43 (1972). w,

38

I64

6.

X-RAY DIFFRACTION

crystallization temperature (for solution-crystallized single crystals). As it happens, one normally observes larger supercoolings in solution than in the melt and therefore shorter fold periods. This in turn means that the peak maxima for melt-crystallized samples will be at smaller angles, closer to the main beam. It is possible, therefore, that overlap with the main beam may produce an apparently broader, more diffuse diffraction profile. This can be readily checked by running a “blank” and removing background scattering. When this is done it can be seen that there remains a strong intrinsic background scatter at low angles. The origin and the exact form of this scatter is not known. Paracrystalline disorder or thermal type fluctuations have both been proposed,26although objections have been raised for each explanation.e0 Alternatively, one might suggest that such scatter may arise from randomly oriented lamella, which are not stacked to give constructive interference and hence discrete diffraction. Whatever the cause of this background scatter, it is possible to produce meltcrystallized samples where there is little effect of this background on the discrete peaks. An additional point is that single-crystal mats are usually made with crystals that have been isothermally grown. At this point we shall presume that such crystals all have the same thickness. In contrast, the majority of studies carried out on melt-crystallized samples have not used isothermally crystallized samples. One would therefore anticipate a distribution of lamella thicknesses and hence a broader, more diffuse diffraction peak than is observed with single-crystal mats. 6.2.3.4. Slit Collimation Effects. Because of more diffuse diffraction peaks, corrections that were largely ignored for crystal mats are of major importance for melt-crystallized samples. Probably the most important of these is for the so-called smearing effect of line or slit geometry in the source or detector systems. Ideally one should use pinhole collimation. This geometry gives information on orientation and most of the scattering equations used (particularly for diffuse scattering) are derived for pinhole collimation. The problem is that pinhole collimation leads to low diffraction intensity. Slit collimation of the source is therefore used to increase the intensity of the diffracted beam. Slits can be thought of as an overlapping sequence of pinholes. Perpendicular to the long dimension of the slit one has a pinhole-type size, which should lead to good resolution. Along the slit one has a large dimension, which should greatly increase intensity. R. Bramer and W. Ruland, Makromol. Chem. 177, 3601 (1976). J. M. Schultz, “Polymer Materials Science.” Prentice-Hall, Englewood Cliffs, New Jersey, 1974. 81

6.2.

CRYSTALLITE SIZE A N D LAMELLAR T H I C K N E S S

FRONT

SIDE

FRONT

I65

SIDE

Kl I

p C O U N T E R TRAVEL

(d 1

I

FILM

(el FILM

Intensity

t

I

I - 28 0 28 FIG.14. (a) Simple pinhole collimation; (b) a highly oriented mat of lamellae; (c) a random orientation of particles, each particle consisting of several lamellae stacked together; (d) the pinhole diffraction pattern from sample (b); (e) the pinhole diffraction pattern from sample (c); (f) using a slit shaped detector on sample (c) radiation is detected at smaller angles than the peak maximum appears using pinhole collimation; (g) dashed line represents the response of samples (b) and (c) with a pinhole detector and sample (b) with a slit detector. Solid line represents the intensity response from sample (c) with a slit detector; see (f).6'

166

6.

X-RAY DIFFRACTION

However, this large dimension interacts with the ideal pinhole pattern, causing distortion or “slit-smearing’’ of the diffraction pattern. The effects of slits on the diffraction patterns are purely geometrical ones. It follows that one should be able to derive geometrical formulas that allow one to desmear the curves. These have been reported in the literature and are given in greater detail in Section 6.2.3.10. It is possible to obtain a simple physical picture of the smearing process produced by slit geometry in the detector in the following way. Consider Fig. 14a, which represents a simple pinhole collimation system with a film detector. Let us examine two types of samples. First, consider an oriented crystal mat with lamellae stacked vertically in the plane of the page. Ideally, in this sample, on traveling from top to bottom one will detect periodic fluctuations in density due to the stacking of crystal cores and fold surfaces (Fig. 14b). Traveling perpendicularly from side to side or front to back through the sample no such fluctuations will be evident. The second sample can be visualized as being made by grinding the first sample such that each particle consists of several lamellae stacked together. The powder is recompressed into a plate producing a system of stacks of lamellae randomly oriented. This will be analogous to a meltcrystallized sample (Fig. 14c). If both samples are run using pinhole collimation, the oriented sample will produce sharp diffraction spots at a particular angle (fold period) and aligned along a particular direction (specimen orientation) (Fig. 14d). In contrast, the random sample should show diffraction at a particular angle (fold period) but in all directions relative to the main beam. That is, a diffraction “cone” will be produced (Fig. 14e). These patterns are analogous to single crystal and powder patterns in wide-angle X-ray diffraction. We now replace the films by a counter with a pinhole entrance and scan the counter from the zero position (main beam) out to higher angles. For both samples we obtain roughly the same information, namely, a diffraction peak at the same 28 value. If the pinhole in front of the counter is replaced by a slit, then the intensity profile obtained is strongly dependent on the sample used. In the case of the oriented sample little change is observed in peak position and intensity profile. However, for the random sample the slit takes thin slices of the diffraction curve (Fig. 14f) and the resulting intensity curve is shown schematically in Fig. 14g. There is a major contribution to the intensity at lower angles than that at which the diffraction maximum appears when using a pinhole in front of the detector. As a result the peak maximum is shifted to lower angles. This is a typical effect of slit smearing resulting from source or detector geometry. Note that application of the Bragg equation to the maximum in the slit-smeared intensity data, of a randomly oriented sample, would result

6.2.

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

167

in apparently larger fold periods than actually exist in the sample. The above description also shows that the smearing effect is much more pronounced in unoriented samples (melt-crystallized) than in oriented samples (single-crystal mats). Several authors have concluded that after slit-desmearing data from melt crystallized samples, the second diffraction maximum occurs at an angle very nearly twice that of the first m a x i m ~ m . ~ *The * ~ implication is that in these samples additional peaks are higher order reflections of the same structure. In contrast, other authors have reported that this is not the case.sQ Two additional corrections have been applied to SAXS curves from bulk polymers, which are routinely applied to wide-angle data.5Q These are the Lorentz and geometric corrections. The Lorentz factor depends on the time a given set of planes reflects X rays under experimental conditions and arises from the lack of true parallelism and monochromaticity of the incident beam. The geometric correction is due to the increase in the Debye-Scherrer ring circumference with diffraction angle. For random samples (melt crystallized) both corrections involved multiplying the measured intensity by 8. For highly oriented samples (crystal mats) only the Lorentz correction is justified.5Q (Note the application of the combined Lorentz geometric factors is still in Only people in the polymer area routinely apply these corrections.) When all these corrections are applied (background, desmearing, Lorentz, geometric) a number of changes appear in the diffraction curve (Fig. 15). Generally the definition of the peak(s) improves dramatically. More importantly, the peak maxima change position, with the low-angle peak moving more than the higher angle peak. However, there still remain differences between the sizes derived from SAXS data and those seen by electron microscopy. In addition, the peaks (apparently) still do not represent higher order reflections from the same structure.5Q 6.2.3.5.Lattice Distortions. Several models have been proposed to explain the latter observations and have been extensively reviewed by Crist.B4 This review is briefly reported here. SAXS of polymers is treated in terms of a disordered one-dimensional lattice. The lattice is composed of lamellar crystal cores, separated by disordered fold surfaces of amorphous regions. The lattice is disordered because the various models allow the crystal core size or the fold surface size to fluctuate in a variety of ways. One can calculate the effects of these size distributions on the structure of the lattice and therefore on the SAXS of the system. G . Kortleve and C. G. Vonk, Kolloid-Z. Z . Polym. 225, 124 (1968). Based on general discussions at the Fourth Int. Conf. Small-angle Scattering of X-Rays and Neutrons, Oct. 1977. &1 B. Crist, J . Polymer Sci.. Polym. Phys. Ed. 11, 635 (1973).

168

6.

X-RAY DIFFRACTION

%

BACKGROUND CORRECTED

0

I

2

D I F F R A C T I O N ANGLE 28. DEGREES

FIG.IS. Log intensity vs. 28 for a quenched sample of polyethylene, showing the effects of various corrections.

With the simpler distributions one can envision that the lattice is constructed in the following way. In two bowls are contained, respectively, the crystal core lengths of a certain distribution and the amorphous lengths with a separate distribution. The lattice is then constructed by chosing at random first a crystal length then an amorphous length from the appropriate bowl. If the average size 7 of the crystal lengths and Z of the amorphous lengths is known, then the average periodicity f of the lattice can be calculated. Intensity functions are then obtained by a Fourier transform of the electron density and size distribution functions. Intuitively, one would anticipate diffraction peaks at a 28 corresponding to R. In fact, according to tlie calculations, the peak maxima are shifted from f . Further, the direction and magnitude of the shift is a function of the model chosen and the magnitude of the shift is different for the first and second peaks. This means that the peaks seen do not appear to be higher order reflections of the same structure. Note that this is a direct result of allowing a distribution of crystal core or amorphous surface thicknesses. The different one-dimension models that have been considered here are the Reinhold, Tsvankin, and Gaussian distributions. 6.2.3.5.1. REINHOLD DISTRIBUTIONS. Reinhold et assumed that all the lamellae are of the same size, and the size distribution of amorC. Reinhold, E. W. Fischer, and A. Peterlin, J . Appl. Phys. 35, 71 (1964).

6.2.

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

I69

phous regions is an asymmetric function as given by

=

0

for y > 0, x < (1 - 2 y ) L a , for y < 0, x > (1 - 2 y ) L a ,

where La is the mean thickness of the amorphous region. The ratio of the peak positions (second to first) depends on the type of distribution function. For a positive skewed distribution ( y > 0), the ratio 2 e 2 / 2 e 1 is greater than 2 and the mean lamellar thickness is larger than that implied by application of the Bragg equation to the observed first peak maximum. The reverse is true for a negative skewed distribution ( y < 0). 6.2.3.5.2. TSVANKIN DISTRIBUTIONS. Tsvankinss assumed that the distribution of amorphous lengths followed an asymmetric, exponential function and the crystalline core is rectangular with width of 2 A . The main characteristics of this model are: 1. The sharpness and intensity of the diffraction peaks decrease as the dispersion of crystal lengths are increased. 2. When the ratio of the average length of crystal and the average amorphous length is larger than 10, then the fold period obtained from the first peak maximum is approximately the average lamellar thickness. This implies that the higher the degree of order within the lamellae, the more accurately will the lamellar thickness determined from SAXS reflect the average thickness. 3. In a special case, no fluctuation of the crystal length was permitted. This model effectively behaves as a type of positive Reinhold distribution. 4. A transition zone of length t was also introduced. In this zone the density decreased linearly from pe to pa, the density of crystal core and amorphous region, respectively. This led to a trapezoidal density profile for the lattice (Fig. 6 ) . Such a transition zone is assumed to be a more realistic model for the interface between crystal core and fold surface than the usual assumption of a sharp interface. The ratio 2OZ/2e1is greater than 2 for a model containing a transition zone. 6.2.3.5.3. GAUSSIAN DISTRIBUTIONS. Blundella7considered the distributions of crystal and amorphous lengths as symmetric Gaussian functions. A distinct two-phase model and one with a transition zone were also included. The effects of the transition zone in this model was just to decrease the peak intensity. The first peak maximum is observed at 88

D. Ya. Tsvankin, Polym. Sci. USSR (Engl. Trans/.) 6, 2304 and 2310 (1964). D. S. Blundell, Acia Crysiallogr., Sect. A 36, 472 and 476 (1970).

170

6.

X-RAY DIFFRACTION

slightly smaller angles than the mean lamellar size (3would imply. The ratio of peak positions, 2e2/2e1, is between 1.85 and 2.0. AND DISCUSSION. Crist has summarized the peak 6.2.3.5.4. RESULTS positions of the first and second discrete diffraction maxima. These are shown in Table VII. Remember s = 2 sin e/x, which means the Bragg equation can be written n = sT. That is, a structure with an average periodicity of R should have maxima at s.t values of 1, 2 etc. As the table shows, this is not always the case for polymer systems where distributions of crystal and amorphous lengths may be present. This implies that simply taking the position of the first peak maximum and using the Bragg equation can lead to erroneous values for “mean lamellar thickness”. Crist has estimated that such an approach can lead to errors as great as k 35% of the mean size. From the table it is apparent that for certain models 2e2/2e1is greater than 2, while for others it is less than 2. Crist suggests that a better approach for analyzing melt-crystallized polymer data is to determine the experimentally observed 2e2/2e, ratio. On the basis of the observed ratio and comparing experimental data with model predictions for half-width, intensity of peaks, etc., it should be possible to select an appropriate model for the system. In a recent paper, one of the few that examines isothermally meltcrystallized polymers, the authors have used SAXS, low-frequency Raman spectroscopy, and electron microscopy.68 The authors of this study conclude that crystallization time, molecular-weight segregation, isothermal lamellar thickening, and nucleation density all play major roles in determining the fold period or periods present in the sample. In general, however, the authors conclude that two kinds of lamellae can exist in TABLEVII. Peak Positions of Scattering Curves Lattice Reinhold vw

0.7- 1 .O 0.9-1.35

1.2-2.0 2.0-2.4

1.7-2.0 2.0-2.2

0.75- 1.05 0.95- 1.5 0.9- 1 .O

1.9- 2.25 2.0-2.5 1.8-2.0

1.95-2.15 2.0-2.4 1.85-2.0

Tsvankinb t = O r>O

Gaussian a

Results depend on crystallinity. Results depend on choice of A.

J. Dlugosz, G. V. Fraser, D. Grubb, A. Keller, J. A. O ’ k l l , and P. L. Goggin, Polymer 17, 471 (1976).

6.2.

171

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

melt-crystallized polymers under certain crystallization conditions. As crystallization time increases then a single lamella thickness is observed, and multiple peaks in SAXS should be interpreted as higher-order reflections of the same entity. 6.2.3.6. Finite Macrolattice Effects. In the previous section we talked of lattice distortion. Strictly speaking, we should refer to a macrolattice, since the crystal core itself contains a lattice. Additional information can be obtained from the line width of the discrete diffraction peak(s).26.6s Considering oriented mats of single crystals that were grown isothermally, one would anticipate little fluctuation in lamellar thickness. The macrolattice containing such lamellae should therefore contain little disorder. Hosemann26 describes two types of so-called paracrystalline disorder (see Chapter 6.1 or Section 6.2.2.4.1.1). In a one-dimensional lattice the units (atoms or lamellae) are displaced from the equilibrium positions prescribed by the ideal lattice points. These are type I defects; the long-range order of the lattice is preserved. In type I1 defects the long-range order of the lattice is rapidly lost as each unit varies in position only in relation to its nearest neighbor. In the various models so far described we have been concerned solely with type I1 defects. Given that such fluctuations may exist in crystal mats the broadening anticipated would be quite &all. However, another source of broadening may be present; namely, that due to the fact that the macrolattice is not infinite. This is analogous to line broadening due to crystallize size in wide-angle X-ray diffraction. Equations have been derived, similar to those used in wide-angle X-ray work, to separate out the two contributions to line broadening, namely, finite lattice size and paracrystalline d i ~ o r d e P * ~ ~ : An/2O1

=

[(l/NZ)+ ( m ~ g , ) ~ ] ' ' ~ .

(6.2.57)

Here An is the width at half height of the nth-order reflection, 2e1the scattering angle of the first-order reflection, N the average number of particles in the lattice, and g, is usually written as Ax/R, where R is the average lattice periodicity (fold period) and Ax represents the displacement from that mean. It follows that a plot of (An/2e1)2 vs. n4 should have intercept at n = 0 of 1/N2 and slope of ,rr4gZ4. One can therefore derive N, the number of lamellae stacked together, and from g, obtain Ax, the mean fluctuation in size. As a further check the equationZ6 n = O.35/gz

(6.2.58)

predicts the number of orders that should be observed for a calculated g , . gs

B. Crist and N. Morosoff, J . Polym. Sci.,Polym. Phys. Ed. 11, 1023, (1973).

172

6.

X-RAY DIFFRACTION

In a recent examplese on a single crystal mat, N = 6.3 and g, = 0.09, representing a mean fluctuation of approximately 10 A. Furthermore n = 0.35/0.09, or approximately four orders of reflection should be visible; three were clearly observed. Note that for some crystal mats as many as ten reflections have been observed, which would imply that the mean fluctuation is only on the order of 4 A. Similar analyses have been carried out on melt-crystallized polyethylene, which was subsequently cold drawn and then examined using pinhole c ~ l l i m a t i o n .With ~ ~ somewhat more sophisticated methods it is possible to calculate both g, and N in the draw direction and at right angles to it. This leads to a model for drawn polymer consisting of fibrils that have a certain periodicity in the fiber/fibril direction. In addition, the data imply that the fibrils themselves have a mean diameter and are grouped into clusters of 1, 4, and 16 fibrils. This latter analysis is based on diffuse scattering from rods with a circular cross section. A more detailed description is beyond the scope of this chapter. One objection that should be stated, however, is that application of the Guinier equation (Section 6.2.3.7.1)is usually only valid at dilute concentrations of one phase in another.?' In addition, Guinier himself has questioned the validity of decomposing a Guinier plot into two or more straight line portions and deriving meaningful sizes from the slopes of the lines.?* 6.2.3.7. Lamellar Thickness from Diffuse Scattering. Let us examine how the information from diffuse scattering may be used to measure fold period or lamella thickness from solution- and melt-crystallized polymers. Diffuse SAXS results from electron density differences that exist between randomly oriented particles and a surrounding medium. Diffuse scattering intensity decreases with increasing diffraction angle. A homogeneous system would not give any small-angle scattering. However, even pure liquids give some SAXS, indicating the presence of inhomogeneities. In a dilute system, particles are randomly dispersed in a second phase and are sufficiently separated so that there is no interparticle interference. The intensities scattered by the various particles are simply additive. At small angles the scattering curve depends on the size and shape of the particles. There are two possible ways in which diffuse scattered intensity data may be useful. As previously mentioned, at low angles there is a major component in the scattering curve from melt-crystallized samples. This R. Bonart and R. Hosernann, KoNoid-Z. Z . Polym. 186, 16 (1962). A. Guinier and G . Fournet, "Small Angle Scattering of X-Rays." Wiley, New York, 1955. 7* A. Guinier, Private communication. 70 71

6.2.

CRYSTALLITE SIZE AND LAMELLAR THICKNESS

173

may be due to the scattering from randomly oriented lamellae. It would be desirable to compute or experimentally determine this low-angle component and remove it from the discrete-scattering profile. This may well change the apparent positions of the diffraction maxima. Further, from the shape of this diffuse component one should be able to determine the fold period of the lamellae. This would act as a check on the dimensions obtained from the discrete diffraction peak(s). When single crystals from solution are sedimented as mats their original structure undergoes some reorganization. This may lead to the observed fluctuation in thickness. It would be desirable therefore to measure the fold period of crystals still in suspension (as formed). Two basic methods are available to do this. 6.2.3.7.1. GUINIER APPROACH.For a thin disk of uniform shape, size, and electron density, the scattering intensity is approximated by7I 21, n2 h2H2 21,n2 l(h) =(hR)2exp ( 7 = 02 ) e w ( - h2Rd2>, f =

(6.2.59)

2H = (12)1’2R,j

where h = 4~ sin e/x, I, is the Thomson coefficient of a single electron, n the total number of effective electrons in the particle, 2R and 2H the diameter and thickness (t), respectively, of the thin disk, and Rd the radius of gyration in the thickness direction. The thickness of the disk can be determined from a Guinier plot: h[1(h)h2] = h[l(h)h2]h+o

-

h2Rd2.

(6.2.60)

From the slope of the line one obtains Rd and therefore 2H, the disk thickness. These equations are valid for all homogeneous platelets provided that the dimensions in two directions are much greater than the third (thickness) direction. 6.2.3.7.2. INVARIANT METHOD.The integral quantity Qh is referred to as the invariant and defined by P ~ r o as d~~ Qh =

lom I(h)h2 dH.

(6.2.61)

The invariant depends on the volume of particles and is independent of their shape:

Qh = 2,rr2Z,p2V,

n = pV,

(6.2.62)

where V is the volume and p the electron density. The thickness of a thin disk can be readily derived from Eqs. (6.2.59)and (6.2.62),assuming that 7s

G.Porod, Kolloid-2. 124, 83 (1951).

I74

6.

X-RAY DIFFRACTION

the disks are of uniform shape, size, and electron density: (6.2.63)

Most platelet-like polymer single crystals will have their large dimensions on the order of 10 pm. This approximates 2R in the previous equations. The thickness ( 2 H ) is close to 100 A. Provided that the ratio 2R/2H is greater than 10-15, the above methods are valid. 6.2.3.7.3. MODIFICATIONS OF THE GUINIER APPROACH.There is, however, a very important restriction on both the Guinier and invariant methods; both assume that the “disks” are of uniform electron density. Based on the usually accepted models of single crystals this is obviously not the case. In addition, Udagawa and Keller7*have studied mats of polyethylene (PE) single crystals. They reported that lamellar thickness increased on addition of potential solvents such as xylene, octane, and decalin to the dried mats. It was proposed that the solvents could penetrate and swell the fold surfaces. The fold surface is envisioned as containing large loose loops and buried folds. Recently it has been reported that the crystallite sizes (mosaic blocks) of uncollapsed lamellae, in paraffin oil suspension, are significantly larger than those observed for crystals in the dried state.75 It was concluded that the morphologies of single crystals in the dried and uncollapsed states are much different. This observation is also in line with observations in the electron m i c r o ~ c o p eand ~ ~ more recent observations made using Fourier transform infrared s p e c t r o s ~ o p y . ~ ~ As previously stated it would be desirable, for a better understanding of the morphology of crystals, to observe or measure the crystals in their original uncollapsed form. This implies that the Guinier equation should be modified to accommodate particles having two distinct phases. In addition, this modified equation should be flexible enough to allow for different degrees of interaction between suspending media and the amorphous fold surfaces. Such a modification has been carried out and the equation applied to polyethylene crystals held in suspension in two different solvents.’* On the basis of these experiments the authors conclude that there is little or

Y. Udagawa and A. KeUer, J . Polym. Sci.. Purr A-2 9, 437 (1971). I. R. Harrison and J. Runt, J . Polym. Sci., Polym. Phys. Ed. 14, 317 (1976). 76 E. L. Thomas, S. L. Sass, and E. J. Kramer, J . Polym. Sci., Polym. Phys. Ed. 12, 1015 74

(1974).

P. C. Painter, J. Runt, M. M. Coleman, and I. R. Harrison, J . Polym. Sci., Polym. Phys. Ed. 15, 1647 (1977). J.-I Wang and I. R. Harrison, J . Appl. Crystallogr. 11, 525 (1978).

6.2.

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

I75

no interaction between crystals in suspension and the suspending liquids. This is true even for the potential solvents, which are known to swell dry mats. It is further concluded, on the basis of the above diffuse scattering studies, that polyethylene crystals in suspension are approximately 90% crystalline. This is some 5-10% higher than the accepted value for such crystals. However, this figure (90%) is in good agreement with values recently derived from heat of fusion measurements of crystals in suspenion.'^ 6.2.3.8.Combination Methods. More sophisticated combinations of discrete and diffuse scattering have been reported recently by a number of a ~ t h o r s . ~ OTypically, -~~ these authors assume a two-phase system and with this assumption it is possible to calculate: 1. the volume percentage crystallinity using a wide-angle technique, 2. the overall fold period and the individual thicknesses of the crystalline and amorphous components, 3. electron density differences between the crystalline core and the amorphous portion, 4. a measure of the “roughness” of the interface between the crystal core and the amorphous part, 5 . the detection of deviations from the ideal two-phase structure, and 6. the range of lamella thicknesses within a particular ample.^^,^^ Undoubtedly, further refinement of these techniques, coupled with the rapid data collection possible with position-sensitive detectors, will help clarify both the nature of a melt-crystallized sample and the crystallization process.8s 6.2.3.9.The Correlation Function. In reviewing the literature the reader will come across the correlation function. This can be defined in one, two, or three dimensions. The correlation function C(r) is defined as the probability that two points j and k , a distance r apart in any direction, lie in the same phase, i.e., the probability that they have the same electron density. It can be expressed as (6.2.64) G(r) = ( APJ A P ~/() b2) 3

I . R. Harrison, J . Runt, L. Stanislow, and D. Bell, J . Polym. Sci., Polym. Phys. Ed. 17, 63 (1979). R. Perret and W. Ruland, Kolbid-Z. Z . Polym. 247, 835 (1971). J. Rathje and W. Ruland, Colloid Polym. Sci. 254, 358 (1976). G. R. Stroble, J . Appl. Crysfulloyr. 6, 365 (1973). Bs W. Ruland, Colloid Polym. Sci. 255, 417 (1977). 0. A. Pringle and P. W. Schmidt, J . Colloid Interface Sci. 60,252 (1977). I. S. Fedorova and P. W. Schmidt, J . Appl. Crystalloyr. 11, 405 (1978). J. M . Schultz, J . Polym. Sci., Polym. Phys. Ed. 14, 2291 (1976). ‘0

176

6. X-RAY DIFFRACTION

where Ap, is the difference between the true electron density p, at pointj from that of the average electron density /T. Thus Ap, = p1 - /T and Apk = p k - i j . The term ( A p 2 ) is the overall average of the square of the density fluctuations. If the two points (jand k ) are at the same position, i.e., r = 0, then their electron densities must be equal and therefore the correlation function at that point has a value of unity [a probability of one; G(0) = 13. As the two points become separated, then the probability of both being of the same electron density diminishes. If r = m, then G(m) = 0. The values of G(r) between these extremes depend on the system under investigation. When a diffraction profile shows no maxima, i.e., diffuse scattering, then a plot of the correlation function vs. the distance ( r ) separating the two points also exhibits no maxima. This type of scattering arises from a random distribution of two distinct phases, each of uniform electron density. Thus, as the two imaginary points become further and further separated, the probability of their lying in the same phase diminishes. However, in discrete scattering, where one observes a periodicity in electron density, the correlation function will have maxima that reflect the positions of the repeat distances. It can be shown that the correlation function is related to the scattered intensity by

Z(h) = K

lorn?

sin hr dr,

(6.2.65)

where h = 4.rr sin O/X and K is the constant for the system, involving electron density and scattering volume. Equation (6.2.65) can be inverted by a Fourier transformation to give G(r) = C

lom

sin hr h2Z(h)hr d h ,

(6.2.66)

where C is a constant. From Eq. (6.2.66) the correlation function can be evaluated by somewhat lengthy procedures, which require the use of a computer. Inspection of Eq. (6.2.66) reveals that it contains the variables h , Z(h), and r. The calculations involve the following steps: 1. A set of r values is selected, e.g., 1, 5, 10, 15, 20, 30, 125, 150, 175 A. 2. For any r value, the entity

sin hr h21(h) -= A hr

. . . , 100,

6.2.

CRYSTALLITE SIZE AND LAMELLAR THICKNESS

177

must be evaluated for each point on the scattering curve [a wide range of Z(h) and h]. 3. Then, A dh is integrated from 0 to ~0 (G(r) 0: J$ A dh). 4. The calculations are repeated for each value of r and finally one can plot the curve of J," A dh vs. r . The values of J," A dh are normalized so that boundary conditions are satisfied, i.e., at r = 0, J," A dh = 1. To achieve the above, one must use desmeared intensity data and the integration in step 3 has to be evaluated accurately, which involves the use of truncation approximations. This appears to be a pointless exercise if all one wants is a measure of the fold period. However, a couple of points should be kept in mind. The correlation function often shows more readily discernible maxima than the original scattering This is particularly true for meltcrystallized samples (see Fig. 16). In addition, the total scattering curve is used, which means that the correlation function contains information from all parts of the scattering system. In theory, it is therefore possible to determine a wide range of sample parameters from this function.88 6.2.3.10. Slit Collirnation Correction. Although theoretical and experimental methods of SAXS have been developed and refined since they were first introduced by Guinier in 1939,88many people are still reticent about employing this technique in their studies. This is in part due to the feeling that the evaluation of SAXS data requires a lot of mathematical manipulation. In particular, the slit length correction or slit-desmearing procedure is somewhat discouraging. In this section slit collimation effects and desmearing methods will be discussed in the hope that the reader will be encouraged to apply the available desmearing computer programs to obtain pinhole intensity data directly from the experimental results. As previously noted the scattered intensity using a pinhole collimation system is often very weak and several hours are needed to record data. In order to increase intensity and decrease experimental time, a slit collimation system is often used. The slits are rectangular in shape; slit width is on the order of 10-500 pm and is much smaller than the length of the slit (10-30 mm). As a result of the slit geometry the measured intensity is an average of the scattered intensity over an appreciable angular range about the true scattering angle. Due to these smearing effects, the measured intensity curve is greatly distorted from the perfect pinhole intenC. G . Vonk and G . Kortleve, Kolloid-Z. Z . Polym. 220, 19 (1967). S. Ergun, Chem. Phys. Carbon 3, 211 (1968). A. Guinier, Ann. Phys. (Leipzig) [5] 12, 161 (1939).

I78

6. X-RAY

DIFFRACTION

-4

0

200

400

600

DISTANCE ( A )

FIG.16. The correlation function calculated from the fully corrected data shown in Fig. 15. The average repeat by this technique is 247 A, based on the position of the peak maximum.

sity curve. A desmearing procedure is therefore required to restore the smeared intensity to that of the pinhole intensity curve for further analysis and interpretation of the experimental data. Note that with partially oriented samples (samples intermediate between those shown in Fig. 14b,c) the desmearing procedure is not valid. As a result, pinhole collimation should be used on this type of sample. As stated, the purpose of the slit collimation systems is to obtain higher scattered intensities. The slits, of course, are collimating the main beam and arranged tG reduce parasitic scattering from slit edges. One such arrangement found in the Kratky camera is shown in Fig. 17.0° In this particular camera, blocklike slits are used to essentially eliminate parasitic scattering on one side of the main beam. Since slit length is much larger than slit width, a much larger “smearing” effect is obtained from slit length compared to slit width. Experimentally, one obtains the scattered intensity as the intensity measured at the detector slit registration plane. The measured scattering angle is the angle between the camera axis and the center of the detector slit. R. W. Hendricks and P. W. Schmidt, Acta Phys. Austriaca 26, 97 (1967).

'\

FIG. 17. An isometric view of the Kratky small-angle collimation system.

180

6 . X-RAY DIFFRACTION

1. Slit width effect. The dimensions of slit width are of the order of microns. As a result the effect on the scattering curve is relatively small. The slit width weighting function can be calculated theoretically or it can be measured (the main beam profile) with no sample in the system. The normalized slit width weighting function of a typical collimated system is shown in Fig. 18a.91 2. Slit length effect. The effects of slit length on the discrete SAXS are as shown in Figs. 14 and 15, namely, (1) the peak maxima of the diffracO1 J. W. Anderegg, P. G. Mardon, and R. W. Hendricks, ORNL-4476. Oak Ridge Natl. Lab., Tennessee, 1970.

ANGLE (milliradians)

FIG.18 (a) Normalized slit-width weighting function, i.e., main beam profile in the r direction (Fig. 17); (b) Normalized slit-length weighting function, i.e., main beam profile in the y direction (Fig. 17).

6.2.

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

181

tion profile are obscured and with the exception of scattering from oriented mats one normally observes shoulders not discrete peaks, (2) the peak maxima tend to shift to smaller angles. A normalized slit length weighting function for a collimated system is shown in Fig. 18b.O' If the slit length weighting function is uniform over the range IyJ s Em,, and the scattering intensity of the sample becomes negligibly small for scattering angles larger than IE,,, 1, then this slit length can be considered infinite. 6.2.3.10.1. THEORY OF SLIT COLLIMATION CORRECTIONS. The measured slit-smeared intensity J ( h) is related to the perfect or pinhole collimation intensity Z(h) by the relation'l J ( h ) = J J w H ( y ) W , ( t ) l [ ( h - r)' +

dt

(6.2.67)

where W Hand W , are the weighting functions of slit length and slit width, respectively, and y and t are variables with the same dimensions as h measured in the directions shown in the receiving plane, Fig. 17. For a slit-collimated system, the slit width dimension is much smaller than slit length. The slit-weighting functions are independent and can be treated separately. By rearranging Eq. (6.2.67), one obtains J(h) = J W , ( t ) F ( h - t ) dt,

(6.2.68)

where F(h

- ?) =

JWH(y)Z[(h -

?)2

+ y']"'

dye

(6.2.69)

The slit width and slit length effects are represented by Eqs. (6.2.68) and (6.2.69), respectively. Distortion due to slit width is small and is often corrected by expanding F(h - t) in Eq. (6.2.68) into a Taylor series about the point h:

where

M K = f t K W , ( t ) dt. As a first approximation, only the first two terms on the right-hand side of Eq. (6.2.70) are considered. Setting M , = 0 implies that the smeared intensity curve and the slit width corrected intensity curve can be set equal. Putting M1 = 0 means that the slit width correction represents a shift of the origin (zero angle) to the fist moment of the slit width weighting function. With a correctly aligned Kratky unit, an approximately symmetric slit width profile can be obtained. The first moment of the profile may then be taken as the position of the peak maximum. After the slit width

182

6.

X-RAY DIFFRACTION

correction, Eq. (6.2.67) becomes J(h) = WH (y)I[(h2

+ y2I1l2dy.

(6.2.71)

6.2.3.10.2. METHODS USED FOR SLITCOLLIMATION CORRECTION. A number of slit length correction methods have been developed by Schmidt el ~ 1 . , @Heine ~ and R ~ p p e r t , @ Kent ~ and Brumerger,e4 M a ~ u r , @ ~ Lake,@6and Vonk.@' These desmearing procedures have been derived based on different concepts and schemes. Only a limited number of the more commonly employed methods will be discussed: 6.2.3.10.2.1. Iterative method. Lakess has developed an iterative procedure in which I(h) is successively approximated by the expression (6.2.72)

For the first iteration, I,(h) is assumed to be equal to J(h), the measured intensity. Then J,(h) is calculated by integrating Eq. (6.2.71) and compared to J(h). For the next iteration, I N + , (h) is modified as described by Eq. (6.2.72). These procedures are repeated until JN+,(h) = J ( h ) . With the availability of high-speed computers the iterative method is a widely used,numerical procedure. 6.2.3.10.2.2. Matrix method. VonkP7approximated the integral of Eq. (6.2.71) by the summation

JW =

x

W ~ ( r j 2- V)'121(rj)AYJ,

(6.2.73)

j

where r

=

(h2 + y2)'12. Then Ayj

=

(r: - ht)lI2 - (rj-:

-

h:)Il2.

(6.2.74)

By selecting Ar = by, Eq. (6.2.73) can be written into a matrix form (6.2.75)

where Aij = WH(r: - h:)1/2[(r,2- h?)'l2 - (rJ-: - h:)1/2].

The desmeared intensity can then be easily solved by matrix methods. P. W. Schmidt, Acfa Crystallogr. 19,938 (1965); J. S . Lin, C. R. von Bastian, and P. W. Schmidt, J . Appl. Crystallogr. 7, 439 (1974). S. Heine and J. Roppert, Acfa Phys. Ausfriaca 16, 144 (1963). P. Kent and H. Brumberger, Acfa Phys. Ausfriaca 17, 263 (1964). O5 J. Mazur, J . Res. Natl. Bur. Stand., Sect. B 75, 173 (1971). J. A. Lake, Acra Crysfallogr. 23, 191 (1967). rn C. C. Vonk, J . Appl. Crystallogr. 4, 340 (1971).

6.2.

CRYSTALLITE SIZE A N D LAMELLAR THICKNESS

I83

However, errors in the experimental data are magnified by the desmearing procedure. This necessitates the use of statistically good data, or a smoothing procedure on the experimental results. Recently VonkW has added some optional schemesggto the original desmearing program to make this revised program more extensive and flexible. The main features of the new program are (i) data manipulations, correcting for background scattering and slit width effect; (ii) tailfitting, fitting the “tail” of smeared intensity by least-squares method; (iii) invariant, evaluation of the invariant from J s J ( s ) ds; (iv) desmearing, slit length correction; (v) correlation function, one- to three-dimensional correlation functions; (vi) particle size distribution functions. 6.2.3.10.2.3. Numerical differentiation method. Guinier and FoumetlOoand DuMond’Ol have derived a solution of Z(h) for infinite slit length by utilizing the derivative of the smeared data:

Z(h) =

2

--

I

=

(y2

J’(Y)

- h2)1/2

(6.2.76)

Schmidtg2evaluated the integral in Eq. (6.2.76) by using the following approximations: 1. The integral is approximated by a summation. 2. In order to evaluate the derivative, the measured intensity J ( h ) is fitted to a polynomial using a least-squares method on six neighboring data points. After some mathematical manipulations, Z(h) can be obtained in the form

(6.2.77) where Ti,is computed from the settings of the collimation system and is the same for all scattering curves measured at the same settings. For a series of slit corrections, this method needs little computer time. The general features of this method are: 1. Equal angular increments are required; the experimental data must either be interpolated or measured at equal increments. 2. Uncertainty in the corrected intensity can be estimated if all the measurements are recorded at equal counting time and the random errors are proportional to the square root of counts recorded.

c. G . Vonk, J . Appl. Crystallogr.

8, 340 (1975).

Vonk, J . Appl. Crystallogr. 9, 433 (1976). IMA. Guinier and G . Fournet, J . Phys. Radium [8] 8, 345 (1947).

en C. G. lol

J. M. W. DuMond, Phys. Rev. 72, 83 (1947).

184

6.

X-RAY DIFFRACTION

6.2.4. Summary

The X-ray techniques described in the previous sections provide a means of determining two basic sizes, so-called crystallite size and lamellar thickness. Wide-angle methods allow one to determine the perpendicular distance through a set of planes that are in register. If several reflections are available one can determine the overall shape of the scattering unit. If the planes in question are normal to the chain axis, then with a lamellar crystal one obtains a measure of the crystal core thickness. Additional information can be obtained if higher orders of the same reflection are present. One can, for example, determine the amount of lattice distortion due to paracrystallinity or strain. Crystallite size determined by either Fourier techniques or using integral breadth allows one to calculate a “number average” or “weight average” size. SAXS techniques allow one to measure the fold period or average thickness of a set of lamellae. This corresponds to the crystal core thickness plus the thickness of two fold surfaces. If the wide-angle method can be used to obtain a crystal core thickness, one can therefore obtain the thickness of the fold surfaces. From the width of the SAXS discrete diffraction peaks it is possible to obtain both a “crystallite size” and a measure of the fluctuation in fold period. Crystallite size by SAXS corresponds to the size of the macrolattice built up of lamellar units. Melt-crystallized samples present a slightly more complex situation. This primarily stems from the fact that not enough work has been reported for isothermally produced samples. As a result it is difficult to separate out the variables that could lead to the experimentally observed broad diffraction peaks and peaks tEat are apparently not simply higher orders of the same reflection. Diffuse SAXS scattering has been used to examine single crystals in their “as-formed’’ state. Such a study supports previous observations that dried-down crystals are significantly different from their as-formed precursors. The treatment of melt-crystallized data by combinations of discrete and diffuse techniques, coupled with the rapid data collection potential of position-sensitive detectors, appears to offer the greatest hope in understanding the structure and morphology of melt-crystallized samples.

7. Electron Microscopy By Richard G. Vadimsky 7.1. Introduction Knowledge of the microstructure of materials was markedly advanced with the advent of the optical and electron microscopes. While the unaided eye can resolve features no smaller than -0.1 mm in size, the optical microscope allows perception of textures 1/500 of this limit. The electron microscope provides a similar improvement in imaging power over even the most advanced optical microscopes. In fact, electronmicroscopic resolutions approaching interatomic distances of some solids (2-3 A) have frequently been demonstrated. Despite its capability of providing microstructural information, however, the electron microscope is often underutilized because of the experimenter’s unfamiliarity with the instrument. In an attempt to correct this situation, this part is written to give the reader a conceptual understanding of the electron microscope, thereby enabling him to recognize its applicability to specific problems in polymer physics. In this field, the electron microscope has been shown to be an indispensable investigative tool. As we shall see later, detailed examination of discrete polymer single crystals is possible only with the electron microscope. Studies of these simplest of polymeric forms have led to an understanding of the fundamental crystallization mechanisms of polymers. Complementary techniques, such as low-angle X-ray diffraction, provided insight into this understanding, but it was selected-area and small-angle electron diffraction along with dark- and bright-field imaging that has lead us to our present-day picture of crystalline polymers. The theory presented in this part should provide a good foundation for further study of electron microscopy, while the details included should assist the researcher embarking on his fist electron microscopic investigation. An elementary discussion of the fundamentals of electron microscopy and electron optics will precede a more detailed, but not rigorous, treatment of instrumental design and operational considerations. One chapter will be devoted to specialized microscopy techniques, including I85 METHODS OF EXPERIMENTAL PHYSICS, VOL. 16B

Copyright 01980 by Academic Press, Inc. All rights of reproduction in any form reserved ISBN 0-12-475957.2

186

7.

ELECTRON MICROSCOPY

scanning-transmission electron microscopy, scanning-electron microscopy, and energy-loss electron microscopy. (The term electron microscopy used in this part refers to work performed on the instrument commonly called the conventional transmission electron microscope. Our discussion is concentrated on this type of instrument because of its widespread use and because knowledge of its operation is basic to a study of the new generation of instruments.) The part will be concluded by examining specific applications in polymer science, which hopefully will adequately demonstrate the instrument’s capabilities. In addition to specific references, a number of general worksl-g are included for background study.

7.2. Fundamentals 7.2.1. Particle-Wave Concept

Our discussion must begin by pointing out that the dual nature concept of light also holds for electrons. That is, an electron beam can be viewed as a bundle of discrete particles of vanishingly small size and, concurrently, as a series of waves. The former concept is needed to explain such phenomena as electron emission and absorption, and is fundamental to the later discussion of lens theory. The wave concept must be invoked to explain the effects of confining electron rays to apertures whose sizes V. K. Zworykin, “Electron Optics and the Electron Microscope.” Wiley, New York, 1945.

* D. Kay, “Techniques for Electron Microscopy.” Blackwell, Oxford,

1961.

R. D. Heidenreich, “Fundamentals of Transmission Electron Microscopy.” Wiley (Interscience), New York, 1964. C. E. Hall, “Introduction to Electron Microscopy,” 2nd ed. McGraw-Hill, New York, 1966.

0. Klemperer, “Electron Optics,” 3rd ed. Cambridge Univ. Press, London and New York, 1971; P. Grivet, M. Y. Bernard, F. Bertein, R. Castaing, M. Gauzit, and A. Septier, “Electron Optics” (transl. by P. W. Hawkes). Pergamon, Oxford, 1965. P. H. Geil, “Polymer Single Crystals.” Wiley (Interscience), New York, 1963. ’ P. Hartman, Phys. Chem. Org. Solid Srare 1,369 (1963); W. J. Dunning, ibid. p. 41 1; H. D. Keith, ibid. p. 461. P. B. Hirsch, A. Howie, R. B. Nicholson, D. W.Pashley, and M. J. Whelan, “Electron Microscopy of Thin Crystals.’’ Butterworth, London, 1965. L. Marton, “Early History of the Electron Microscope.” San Francisco Press, San Francisco, California, 1968. On L. Marton, C. Marton, and W. G. Hall, “Electron Physics Tables,” Circ. No. 57. U.S. Natl. Bur. Stand., Washington, D.C., 1956. * L. Marton, Lab. Invest. 14, 739-745 (1965).

7.2.

FUNDAMENTALS

187

are comparable to the wavelength of the rays.? Wave theory also provides a foundation for the study of electron diffraction by solids. 7.2.2. Image Formation$

Images produced in an electron microscope arise from the interaction of a high-energy electron beam with a thin-film specimen. This interaction manifests itself in various forms of electron scattering that are primarily dependent on material characteristics, e.g., thickness, density, crystallinity. The scattering processes include elastic scattering, inelastic scattering, and absorption. Let us examine these processes, their dependence on material characteristics, and their contribution to image formation. 7.2.2.1. Electron Scattering§. When a fast electron enters a solid specimen it falls under the influence of the atomic charges of the host atoms. A simplistic representation of what occurs appears in Fig. 1. In the first case, a fast electron may be deflected from its original trajectory with negligible loss of energy. The magnitude of the deflection of this elastically scattered electron is dependent on the size and charge of the participating atom as well as the original trajectory and energy (velocity) of the beam electron. In general, few electrons traverse the normal transmission specimen and experience only elastic scattering. Most electrons will be inelastically scattered. As seen in Fig. 1, this means that a fast electron can pass close enough to a core electrony (assumed to be unaffected by binding) to transfer energy to it and thereby cause the atomic electron to be either ejected from the atom or forced into an allowed excited state. In addition to losing some of its energy as a result of this “collision,” the beam electron also assumes a new trajectory. The total amount of inelastic scattering obviously depends upon the severity and number of such electron-atom interactions. These factors are in turn dependent on the mass thickness (density and thickness) of the specimen. A thick and/or dense sample, which scatters electrons readily, is said to possess a high scattering power. t The wavelength of the electron is determined from the relation A = 12.25/V2, where A is the wavelength in angstroms and V the voltage (energy) of the electron. (See also Marton er u/.Oa) $ See also Martomgb 5 See also Vol. 7B of this series, Chapter 9.2. ll Interaction of a beam electron with a valence electron of the host atom does occur, but will be considered negligible for our purposes.

7.

188

ELECTRON MICROSCOPY BEAM ELECTRONS

\

\

\

EL AS T IC AL LY

\'

/

J

SCATTERED

FIG.1. Elastic and inelastic scattering of beam electrons by a host atom. The elastically scattered electron loses little or no energy as it is deflected. The inelastically scattered electron suffers a change in momentum as well as direction.

Figure 2 illustrates the distribution of scattered electrons as a function of scattering angle for both elastically and inelastically scattered electrons. It will be noticed that, in addition to being fewer in number, the elastically scattered electrons are scattered more widely than the inelastically scattered electrons. The relevance of these observations will become apparent later. 7.2.2.2. Order in Material. Before discussing how scattered electrons contribute to image formation, we should consider briefly the concept of order in materials. An ordered, or crystalline, material exhibits a regular packing of atoms or molecules. The atoms or molecules of a disordered or amorphous material are generally distributed randomly throughout the material.t t When utilizing instruments capable of determining atom structure, this definition must be clarified. In electron microscopy, for example, an amorphous material is defined as one in which order, if it exists, is on a scale below the resolving power of the instrument.

7.2.

FUNDAMENTALS

189

When an electron beam passes through an amorphous specimen random electron scattering occurs, i.e., electrons are scattered in all directions in a statistically predictable manner. Ordered or crystalline materials, when oriented favorably relative to the beam, tend to scatter electrons in a more regular fashion. Specifically, the ordered array of host atoms acts like a grating and reflects or diffracts electron waves just as a grating in an optical system diffracts light waves. The subsequent recombination and interference of the primary and diffracted beams provides contrast for imaging crystal lattices or a diffraction pattern for determining the configuration of the atoms in the sample. 7.2.2.3. Contrast. Whether an amorphous or crystalline material, then, scattering results in electrons deviating from their original trajectories. Upon exiting the specimen, some of the electrons will have been scattered at angles large enough to cause them to fall outside limiting or selecting apertures in the microscope. These electrons cannot, of course, contribute to image formation. Consequently, a region of high scattering power, which produces many widely scattered electrons, appears darker than a region of low scattering power. This intensity variation, or contrast, is useful then in observing variations in thickness and/or density in an amorphous sample and variations in crystallographic orientation in a crystalline sample,

\ m

z

0

a I-

/

INELASTICALLY SCATTERED ELECTRONS

0

w

J W

LL

0

a W

m

4z

cSCATTERING ANGLE

FIG. 2. Distribution of elastically and inelastically scattered electrons as a function of scattering angle.

190

7.

ELECTRON MICROSCOPY

7.2.2.4. Image Degradation. It can be seen, of course, that some beam electrons can undergo multiple scattering and still pass through the aperturing system. Such electrons are detrimental to image formation, contributing to what is commonly called a backgroundfog. Further, the reduction in energy of electrons inelastically scattered on their way through a specimen results in their contributing to chromatic aberration, another image-degenerating phenomenon. In Section 7.3.3. we shall examine these scattering-related problems as well as other factors that tend to degrade electron-microscopic images. 7.2.2.5. Absorbed Electrons. If a specimen is very thick or possesses a very high scattering power, entering electrons may experience so many collisions that they will lose all of their energy before they can escape the sample. Such absorption of beam energy results in heating of the specimen. This phenomenon is a most important consideration with organic polymers because of their heat sensitivity and poor heat conductivity. Needless to say, absorbed electrons cannot contribute to image formation. Areas of total absorption will appear dark in the normal transmission image. It should be noted that absorption is negligible in the usual thin specimen examined in the transmission electron microscope. 7.2.3. I mage Interpretat ion

We have seen that the different scattering processes produce contrast in the image, which can be related in general to specimen thickness, density, and/or crystallinity. Let us now take a closer look at electronmicroscopic images and determine what they can tell us about the specimens being examined. 7.2.3.1. Bright-Field Images. The images we have described thus far, i.e., bright areas representing regions of low scattering power, darker areas those of high scattering power, are referred to as bright-field images. Undoubtedly the most widely used mode of operation, brightfield microscopy allows direct observation of sample morphology or form. Figure 3a is a bright-field electron micrograph of a polyethylene single crystal grown isothermally from a dilute solution. The dark areas on the crystal represent diffraction contrast. As indicated earlier, this means that planes of atoms in the crystal are diffracting beam electrons out of the microscope’s collecting apertures. Continued electron irradiation of such polymer crystals completely destroys their crystalline order, primarily by radiation damage and crosslinking. Under normal operating conditions this destruction is almost instantaneous (Fig. 3a was photographed with greatly reduced illumination). However, this disordering occurs on a molecular level, allowing continued examination of the overall morphology of the crystal. As seen

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191

FIG.3. Polyethylene single crystal grown from dilute solution. (a) Bright-field image exhibiting diffraction contrast, (b) bright-field image following molecular disordering by continued electron beam irradiation.

in Fig. 3b, the contrast between the crystal and its thin carbon support film is now quite poor. From what we know of electron scattering, however, this is not unexpected, since the crystal and support film have similar densities and thicknesses. To enhance their bright-field image, such samples are given a thin coating of a heavy metal. The overall improvement in contrast, resulting from the increased scattering, is illustrated in Fig. 4. Evaporated obliquely to the surface, the heavy metal coating provides an additional enhancement of topographical perturbations by producing shadows. This shadowing technique will be described in more detail later. For those unfamiliar with electron micrographs, it should be noted that the observed contrast is the reverse of that accustomed to; i.e., the shadow is light instead of dark. The scattering process clearly accounts for this; heavy metal built up on the leading edge of a protrusion will scatter electrons more widely, and hence appear darker, than the trailing side of the protrusion, which receives no metal.

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FIG.4. Polyethylene single crystal shadowed at tan-' 4 with Pt-C.

Before moving on, a comment should perhaps be made about the pleat traversing the crystal's short axis in Fig. 4. It was produced by the collapse of the originally pyramidally shaped crystal as it dried down on the support substrate. This is a good example of how a specimen may undergo some physical alterations during preparation, but can still provide useful information about its prepreparative condition. Perhaps the best demonstration of this, however, appears in a study of three-dimensional crystal associations by Lotz et al. lo 7.2.3.2. Diffraction Patternst. The foregoing micrographs demonstrate that bright-field images represent the physical shape of the specimen. The electron microscope can also be used to reveal the atomic structure of many specimens. In the previous section on image formation we saw that ordered specimens can produce diffracted waves of elect An excellent presentation of the theory and practice of electron diffraction is given in Hirsch er aLB lo B. h t z , A. J. Kovacs, and J . C. Wittmann, J . Polym. Sci., Polym. Phys. Ed. 13, 909 (1975).

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trons. By properly adjusting the microscope, as described in Section 7.4.4., we can image the plane on which these scattered waves are focused. The image so obtained is a series of spots or rings. By using classical crystallographic techniques, we may decipher these diffraction patterns and determine the atomic spacings and orientation of the diffracting specimen. Polycrystalline samples, in which many small crystals assume a variety of orientations, produce a diffraction pattern that exhibits a central spot surrounded by a number of rings (Fig. 5 ) . In classical crystallographic jargon, the radii of these rings represent reciprocal lattice spacings. Single-crystal diffraction patterns consist of spots instead of rings (Fig. 6). Again classical crystallography, as developed in Chapter 6.1 (this volume), is employed and the experimenter may determine from the location of the spots the dimensions and orientation of the unit cell with respect to the physical crystal. When interpreting diffraction patterns, or any other electron-microscopic images, one must always consider possible alterations in the specimen during preparation. For example, since we know the crystal that produced the pattern in Fig. 6 had collapsed during dry-

FIG.5. Electron diffraction pattern from polycrystalline sample of thallium chloride.

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FIG.6. Diffraction pattern from single crystal of polyethylene. Insert indicates portion of crystal that produced diffraction pattern.

down, our interpretation must take into account that the atomic planes are tilted somewhat from their as-grown orientation. It should perhaps be noted that the electron beam will be diffracted by only those crystal lattice planes very nearly parallel to the incident beam [i.e., (hkO) planes in polyethylene single crystals]. This is because the short wavelength of the electrons causes the radius of the Ewald sphere of reflection to be very large compared with reciprocal lattice spacings. For example, the electron wavelength at 100 kV is 0.037 A, giving an Ewald Crystal spacings of the order of 2 8, have recipsphere radius of 27 rocal lattice vectors of magnitude 0.5 A-l. It can be seen, then, that the Ewald sphere is very nearly a plane section through the reciprocal lattice. In fact, the electron diffraction pattern is often considered to represent a “cut” or section of the reciprocal lattice, which is normal to the incident beam. 7.2.3.3.Dark-Field Images. If the microscope is adjusted to produce a physical image of the sample using only the electrons forming one of the

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FIG.7. Dark-field image of a polyethylene single crystal,

diffraction pattern spots, a dark-jeld image is obtained (Fig. 7). Bright regions in this case represent ordered areas possessing the same orientation. It can be seen that dark-field microscopy is useful in studying long-range order. The one operational precaution that must be heeded whenever we are trying to observe effects of crystallinity in organic polymers, as we are in the dark-field and diffraction modes, is that of reduced illumination. Beam-sensitive polymers dictate that we work quickly and with as low a beam intensity as possible. The mechanics for reducing beam intensity and performing dark-field microscopy are presented in Section 7.4.

7.3. Electron Opticst Familiarity with the basics of electron optics is prerequisite to a discussion of the electron microscope. Since space limitations preclude an extensive development here, a synopsis of the subject is presented, with t See also Vol. 4A of this series. Section 1.1.8.

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details on only those factors of direct concern to the microscopist. Accordingly, discussion of a number of classical lens aberrations, typically included in electron optics treatises, is strictly limited or deleted. This action is not unwarranted, however, since these aberrations are totally compensated for by present-day instrumental design or operational techniques.

7.3.1.Lens Theory A brief examination of the fundamentals of lens theory will provide an important foundation for later discussions. Figure 8 illustrates the basic principle of a lens. A ray originating from object point Po is deflected by the radial field of the lens and intercepts the optic axis at image point Pi. Obviously, for all rays originating from Po to be imaged at Pi, the angular deviation a of the ray must be proportional to r . The proportionality constant l/fis related to the object and image distances in the elementary lens relation,

l / f = I/.& + 1/h*

(7.3.1.)

Rewriting to obtain an expression for a,we obtain a / r = I/.& + l/h

or

a = r(l/fo

+ l/fl).

(7.3.2.)

We see then that the angular deviation of a ray passing through a lens is dependent on the radial distance the ray is from the axis and on the object and image distances. Physically, the field strength of the lens must increase radially according to a law of the form E = Kr,

(7.3.3.)

where K is a constant. For electromagnetic lenses the magnetic field exerts an additional LENS

OPTIC

FIG.8. Basic principle of a lens. Po and P,are object and image points, respectively,& and J; the object and image focal lengths, respectively.

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FIG.9. Path of an electron through a magnetic lens.

force on the electron at right angles to the field. As a result the electrons describe a helical path through a magnetic lens (Fig. 9).

7.3.2.The Ideal Lens Having examined the basics of the focusing action of a lens, let us now take a closer look at a perfect optical lens. We shall see later how a real system deviates from this ideal. Figure 10 illustrates a useful way of describing a lens. To the left of the midplane of this “thin” lens lies object space, to the right image space. All rays (or electron trajectories) passing through the object focal point F, are refracted by the lens and traverse image space parallel to the axis. A paraxial ray in object space will be refracted by the lens and converge to image focal ‘point F, . More importantly, any point lying in the plane perpendicular to the axis and containing P, is imaged in a conjugate plane containing Pi.Magnification of this image depends on the location of the object plane and the object and image focal lengths f , and J.; , respectively. Specifically, the lateral magnification Y’lY =fix, = X i / A .

(7.3.4.)

It can be seen that changes in the focal lengths alter the size of the image y’ . In practice such changes are accomplished by appropriately adjusting the lens current. The foregoing basic thin-lens description, although useful, does not accurately represent electron microscope lenses. Electron lenses are gen-

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OPTIC AXIS

y

IMAGE SPACE OBJECT SPACE

FIG.10. Schematic of a thin lens.

erally “thick,” meaning that the fields of one electrode penetrate into the other electrode. As a result, a region of varying refractive index extends over a considerable distance. A ray-tracing diagram describing a thick electron lens employs two principal planes, both lying on one side of the lens midplane. Figure 11 typifies a two-electrode lens exhibiting a lower voltage or smaller refractive index on the left. Depending on the shape and strength of the lens, such diagrams can become quite complicated; e.g., the object plane could lie within the lens. OBJECT PLANE

I

I I

M Hi

H,

I I

IMAGE PLANE

I

I

I I Pi

IMAGE SPACE OBJECT SPACE

FIG.11. Schematic of a thick lens, where a lower voltage or smaller refractive index is on the left.

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I99

A detailed examination of the optics of such lenses is unnecessary for our purposes, however, especially since equivalent thin-lens diagrams can be constructed in most cases. We shall instead move on to examine how the ideal image is degraded.

7.3.3. Irnage-Degrading Factors

The foregoing discussion of electron optics pertained to an ideal lens. A real lens, however, is not this perfect; its image is not a faithful reproduction of its object plane. This is so because a physical system inherently contains a number of image-degrading factors. Specifically, the image formation process in the electron microscope is complicated by geometrical aberrations, chromatic aberration, and space-charge effects. It is useful for the microscopist to understand how these factors degrade the image and what can be done to minimize their effects. 7.3.3.1. Spherical Aberration. Spherical aberration is the principal imaging error of a group called geometrical aberrations and is the only one that causes unsharpness of the image on the optic axis. It occurs because real lenses produce fields that vary nonlinearly about the axis. That is, Eq. (7.3.3.) becomes E

=

K(Z)r,

(7.3.5.)

with K ( Z ) a function of Z alone. The result is that rays passing through the outer portion of a lens are deflected more strongly than would be predicted in the ideal case. Such strongly deflected rays do not cross the optic axis at the calculated focal point but at a point closer to the lens. Inner rays exhibit a similar, but less severe, deviation from the ideal. Figure 12 illustrates the effect. One ray, leaving object point Po at an angle p Z ,is more strongly refracted by the lens, crosses the axis closer to

FIG.12. Schematic depicting the origin of spherical aberration.

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the lens, and is imaged at a distance Ar, from the axis, at Pi2.All rays leaving point Po between angles p1 and p2 will fill the region between Pi1 and P:, producing a circle of confusion about Pi. Point Po will therefore have an apparent diameter of A r , / M , where M is the magnification. This circle of confusion is a constant for a particular lens and focal length, and is proportional to the cube of the off-axis distance of a ray, i.e., Ar, a rp3.

(7.3.6.)

For the very important objective lens of an electron microscope, p is small, and iff, is fixed, rp p. Therefore Ar,

a

p".

(7.3.7.)

If we designate C, as the constant of proportionality, we obtain Ar, = C,@.

(7.3.8.)

C, is commonly referred to as the spherical aberration coefjcient. The optical quality of a lens is often judged by examining the dimensionless ratio of its aberration coefficient to its paraxial focal length i.e.,

C,/f= Ar/fP3.

(7.3.9.)

For a typical electron microscope objective lens C,/f= 1.00. It should be stressed that C, is not a lens constant, however, but is a function of the object and image distances. Further, if the aberration and beam aperture are measured on different sides of the lens, the magnification must be considered, specifically,

W '= y'/yCSp3,

(7.3.10.)

where p is the semiaperture angle on the object side. In considering the above described defects of a nonlinear field, one might intuitively perceive that beam broadening due to spherical aberration would be reduced if the widely scattered rays could be prevented from contributing to the image. Indeed, this is exactly what Eq. (7.3.8.) tells us. The microscopist routinely accomplishes this by inserting into the beam a limiting aperture, the objective aperture. The selection of the appropriate size objective aperture, however, demands close attention. 7.3.3.2. Aperture Effect. If, in an attempt to reduce spherical aberration, we were to begin using smaller and smaller limiting apertures, we would find that the amount of beam spreading would, after an initial decline, begin to increase. This is because, as in light optics, passage of a wave through a circular aperture results in the wave being diffracted. The smaller the aperture becomes, the more severe the diffraction. This

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20 I

is evident from the diffraction error equation, Ard

A/@,

(7.3.11.)

where Ard represents the amount of beam spreading, A the wavelength of the radiation, and /3 the limiting ray angle, or effective apertfire. Eventually the diffraction effects would overwhelm the spherical aberration effects. To minimize these two opposing phenomena then, we must optimize the limiting aperture. The optimum aperture size is that which minimizes the sum of the spherical error Ars, and the diffraction error Ar, . It has been found that the minimum radius of confusion is produced when these two errors are of the same order of magnitude. Thus, (7.3.12.) (7.3.13.) This tells the microscopist that, should he change his accelerating potential or electron wavelength, his optimum aperture for best resolution will be changed. Concurrently, a change in the spherical aberration coefficient of a lens, effected by a change in the object or image distance, will similarly alter the optimum aperture value. In practice, the physical aperture selected is often somewhat smaller than the optimum if the specimen is a thin film of low contrast, and somewhat larger than the optimum if a thick, high-contrast film is examined. In the former case, the microscopist is concerned with obtaining sufficient contrast to study his specimen. In the latter, he has contrast to spare and works at increasing the brightness of the image by collecting widely scattered electrons. 7.3.3.3. Chromatic Aberration. It was mentioned earlier that an electron’s trajectory is altered upon passing through an electric or magnetic field and that the magnitude of the deflection is in part dependent on the electron’s velocity. Faster electrons spend less time in the field and hence exhibit less pronounced deviations. This phenomenon is the source of another image-degrading factor, chromatic aberration. We learned in our discussion of an ideal lens that parallel rays entering a lens are focused to one point on the axis. But electrons of different velocities deviate from this ideal. Figure 13 illustrates what occurs. Two electrons el and e, of different velocities u1 < u, are traveling parallel to the axis when they enter the lens. Electron e, is deflected and intercepts the axis at F. The slower electron el is deflected more strongly, crosses the axis closer to the lens, and is imaged at F‘. If zll and zl, mark the range

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I

FIG. 13. Schematic depicting the origin of chromatic aberration

of energies of the electrons in a particular beam, then electrons with velocity u, such that u2 < u < u l , will fill the area Arc. Once again we see spreading of the electron beam about the desired point. The equation describing the magnitude of the chromatic error takes a form similar to that of the spherical error, i.e., Arc = K C c %

(7.3.14.)

Arc represents the amount of beam spreading, p the ray angle, C, the chromatic aberration coeficient, and K a constant that encompasses the variations in electron energies and in the lens energizing current. C, , like C,, is frequently described in terms of focal length; i.e., as C C / J For a weak magnetic lens, Cc/f= 1 ; for the stronger objective lens, C,/f = 0.8. Near the optic axis chromatic aberration is hardly noticeable. Marginal rays, however, are drawn into a spiraling rotation, causing picture points to appear as small lines. To minimize this electronic aberration it is obvious that we must reduce the energy spread of the electrons. The microscope designer makes a major contribution toward this end by designing an instrument with tolerances on the accelerating voltage and lens currents of 0.01-0.001%. Another instrumentally related improvement arises from using a field emission cathode instead of the more common thermionic emitter. As will be described later, the smaller source size of the field emitter is responsible for its ability to produce a more nearly monoenergic electron beam. The foregoing are engineering attempts aimed at reducing the energy spread of the electrons that reach the specimen. As pointed out earlier, however, inelastic scattering of electrons in the specimen causes a reduction in electron energy. Consequently, even a beam of monoenergetic electrons would exit a specimen as a nonmonochromatic beam. Obviously, chromatic error arising from the electron beam-specimen in-

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203

teraction cannot be totally eliminated. Its effects can be reduced, however, by preparing specimens as thin as possible. 7.3.3.4. Distortion. Distortion is a geometrical aberration that does not affect the sharpness of the image but only its geometrical faithfulness. That is, rays from object points reunite in the image plane to form conjugate image points, but the magnification varies across the entire plane. Like spherical aberration, the magnitude of the error is proportional to the cube of the off-axis distance of a ray, ArD = C D ( y ' / y ) r 3 .

(7.3.15.)

Here, ArD represents the distance of an actual marginal point from its ideal position. Figure 14 illustrates the case when the actual image points are at a greater distance from the axis than the ideal image points. In the geometrical analysis in Fig. 14a, we see that the image principal plane Hi is a

a

b FIG.14. Origin and observed effect of pincushion distortion.

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curved surface, concave toward the image focal point F, . Similarly, H,, (not shown) is concave toward the object focal point. This distortion of the principal planes causes rays originating from object points P, and P,, where y, = 2y,, to be imaged at points PI’ and P,‘,where y,’ > 2y1’. Figure 14b represents the image of a square grid produced under these conditions and illustrates why this imaging error is dubbed pin-cushion distortion. Such field aberrations that affect off-axis image points depend in part on the lens field utilized for image projection. For example, field aberrations can be changed by shifting an aperture stop along the axis. The conditions that produced the above-described pin-cushion distortion will produce barrel distortion if the limiting aperture is moved from behind the lens to in front of the lens. Figure 15 illustrates the geometrical analysis and square grid image corresponding to barrel distortion. Magnetic lenses in the electron microscope produce another imaging error, rotational, or anisotropic distortion. Here, real image points are displaced tangentially from their ideal image points. In this case image elements appear to spiral away from their intended location (Fig. 16) in a Hi

a

b FIG. 15. Origin and observed effect of barrel distortion.

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205

FIG.16. Observed effect of anisotropic distortion.

direction determined by the polarity of the lens field. Once again the degree of rotation varies with the off-axis distance of the imaged elements. Distortion becomes apparent when long-focal-length lenses image large fields of view. Adjustment of a microscope’s projector lens to minimize magnification for purposes of locating a small feature on a large field produces a useful, though markedly distorted, image. Of course, most electron microscopy is performed at relatively high magnifications, and so distortion is generally not a concern. However, the microscopist should be aware of its effects so that low-magnification images will not be misinterpreted. 7.3.3.5. Astigmatism. Like distortion, astigmatism is generally easy to identify and minimize. Unlike distortion, its presence is more evident at higher magnifications. Asymmetrical images result from a lack of axial symmetry of a lens field. Such deviations from symmetry can be effected by a very slight misalignment of the components of the microscope. This sensitivity to .alignment, however, is used to advantage to optimize the electron optical system, as we shall see later. 7.3.3.6. Space-Charge Effects. The final image-degrading factor we shall consider is space-charge effects. If the density of electrons in a beam becomes very high, their mutual repulsion will cause a spreading of the beam. High electron densities occur in the electron microscope at every focal, or crussover point (section A-A in Fig. 12). However, space-charge effects in the typical instrument are generally small enough to be neglected. In scanning-transmission electron microscopy, on the other hand, space-charge effects may become significant, especially if the instrument is equipped with a field emission source and is operated at high current levels. The resolution limit in scanning-transmission microscopy is partially limited by the size of the electron beam probe impinging upon the sample and space-charge effects work to increase this probe size.

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7.4. The Instrument Conceptually, the electron microscope is not a complicated instrument. It is composed basically of an illuminating system, a specimen holder, an objective lens, and a projection system, all held under vacuum since electrons are highly absorbed by air. The illuminating system consists of an electron source, which supplies a copious amount of electrons of a selected wavelength or energy, and condenser lenses, which deliver a collimated electron beam of selected size and intensity to the specimen. The specimen holder provides for mounting and manipulating the specimen. The objective lens is used for focusing the image and determines its resolution and contrast. The projection system contains several lenses for magnifying and projecting the image for viewing or recording. Let us take a closer look at these systems, proceeding from the electron source to the observed image. 7.4.1. The Illuminating System 7.4.1.l.The Electron Source. Most electron microscopes today employ a version of the self-biased “gun” as their electron source. Pictured schematically in Fig. 17, such a gun consists of a shielded v-shaped filament and an anode. To operate, the tungsten hairpin filament is resistively heated until thermionic emission from its tip is stabilized (at = 2650 K). A high negative voltage, generally 40-100 kV, is then applied to the

FIG.17. Schematic of self-biased electron gun.

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207

apertured shield (Wehnelt cylinder) and, via a bias resistor, to the filament. The large voltage gradient between the filament and the grounded anode accelerates the electrons (shaded area) to and through the aperture in the anode. The potential difference between the shield and filament (-200 V), arising from the voltage drop across the bias resistor, results in the shield aperture acting as a very strongly converging lens. Consequently, the cloud of electrons from the tip is highly concentrated around the axis. This effectively creates a virtual source near the aperture. That is, electrons will appear to originate from this space charge area and not from the tip. The importance of this phenomenon is that the effective source is smaller than the actual emission area and exhibits reduced sensitivity to inhomogeneities of the tip surface. However, since the electrons are emitted from a tip of finite size, electron path lengths through the accelerating field will differ and final kinetic energies will vary by as much as 2 3 eV about the selected value. As indicated earlier, a spread in electron energies leads to chromatic aberration in the optical system, which, in turn, degrades resolution. Over the years, experiments with various gun designs and filament materials have been undertaken. The goal of the researchers has been to produce a more stable, brighter, and more nearly monoenergetic electron source. One promising innovation seeing increasing use is thefield emission source. In this design a piece of fine oriented tungsten wire is welded to the tip of a common hairpin filament and chemically etched to produce a point with a 1000 8, radius. In a good vacuum (better than lo-@ torr), application of a strong electric field in the vicinity of the tip results in a copious amount of electrons being extracted-even at room temperature. Emission current densities of the order of lo7 A/cm2-srad are not uncommon for field emission cathodes. And since the area of emission is relatively small, more coherent beams are produced-variations of only 0.3 eV are typical. 7.4.1.2. The Condenser Lenses. Upon exiting the emission chamber through the aperture in the anode, the electron beam falls under the influence of the condenser lenses, which determine the intensity and size of the beam falling on the specimen. In single-condenser operation, beam intensity at the specimen is proportional to beam size, decreasing with a less focused or larger beam. With a double-condenser system, however, the intensity is independent of beam diameter. Hence, for beamsensitive samples, such as organic polymers, the beam can be restricted to the area of observation and the intensity can be reduced to a safe level. A second advantage of the double-condenser system is that the electron beam is made more nearly perpendicular to the specimen surface, or as it is more commonly stated, the aperture of illumination (the angle the beam

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ELECTRON MICROSCOPY

makes with a normal to the specimen surface) is very small (Fig. 13). This tends to minimize the electron path lengths through the sample and thereby reduce chromatic aberration effects. 7.4.2. The Specimen Holder

Specimen holders typically accommodate a 3 mm diam specimen. Inserted into the specimen chamber through a vacuum air-lock, the specimen can be manipulated in anx-y direction and usually can be tilted a few degrees off axis (necessary for stereomicrography). Special stages are available that allow thermal or mechanical stressing of the specimen during examination. A common problem facing electron microscopists is specimen contamination. Organic molecules, finding their way into the column from the pumping system, are polymerized by the electron beam and form deposits on the surface of samples. This not only reduces the resolution of the instruments, but can create artifacts that may mislead the investigator. To partially overcome this problem, anticontamination blades may be inserted around the specimen. Cooled to liquid nitrogen temperatures, these blades act as a trap for the contaminating species and greatly reduce specimen contamination. Another attempt to reduce contamination in electron microscopes has been to replace the usual fluids used in vacuum diffusion pumps, such as hydrocarbons, silicones, and polyphenyl ethers, with a perfluorinated polyether.ll Vapors from this fluid are not polymerized by the electron beam and therefore do not deposit on the specimen. While a clean vacuum system can be obtained by using such a fluid, care to properly exhaust the roughing pump is mandated, since the decomposition products interact to produce toxic gases. 7.4.3. The Objective Lens

If the functional parts of the electron microscope were listed in order of importance, the objective lens would command a high rank. This is because the resolution, or imaging precision, of the entire instrument is heavily dependent on the precision of the objective lens. Fortunately for the microscopist, much of the burden of optimizing the objective lens is born by the designer and the instrument maker. The experimenter uses his knowledge of lens theory and electron optics to guide him in selecting and centering the objective lens aperture to optimize contrast, adjusting L. Holland, L. Laurenson, P. N. Baker, and H. J . Davis, Nature (London) 238, 36 ( 1972).

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209

the lens current to optimize focus, and adjusting the stigmators to compensate for any asymmetry that may appear in the image. Let us examine each of these operations, since they must be mastered by both the novice and the expert microscopist. We noted earlier that one often uses an objective aperture larger than optimum in order to allow more widely scattered electrons to contribute to image formation. Since such action concurrently causes a loss of resolution, the microscopist must select the aperture that best suits his particular investigation. If relatively low magnification microscopy is being carried out, ultimate resolution is not a concern, and contrast improvement can be sought. Where very fine structure is to be examined, however, a close-to-optimum size aperture must be employed to allow separation of closely spaded image points. For electron diffraction, as will be seen shortly, the objective aperture is removed entirely from the beam. In a later section, a technique called topographical contrast imaging is described. In this instance the objective aperture is moved from its normal axially centered position to a position where the aperture just begins to cut off the beam. The phase differences set up by the electron waves passing close to the edge of the aperture result in contrast enhancement of low-contrast samples. Although not widely applicable, the technique has shown merit in certain studies. The second operation concerning the objective lens sounds like a trivial exercise-adjusting the lens current to focus the image. However, a series of micrographs must often be taken to ensure that the optimum image is recorded. Figure 18 represents a through-focus series of micrographs. While Fig. 18c represents a perfectly focused image, the slightly underfocused image in Fig. 18b has an apparent sharpness that makes it a more pleasant rendition of the object. It is probable that this micrograph would be selected for presentation. Such defocusing must, of course, be accompanied by a loss in resolving power. So once again the microscopist must make a judgment as to what instrumental condition he should employ to obtain the most information from his specimen. Since a through-focus series of micrographs is generally taken only for high-magnification examination of a critical field, the microscopist must quickly learn what objective setting is best for the routine work where he will be recording only one micrograph. The final aspect of the objective lens to be considered is that of stigmating the image. If the magnetic fields set up by the objective become slightly distorted, due to misalignment of lens components for example, the image produced will appear asymmetrical. To eliminate minor disparities, special stigmator coils are built into the column, which allow the microscopist to produce appropriate compensating fields. Failure to stig-

FIG. 18. Through-focus series of a holey film. (a,b) Underfocused, (c) focused, (d,e) overfocused. By adjusting the objective lens current, the objective focal length was changed 500 8, between micrographs.

7.4.

T H E INSTRUMENT

21 1

mate before beginning to work can result in misleading artifacts being recorded. Symmetrical Fresnel fringes, as will be described later, indicate a properly aligned instrument. 7.4.4. The Projection System

Immediately following the objective lens is the projection system. Composed of an intermediate lens, or lenses, and a projection lens, it is this part of the instrument that magnifies the image to a size convenient for viewing and recording. The desired magnification is generally obtained by manipulating a single control on the microscope, which adjusts the currents energizing the various lenses of the projection system. 7.4.4.1. Modes of Operation. The microscopist determines which imaging mode he will use by appropriately adjusting the intermediate lens current alone. Figure 19 illustrates how this is accomplished. For

FIG. 19. Schematic depicting how adjustment of the intermediate lens current determines the mode of operation. (a) Normal bright-field imaging, (b) electron diffraction produced by weakening intermediate lens, (c) dark-field imaging, illuminating beam tilted to allow diffracted electrons to produce image.

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7.

ELECTRON MICROSCOPY

normal bright-field microscopy (Fig. 19a) the intermediate lens is adjusted to produce an image at its image plane and subsequently on the viewing screen. Electrons widely diffracted by the specimen are intercepted by the objective aperture. This, as we have seen, improves contrast in the observed image. To perform electron diffraction (Fig. 19b), the intermediate lens is weakened to produce a cross-over (focus) at the intermediate image plane. The observed image in this case is a diffraction pattern. As can be seen, the objective aperture is removed to perform electron diffraction. This allows the higher orders of diffracted rays to contribute to the image. To obtain a diffraction pattern from a portion of the imaged field, the microscopist substitutes an appropriate size selected-area aperture for the normal, relatively large, diffraction aperture. To perform dark-field microscopy (Fig. 19c), the intermediate lens is energized as for bright-field microscopy, and the objective aperture is used. In this instance, however, deflection coils are used to tilt the electron beam before it enters the specimen, and a diffracted beam is used to produce the image. By imaging different diffracted beams, the dark-field image indicates which areas possess like crystallographic orientations. It is possible to perform dark-field microscopy without tilting the beam. This is accomplished by moving the objective aperture to accept only the electrons from a specific spot (diffraction mode). While a useful darkfield image will result, the off-axis position of the objective aperture leads to increased spherical aberration and astigmatism.

7.5. Operational Considerations In this chapter we shall examine a few topics in some detail to better prepare the experimenter embarking on his first microscopic investigation. 7.5.1. Specimen Preparation

Before any electron microscopic study can begin, the “obstacle” of specimen preparation must be overcome. In many instances preparing the specimen can be the most difficult part of the investigation. The microscopist may face the task of mounting minute samples, such as polymer single crystals, which can be only a few thousand angstroms across and 100 A thick. Here a suitable support film is required. In the case of bulk materials, the experimenter will work to minimize the thickness of his thin-fdm specimen in order to avoid significant absorption of the electron beam. Sectioning or replication would be the techniques employed here.

7.5.

OPERATIONAL CONSIDERATIONS

213

7.5.1.1. Support Grids. The specimen holder of the electron microscope accommodates a thin, metal wire mesh, generally a few millimeters in diameter, as the basic sample support. The thin-film specimen is supported by the crosswires of the mesh and examined in the open areas. A common grid size is 200 mesh/in., but the microscopist may select any of a variety of sizes and designs to suit his particular application. A more open grid may be employed if the specimen is relatively strong, while a finer mesh should be used for extrathin, delicate specimens. The support grids may be purchased precut to the appropriate size or, alternatively, cut from a large piece of mesh. Employed when the specimen is originally larger than the precut grid size, the latter technique allows the microscopist to preview his sample in a light microscope and select specific areas of interest for electron microscopic examination. A special punch is used to extract the selected area. 7.5.1.2. Support Films. Many samples require an additional support film. Polymer single crystals, as mentioned, need support merely because of their size. Thin (100 A) carbon films serve well as support films, exhibiting exceptional strength and a high transparency to the electron beam. Preparation of such films begins by vacuum evaporating, or sputtering, carbon onto a smooth, clean surface, generally freshly cleaved mica or a freshly fractured NaCl crystal. By slowly lowering the coated substrate into clean water, the carbon film floats off onto the water surface. The metal support grid is brought up from beneath the surface to catch the carbon. Subsequently the specimen is deposited onto the carbon film. (If an oil-free vacuum system is available, carbon films as thin as 10 A can be prepared.12) A variation of the above procedure is useful in preparing thin, meltcrystallized,samples as well as single crystals. In this case the specimen is deposited directly onto a piece of freshly cleaved mica. The carbon film is then vacuum-deposited and stripped off as previously described. The adhesion of the sample to the carbon film is generally greater than that of the sample to the mica, and so the stripped film will contain the specimen. Plastic films have also been used as specimen supports. Many plastics, of course (and indeed many polymer specimens), degrade and deform when exposed to an electron beam. A few, however, have demonstrated ~ ~ phenylated remarkable radiation resistance. Isotactic p o l y ~ t y r e n eand p~lyphenylenes,'~ for example, cross-link under electron beam irradiation

'' A. S . Baev, A. A . Aleksandrov, and A. G . Kiselev, Instrum. Exp. Tech. (Engl. Trans].) No. 2, p. 578 (1970). l3 W. Baumeister and M. Hahn, Nururwissenschufren 62, 527 (1975). I4 J. K. Stille, Mukromol. Chem. 154, 49 (1972).

214

7.

ELECTRON MICROSCOPY

and thereby acquire a high degree of stability. Such films exhibit a small mass loss (,

(9-6.111

where AH,,, is the heat capacity change between T , and T k , and the subscript 0 denotes temperature T k . Allowing the crystals to melt at T k :

AHfo =

xi

AH!.

(9.6.12)

Next, the polymer melt is heated from Tk to T,, and

(9.6.13) AH2.0 = Ha,, - Ha.0. It follows that AHo,, + AHfo + AH,,0 = AH,,,. In Fig. 25 the area ABEF is equivalent to AHo,,, the dashed line being the extrapolated line from the low-temperature region, where it is assumed that no melting is taking place. is the area DHGF and the remainder BCDE must then be

3 24

9.

THERMAL ANALYSIS OF POLYMERS C

FIG.25. DSC polymer melting curve and instrumental baseline; the division of areas applies to the crystallinity determination using the perfect crystal heat of fusion at the perfect crystal melting point (from Gray124).

AHf,,. x1 can then be determined. However, for polyethylene the extrapolated dashed line will not be that shown in Fig. 25. Arguments have been presented to show that the “correct” baseline for any polyethylene is found by passing a straight line through the trace at 140°C and to the curve at a temperature where it is assumed that no melting is taking place.lz4 This is valid only in the absence of, or correction for, instrumental baseline curvature. Many others have proposed alternative baseline c ~ n ~ t r ~ c t i o n ~ . ~ ~ Most recently Guarini et al. 134 proposed a construction that compensates for changes in sample thermal emissivity for reactions of the type ASOLID BSOLIO + CGAS. +

Almost no data comparing the proposed baseline methods have been pubdid present limited data on the lished to date. However, Brennan et a/.127 decomposition of polyvinyl alcohol comparing the kinetic parameters determined using three different baselines. It will require much further study on a variety of polymers to determine which method arrives at the “best” AHfl. Assuming that AHfl can be found accurately, AHP, must then be determined in order to find the weight fraction crystallinity. However, only 131

G . Adam and F. H. Muller, Kolloid-2. & Z. Polym. 192, 29 (1963). A. Engelter, Kolloid-2. & 2. Polym. 205, 102 (1964). G . G . T. Guarini, R. Spinicci, and D. Donati, J . Therm. Anal. 6, 405 (1974).

9.6.

QUANTITATIVE METHODS

325

limited data on AH& exist in the literature for most polymers. One notable exception is polyethylene. Many different methods have been used to determine this quantity for polyethylene, all of which arrive at roughly the same value (70 cal/g). Historically, AHP, was first derived from data on n-alkanes of suitable ~ t r u c t u r e . ~ ~For J ~ instance, ~ - ~ ~ ~ the heats of fusion of a series of n-paraffins can be determined and plotted as a function of the number of carbon atoms in the chain. The resulting plot can then be extrapolated to l/n + 0, where n is the number of main chain carbons. The intercept is then taken as AHP, . Heat of fusion data for polyethylene single crystals of varying fold period (I) can be used to determine AHfl.140J41 By plotting the samples’ heat of fusion vs. 1/I and extrapolating to 1/I 4 0, one can find AH&. Alternatively, heat of fusion data can be plotted vs. specific volume for samples of varying densities.s5,140.142-145 Extrapolation to the specific volume of 100% crystalline polyethylene (1.00 cm3/g)t46yields A m , . Aml has also been estimated by melting point depression techniques ,147J48 from high-pressure and from heat of fusion data on extended-chain polyethylene Agreement between the degree of crystallinity calculated from calorimetric data (xc)and from other techniques has generally been good. Several have found reasonable accord between xc and that 135

M. Dole, W. P. Hettinger, N. R. Larson, and J. A. Wethington,J. Chem. Phys. 20,781

( 1962).

F. W. Billmeyer, J . Appl. Phys. 28, 1 1 14 (1957). M. G. Broadhurst, J . Res. Nail. Bur. Stand., Sect. A 67, 233 (1963). 138 B. Wunderlich and M. Dole, J . Polym. Sci. 24, 201 (1957). 130 C. M. L. Atkinson and J. J. Richardson, Trans. Faraday SOC. 65, 1749 (1969). I4O E. W. Fischer and G. Hinrichsen, Kolloid-Z. & 2. Polym. 247, 858 (1971). 141 L. Mandelkern, A. L. Allou, and M. Gopalan, J . Phys. Chem. 72, 309 (1968). 142 E. W. Fischer and G. Hinrichsen, Polymer 7, 195 (1966). H . Hendus and K. H. Illers, Kunststofle 57, 193 (1967). 144 F. Hamada, B. Wunderlich, T. Sumida, S. Hayashi, and A. Nakajima,J. Phys. Chem. 72, 178 ( 1968). 145 D. A. Blackadder and T. L. Roberts, Angew. Makrornol. Chem. 27, 165 (1972). 14’ C. W. Bunn, Trans. Faraday SOC. 35, 482 (1939). 147 F. A. Quinn and L. Mandelkern, J . A m . Chem. SOC. 80, 3178 (1958). 148 F. E. Karasz and L. D. Jones, J . Phys. Chem. 71, 2234 (1967). 140 J. Osugi and K. Hara, Rev. Phys. Chem. Jpn. 36, 28 (1966). lJ0 B. Wunderlich and C. M. Cornier, J. Polym. Sci., Part A-2 5, 987 (1967). 151 M. Dole, J . Polym. Sci., Part C 18, 57 (1967). 152 R. L. Blaine, Appl. Brief TA-12, DuPont Corp., Wilmington, Del. (1975). IMS. Y. Hobbs and G. I. Mankin, J. Polym. Sci., Part A-2 9, 1907 (1971). 154 A. Peterlin and G. Meinel, Appl. Polym. Symp. 2, 85 (1966). lS5 D. A. Blackadder, J. S. Keniry, and J. J. Richardson, Polymer 13, 584 (1972). L. Mandelkern, “Crystallization of Polymers,” p. 306. McGraw-Hill, New York, 1964. 138

13’

326

9.

THERMAL ANALYSIS OF POLYMERS

derived from density data. For instance, Peterlin and Meinell" determined the degree of crystallinity for polyethylene samples of different mechanical and thermal histories from calorimetric, density, and nitric acid weight loss data. Sample oxidation by nitric acid attack takes place in two s t a g e ~ . ' ~ ' J First, ~ ~ the fold surfaces and other noncrystalline material are destroyed and the crystalline core exposed. Then there is oxidative attack on the crystalline core.? From the inflection point of the weight loss vs. time-of-treatment curve, the degree of crystallinity can be calculated. It was found that the crystallinity values determined from this type of analysis agree well with those determined from density and calorimetric studies. Dole151found a correlation between crystallinities determined by calorimetric and dilatometric, X ray, and infrared proce~ ~compiled data on dures if Awl is taken as 70 cal/g. M a n d e l k e ~ dhas natural rubber to show that agreement exists between crystallinity values calculated from calorimetric, X ray, density, and expansivity measurements. He suggests that the small observed differences in calculated crystallinity values could reflect technique sensitivity and the differing contributions of imperfections. Salyer and K e n y ~ n reported l~~ good accord between X ray and calorimetric crystallinities for ethylene -vinyl acetate copolymers, On the other hand, Haberfeld and RefnerlSofound poor agreement for chemically crosslinked polyethylene. Wakelyn and Younglgl also compared crystallinities derived from X ray and calorimetric data. MacKnight et ~ 1 . compared calorimetric, X ray, and infrared (xl) crystallinities for poly(ethy1ene-m-acrylic acid) and poly(ethy1ene-co-methacrylic acid) with a variety of comonomer ratios. They found values derived from DSC lower than the others. Correction of Awl resulted in excellent correlation between xc and xI. Consistently low xc values relative to density and X-ray crystallinities were also observed by Ver Strate and WilchinskylMfor ethylene-propylene copolymers.

t See also Part 8. A. Peterlin and G . Meinel, J . Polym. Sci., Parr B 3, 1059 (1965). A . Peterlin, G . Meinel, and H. G . Olf, J . Polym. Sci., Part B 4, 399 (1966). lag I. 0. Salyer and A. S. Kenyon, J . Polym. Sci., Part A-1 9, 3083 (1971). lSl) J. L. Haberfeld and J. A. Refner, Thermochim. Acra 15, 307 (1976). N. T. Wakelyn and P. R. Young, J . Appl. Polym. Sci. 10, 1421 (1966). W. J. MacKnight, W. P. Taggart, and L. McKenna, J . Polym. Sci., Parr C 46, 83 (1 974). G . Ver Strate and Z. W. Wilchinsky, J . Pulym. Sci., Part A-2 9, 127 (1971). la'

lW

~

6

~

9.7.

OTHER APPLICATIONS

327

9.7. Other Applications 9.7.1. Phase Changes

It is well known that some polymers can exist in two or more different crystalline modifications. By employing DTA/DSC it is possible to observe the melting of the various forms and quantify these transitions. The data derived can be helpful in determining a phase diagram for the particular polymer. Many early studies dealt with the three crystalline forms of polybutene-1.184-166 Later, evidence for polymorphism was found for a variety of polymers including poly(trans- 1,4-i~oprene),~~' poly-p-xylylene,168and isotactic p~lypropylene.'l-'~ 9.7.2. Glass Transition

The glass transition temperature ( T . )is an important factor in the mechanical behavior of polymers. It is usually defined as the temperature of onset of main-chain segmental motion in amorphous regions of polymers and is accompanied by abrupt changes in modulus and expansion coefficient. Typical values of Tg for a number of polymers are given in Table I. Tg is not associated with an enthalpy change with temperature but with a sudden change in specific heat. It is usually determined to be either the inflection point of the Tgcurve or the extrapolated onset temperature (Fig. TABLEI. The Glass-Transition Temperature of Some Selected Polymers ~~

Polymer Polystyrene Polyacrylonitrile Poly(viny1 chloride) Poly(viny1 alcohol) Poly(viny1idene chloride) Poly(viny1 fluoride) Poly(viny1idene fluoride) Pol ythioethylene Pol ychloroprene Poly(ethy1ene oxide) Pol yox ymethylene Poly(dimethy1 siloxane)

Monomeric Unit -CHz-CH(CeHs)-CHz-CH(CN)-CH,-CHCI-CHz-CH(OH)-CHZ-CCl2-CHz-CHF-CHz-CFz-S-CHz-CHZ-CHz-CCI=CH-CHz-0-CHZ-CH2-0-CH2-Si(CH3)2-O-

100 96.5 87 85 - 17 -20 -39 -50 -50 -67 -85 -123

F. Danusso and G . Gianotti, Makromol. Chem. 61, 139 (1963). C. Geacintov, R. S. Schotland, and R. B. Miles, J . Polym. Sci., Part B 1, 587 (1963). lea V. A. Era and T. Jauhianinen, Angew. Makromol. Chem. 43, 157 (1975). Ie7 E. G. Lovenng and D. L. Wooden, J . Polym. Sci., Part A-2 7 , 1639 (1969). lea W. D. Niegisch, J . Appl. Phys. 37, 4041 (1966). lBp

le5

328

9.

THERMAL ANALYSIS OF POLYMERS

1

TEMPERATURE

--t

FIG.26. Two manners in which the glass transition temperature has been defined. In A the glass transition temperature is taken to be the inflection point of the curve (TgA);in B, T, is chosen to be the extrapolated onset temperature (TgB).

26). Some thermal analysts select other points, and there is no general agreement as to the correct method. Ideally, a peak should not be seen in the DTA/DSC curve at this temperature but one should observe a baseline shift (Fig. 27, curve a). However, a variety of other shapes for the curve through T, have been reported (Fig. 27, curves b,c,d). Shapes like curve (b) are most common while ( c ) is seen less frequently. It has been suggested that the endothermic peak of curve (b) is a result of the disruption of ordered-chain segments that are present below Tg.160-171However, others have reported that the shapes of (b) and (c) are governed by the rate at which the polymer is cooled from the melt and subsequently have observed the anoheated through T,.172J73 Roberts and Sher1ike1-I~~ malous curve shapes in (d) for a variety of polymers. It was suggested that this step change in the exothermic sense could be due to the relaxation of internal strains within the samples. M. S. Ali and R. P. Sheldon, J . Appl. Polym. Sci. 14, 2619 (1970). P. V. McKinney and C. R. Foltz, J . Appl. Polym. Sci. 11, 1189 (1967). M. I. Kashmiri and R. P. Sheldon, J . Polym. Sci., Part B 7, 51 (1969). B. Wunderlich, D. M. Bodily, and M. H. Kaplan, J . Appl. Phys. 35, 95 (1964). A. E. Tonelli, Macromolecules 4, 653 (1971). R. C. Roberts and F. R. Sherliker, J . Appl. Polym. Sci. 13, 2069 (1969).

lBO

170 171 172 173

17*

9.7.

OTHER APPLICATIONS

329

The effect of heating rate on Tg has been studied by a number of authors. Strella175has shown theoretically that the observed Tg should increase with increasing heating rate and his results on atactic polypropylene and poly(methy1 methacrylate) supported this. Strella also postulated that a plot of the logarithm of the observed Tgvs. heating rate should be linear and can be extrapolated to zero rate to give the “correct” Tg. other^^^^-^^@ have also found that Tgincreases with heating rate. Alternatively, it has been reported that Tg is either independent of heat ratelB0or increases with decreasing rate.lB1 Millerls2 has shown that if small samples (2-5 mg) are used, there is little dependence of Tg on heating rate. He postulated that the variation of Tg with rate is a consequence of the relatively large sample sizes used in the previous studies.

TEMPERATURE

FIG. 27. Examples of DTA curves in the glass transition region (from Roberts and Sherliker’?’).

S. Strella, J . Appl. Polym. Sci. 7 , 569 (1963). R. F. Schwenker, Jr. and R. K. Zuccarello, J . Polym. Sci., Part C 6, 1 (1964). 17? E. Wiesener, Faserforsch. Textiltech. 21, 442 (1970). 178 J. M. Barton and J. P. Critchley, Polymer 11, 212 (1970). l’@ J. M. Barton, Polymer 10, 151 (1969). lE0 J. J. Keavney and E. C. Eberlin, J. Appl. Polym. Sci. 3, 47 (1960). J. J. Maurer, Anal. Calorim. Proc. A m . Chem. Symp., I968 Vol. 1 p. 107 (1968). G. W. Miller, J . Appl. Polym. Sci. 15, 2335 (1971).

330

9.

THERMAL ANALYSIS OF POLYMERS

The influence of crystallinity on Tg is difficult to predict. It is generally thought that crystallites tend to restrict amorphous chain mobility and hence increase T g . An increase of Tg with increasing crystallinity has been reported by a number of author^.^^^-^^^ However, others have ~ ~ ~ for ~ ~ ~poly(4~ found that Tg is independent of ~ r y s t a l l i n i t yand, methyl-pentene-1), that Tg decreases with increasing c r y ~ t a l l i n i t y . ~ ~ ~ ~ ~ ~ This latter observation was explained on the basis of amorphous freevolume considerations. lD2 The effect of molecular weight (M) on Tg has been well documented. It has been found that Tg increases rapidly with M up to a certain critical molecular weight and then remains (Fig. 28). Mathematically, this relationship has been expressed as105

Tg = Tgm- A I M n

(9.2.1)

where M, is the number average molecular weight, Tgmthe glass transition temperature for a polymer of infinite molecular weight, and A a constant for the particular polymer. The value of A for polystyrene is approximately 109. This type of behavior can be rationalized in the following way. Since chain ends are not restrained to the same degree as the middle segments of a polymer molecule, the mobility of the ends will be greater than the chain middle at aparticular temperature. Thus, the more chain ends in a material, the further one has to cool the sample to reach the point at which the average relaxation time is the same as that of the sample containing no chain ends; Tg decreases with decreasing molecular weight. However, chain entanglements become important above a critical molecular weight, the mobility of a chain being largely determined by the mobility of the chain segments between entanglements. This remains approximately constant as molecular weight increases above the molecular weight for chain entanglements. B. E. Read, Polymer 3, 529 (1962). W. Woods, Nature (London) 174, 753 (1954). laa S. Newman and W. P. Cox, J . Polym. Sci. 46, 29 (1960). lSe J. A. Faucher, J . V. Koleske, E. R. Santer, Jr., J. J. Stratta, and C. W. Wilson, III,J. Appl. Phys. 37, 3962 (1966). la' Y. Uematsu and I. Uematsu, Rep. Prog. Polym. Phys. Jpn. 2, 27 (1959). 188 J. V. Koleske and R. D. Lundberg, J . Polym. Sci., Parr A-2 7, 795 (1969). F. C. Stehling and L. Mandelkern, J . Polym. Sci., Part B 7 , 255 (1969). l9O J. D. Hoffman and J. J. Weeks, J . Res. Narl. Bur. Stand. 60,645 (1958). lS1 J. H. Griffith and 8 . G. Rinby, J . Po/ym. Sci. 44, 369 (1960). Io2 B. G. Rinby, K. S. Chan, and H. Brumberger, J . Polym. Sci. 58, 545 (1962). M. C. Shen and A. Eisenberg, Rubber Chem. Techno/. 43, 95 (1970). B. Ke, J . Polym. Sci., Part B 1, 167 (1963). T. G. Fox and P. J. Flory, J . Polym. Sci. 14, 315 (1954). 185

l c D. ~

9.7.

OTHER APPLICATIONS

33 1

FIG.28. Glass transition temperature of atactic polypropylene as a function of molecular weight (from Ke'").

DTA/DSC has also been used to study other molecular relaxation mechanisms. For instance, Boyer et al.1n6-1nnhave reported the existence of two liquid-liquid transitions above Tg for a variety of amorphous polymers. These transitions have been designated T,, and T;,. T,, is thought to occur at a temperature of (1.2 2 O.05)Tg and Ti, at approximately 40°C above T i ,.lSn The nature of these transitions is unclear, but it has been postulated that the T,, transition involves the coordinated rotational motion of a large portion of a molecule in the amorphous state.lns O However, in a recent study on atactic polystyrene, Patterson et U ~ . ~ O have found no evidence for a well-defined thermodynamic transition above T,.

9.7.3.Polymerization In general, polymerization is an exothermic process where the amount of heat evolved is proportional to the extent of reaction. It has been shown that DTA/DSC provides a fast and convenient method for the study of the heats and kinetics of polymerization, curing reactions, and the effect of polymerization catalysts. However, the application of DTA/DSC to polymerizations has not been extensively reported until recently. S . J . Stadnicki, J. K. Gillham, and R. F. Boyer, J . Appl. Polym. Sci. 20, 1245 (1976). J. B . Ems, R . F. Boyer, and J. K. Gillham, Polymer (to be published). J. B. Enns and R. F. Boyer, to be published. l W J. B. E m s and R. F. Boyer, to be published. 2w G. D. Patterson, H. E. Bair, and A. E. Tonelli, J . Polyrn. Sci., Part C 54,249 (1976). lB7

332

9.

THERMAL ANALYSIS OF POLYMERS

and heats of polyQuantitative information on reaction m e r i ~ a t i o nhave ~ ~ been ~ ~ ~determined ~~ for various monomers. For example, Malavasic et aLZ1Ostudied the course and kinetics of the isothermal, free-radical polymerization of methylmethyacrylate by DSC. It was found that the polymerization was first order with respect to monomer concentration in the early stages of reaction. If it is assumed that the initiator concentration is independent of time for low degrees of conversion, ln[Mo]/[M] = k't,

(9.7.2)

where [MO]and [MIare the initial monomer concentration and monomer concentration at time t , respectively, k' is a composite of initial rate constants, and t is time. Note that the total area (A) under the exotherm can be related to the total heat of polymerization.'O It it is assumed that the heat of reaction evolved at any time is proportional to the number of moles of monomer consumed, then the area a (Fig. 29) is proportional to the amount of monomer reacted up to some time t . Thus ln[A/(A

-

a)] = k ' t .

(9.7.3)

ln[A/(A - a)] can then be plotted vs. time and k' determined. This was then performed for isothermal polymerization at several other temperatures (Fig. 30). The calculated rate constants were then assumed to conform to the Arrhenius equation k'

= Ze-EIRT,

(9.7.4)

where Z is the preexponential Arrhenius constant, R the universal gas constant, T the absolute temperature, and E the activation energy. A plot of k' vs. 1/T will then yield the overall activation energy for polymerization. Other methods have also been proposed to obtain kinetic information from calorimetric data.211 Each method offers some particular adF. Delben and V. Crescenzi, Ann. Chim. (Rome) 60, 782 (1970). K. E. J. Barrett and H. R. Thomas, J . Polym. Sci. Part A - f 7, 2627 (1969). uL1 K . E. J. Barrett, J . Appl. Polym. Sci. 11, 1617 (1967). 2w P. Godard and J. P. Mercier, J . Appl. Polym. Sci. 18, 1493 (1974). 205 A. Moze, I. Vizovisik, T. Malavasic, F. Cernee, and S. Lapanje, Makromol. Chem. 175, 1507 (1974). 206 J. R. Ebdon and B. J. Hunt, Anal. Chem. 45, 804 (1973). 207 P Peiper and W. D. Bascom, Anal. Calurim. Proc. Third Symp. 1974 Vol. 3 , p. 537 ( 1974). K. Hone, I. Mita, and H. Kambe, J . Polym. Sci., Part A-I 7 , 2561 (1969). 209 C. H. Klute and W. Viehmann, J . Appl. Polym. Sci. 5, 86 (1961). 210 T. Malavasic, I. Vizovisik, S. Lapanje, and A . Moze, Makrornol. Chem. 175, 873 (1974). 211 E. P. Manche and B. Carroll, Phys. Methods Macromol. Chem. 2, 239 (1972). 201

2oz

9.7.

OTHER APPLICATIONS

333

I I

I I I I

I I I I

t TIME

+

FIG.29. Typical DSC trace of an isothermal polymerization or crystallization. Area a is proportional to the amount of monomer reacted or fraction of polymer crystallized up to time t .

5

TIME (MINUTES)

FIG.30. Determination of initial rate constants for isothermal bulk polymerization of methyl methacrylate in the presence of 5.2 x lo-* mol/dm3 of 2,2'-azoisobutyronitrile (from Malavasic et a/.210).

9.

334

THERMAL ANALYSIS OF POLYMERS

vantage in obtaining numerical values for the rate constant, the order of reaction, the activation energy, and the preexponential factor Z . DTA/DSC has also found wide application in the study of curing reactions.zlz-zls Hess et a1.217*218 determined the relative degree of cure of an unsaturated polyester-styrene system by assessing the residual cure remaining in the system. The ratio of the heat of postcure to the heat of polymerization of an uncured polyester -styrene sample was defined as the degree of cure. This approach has also been applied to other sysBarton2z0applied DSC to the study of the curing reaction of bisphenol A diglycidyl ether with 4,4'-diamino-diphenylmethane. In the course of this study a method was developed to obtain a profile of resin cure characteristics and a prediction of isothermal curing curves from to dynamic DSC experiments. Andersonzz1 has applied DTA to the study of epoxides both reacted and unreacted with various polymerizing agents. It was shown that heating rate, concentration of polymerization catalyst, and extent of cure effect the resulting DTA curve. The effect of several organic peroxide initiators on the polymerization of diallyl-o-phthalate was reported by Willard.z2z Heats of reaction and reaction rate constants were calculated for each. Others have also reported the application of DTA/DSC to study the effects of polymerization catalysts.~~~"~4 9.7.4. Identification

Polymer melting points can be used for identification of components of incompatible polymer mixtures on the basis that each shows its characteristic melting endotherm in the mixture. This was demonstrated by Chiu (Fig. 31). Quantitative analysis reon a mixture of seven sulting in the percentage of each component in the mixture can also be performed.zz6 'lS

D. H. Kaelbe and E. H. Cirlin, J . Polym. Sci., Part C 35, 79 (1971). K. Hone, H . Huira, M. Sawada, I. Mita, and H. Kambe, J . Polym. Sci., Part A-I 8,

1356 (1970). '14 E. Sacher, Polymer 14, 91 (1973). '15 R. A. Fava, Polymer 9, 137 (1968). z18 J. M. Barton, Makromol. Chem. 171, 247 (1973). '17 P. H . Hess, D. F. Percival, and R. R. Miron, J . Polym. Sci., Part B 2, 133 (1964). G . B. Johnson, P. Y.Hess, and R. R. Miron, J . Appl. Polym. Sci. 6 , 519 (1962). '18 C. B. Murphy, J. A. Palm, C. D. Doyle, and E. M. Curtiss,J. Polym. Sci. 28,47 (1958). '"J. M. Barton, J . Macromol. Sci.,Chem. 8, 25 (1974). H. C. Anderson, Anal. Chem. 32, 1592 (1960). P. E. Willard, J . Macromol. Sci., Chem. 8, 33 (1974). K. L. Paciorek, W. G . Lajiness, and C. T. Lenk, J . Polym. Sci. 60, 141 (1962). 224 C. B. Murphy, J. A . Palm, and C. D. Doyle, 1.Polym. Sci. 28,453 (1958). 225 J. Chiu, J . Macromol. Sci., Chem. AS, 3 (1974). P. S. Gill, Can. Res. Dev. 1, (1974).

'"

9.7.

I

I

I

0

50

100

335

OTHER APPLICATIONS

150

1

I

200

250

I

300

I

350

400

TEMPERATURE (OC) Fic. 31. DTA curve of a seven-component polymer mixture (polytetrafluoroethylene (PTFE), high-pressure polyethylene (HPPE), low-pressure polyethylene (LPPE), polypropylene (PP), polyoxymethylene (POM), nylon 6 , and nylon 6,6). Sample weight, 8 mg; heating