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/<, ,$/A

=

[ha* + kb*I,

(6.1.59) (6.1.60) (6.1.61)

The geometrical relationships between the film coordinates x and y (see Fig. 30) for a given diffraction spot and the reciprocal lattice coordinates 5 and ,f are given in Table VII, as given by B ~ e r g e r . ~ ~ The main problem in indexing fiber patterns thus reduces to identifying the proper values of h and k to be associated with each value of 5. This t It is assumed that the reciprocal lattice is dimensioned in reciprocal angstrom units as previously discussed. If the reciprocal lattice is taken to be dimensionless then the coordinates are simply 6 and 6. In this case the radius of the Ewald sphere is unity. 33 M. J . Buerger, "X-Ray Crystallography," pp. 137ff. Wiley, New York, 1942.

6.1. U N I T CELL AND CRYSTALLINITY

73

TABLEVLI. Equations for Conversion from Film Coordinates to Reciprocal Space Coordinates A.

Cylindrical filma

6

=

y/(rF2 + y2)1i2

where r, is the radius of cylindrical camera. B. Flat-plate film" 6 = y / ( P + xp + 4)"'

6

=

[2 -

p

- 2(1 -

5")"'

D

1/I

(02 + x2)"d where D is the specimen-to-film distance. By convention 5 and 6 are dimensionless. If the reciprocal lattice is expressed in reciprocal angstroms, the coordinates analogous to 6 and 6 are (/A and ,$/A, where X is the wavelength of radiation.

necessarily involves trial and error procedures. One ordinarily begins by trying to find an hkO reciprocal lattice net, i.e., a parallelogram with edges a* by b" that accounts for the values determined from the equatorial reflections on the film. The smallest value of (will ordinarily correspond to a low index such as 100, 010, or 110. Once an appropriate trial net has been chosen, each value of 6 can be associated with a specific reciprocal lattice point. An example illustrating the indexing procedure for a cylindrical film pattern of polyethylene is given in Figs. 38 and 39 and Table VIII. Figure 38 shows the x-ray pattern made with Cu K a radiation. Table VIII lists the 2x values for each spot on the equator (or zero layer) and on the first layer. Also listed are the 6 and 6/h values for each line, which are computed from the x and y values for each spot using the equations of Table VII. Note that y = 0 for each equatorial spot and y = 22.77 mm for each spot on the first layer. Since the camera radius r, is 30 mm, the latter value also gives c =

/A sin[tan-'(y/r,)]

-L A = 2.55

5

A.

(6.1.62)

The values of d are obtained from Eq. (6.1.61), and 28 values are then readily obtained via Bragg's law. Figure 39 shows a series of circles with radii equal to the values of ( for the equatorial reflections (drawn to an appropriate scale). An a* by b* net must be selected so that a reciprocal lattice point lies on each of these circles, thereby indexing the responsible reflection. For a crystal whose

74

6.

X-RAY DIFFRACTION

FIG.38. Cylindrical film fiber pattern of polyethylene. Fiber axis vertical.

structure has not previously been determined, neither the magnitudes of a* and b* nor the angle between them is known at this point in the analysis. We first assume that they are perpendicular to each other. If, for example, we further assume that the first reflection is the 110 and the second is the 200, then 1a*I = 1/2dzoo= 0.135 A-' and Ib*l = 0.202 A-l. The net defined by these values is constructed as shown in Fig. 39. In this case all equatorial reflections can be indexed on this basis with the results shown in Table VIII. If some other assumptions had been made, the reciprocal lattice points defined by the net would not have corresponded with each of the circles of radius (/A, and the equatorial reflections would not have been indexed. In order to facilitate the trial and error procedure needed to identify the first few reflections and establish the a* by b* net, graphical techniques described e l ~ e w h e r eare ~ ~often . ~ ~ useful. 3r( L. E. Alexander, "X-Ray Diffraction Methods in Polymer Science," pp. 54-56. Wiley, New York, 1969. W. Bunn, "Chemical Crystallography, "2nd ed. pp. 153-154 and 460. Oxford Univ. Press. London and New York, 1961.

6.1.

75

U N I T CELL A N D CRYSTALLINITY

TABLEVIII. Indexing of Cylindrical Film Fiber Pattern of Polyethyleneo

Reflection

2x (mrn)

5

[/A

d

2e

hkl

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

22.45 24.95 31.55 38.10 40.15 42.75 46.20 51.55 55.05 60.15 64.50 68.95 72.05 78.20 81.30 93.00

0.372 0.413 0.520 0.624 0.657 0.698 0.75 1 0.833 0.886 0.961 1.024 1.087 I . 130 1.213 1.254 1.399

0.241 0.268 0.337 0.405 0.426 0.452 0.487 0.540 0.574 0.623 0.664 0.705 0.733 0.787 0.813 0.908

4.14 3.73 2.97 2.47 2.35 2.21 2.05 I .85 1.74 1.61 1.51 1.42 1.37 1.27 1.23 1.10

21.44 23.82 30.13 36.38 38.34 40.82 44.12 49.23 52.57 57.44 61.59 65.84 68.80 74.67 77.64 88.81

110 200 210 020 120 310 220 400 320, 410 130 230 510 330 520 040, 600 530

I

16.05 21.30 24.45 32.45 42.95 60.30 64.55 72.05 76.65

39.81 41.68 43.01 46.93 53.06 64.72 67.77 73.26 76.69

0.11 111 20 1 211 121 321, 411 03 1 23 1 51 1

First layer (y = 22.77 mm) 2 3 4 5 6 7 8 9

' Zero layer (equatorial), r,

=

0.313 0.375 0.415 0.578 0.658 0.883 0.937 1.029 1.083

0.203 0.243 0.269 0.336 0.426 0.573 0.608 0.667 0.703

30.0 mm, A

=

2.27 2.17 2.10 1.93 1.73 1.44 1.38 1.30 1.24

1.5418 A.

A similar procedure allows us to index the reflections on the first layer using the same a* by b* net, and assuming that c* is perpendicular to a* and b* (orthorhombic cell) so that the 001 reciprocal lattice point is at the center of the circles of radius e / h for the reflections on the first layer. These results are also given in Table VIII. The final results of the indexing procedure are that polyethylene is ort horhombic with

a

=

7.41 A,

b

=

4.94 A,

c'

= 2.55

A.

In general the reciprocal lattice section determined from the equatorial reflections must be considered tentative until all the reflections on the pattern have been indexed. If c* is perpendicular to a* and b* as in the example of polyethylene, then the procedure for indexing the reflections on the first and higher layer lines is identical to that used for the zero layer. But if c* is not perpendicular to a* and b*, then the situation is compli-

76

6.

X-RAY DIFFRACTION

I

FIG. 39. Indexing the equatorial reflections of polyethylene by the reciprocal lattice method.

cated by the fact that the point about which the section rotates is no longer easily located. The position of this point is affected by the angle that c* makes with the a* by b* plane and the magnitude of c". 6.153. Determination of Symmetry from Systematic Extinctions. It was pointed out previously that the existence of lattice centering can be detected from the systematic absences that occur in the x-ray reflections. These absences or extinctions are caused by the translations that produce the lattice centering. Provided there are a reasonable number of indexed x-ray reflections of the general type hkl it is always possible to decide

6.1.

77

U N I T CELL A N D CRYSTALLINITY

whether the unit cell is centered or not by examining which reflections are present and which are absent. The conditions for presence of the various classes of reflections are given in Table IX for each lattice type. It should, of course, be realized that a given reflection may be absent for a variety of reasons including poor measurement sensitivity. Therefore, TABLEIX. Extinction Criteria for Lattices and Symmetry Elements' Class of reflections hkl

Condition for presence

Lattice type: primitive + 1 = 2n body-centered =2n centered on the C face centered on the A face k+1=2n +1=2n centered on the B face

none h +k h+k k

h+k

centered on all faces k+1=2n +1= =22nn 1 + k 1 = 3n rhombohedral, obverse - k 1 = 3n rhombohedral, reverse h = 2n glide plane 11 (001) k = 2n h+k =2n h+k =4n k = 2n glide plane 11 (100) 1 = 2n k+1=2n k+I=4n glide plane 11 (010) h = 2n 1 = 2n h +1=2n h +1=4n glide plane 11 (170) 1 = 2n h = 2n h+k =2n 2h + 1 = 4n 1 = 2n screw axis 11 c 1 = 3n 1 = 4n 1 = 6n h+k =2n screw axis 1) a h = 4n k = 2n screw axis 1) b k = 4n h = 2n screw axis 11 [ 1101

h -h h hkO

Okl

hOl

hhl

001

Mw)

OkO

hh0 a

Interpretation

A&er Buerger.P*

+ +

Symbol

P I C A B F

Occurrence in lattice type

78

6.

X-RAY DIFFRACTION

conclusions can only be based on the classes of reflections that are present and by inference the classes that are absent. Other types of symmetry operators that involve translations also lead to extinction of certain classes of reflections for the same reason that the lattice-centering translations cause extinctions. Such symmetry operators are glide planes and screw axes. Thus it is often possible to establish the existence and nature of these symmetry elements from the extinctions they cause. The conditions for presence of the various glide planes niques. The x-ray technique does provide a method for fully establishing existence of various symmetry elements it is necessary to first determine the lattice type and then proceed to glide planes and finally to screw axes. Extinctions caused by lattice centering should not be used to infer additional symmetry and extinctions caused by glide planes should not be used to infer the presence of screw axes. It is clear that the general class hkl gives information about lattice centering, the classes with two independent indices such as hkO give information about glide planes, while information about screw axes is derived from the classes of reflections with only one independent index such as hOO. With the knowledge of the lattice symmetry gained from indexing the diffraction pattern (size and shape of the reciprocal lattice unit cell) together with the information gained from the systematic extinctions and other physical data, it is sometimes possible to establish the crystal’s space group. This is facilitated by the use of a “diffraction symbol,” which consists of the Laue or Friedel group, followed by the lattice symbol ( P , C , I, F), followed by symbols that indicate the presence or absence of glide planes and screw axes. The latter are written in the same order as the symbols in the Laue group or space group, a dot or dash being indicated if no glide plane or screw axis corresponds to a particular location. For example, Corradini and GanisZ6found that the fiber pattern of poly-urtho-methylstyrene could be indexed for a tetragonal unit cell with a = b = 19.01 A, c = 8.1 A. They noted systematic absences in the hkl reflections when h + k + I = 2n + 1, in the Okl reflections when I = 2n + 1, and in the hhl reflections when 2h + I = 4n + 2. According to Table IX these absences indicate a body-centered lattice with a c-glide plane parallel to (100) and a d-glide plane parallel to (li0). In the tetragonal system the order of listing symmetry elements in the space group is ca[lIO]. Since only two Laue groups 4/m or 4/mmm occur in the tetragonal system, the diffraction symbol is either 4/m I . cd or 4/mmmZ . cd. The space groups consistent with a given diffraction symbol can be obtained from Table 4.43 of the International Tableslo (Vol. I, p. 349). In general there are only a few space groups consistent with a given diffraction symbol, and in the case of 50 of the 120 diffraction symbols, the space

6.1.

UNIT CELL A N D CRYSTALLINITY

79

group is uniquely determined. The above case corresponds to one of these unique cases and the space group is found to be 14,cd. 6.1 5 4 . Routine Investigation of Molecular Structure. Often the configuration of the molecule is known or at least limited to a relatively small number of possibilities based on prior chemical data and other techniques. The x-ray technique does provide a method for fully establishing or checking proposed configurational models since any determination of the crystal structure will require that the molecular configuration also be determined. We return to consider the use of x-ray data to establish configuration in a subsequent example. Assuming that the configuration is already known, one of the major goals of any x-ray study of the structure of a polymer is to determine the conformation of the polymer chain, i.e., the repeat unit of the chain. This can often be done even if the chains do not pack together systematically to form a well-defined three-dimensional lattice. In the case that a lattice and its corresponding unit cell can-be defined and a reasonably large number of x-ray reflections are available for measurement, the most accurate approach is the classical method of comparing the calculated intensities for a trial structure to the experimentally observed intensities. In this case the configuration, conformation, and packing of the chains are determined simultaneously. The main problem then reduces to obtaining a reasonable trial structure. One of the simplest quantities to measure is the chain repeat distance. If the chains are assumed to lie parallel to the fiber axis, the repeat distance may be determined directly from the distance between layer lines on a fiber pattern using Eq. (6.1.51). Knowing the repeat distance, the configuration of the molecules, and the accepted values of bond lengths and angles, the conformation can often be reasoned out. For example, knowing that for polyethylene the repeat distance is 2.55 A, that it consists of+CH,+groups, that the C-C single bond length is about 1.54 A and that each carbon atom tries to maintain approximately tetrahedral bonding, the planar zigzag arrangement is found to be the only reasonable conformation. has noted that certain characteristic repeat distances are more likely in single-bonded carbon chain polymers. This is because of the tetrahedral bonding about each carbon atom and because of the principle of “staggered bonds,” which holds that certain bond rotation angles are more likely because they result in lower potential energy states because they reduce steric hindrance. Minima in potential energy tend to occur at rotation angles of 180, + 60, and - 60, as illustrated in Fig. 40. These rel36

C.

W.Bunn, Proc. R .

Sac. London. Ser. A 180, 67 (1942).

80

6. X-RAY DIFFRACTION X

X

X

X

H

li

T

G

6

FIG.40. Common bond rotation angles in single-bonded carbon chains. FromTad~koro.~'

ative bond rotation sequences are also called rrans ( T ) , left gauche ( G ) , and right gauche respe~tively.~' Repeat units thus tend to be made up of simple combinations of these bond sequences with characteristic repeat distances as illustrated in Table X and Fig. 41. The T2 symbol, for example, indicates that two adjacent trans sequences will produce a repeat unit, which in this case is identical with the fully extended chain. Alternating TG TG TG . . . sequences repeat after three alternations, i.e., (TG),. This chain has the symmetry of a 3/1 helix. Bond rotation angles differing somewhat from 180', and k60' also occur. With bulky substituent groups the helix may open somewhat to provide more space. It should, of course, be understood that substituent groups such as those occurring in vinyl polymers may require the tetrahedral bond angles in the chain to open somewhat to sterically accommodate the substituent groups. This latter effect would tend to lengthen the repeat distance slightly compared to the ideal case shown in Table X. The observation of a 6.5 8, repeat distance for isotactic polypropylene thus strongly suggests that the isotactic polypropylene molecule has a helical conformation corresponding to (TG),or a 3/1 helix in which the tetrahedral bond angles have been opened slightly compared to the ideal 109.5'.

(z),

37

H. Tadokoro, J . Polym. Sci., Purr C 15, 1 (1966).

TABLEX. Repeat Distance for Bond Sequences Obeying the Principle of Staggered Bonds" Identity period Mode

(A)

T2

2.5 5.0 4.4 3.6 a

After B ~ n n . ~ ~

Identity period Mode

('4 6.2 6.2 8.8 8.5

6.1.

UNIT CELL A N D CRYSTALLINITY

81

FIG.41. Chain conformations and repeat units based on the principle of staggered bonds. After B ~ n n using ~ ~ , the notation of T a d ~ k o r o . ~[H. ' Tadokoro, J . Polym. Sci. C , pp. 1-25 (1966)l.

The same type of argument may be useful in deriving information about the configuration of the chain as well as its conformation. The classical case is that of the difference between isotactic and syndiotactic vinyl polymers, e.g., the crystalline polypropylenes. Syndiotactic vinyl polymers cannot have a (TG), conformation with a repeat near 6.5 8, because the configurational repeat unit is approximately double that for the isotactic case. Several syndiotactic vinyl polymers have extended-chain conformations corresponding to T4 with repeat distances of approximately 5.0 A. The measured repeat distance for syndiotactic polypropylene is about 7.4 A, indicating that the molecule has neither an extended the synchain nor a 3 / 1 helix structure. According to Corradini et diotactic polypropylene chain exhibits a ( T2G2)2conformation. This corresponds to a 2/1 helix with four backbone carbon atoms (two monomeric units) per motif. As a final example of the use of the repeat distance to gain information about chain conformation consider nylon 6. The common a form of nylon 6 has a fully extended chain conformation with a chain repeat distance of 17.24 A as shown in Fig. 42. This chain repeat distance contains two chemical formula units. The unit cell of a nylon 6 as determined by 38

P. Corradini, G. Natta, P. Ganis, and P. A. Temussi, J . Polym. Sci., Purr C 16, 2477

(1%7).

82

6.

X-RAY DIFFRACTION

t

1

I

FIG.42. a extended chain structure of nylon 6 according to Holmes er a/.39[J. L. White and J. E. Spruiell, J . Appl. Polym. Sci.: Appl. Polym. Symp. 33, 91-127 (1978).]

Holmes et ~ 1 is monoclinic . ~ ~ with parameters a = 9.56 A, b = 17.24 A (chain axis), c = 8.01 A, p = 67.5'. The packing of the chains to form this cell is considered later. The designation of the chain axis as the b axis is required by the monoclinic unit cell. A second form of nylon 6 called y (or sometimes p ) occurs under certain conditions. The y form differs from the (Y form in both chain packing and repeat distance. The repeat distance for the (Y form is 16.88 A. This clearly indicates a small contraction from the fully extended chain length that must be caused by some twist or kink in the chain. Based on this reasoning and other data, A r i m ~ t proposed o~~ that the yform contains extended chain segments with kinks caused by rotation by about 60" of the group between C and C'. 0

II

C

\

N

/

I

c

\

C'

H 39

*'

D. R. Holmes, C. W. Bunn, and D. J . Smith, J . Polym. Sci. 17, 159 (1955). H. Arirnoto, J . Poly. Sci., Part A 2, 2283 (1964).

6.1.

83

U N I T CELL A N D CRYSTALLINITY

6.1 5 5 . Analysis of Helical Structures. The conformations of several helical polymer molecules have been determined by considering the particular effects on the diffraction patterns caused by the helical geometry. The x-ray diffraction effects caused by helical structure were first developed by Cochran et a / .4 1 and were extended by Franklin and K l ~ gClark , ~ ~ and MU US,^^ and Davies and The basic approach involves calculation of the Fourier transform, i.e., the structure factor, of atoms arranged on a helix. Although the packing of the helical chains together to form the three-dimensional structure of the unit cell must ultimately be considered, important details of the chain conformation can be learned from an investigation of the scattering from a single helical molecule. Aside from a proportionality constant this is equivalent to the scattering from a group of identical molecules oriented in a parallel array and scattering independently. In order to understand the significant results consider, following Cochran et the Fourier transform of a continuous helix of uniform unit electron density. As shown in Fig. 43, a point x , y, z in direct space can also be described by the cylindrical-polar coordinates r , 4, z; a point at x, q , { in reciprocal space can also be described by the cylindricalpolar coordinates 6, $, {. The helix of radius rand pitch P is defined by x = r cos(2~z/P),

y =

Y

sin(2~z/P),

z

=

z.

The value of the Fourier transform in reciprocal space is F(x,7 , 5) = .fexp[2~i(xx+ y q

+ zg)l d V ,

(6.1.63)

where the integration is over the helix, i.e., dV is the volume element of the helix. Introducing the above relations for x, y, z, W. Cochran, F. H. C. Crick, and V. Vand, Acfa Crysrallogr. 5, 581 (1952). R . E. Franklin and A. Klug, Acfa Crysrallogr. 8, 777 (1955). E. S. Clark and L. T. Muus, Z. Kristallogr., Krisrallgeom.. Kristallphys., Krisrallchem. 117, 108 (1962). 44 D. R . Davies and A. Rich, Acfa Crystallogr. 12, 97 (1959). IZ

''

a

b

FIG.43. (a) Cartesian (x, y , z ) and cylindrical-polar ( r , 4, z ) coordinates of a point on a helix. (b) Corresponding coordinates of a point in reciprocal space.

6.

84

=

lop

X-RAY DIFFRACTION

[

z

+ rr) sin 27r-PZ + z{

exp 27ri (YX cos 27rP

In cylindrical-polar reciprocal space coordinates this result is

F(5,

+, 5) =

I”

{ [

exp 27ri 5r cos

0

(27r;

-

+) + z t ] )

dz. (6.1.65)

This integral vanishes unless 5 = n / P , where n is an integer. When 7 1 show . ~ ~ that

5 = n / P , Cochran et ~

(6.1.66)

where J,, is the Bessel function of order n. These results show that the scattering from a helix with an identity period P is confined to layer lines at heights 5 = n / P in reciprocal space, a feature that is anticipated for any periodic structure of period P . The amplitude of the x-ray scattering and hence the intensity on the nth layer line is controlled by the nth-order Bessel function. A characteristic of Bessel functions is that maxima in J J X ) occur at increasing values of x with increasing order n. This leads to a characteristic X shape to the intensity distribution scattered from a continuous helix of given radius Y as shown in Fig. 44.45 The situation is a bit more complicated for a discontinuous helix. For a helical structure with pitch P and identical scattering centers located at points separated by the interval p , Cochran et ~ l . show ~ * that the transform vanishes except on planes at height

c = - n+ -

m (6.1.67) p P’ provided P / p can be expressed as a ratio of whole numbers. In Eq. (6.1.67)m ,like n, is any integer value. If the crystallographic repeat distance along the helix is c, then we must have

5 = UC, where 1 is the layer line number. Since t can be rewritten as 1 = tn

(6.1.68) =

c / P and u = c / p , Eq. (6.1.67)

+ um.

(6.1.69)

On a given layer line 1, the Bessel functions of order n satisfying Eq. M. Kakudo and N. Kasai, “X-Ray Diffraction by Polymers,” pp. 304ff. Kodansha Ltd., Tokyo, 1972.

6.1.

U N I T CELL A N D CRYSTALLINITY

85

Laler line (Ihc lipiircr arc Ihc order number$)

I

3 2

I 0

-1

7

-2 -3

-2

’

-4

-3 -6

..

(a)

6 5 4 3 2 1 0 1 2 3 4 5 k

(b)

Order of Bcrrcl fmclian

FIG. 44. Characteristics of diffraction pattern from a continuous helical structure (a) Bessel function distribution for n = I (The intensities of various layer lines are influenced by J.). (b) The angle a is roughly equal to the pitch angle of the continuous helix. [M. Kakudo and N. Kasai, “X-Ray Diffraction by Polymers,” Kodansha Ltd., 12-21, Otowa 2-Chome, Bunkyo-ku, Tokyo 112 (1972)l.

(6.1.69) contribute to the transform, which is given by (6.1.70) The summation runs over values of n satisfying the selection rule, Eq. (6.1.69). For an actual molecule there will be atoms of different types, each lying on helices of different radii and exhibiting different scattering factors. The extention of Eq. (6.1.70) to this general case gives

fiJn(2.rrtr1)exp

= n

J

JI

- +I

+ “) 2 +

%I},

(6.1.71)

wheref, is the scattering factor of thejth atom, r1the radius of the helix on which it lies, and and zj rotational and translational shifts of the atom, which arise because the first atom on t h e j t h helix may not start off at x = r , y = 0, z = 0. Franklin and K l ~ gClark , ~ ~and and Davies and give cylindrically averaged expressions based on Eq. (6.1.71) from which the intensity distribution on each layer line of a fiber pattern can be computed for a single molecule. The appropriate equation for the averaged x-ray intensity scattered by a helical polymer chain is

86

6.

X-RAY DIFFRACTION

+ [C frJn(2m.frj)sin (%C

- n4,)I2}.

(6.1.72)

J

Such calculations can be useful, but are lengthy and are best carried out by computers. An example is shown in Fig. 45 for the proposed 13/6 helix of polytetrafluoroethylene (PTFE), where it is also compared to the actual fiber pattern. In considering the comparison it must be remembered that the crystalline chain packing in PTFE causes intermolecular interferences, which result in discrete spots on each layer line. One can consider that the fiber pattern of a crystalline material samples the molecular transform only at the reciprocal lattice nodes. The full calculation of the layer line intensities for a given trial molecular structure is probably justified only if it is needed to distinquish between two similar but different trial structures. A somewhat simpler, more qualitative approach based on the selection rule, Eq. (6.1.69), is often sufficient to distinguish between possible trial structures. The basis of this approach is that in the case of most synthetic polymers the contributions to the fiber pattern from high-order Bessel functions are negligible and only low-order Bessel functions with n less than about 7 need to be considered. Further, the lower the order of the Bessel functions contributing to a given layer line, the stronger will be the average intensity of scattering on the layer line and the nearer to the meridian will the strongest maximum lie. These ideas can be readily illustrated in the case PTFE.

FIG.45. (a) Averaged x-ray intensity calculated for a 13/6 helical molecular model and (b) fiber pattern of polytetrafluoroethylene (0°C).

6.1.

87

U N I T CELL A N D CRYSTALLINITY

0 0

12

0

0

0 0

0

10

0

0

0

0

8

0 0

0 6 0

0

0

0

0

4

0

0

0

0

2

0

0 a

Figure 46 shows the solution of the selection rule for the 13/6 helical conformation ( u = 13, f = 6). In Table XI the low-order Bessel functions contributing to each layer line are given and compared to the estimated average layer line intensity from the fiber pattern of Fig. 45b. The agreement is reasonably good. A comparison to the more quantitative calculations shown in Fig. 45a is also interesting. Note also the distance from the meridian of the maxima in Fig. 45a and compare to the principal order of Bessel function contributing, Table XI. 6.1.5.6. The Packing of Chains. The crystal structure of a polymer is determined by the conformation of the molecules and the lateral packing of the chains. In general, the molecules are packed together in a manner that tends to minimize the potential energy of the structure, a fact that can be interpreted more specifically to mean that the molecules tend to pack TABLEXI. Selection Rule for 13, Helix (Polytetrafluoroethylene) I

Intensity

n

I

Intensity

n

0 1 2 3 4 5 6

medium weak medium weak very weak very weak weak strong

0 -2 -4 -6

7 8 9 10 11 12 13

very strong not observed not observed not observed not observed not observed not observed

-1 -3

+5

+3 +I

-5 +6 +4 +2 0

88

6.

X-RAY DIFFRACTION

so as to minimize the volume occupied while maintaining appropriate distances for van der Waals bonds between adjacent chains. The packing controls the precise size and shape of the unit cell and hence the location of the spots on each layer line of the fiber pattern. Conversely, if the size and shape of the unit cell can be determined by indexing the fiber pattern, the number of molecules threading through each unit cell can be established. Systematic extinctions will give some information about the symmetry of the packing arrangement; consequently, it may be possible to reason out one or more possible packing arrangements. Scale models of the molecules are very helpful at this stage. These packing arrangements can then serve as trial structures against which to compare the experimental intensity data. Refinement of this structure then proceeds in a manner similar to that for any other crystal structure as described in the next section. As a specific example consider the case of poly-ortho-methylstyrene, whose space group is 14,cd, as discussed previously. The molecular weight per chemical repeat unit of this polymer is 117.9 and its experimental density is about 1.07 g/cm3. Application of Eq. (6.1.57) gives 16 chemical repeat units in each unit cell. The crystallographic repeat along the chain of 8.1 8, suggests four chemical repeats per crystallographic repeat with a conformation approaching (TG)z(TC)z(see Table X). Thus, there are four chain stems threading through each unit cell. Figure 47 shows the space group and the structure model that Corradini and Ganis2s determined for poly-ortho-methylstyrene. Note that the only reasonable locations for the helical molecules are coaxial with the 4, screw axes in the space group. The molecule in the upper right-hand corner, for example, is related to the molecule in the lower left-hand corner by the body-centering translationja + j b + Jc. The c-glide planes require that each right-handed helical chain is surrounded by four left-handed chains and vice versa. A few additionalpoints concerning the packing of chains are worth special note. First, the possible existence of hydrogen bonding, as between NH and OC groups on adjacent chains in polyamides and polypeptides, will often provide a key to the packing arrangement. For example, the molecules of many extended-chain polyamides and polypeptides occur in hydrogen-bonded sheets. This is illustrated for the a form of nylon 66 in Fig. 48. The stacking of these hydrogen-bonded sheets to form the structure must then be considered, with the result for nylon 66 illustrated in Fig. 49.46 As shown by Bunn and Gamer," there is one chemical repeat per triclinic unit cell. E. S. Clark and F. C. Wilson, in "Nylon Plastics" (M. 1. Kohan, ed.). p. 276. Wiley,

New Yo&, 1973. I7 C. W. Bunn and E. V. Gamer, Proc. R.

London. Srr. A 189, 39 (1947).

6.1.

U N I T CELL A N D CRYSTALLINITY

89

FIG.47. Projections on the (001) plane of (a) the space group, and (b) the structure model of polyurtho-methylstyrene. [P. Corradini and P. Ganis, Nuovo Cimento. Suppl. 15, 96

(19W.l

FIG. 48. Arrangement of nylon-66 chains in crystals of a-form according to Bunn and Gamer. '* [ J . L. White and J. E. Spruiell, J . Appl. Polym. Sci.: Appl. Polym. S y m p . 33, 91 - 127 (1978).]

90

6. X-RAY DIFFRACTION

FIG.49. Perspective drawing of a unit cell of nylon-66. The viewpoint is 11A up, 10A to the right and 40A back from the lower left comer of the cell. (b) Principal crystallographic planes of nylon-66. [E. S. Clark and F. C. Wilson, in “Nylon Plast’cs,” (M. I . Kohan, ed.). Wiley, New York, 1973.1

If the chain does not possess a center of symmetry, but is directional, there is the possibility of either parallel or antiparallel packing. Alternate chains in the hydrogen-bonded sheets of the (Y form of nylon 6 are antiparallel39as shown in Fig. 42. This antiparallel arrangement more readily produces complete hydrogen bonding than does a parallel arrangement of extended chains. Pauling and core^*^ have pointed out that strong hydrogen bonding can occur with parallel chains in a so-called pleated sheet structure as illustrated in Fig. 50 for an arbitrary polypeptide chain. Notice, however, that the chains do not have fully extended conformations in this structure. The y form of nylon 6 appears to be a variant of this pleated sheet structure. I8

L. Pauling and R. B. Corey, Pruc. Narl. Acud. Sci. U.S.A. 37, 251 (1951).

6.1.

U N I T CELL A N D CRYSTALLINITY

91

6.15 7 . The Trial Structure and Refinement Techniques. The major problem in crystal structure analysis is to devise a trial structure that is a sufficiently good approximation to the true structure to be worthy of refinement. As already discussed, one approach is to simply make an “educated guess” at an appropriate structure based on the available information. Considering Eqs. (6.1.44)and (6.1.31),this problem is equivalent to evaluating the phases associated with each structure factor. It should be realized that there is no general approach that is guaranteed to solve this problem. Thus the ingenuity of the investigator is often an important ingredient for finding a trial structure. In addition to the basic principles already discussed there are some other guidelines, strategies, and data-handling techniques that can be used to aid the crystallographer in his search for a trial structure. Many of these are beyond the scope of the present review, but a few simple principles can be described here. Perhaps the most important of these occurs in the case of centrosymmetric crystals, i.e., crystals that contain a center of symmetry. For such crystals p(x, Y ,

2) =

p(%

(6.1.73)

7, Z).

It is readily shown that this requirement necessitates that the phases of the Fhkl for such a crystal are either 0 or T,and the problem reduces to establishing whether the modulus IFkkll is to be accompanied by a positive or negative sign (eo = 1 , eirr= - 1). Also Eq. (61.73) implies that p ( x , y, 2 ) is an even function; thus

CFR

C\” R

\

\

\

\

CHR

C,HR

C,HR

C,H R

CHR

\

CpR

C,HR

\CHR

‘CHR

‘Cp R

\cHR

\cHR

‘CHR

\ \ \ FIG. 50. Pleated sheet model of Pauling and Carey.* [J. L. White and J. E. Spruiell, J . Appl. Polym. Sci.: Appl. Polym. Symp. 33, 91-127 (1978).]

6.

92

X-RAY DIFFRACTION

A second principle is that the contribution from a heavy atom will often dominate the phases of the Fhkl values computed for the entire unit cell. Thus if the precise position of a heavy atom (or atoms) can be determined, the phases for the heavy atom contribution can be computed and used together with the experimental values of IFhkll to obtain a zeroth approximation to the electron density function p ( x , y , 2). This function will often reveal additional information about other atoms in the structure and allow further refinement of the computed phases. Other principles and strategies for finding a trial structure include the replaceable atom method?* the use of relations between structure factors to determine their p h a s e ~ and , ~ the ~ ~ use ~ ~of the Patterson f u n c t i ~ n . ~ ~ . ~ The interested reader is referred to excellent discussions of these methods found elsewhere. 54-57 Once a trial structure worthy of refinement has been found, it is a relatively straightforward procedure to refine the structure to the limit allowed by the accuracy of the experimental data. Two basic approaches are the primary ones used for refinement. The first is the application of successive Fourier syntheses. In this approach the phases of the Fhkl calculated from the initial trial structure are used with the experimental moduli IFhkll to synthesize the electron density function. It will often turn out that a study of this synthesized electron density function will reveal that certain atoms are not at precisely the same locations assumed in the trial structure. Relocating these atoms allows a new set of phases to be computed, and a second Fourier synthesis is carried out. This process can be continued until no further improvement in the model occurs between successive Fourier syntheses. The second approach to refinement is to use least squares multiple regression. This approach treats the unknown atomic coordinates as the Is

D. Harker, Acra Crysrallogr. 9, 1 (1956). D. Harker and J. S. Kasper, J . Chem. Phys. 15, 882 (1947); Acra Crysrallogr. 1, 70

( 1948).

'' R. K. Bullough and D. W. J . Cruickshank, Acra Crystallogr. 8, 29 (1955). sz A. L. Patterson, Phys. Rev. 46, 372 (1934); 2. Kristallogr., Mineral., Retrogr., Abr. A

90, 517 (1935). Js

D. Harker, J . Chem. Phys. 4, 381 (1936). H. Lipson and W. Cochran, "The Determination of Crystal Structures." Bell, London,

1953.

M. J. Buerger, "Crystal Structure Analysis." Wiley, New York, 1960. M . M. Woolfson, "Direct Methods in Crystallography." Oxford Univ. Press, London and New York, 1961. " M. J. Buerger, "Contemporary Crystallography." McGraw-Hill, New York, 1970.

6.1. UNIT

CELL A N D CRYSTALLINITY

93

constants to be evaluated in a series of equations for the Fhkl. An advantage of this method is that unknown parameters of the temperature factor can be found as well as the mean atomic locations. Extensive discussions of refinement techniques can be found in B ~ e r g e r ~ ~and *~' Woolf~on.~~ 6.15 8 . Application of Structure Analysis Techniques. An outstanding example of the application of many of the foregoing principles to determine the crystal structure of a polymer was recently provided by Tanaka, Chatani, and Tadokoro's analysis5* of polyisobutylene (PIB). The PIB molecule contains two methyl groups on alternate carbon atoms, 1.e..

which cannot be spatially accommodated by a fully extended chain conformation. Thus, when stretched, PIB crystallizes with the molecules in a helical conformation. Several previous investigators had studied the structure of PIB including Fuller et a / ., 59 Liquori ,80 Allegra et ul., and Bunn and Holmes.82 Tanaka et a1.58made fiber patterns of samples stretched eleven times their original length and estimated the intensities of all but the weak reflections using a microphotometer. The x-ray patterns were indexed assuming that PIB belongs to the orthorhombic crystal system. All reflections could be indexed on this basis with a unit cell of dimensions u = 6.88, b = 11.91, and c (fiber axis) = 18.60 A. The observed systematic absences were (hOO)when h is odd, (oko)when k is odd, and (001)when 1 is odd. The lattice is thus primitive and, according to Table IX, the cell contains a 2, screw axis parallel to each crystallographic axis. This is sufficient to establish the space group uniquely as P2,2,21. The chain repeat distance and the distribution of intensity in the layer lines suggested that the repeat unit contained eight monomeric units. In order to obtain a reasonable calculated density, it was necessary to double this number of monomeric units by assuming that two molecules 68

T. Tanaka, Y. Chatani, and H . Tadokoro, J . Polyn. Sci., Polym. Phys. Ed. 12, 515

( 1974). as 'I

C. S . Fuller, C. J. Frosch, and N . R. Pape, J . Am. Chem. Soc. 62, 1905 (1940). A . M. Liquori, Acra Crystullogr. 8, 345 (1955). G. E. Allegra, E. Benedetti, and C. Pedone, Macromolecules 3, 727 (1970). C. W. Bunn and D. R . Holmes, Discuss. Furaday Soc. 25, 95 (1958).

6.

94

X-RAY DIFFRACTION

TABLEXII. Comparison of Layer Line Intensities to Orders of Bessel Functions for Possible Polyisobutylene Helices

Layer line 1

Estimated average intensity of the layer linea ~

~

0 1 2 3 4

5 6

7 8 9

Orders of Bessel functions, n 8/1 Helix

8/3 Helix

8/5 Helix

8/7 Helix

~~

Very strong Medium weak Medium Strong Medium Strong Medium Weak Strong Very weak

-8, 0, 8

-7, -6, -5, -4, -3, -2, -1, -8, -7,

1 2 3 4 5 6 7 0, 8 1

-8, -5, -2, -7, -4,

0, 8 3 6 1

4

-1, 7

-6, 2 -3, 5 -8, 0, 8 -5, 3

-8, 0, 8 -3, 5 -6, 2 -1, 7 -4, 4 -7, 1 -2, 6 -5, 3 -8, 0, 8 -3, 5

-8, -1, -2, -3, -4, -5, -6, -7,

0, 8 I 6 5 4 3 2 1 -8, 0, 8 -1, 7

Estimates given by Liquon.m

thread through each unit cell. The calculated density on this basis, 0.972 g/cm3, was in reasonable agreement with the experimental density of 0.916 g/cm3 for the unstretched, amorphous sample. These results were in good agreement with the earlier study by Fuller et ~ 1 . ~ ~ The general nature of the helical conformation taken by the molecules was considered next. As previous investigators had noted, certain features of the chain conformation can be established and certain possibilities eliminated using qualitative considerations alone. Only 8/1, 8/3, 8/5, and 8/7 helices need be considered, since an even number of turns with an even number of monomer units would have the effect of generating a reduced repeat distance. LiquorPo had also argued that the 8/1 and 8/3 conformations were not stereochemically feasible, but later studiesB1did not seem to bear this out. A more direct approach to further limit the possibilities is the qualitative application of the helical transform theory based on the selection rule, Eq. (6.1.69). Table XI1 shows a comparison for each helix type between the low-order Bessel functions contributing to each layer and the estimated average intensity of the layer line. It is clear from Table XI1 that this type of qualitative analysis does not distinguish between the 8/1 and 8/7 and between the 8/3 and 8/5 helices. On the other hand, both the 8/1 and 8/7 helices can be eliminated from further consideration because the Bessel function orders contributing to each layer line do not compare favorably with the experimental intensities while those for the 8/3 and 8/5 helices do. Tanaka et ~ l . ~went * a step further and calculated the cylindrically averaged x-ray intensity scattered from a single chain from Eq. (6.1.72)

6.1. U N I T CELL A N D CRYSTALLINITY

95

30 Liquori

20 8 t h layer

10

i . .-..,,

C

1 0

6

E

E

lo

o

__

L

g

1

- . L h I a y e '

1 -L l-., 6th'ayer ......

i!L ...

0

30

......

./.,.. . . ... .. .. .

: I . .

, . . :,. ,:.... . . . .4 t h. layer '

42001

,

02

04

Wd')

>.

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.

6.2.

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

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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

7.2.

FUNDAMENTALS

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.

192

7.

ELECTRON MICROSCOPY

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).

7.2.

FUNDAMENTALS

193

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.

194

7.

ELECTRON MICROSCOPY

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

7.3.

ELECTRON OPTICS

195

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.

7.

I96

ELECTRON MICROSCOPY

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.

7.3.

ELECTRON OPTICS

197

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-

7.

198

ELECTRON MICROSCOPY MID PLANE (PRINCIPAL PLANE 1 H

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.

7.3.

ELECTRON OPTICS

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.

200

7.

ELECTRON MICROSCOPY

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

7.3.

ELECTRON OPTICS

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

202

7.

ELECTRON MICROSCOPY LENS "1

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-

7.3.

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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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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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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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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.

212

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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.

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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).

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and thereby acquire a high degree of stability. Such films exhibit a small mass loss (<5%) and negligible shrinkage, even after very high electron doses. For very high resolution work, where the size of the structure to be studied approaches that of the supporting substrate, a holey film support is employed. The specimen is supported on a thin film which contains a network of holes and is examined where it bridges a hole so that electron scattering is attributable only to the specimen. While various techniques for producing holey films have been described, particularly good results are obtained using the technique of Fukami and Adachi.I5 Production of their “self-perforated microgrids” involves applying a water repellent to a glass slide, forming numerous minute water droplets (by condensation), spreading a dilute solution of plastic over the droplets, and stripping the film formed by evaporation of the solvent. Hole sizes of 0.2- 10 pm can be prepared by altering the size of the water droplets, by selecting the appropriate water repellent and/or adjusting the amount of substrate cooling. Such holey films are often stabilized by application of an evaporated carbon film. 7.5.1.3. Shadowing. Electron microscopy of organic polymers is somewhat hampered by the low-contrast images they produce. We learned earlier that image contrast arises from electrons being scattered outside limiting apertures in the microscope. It should not be unexpected then that materials exhibiting a low scattering power (low density), such as organic polymers, would produce low contrast images. Figure 3b illustrates the problem. Here a polyethylene single crystal is seen to have little more contrast than its carbon support film. Since electron scattering is also dependent on electron beam energy, a slight contrast improvement could be obtained by reducing the accelerating voltage. However, a more dramatic increase will accrue from shadowing the specimen. Vacuum evaporation of a heavy metal onto such a sample produces a very highly scattering layer, which greatly improves contrast (see Fig. 4). Applied at an angle, shadowing also accents surface perturbations. If the angle is known, the technique provides a means for determining thickness, as shown in Fig. 20. Carbon-platinum is perhaps the most commonly used shadowing material because it is easily applied and exhibits a fine grain structure that allows high-resolution microscopy to be performed. Recent work, however, indicates that tungsten might be preferred for ultrahigh-resolution work.Ig

Is

A. Fukami and K . Adachi, J . Electron Microsc. 14 (2). 112 (1965). H. S . Slayter, Wtrumicroscopy 1, 341 (1976).

7.5.

OPERATIONAL CONSIDERATIONS

a = b tan

/

215

SHADOWI NG ANGLE

Q

SPECIMEN

FIG.20. Shadowing used to determine thickness.

It should be noted that contrast enhancement by shadowing is only applicable to bright-field imaging. For electron diffraction or dark-field microscopy, the diffracted beams from the metal coating would overwhelm those from the sample. Further, the shadowing layer must be applied directly to the specimen to be meaningful. That is, it must precede application of a carbon support film. In those instances where the sample is already mounted on a carbon film, such as when a diffraction specimen has been prepared and examined, the shadowing step to improve contrast must be performed on the specimen side of the film. This point is stressed because in some cases the sample will be on the side of the carbon film closest to the metal support grid, and the microscopist will have to remember to perform his shadowing from underneath or through the grid. 7.5.1.4. Sectioning. If the internal structure of a bulk sample is of interest, the specimen may be sectioned. It is particularly difficult to prepare thin sections of organic polymers, since they tend to deform or bend away from the cutting edge. An automatic microtome equipped with a diamond cutting blade is recommended for sectioning polymers. For the softer or lower molecular-weight polymers, it may be necessary t o cool the sample or reinforce it by impregnating it with a hard material o r potting it in a harder matrix. One study involved infusing liquid sulfur into crazes in the sample, quenching, sectioning, and removing the sulfur by sublimation in vacuum.'' Once the section is prepared, the problem of obtaining meaningful contrast again arises. The surfaces will possess cutting marks, of course, and so the shadowing technique is not applicable. The researcher generally sections samples that have constituents whose differences can be accented by staining or etching techniques. R . P. Kambour and A. S. Holik, .I. Polym. Sri.. Part A-2 7, 1393 (1969).

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7.5.1.5. Staining. Staining a sample means chemically treating it with an agent that will affect one of its constituents to a greater degree than any others. A notable demonstration of the technique arose from studies of polybutadiene rubber in ABS plastic. Exposing a thin section of this material to an osmium tetroxide vapor, or dilute aqueous solution, results in the rubber particles absorbing the osmium.1* This enhances their electron scattering ability and thereby improves their contrast. A modified osmium tetroxide stain, using solvents, has shown success on this and similar systems.lD 7.5.1.6. Etching. In the case where a sample is a blend of two polymers, neither of which has a known staining agent, or where different phases of the same polymer exist because of variations in molecular weight or crystallinity, an etching technique may be employed to accent the physical differences. Preferential attack of low-molecular-weight species in polyethylene by nitric acid fumes is one well-known demonstration of this technique.20 More recently, a gentler, more discriminating etchant was used to clearly reveal lamellar detail in a number of olefins.21 Unlike nitric acid, the new permanganic reagent does not penetrate the interior of the sample. It appears to remove disordered material preferentially and is sensitive to crystallographic (lamellar) orientation. Polymer solvent etchants are less severe than oxidizing agents, but they can promote swelling of the specimen. Since etching is generally a surface treatment of relatively thick material, samples so treated may be too thick to be placed directly into the microscope. Replication, a specimen preparation technique to be described, provides the means for studying such etched surfaces. Extreme care must be taken when interpreting images of etched samples. The specimen can easily be deformed during the etching or subsequent handling stages, so that conclusions about the original structure should be substantiated by other investigative techniques. 7.5.1.7. Fracturing. In addition to sectioning, the internal structure of a bulk specimen can be revealed by fracturing; i.e., cooling the specimen to liquid nitrogen (or liquid air) temperatures so that it becomes brittle, and then breaking it. Since the fractured specimen is still thick, electron-microscopic examination can only be performed by preparing a replica or, if the surface is too rough for this, by employing a scanning electron microscope. K . Kato, J . Electron Microsc. 14 (3). 219 (1965). C. K. Riew and R. W. Smith, J . Polym. Sci., Part A-1 9, 2739 (1971). lo R. P. Palmer and K. Sakaoka, Makromol. Chem. 74, 174 (1964). '' R . M. Olley, A. M. Hodge, and D. C. Bassett, J . Polym. Sci.-Polym. Phys. Ed. 17,627 Is

(1979).

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217

For organic polymers, fractography is a particularly difficult technique to master. The fracture surface is complicated with structure that at times is dependent more on the way the sample was fractured than on the composition of the material. With care, however, valuable information can be extracted from fractured sample.22 7.5.1.8. Replication?. In some instances it is desirable to examine a surface of a bulk, fractured or etched sample. In this case a replica of the surface is prepared. A common technique is to coat the surface with a collodion solution, typically 1-5% in amyl acetate. Following evaporation of the solvent, the plastic replica is peeled off, shadowed, and coated with a carbon film. It is then placed on a metal support grid, plastic side down, and exposed'to solvent fumes that slowly dissolve the plastic, leaving only the shadowed carbon replica. For polymers in particular, much care is required in preparing replicas and interpreting their images. The replication procedure may deform the sample during stripping, leading to artifacts in the final carbon replica. Or, too thick a replica may swell during dissolution of the plastic, fracturing the carbon replica. It should be noted that the attainable resolution of a replica never approaches that of a directly observed specimen, somewhat limiting the technique's applicability. 7.5.1.9. Gold Nuclei Decoration. The last specimen preparation technique to be considered, gold nuclei decoration, has limited applicability but is worth noting. Some specimens exhibit very small steps, perhaps a few unit cells high, on their otherwise flat surfaces. The usual shadowing technique will fail to reveal these. Such samples are given a verv thin coating of gold and are heated slightly.23 The gold tends to migrate to the steps and clearly delineate their boundaries. This technique may become obsolete with the advent of topographical contrast imaging, a relatively new method of improving contrast in high-resolution electron microscopy (see Section 7.6.2.). 7.5.2. Focusing Focusing of the electron microscopic image was touched upon earlier when a through-focus series of micrographs was described. To produce such a series, which records over- and underfocused images about the optimum focus, the microscopist must, of course, be able to recognize a near-focus condition. Let us consider ways of obtaining a near-focus and focused image.

t

See also Vol. 11 of this series, Section 13.5.1. F. R. Anderson, J . Appl. Phys. 35 ( I ) , 64 (1964). G . A . Bassett, Philos. M a g . [8] 3, 1042 (1958).

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7.5.2.1. Wobbler Focusing. For low-magnification imaging, a technique called wobblerfocusing is commonly employed. Most current-day instruments contain deflection coils in the condenser system that, when excited by an alternating current, effectively oscillate the illuminating beam between two different incident directions. The result is that the image is blurred for all settings of the objective lens except where it is focused on the object plane. Wobbler focusing is very useful in bringing a badly out-of-focus image close to focus. Seldom, however, can an exactly focused image be produced by this technique. 7.5.2.2. Minimum-Contrast Focusing. More accurate focusing of low-magnification images can be performed using minimum-contrast f o cusing. Here, the objective aperture is removed and the objective lens current is rapidly varied. The microscopist finds that setting at which minimum contrast is obtained. The objective aperture is reinserted before photographing the image. Minimum-contrast focusing, although not widely used, does produce good results at up to perhaps 10,000~ magnification. 7.5.2.3. Fresnel Fringes. An aid in determining the relative focus of highly magnified images is to observe their Fresnel fringes. These are a series of intensity maxima that appear in out-of-focus images and arise from the interference of out-of-phase rays. A holey film is a good specimen for studying Fresnel fringes. We know that an electron beam is retarded as it passes through thin-film specimens and that the amount of retardation is dependent on the specimen’s density. A beam passing through a hole in the carbon film is not retarded, while that passing through the surrounding film experiences a phase delay. If the image of such a sample is perfectly focused, the rays passing through the hole and those passing through the film combine and the hole appears in simple contrast with the film. If, however, the lens is focused above the object (overfocused), the rays will interfere and produce a bright fringc around the hole. Similarly, focusing below the object (underfocusing) produces a series of more distinct virtual fringes of reversed contrast within the hole (see Fig. 21). By observing the contrast of these fringes then, the microscopist can determine where he is relative to optimum focus and may adjust the objective lens to obtain a desired effect for a single micrograph or may proceed to record a through-focus series. It happens that Fresnel fringes are also extremely sensitive indicators of the operating condition of the microscope. For example, a very slight deviation from the axial symmetry of the fields in the objective lens causes the Fresnel fringes to appear asymmetrical. By routinely adjusting special stigmating coils to produce symmetrical fringes, the microscopist optimizes the objective lens fields.

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FIG.21. Fresnel fringes of underfocused image of a hole in a carbon film (4.85 p n underfocused).

Further, since very slight changes in focusing affect Fresnel fringes, any instability in those components responsible for maintaining focus will be readily apparent. Specifically, instability in the objective lens current or the high-tension supply t o the gun will affect a blurring of the fringes. Vibration of the object, arising from ineffective vibration isolation of the microscope, causes a similar fringe blurring. 7.5.2.4. Defocusing Contrast (Phase Contrast). The same mechanism responsible for producing Fresnel fringes is used to advantage in certain high-resolution studies. The technique pertains to the examination of thin-film specimens containing microvoids (<50 A) or similarly sized particles of a different density than the film. In either instance, diffraction contrast is insufficient to image the small structures. However, phase shifts, or temporary changes in wavelength of the electron waves, can provide observable contrast. A small void in the specimen will retard the electron beam less than the film. As a result, voids will appear as dark spots in an underfocused image and light spots in an overfocused image. A particle with a density higher than that of the film will exhibit

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the opposite effect-light in an underfocused image, dark in an overfocused image. Such voids and particles may not be visible at all when the image is perfectly focused. A through-focus series of micrographs is generally required to study these phenomena.

7.5.3.Resolution We earlier examined factors leading to electron beam spreading and the subsequent reduction in instrument resolution. Since many studies require that the ultimate resolving power of a particular instrument be employed, let us establish a clearer picture of what resolution means and indicate physical steps the microscopist can take to ensure optimum performance of the microscope. Figure 22 represents an intensity profile across two closely spaced particles in a bright-field image. The light curves represent the profiles of the individual particles, the dark curve that of their sum. The hillock between the minima will vary in height depending on the separation of the particles, being smaller for closer particles. If the intensity difference between this local maximum and the minima becomes too small, the particles will appear as one object. Two particles are said to be resolved if their minima are separated by a distance equal to the width of their individual profiles at half their maximum intensity change. Although this half-width criterion is selected arbitrarily, in practice it is extremely difficult to resolve two images that are separated by less than this half-width distance. The phenomena responsible for the beam spreading depicted by these profiles have been noted. Let us briefly mention what practical steps can be taken to minimize resolution-limiting factors. Perhaps the most important action the microscopist can take to maintain a high-resolution capability is to be sure the microscope is performing at its design capability. Specifically, the stability of the high-voltage supply and the lens current stabilizers should be checked routinely.

t -I+ FIG. 22. Intensity profile across two closely spaced particles (bright-field image).

7.6.

OTHER MICROSCOPY TECHNIQUES

22 1

Alignment of all components of the electron optical column must be maintained. The instrument’s isolation from external vibration should be verified. As noted earlier, these resolution-limiting factors can be checked by studying the Fresnel fringes around a hole in a carbon film. Contamination buildup on the specimen, due to polymerization of residual gas molecules, must also be identified and minimized. Its effects can also be observed with the aid of a holey film. If contaminating species are being deposited on the specimen, a hole will shrink in size and eventually close. By monitoring this process the contamination rate can be calculated and, if found unacceptable, reduced by appropriate anticontamination techniques, as described earlier. Contamination is not restricted to the specimen. Any surface in the microscope that “sees” the electron beam is susceptible to a build-up of polymerization products. Surfaces of particular concern are the various fine, limiting apertures, especially those in the condenser lenses and the objective lens. An insulating layer deposited here will become charged by the electron beam and deflect or distort the beam. Regular cleaning, or use of thin metal “self-cleaning” apertures where possible, is prerequisite to performing high-resolution microscopy. Should the microscopist be planning a very high resolution study, consideration must also be given to the structure of the support film, the sample coatings, and even the recording film. 7.5.4. Magnification Calibration

The magnification of the electron microscopic image is known from the settings on lens control knobs or from magnification meters. Both of these indicators, however, cannot be expected to be accurate to better than ?5%. If a more precise determination of the magnification is required, use of an “internal” calibration standard is required. Such standards are substances of known size, which are deposited on top of or along with the specimen. Two common internal standards are latex spheres and ruled grating replicas. The former can be purchased commercially in various sizes from 0.091 to 100 pm. The latter is available with 1134 or 2160 lines/mm. Direct measurement of either of these standards on one’s micrographs allows accurate determination of magnification.

7.6. Other Microscopy Techniques In this chapter, we examine methods of performing special microscopic investigations. We begin with three techniques employed on the transmission electron microscope. The remaining methods require the use of a different kind of electron microscope.

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7.6.1. Small-Angle Electron Diffraction

Some ordered materials, including many polymer samples, exhibit large repeat distances (20-20,000 A). Specialized operation of the electron microscope allows determination of these spacings by small-angle electron diffraction. The technique calls for adjustment of the condenser lenses to focus the electron beam on the specimen. The objective lens is not energized, and the diffraction pattern is focused on the viewing screen by use of the diffraction and projection lenses. The greatly enlarged image allows observation of spacings up to several thousand angstroms. Bassett and KellerZ4applied the technique to the study of stretched polyethylene, publishing diffraction patterns exhibiting spacings between 1000 and 2000 A. Assuming the required thin-lilm specimen can be prepared, small-angle electron diffraction offers advantages over conventional X-ray smallangle diffraction. Patterns are produced more quickly and easily, and larger spacings are obtainable. Further, the area that produced the diffraction pattern is also imaged and recorded, allowing for a more accurate structure analysis. 7.6.2. Stereomicrography

In those instances where a specimen possesses a complicated or multilayered structure, production of stereomicrographs may prove useful. Electron microscopes generally provide a means for tilting the specimen a few degrees (-3-9") about its normal position. A stereopair of micrographs is taken by recording the same field at both positive and negative tilt angles. When properly mounted and viewed through a stereo viewer, the field becomes three dimensional, greatly simplifying interpretation. Figure 23 is an example of stereomicrography. 7.6.3. Topographical Contrast Imaging

It has been noted that the microscopist may frequently encounter specimens that produce low-contrast images. A relatively new technique, topographical contrast imaging,25shows much promise toward improving this situation, especially in the area of high-resolution electron microscopy.26 G . A. Bassett and A. Keller, Philos. M a g . [S] 9, 817 (1964). S. Nakahara, D. M. Maher, and A . G . Cullis, Electron Microsc., Proc. Eur. R e g . ConJ Electron Microsc., 6th, 1976 p. 85 (1976). a8 A. G . Cullis and D. M. Maher, Philos. M a g . [S] 30 (2), 447 (1974). *I

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FIG.23. Stereomicrograph of intercrystalline links. Shadowed at tan-'

223

4 with Pt-C.

An adaptation of the Foucaultz7 method is used to produce the image contrast. Specifically, electrons deflected through small angles ( rad) by potential gradients within the specimen are selected by a specially positioned objective aperture and produce a bright-dark contrast (see Fig. 24). The said gradients exist at clear structural boundaries. One such boundary, the specimen -vacuum interface, will provide topographical information. Within the sample, voids or different phases exhibiting distinct boundaries will also produce informative, contrasty images. To select the pertinent imaging electrons, the objective aperture is displaced axially to a position slightly below the back focal plane (BFP) of the objective, or is moved off axis while remaining in the BFP. We shall confine our discussion to the latter, more common technique. The microscope is set up for normal bright-field microscopy and the image focused. The objective aperture is then displaced from its normal axial position to one where it blocks half the image. The displaced aperture effectively removes approximately half the spectra in the diffraction plane of the objective lens. Specimen structure in the region of the dark-light transition exhibit the described bright-dark contrast. 27

J. Faget, M. Fagot, and C. Fert, Proc. Eur. Reg. Conf. Electron Microsc., 2nd, I960 p.

18 (1961).

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FIG.24. Topographical contrast of a holey film produced by special positioning of the objective aperture.

Perhaps the most important advantage of the technique arises from the fact that images are recorded in focus. As a result, size determinations can be made of very small features. It will be remembered that in normal bright-field microscopy, images of low-contrast specimens are usually recorded slightly underfocus. The fringes that provide the contrast in this case confuse the outline of objects and can result in large errors in measured size distribution and hence estimates of, e.g., volume fraction.28 Further, the resolution of images produced under the optimum focus condition of topographical contrast imaging is limited primarily by instrument aberration. As a result, resolution approaching the atomic scale has been demonstrated.*@ 7.6.4. Scanning Electron Microscopy

Conventional transmission electron microscopy (CTEM), as we have seen, involves flooding a thin-film specimen with a beam of electrons and M . Riihle, Proc. I n t . Conf. Radiat.-Induced Voids Met.. 1971 p. 255 (1971). A. G. Cullis and D. M. Maher, Ultramicroscopy 1,97 (1975).

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analyzing the images produced by a perturbance of the electron waves as they pass through the specimen. It was noted, however, that sometimes a sample cannot be made thin enough to allow transmission of electrons. Either a replica of the surface is made for a transmission study, or a scanning electron microscope (SEM) is employed. In a SEM the electron beam is focused on the specimen. Scanning coils built into the column move the focused spot in a rastering pattern across the sample. The interaction of the electron beam with the specimen results in the production of inelastically (secondary) scattered electrons. Many secondaries escape the surface and some of these are collected by nearby detectors. After electronic processing, the intensity of a cathode ray tube (CRT) is modulated in proportion to the number of collected electrons. The deflection coils of the CRT are synchronized with the scanning coils in the column. As the beam is scanned across the specimen then, a conjugate image is built up on the screen of the CRT. Since the viewing monitor size is fixed, changing the area scanned on the specimen effectively changes the magnification. The secondary electron imaging referred to above is by far the most popular SEM imaging mode. For this reason, we shall confine our discussion to secondary electron imaging only. Secondary images are relatively easy to interpret, owing to their familiar three-dimensional appearance (see Fig. 25). They are principally topographical images since secondary electrons have relatively low energies (<50 eV, most probable - 4 eV) and cannot escape from depths greater than 50-100 8, below the surface. The lateral resolution of secondary images is slightly greater than the size of the incident beam. Commercail instruments are currently available with a guaranteed secondary electron image resolution of 70 A. It should be noted, however, that most SEM studies have been performed on instruments providing only 100-300 8, resolution. There are a number of limitations that prevent SEM resolution from approaching that of CTEM. The first of these is electron scattering. If an infinitely small electron spot could be focused on a specimen, the scattering within the specimen would result in the collection of electrons from a larger area. Of course, lens aberrations, aperture diffraction effects, and size and brightness of the source dictate that the probe will have some finite size. The typical SEM equipped with a thermionic source has a theoretical probe size of -40 A. A field emission equipped instrument has an apparent lower limit of 10 81.30 Organic polymers offer special obstacles to SEM investigation. Being nonconductors, they quickly become charged under electron beam irra-

-

3o

H. C. Pfeiffer, Scanning EIecrron Microsc. Part I, p. 113 (1972).

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FIG.25. Scanning electron micrograph of a fracture surface of specially treated polycarbonate (sample courtesy of H. E. Bair).

diation. This can cause a distorted image or even a loss of image if the surface becomes charged enough to deflect the beam off the specimen. To overcome this problem, such samples are given a thin (-50 A) conductive coating with a path to ground. We saw earlier that electron beam irradiation can also destroy the crystallinity of polymer samples. Thermally sensitive specimens can experience even greater damage, often exhibiting a “burnt in” raster image where high magnification SEM observations were attempted. A number of SEM manufacturers have recently added an interesting option to their product line. It is now possible to carry out live-time stereo scanning electron microscopy. The scanning system is altered so that the electron beam can be made to impinge upon the specimen from two slightly different angles. Odd-number frames scanned are synchronized to one of the “source” directions and the even-number frames to that of the other “source.” By rapidly switching between the two images in this manner, and with the aid of a special viewer, three-dimensional images are produced. Stereo scanning images are useful for studies in-

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volving micromanipulation of the specimen, or where complicated opened structures are examined. Other SEM operational modes are possible, e.g., backscatter electron imaging, specimen current imaging, Auger electron spectroscopy, X-ray analysis and mapping, voltage contrast imaging, cathodoluminescence imaging, and magnetic contrast imaging. These are specialized techniques, however, and generally offer little or no applicability to polymer studies. 7.6.5. Scanning-Transmission Electron Microscopy

If we combine the scanning mode of operation of SEM with the thinfilm techniques of CTEM we broaden our investigative capabilities. The advantages of scanning-transmission electron microscopy (STEM) over CTEM arise from the fact that the former contains no image forming lenses following the specimen (objective lens in CTEM). This effectively eliminates chromatic aberration arising from multiple-scattering processes in the specimen. As a direct consequence of this, thicker specimens or lower accelerating voltages may be employed with no loss of re~olution.~ Further, ~ restriction on the size of the collection angle for electrons escaping the sample (effective aperture) are lessaevere in the STEM, allowing greater latitude in selection of aperture size to maximize contrast. Another advantage is the ability to perform microelectron diffraction on areas as small as 30 A.32 Some of the present drawbacks of STEM include its requirements of a bright source to minimize noise in the electronically produced image; the possibility of undersirable space-charge effects from the high-current, small-diameter beams; and its tendency to produce poorer diffraction contrast images than the CTEM. 7.6.6. Low-Loss Electron Microscopy

In our discussion of the SEM it was mentioned that reflected or back-scattered electrons can be used for imaging. If an energy analyzer is added to the microscope, electrons of a specific range of energies can be “filtered” from the back-scattered beam and used to form unique images.33 In low-loss electron microscopy, electrons having lost more than 400 eV (from a 15 KeV incident energy) are rejected and those remaining are collected to form the image. While only 0.1% of the incident T. Groves, Scanning Electron Microsc. Part 11, p. 311 (1976). R. H. Geiss, Scanning Electron Microsc. Part 11, p. 337 (1976). 33 0. C. Wells, Appl. Phys. L e f t . 19 (7), 232 (1971).

31

3*

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ELECTRON MICROSCOPY

beam current is typically collected in this case, the image contrasts are so strong that good image formation is still possible. The lateral resolution of low-loss images can be 2.5-10 times better than the conventional secondary electron image.34 Low-loss signals are obtained from a surface layer perhaps 50-100 8, thick. The resulting image then is useful in examining surface phenomena where the underlying structure may complicate interpretation. While the technique has not yet been applied to the study of polymers, Broers' success with biological specimens34 indicates that low-loss imaging may also be useful for certain polymeric materials.

7.7.Applications We conclude this part with a small sampling of polymer science research studies in which the electron microscope played a major role. An attempt was made to select those studies that offer some unique or useful preparative technique or that produced noteworthy experimental findings. 7.7.1. Single Molecules A number of investigators have employed the electron microscope to image individual polymer molecules in an attempt to determine their molecular weight. Their common technique involved dissolving the polymer in a poor solvent so that the molecules would assume a more coiled conformation-this reduces the liklihood of molecular entanglements. (For polyethylene, a 0.005% n-hexadecane solution has proved ~ u c c e s s f u 1 . Studies ~~ with polystyrene reveal that n - b ~ t a n oand l ~ ~cycloh e ~ a n e , ~in' concentrations as small as 1 ppm, produce satisfactory results.) The dilute solution is then sprayed onto a carbon or plastic support film. Upon evaporation of the solvent, the now discrete molecules are shadowed with a heavy metal and examined. Aside from the uncertainties associated with shadowing, as noted earlier, the density of the polymer molecule must be estimated in making the molecular weight calculation. It is recommended, therefore, that the technique be employed only where classical molecular weight determination methods are ineffective or very time consuming. This effectively restricts its application to polymers of molecular weights M , > 106.37 34

A. N. Broers, Scanning Electron Microsc. Part I, p. 233 (1975).

'' M. Furuta, J . Polym. Sci., Polym. Phys. Ed. 14, 479 (1976). 36

M. J. Richardson, Proc. R . SOC.London, Ser. A 279, 50 (1964). B. M. Siege], J . Polym. Sci. 5 (l), 111 (1950).

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Direct observation of an extended chain hydrocarbon molecule is, of course, not possible. The scattering from the atoms of such a molecule is insufficient to produce detectable contrast.

7.7.2. Polymer Single Crystals Polymer single crystals have been the subject of numerous research studies. Their relative simplicity makes them unique specimens on which quantitative structure analysis may be performed and from which insight into crystallization mechanisms can be obtained. As we have seen, the electron microscope provides the means by which such investigations can be carried out on individual polymer crystals. We have already seen micrographs of single crystals of polyethylene, perhaps the most widely studied polymer. A most comprehensive study of such crystals was undertaken by Bassett and Keller. They examined the effects of temperature on m o r p h o l ~ g y~, ~e c~ t o r i z a t i o nand , ~ ~crystal~~~ lization habit^.^^-^^ Dark-field studies of twinned polethylene crystals have been performed by White.44*45He also produced a useful guide for those undertaking such studies .44,46 Kovacs and c o - w ~ r k e r sskillfully ~~ employed dark-field and electron diffraction to study multiple twinning in polyethylene oxide and di-block copolymers of ethylene oxide and styrene. While these and similar studies provided much toward establishing an understanding of chain folding and molecular packing in single crystals, questions about fine detail persisted. One such question pertained to the exact nature of the fold surface. Considerable experimental data suggested that adjacent reentry predominated and that the surface was composed of sharp, regular folds. Concurrently, a like amount of evidence supported the opposing view that the fold surface was essentially amorphous with loops randomly connecting molecules in the crystalline lattice. A single model to explain all the then existing experimental evidence was proposed by Blackadder and It called for regular adjacent D. C. Bassett and A . Keller, Philos. M u g . [8] 7 , 1533 (1962). D. C. Bassett and A. Keller, Nature (London) 184, 810 (1959). I0 D. C. Bassett, Philos. M u g . [8] 12, 907 (1965). " D . C. Bassett and A . Keller, Philos. M a g . [8] 6, 345 (1961). " D. C. Bassett, F. C. Frank, and A. Keller, Philos. M u g . [8] 8, 1739 (1963). a D. C. Bassett, F. C. Frank, and A. Keller, Philos. M a g . [8] 8, 1753 (1963). 91 J. R. White, J . Marer. Sci. 9, 1860 (1974). J . R. White, J . Polym. Sci., Polym. Phys. Ed. 12, 2375 (1974). J. R. White, Polymer 16, 157 (1975). A. J. Kovacs, B. Lotz, and A. Keller, J . Mucromol. Sci., Phys. 3 (3), 385 (1969). 48 D. A . Blackadder and T. L. Roberts, Makromol. Chem. 126, 116 (1969). 38

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reentrant folds but with a surface layer of lower density than that of the crystalline core. The density difference was attributed to the fold surfaces being inclined relative to the c axis of the crystal. While this latter concept has yet to be directly confirmed, support for the regularity of the folds has come from Georgiadis and M a n l e ~ who , ~ ~ epitaxially crystallized nascent polyethylene onto the fold surfaces of polyethylene seed crystals, and from Harrison,5owho studied the melting and morphology of polyethylene single crystals in suspension. A similar study by Roe et al. 51 contributed toward a further understanding of polyethylene single crystals. Studying the lamellar thickening of crystals annealed at low and high pressure in nonsolvents, they found that the normally pyramidally shaped polythylene single crystals flatten when exposed. to 6 kbar pressure. They theorized that since the “amorphous” region is more compressible than the crystalline lamellae, application of high pressure essentially eliminates any differences in specific volume, and flat crystals become favored as a state of lower energy. Two aspects of this last work are noteworthy. First, the fact that the crystals were annealed while suspended in solution precluded questions about the effects of a supporting substrate (a problem facing most annealing studies). Second, it was found that the annealing effects of two or more crystals in intimate contact were more severe than monolayer crystals (see Fig. 26). This implies that direct extrapolation of single crystal results to bulk polymers may at times be imprecise. For the reader embarking on single-crystal studies it is recommended that he familiarize himself with the self-seeding preparative technique after Blundell et al. 52 7.7.3. Melt-Crystallized Polymers

With few exceptions (see Patel and Patel,53for example) polymer single crystals are grown from dilute solutions. Crystallization of polymers from the melt leads to the production of the more complicated spherulitic structures. Most early electron microscopic investigations of meltcrystallized samples involved replication of a free, etched, or fractured surface. Such specimens revealed little about internal structure so that models proposed to explain crystallization mechanisms were generally based on other than electron microscopic observations. T. Georgiadis and R. St. J . Manley, J . Polym. Sci., Purr E 9, 297 (1971). I. R. Harrison, J . Polym. Sci., Polym. Phys. Ed. 11, 991 (1973). 51 R. J. Roe, C. Gieniewski, and R. G. Vadimsky, J . Polym. Sci., Polym. Phys. Ed. 11, 1653 (1973). 6* D. J. Blundell, A. Keller, and A. J . Kovacs, J . Polym. Sci., Purr E 4, 481 (1966). 55 G. N. Patel and R. D. Patel, J . Polym. Sci., Purr A-2 8, 47 (1970).

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23 I

FIG.26. Polyethylene single crystals annealed while suspended in silicon oil. Note that reorganization is more pronounced where the two crystals were in intimate contact during the annealing process. Pt-C shadowed at tan-' t .

Keith and c o - ~ o r k e r s , " *however, ~~ employed a special preparative technique that revealed the internal structure of their specimens and led to their direct observation of previously conjectured intercrystalline links. They capitalized on the fact that during melt-crystallization lowmolecular-weight species segregate between growing spherulite lamellae. Their procedure involved mixing polyethylene fractions with very lowmolecular-weight material, which they dissolved away following crystallization. The resulting open skeletal structure revealed the intercrystalline links (see Fig. 27). To dispel suggestions that the observations were artifacts arising from shrinkage of the material upon cooling, Keith et deposited additional

'' H. D. Keith, F. J. Padden, Jr., and R. G. Vadimsky, Science 150, 1026 (1965). This article shows only an example of polymer drawing. It contains no existence of intercrystalline links. 55 H. D. Keith, F. J. Padden, Jr., and R. G. Vadimsky, J. Polym. Sci., Part A-2 4, 267 (1966). H. D. Keith, F. J. Padden, Jr., and R. G. Vadimsky, J . Appl. Phys. 37, 4027 (1966).

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FIG.27. Intercrystalline links revealed by dissolving away low-molecular-weight species segregated between growing spherulites in a melt crystallized sample. Pt-C shadowed at tan-’ 2.

polymer on top of specimens following the initial crystallization but before cooling. This material crystallized epitaxially on the links, conclusively marking them as growth features (see Fig. 28). This “labeling” of links was used to advantage later in deformation studies5’ to help distinguish between growth and deformation features (see Fig. 29). In later work, Kanig developed a technique to increase contrast in bulk semicrystalline p ~ l y e t h y l e n e . ~A~sample * ~ ~ is first treated with chlorosulphonic acid, which diffuses selectively into the amorphous regions and reacts with the polymer, resulting in an increase in the effective density of the amorphous zones. As we have seen, this will cause an increase in electron scattering and thereby produce an observable contrast between amorphous and crystalline regions. The acid treatment concurrently s7 R. G . Vadimsky, H. D. Keith, and F. J. Padden, Jr., J . Polym. Sci., Part A-2 7, 1367 (1969). JB G. Kanig, Kolloid-2. 2. Polym. 251, 782 (1973). G. Kanig, Prog. Colloid. & Polym. Sci. 57, 176 (1975).

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causes some cross-linking to take place. This makes for more rigid samples, which are less affected by sectioning and are less susceptible to beam damage. More recently, Olley and co-workers have developed a new, controlled etching technique capable of revealing fine detail in a number of (perhaps all) olefins2I. First results obtained with the method promise to stimulate renewed interest in bulk polymer studies. 7.7.4. Oriented Polymers

The term “oriented polymers” generally pertains to polymers exhibiting long-range molecular order. Such samples are produced by deforming a solid or by extruding a melt. An interesting study of the former type was recently reported by Tar‘in and Thomas.so Their study of the microfibrillar structure of deformed polyethylene spherulites is noteP. M. Tailn and E. L. Thomas, Polym. Eng. Sci. 18 (6), 472 (1978).

FIG.28. Lamellar crystals grown epitaxially on intercrystalline links. Shadowed at tan-’

4.

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FIG. 29. A uniaxially deformed melt crystallized specimen. Note that intercrystalline links "marked" by epitaxial overgrowth facilitate recognition of predeformation boundaries. Shadowed at tan-' 4.

worthy because of their gold marking technique, which enabled them to perform a quantitative analysis of the deformation process. The first step in their sample preparation procedure involved deposition of gold microdots on their undeformed films. These were used to determine the strain and the draw ratio of the material. Following uniaxial deformation of the films, the entire surface received a gold decoration coating. To study the deformation process of the microfibrillar structure, some samples were deformed several times and gold-decorated after each deformation. The results of this study confirmed earlier work that demonstrated that the microfibrils are considerably stronger than the untransformed lamallae. In the area of melt extrusion, interesting results have accrued from the numerous studies of regular block copolymers. It has been shown that segregated microphases can be formed in a periodic arrangement and extend over a whole macroscopic specimen. By using a special extrusion

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method,61 Dlugosz and collaborators prepared a plug of a tri-block copolymer SBS (polystyrene-polybutadiene-polystyrene), which possessed a “single-crystal” character.62 The sample consisted of a hexagonal array of polystyrene cylinders, which were embedded in the polybutadiene matrix and whose axes were parallel to the extrusion direction. Preparation of such samples for electron microscopic examination involves sectioning and staining. In the above-referenced work, the sample and the microtome knife were cooled with liquid air.6z The sections were stained with osmium tetroxide vapor, which reacts with the rubber microphase. These areas then scatter more electrons and appear dark in an image of the section.

M. J . Folkes, A. Keller, and F. P. Scalisi, Kolloid-Z. 251, 1 (1973). J. Dlugosz, M. J . Folkes, and A. Keller, J . Polym. Sci., Po/ym. Phys. Ed. 11, 929 ( 1973).

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8. CHEMICAL METHODS N POLYMER PHYSICS

By G.N. Patel 8.1. Disorder in Polymer Crystals and Chemical Methods When single crystals of polymers were first dis~overedl-~ and studied, there was no need to suspect the existence of appreciable disorder within them. Hence, a regular fold structure with adjacent reentry was proposed by Keller.3 However, macroscopic measurements on sedimented mats of single crystals began casting doubt on the perfection of foldedchain crystals. The relevant measurements include the determination of den~ity,~ heat - ~ of f ~ s i o n ,amorphous ~.~ scatter in X-ray pattern^,^ mobility by NMR,gJo intensity of crystallinity-sensitive infrared b a n d ~ , ~ and Jl absolute intensity of low-angle X-ray reflections.12 The results of these measurements indicated that there is an appreciable amount of amorphous material in the crystals. The next question raised was the location of the amorphous material. Moire patterns4.l3observed from the overlapping single crystals suggested that there is no sizable disorder in the crystal lattice. Some chain ends within the lattice can produce a variety of defects14 and contribute to the deficiency in crystallinity, but the deficiency in crystallinity could not be due to them alone. Accordingly, such P. H. Till, J . Polym. Sci. 24, 301 (1957). E. W. Fischer, 2. Naturforsch., Teil A 21, 753 (1957). A. Keller, Philos. Mag. [8] 2, 1171 (1957). E. W. Fischer and R. Lorenz, Kolloid-2. 2. Polym. 189, 97 (1963). J . B. Jackson, P. J. Flory, and R. Chiang, Trans. Faraday Soc. 59, 1909 (1963). G. M. Martin and E. Passaglia, J . Res. Natl. Bur. Stand., Sect. A 10, 221 (1966). H. Hendus and K. H. Illers, Kunststoffe 57, 193, (1967). * E. W. Fischer and G. Hinrichsen, Kolloid-2. 2. Po/ym. 213, 93 (1966). E. W. Fischer and A. Peterlin, Makromol. Chem. 74, 1 (1964). lo K. Bergmann and K. Nawotki, Kolloid-Z. 2. Polym. 219, 132 (1967). l1 T. Okada and L. Mandelkern, J . Polym. Sci., Part B 4, 929 (1967). l2 E. W. Fischer, H. Goddar, and G . F. Schmidt, J . Polym. Sci., Part B 5, 619 (1967). lS B. A. Lotz, A. J. Kovacs, G . A. Bassett, and A. Keller, Kolloid-Z. 2. Polym. 209, 115 (1966). l4 P. Predecki and W. 0. Statton, J . Appl. Phys. 37, 4053 (1966). 237 METHODS O F EXPERIMENTAL PHYSICS, VOL. 168

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

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amorphous material should lie outside the lattice and therefore on the fold surface of the crystals. Electron microscopy of melt-crystallized polymers has shown that the spherulites are composed of lamellae radiating from a point in all direction^,^^,'^ and measurements of physical properties like density, heat of fusion, and mobility by NMR suggests that melt-crystallized polymers have considerable amount of amorphous material.17-19 The amorphous material, therefore, should be between the lamellae. Once it was established that polymer chains are folded and amorphous material is located on both the sides, top and bottom of the lamellar crystals, a number of other questions were Are the folds of the adjacent reentry type? Are the folds loose or sharp? Are the folds of switchboard (i.e., random reentry) type? Is the surface of the crystals smooth or rough? What is the distribution of folds? Are the chain ends excluded from the lattice? How are the branches distributed in the crystals? Evidence presented by the physical methods was insufficient to answer the above questions satisfactorily. Therefore, a number of chemical methods like plasma etching,24a5 halogenati0n,2~*~~ dehydrohalogena~ ~ , developed ~~ simultaneously to tion,28hydrolysis ,29,30 and ~ x i d a t i o nwere study surfaces of polymer crystals. Some of these methods like halogenation and dehydrohalogenation modify the crystal surfaces, while the A. Keller and D. C. Bassett, J . R . Microsc. Soc. 79, 243 (1960). P. H. Geil, J . Polym. Sci. 47, 65 (1960). l7 W. P. Slichter, J . Appl. Phys. 31, 1865 (1961). R. L. Miller, in “Crystalline Olefin Polymers” (R. Raff and K . W. Doak, eds.), p. 577. Wiley (Interscience), New York, 1965. P. H. Geil, Polym. Rev. 5, (1963). 2o P. J. Flory, J . A m . Chem. Soc. 84,2857 (1962). 21 L. Mandelkern, “Crystallization of Polymers.” McGraw-Hill, New York, 1964. 22 A. Keller, Prog. Phys. 31, 623 (1968). L. A. Keller, Kolloid-Z. Z . Polym. 231, 386 (1969) F.R. Anderson and V. F. Holland, J . Appl. Phys. 31, 1516 (1960). l5 F. R. Anderson, J . Appl. Phys. 34, 2371 (1963). 2e A. Keller, W. Matreyek, and F. H. Winslow, J . Polym. Sci. 62, 291 (1962). 27 I. R. Harrison and E. Baer, J . Polym. Sci., Part A-2 9, 1305 (1971). 28 I. R. Hamson and E. Baer, J . Colloid Interface Sci. 31, 176 (1969). 2a R. F. Nickerson and J. A. Habrle, Ind. Eng. Chem. 39, 1507 (1947). 30 A. Sharples, Trans. Furaday Soc. 52, 1003 (1956); 53, 913 (1957). R. P. Palmer and A. J. Cobbold, Mukromol. Chem. 74, 174 (1964). 32 D. J. Priest, J . Polym. Sci., Part A2 9, 1777 (1971). l5

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239

others like hydrolysis and oxidation cut folds and remove the amorphous surfaces. Physical methods such as infrared spectroscopy, NMR, lowand wide-angle X-ray analysis, differential scanning calorimetry, and gel permeation chromatography are employed to analyze the chemically treated crystals. Some techniques like selective oxidation by nitric acid and ozone are developed to such an extent that they can answer all the above questions. Depending upon their uses, chemical methods can be divided into two classes: (1) techniques used to reveal morphology of crystals buried in the polymer matrix, and ( 2 ) techniques used to study nature of the crystal surfaces. Class 1 includes two types of techniques: (1) solvent-etching and (2) plasma-etching. Both techniques are used to reveal crystals embedded in the amorphous polymer matrix. Class 2 also includes two types of techniques: (1) surface removal techniques like oxidation of polyolefins by HN03 and 0 3 ,and hydrolysis of cellulose, polyamides, and polyesters, and (2) surface modification techniques like halogenation of polyolefins and dehydrohalogenation of poly (vinylidene chloride). Each technique. is described in two parts: the technique itself and the analysis of the results that follow. Physical methods used to study the products of the reactions are not described in this part. The reader is referred to the appropriate chapters in this book or references quoted therein. Selective halogenation and oxidation have some useful applications apart from their use as analytical techniques. Regular block copolymers can be prepared by selective halogenation of crystal surfaces. Certain properties like flame, chemical, and solvent resistivity, thermal stability, and dyeability of a polymer can be increased by halogenation. Longchain dicarboxylic acids and paraffins of uniform molecular weight can be prepared by selective oxidation of polymer single crystals. The selective oxidation technique has also been used to study the location of radiation-induced chemical changes in polymers. This is described in Chapter 8.6.

8.2.Solvent-Etching The solvent-etching technique is based on the principle that amorphous polymer chains are more soluble than the crystallized ones. A solvent will remove molecules as a whole from the amorphous regions without breaking covalent bonds. When a polymer is crystallized from the melt, crystalline structures like spherulites are covered by a thin layer of

240

8.

CHEMICAL METHODS IN POLYMER PHYSICS

almost amorphous chains. The crystalline structures can therefore be revealed by selective dissolution of the amorphous material. Reding and Walter33etched polyethylene surfaces with boiling carbon tetrachloride: The etching apparatus consisted of a flask of boiling solvent fitted with a large diameter, straight-tube reflux condenser. The solvents used were carbon tetrachloride for lowdensity polyethylene and toluene or benzene for high-density polyethylene. With the solvent refluxing, a strip of sample was lowered on a wire from the top of the condenser to a point where the refluxing solvent vapor wet the surface of the plaque. The sample was then slowly withdrawn and examined under reflected light. If a loss in surface gloss was apparent, the sample had been sufficiently etched. If not the immersion in solvent-vapor was repeated. The contact time, during immersion in solvent vapor, was kept very short to avoid swelling or annealing of the surface. The constant flushing of the surface by fresh solvent vapor effectively removed solvated resin, so that essentially no reprecipitation occurred on the surface of the sample.

Etched samples were dried to remove any adsorbed solvent before the surface was replicated for electron microscopy. Mackie and R ~ d i nusing , ~ ~ the solvent-etching procedure described by Reding and Walter, observed that etch surfaces of a low-density polyethylene were obscured by an array of small fibers due to precipitation of the polymer. Clean, etched polyethylene surfaces were obtained by modifying the original technique as follows: Carbon tetrachloride was added to a flask of water at 80°C so that the top half of the flask was filled with solvent vapor. Molded polyethylene was lowered through etchant vapor into the water. In this way any solvent that condensed on the specimen surface was maintained at about 80°C and quickly dispersed in the water. Reprecipitation was unlikely to occur as a result of evaporation of condensed solvent. A thin film of polyethylene generally spread on the water surface after the specimen had been immersed.

The modified etching technique was generally satisfactory, although a few instances of apparent surface deformation (possibly melting) were encountered. Bailey35used xylene for revealing spherulitic structures in polyethylene and polypropylene. The samples were etched in the vapor for only a few seconds. Benzene has also been used to etch polyethylene and polyproThe etched p ~ l e n e . ~ ~Nylon . ~ ' 6 has been etched by dilute acetic surfaces were studied by optical and electron microscopy. F. P. Reding and E. R. Walter, J . Polym. Sci. 38, 141 (1959). J. S. Mackie and A. Rudin, J . Polym. Sci. 49, 407 (1961). 35 G. W. Bailey, J . Polyrn. Sci. 62, 241 (1962) 36 D. F. Kagan and L. A. Popova, Vysokomol. Soedin., Ser. A 12, 2774 (1970). 37 L. S. Li and V. A. Kargin, Vysokomol. Soedin. 3, 1102 (1961). 38 K. Kubota, J . Polym. Sci., Part E 3, 545(1965). 33 34

8.3.

PLASMA-ETCHING

24 1

The solvent-etching technique is used to reveal spherulitic structures at polymer surfaces. A solvent will remove molecules as a whole from the surfaces. It is therefore useful if there are amorphous molecules present on the surface that can be dissolved as a whole. The response of a particular specimen to the solvent-etching treatment is dependent on the nature of polymer, the type of solvent applied, and the exposure time and temp e r a t ~ r e .Thus, ~ ~ a necessary condition for attaining the intended “etching” may not always be found. Only a very thin layer of polymer surface is etched away by this technique. The plasma-etching technique, described in the following chapter, is used for revealing structures buried deeper in a film or fiber.

8.3. Plasma-Etching Plasmas are sometimes called “the fourth state of matter.” In plasmas many of the atoms and molecules are charged while others take the form of free radicals. They also contains neutral and excited molecules. Plasmas used for etching and modification of polymer surfaces are often referred to as low-temperature plasmas, which may be characterized as “less ionized” compared to highly ionized hot plasmas. Gas plasmas have been used to polymerize a large number of monomers and to treat surfaces of polymers to improve wettability and bonding capabilities and to study the morphological structures of the surfaces. Plasma-etching of polymers is not selective; it shaves off both crystalline and amorphous regions. The technique is used to reveal the interior morphology of polymer films, fibers, and thick solution-grown crystals. A number of plasma reactor~~O-~l have been used for plasma polymerization and modificatibn of polymer surfaces. All these reactors can be M. M. Millard, J. J . Windle, and A. E. Pavlath, J . Appl. Po/ym. Sci. 17, 2501 (1973). H. Yasuda and C. E. Lamaze, J . Appl. Polym. Sci. 15, 2277 (1971). 4 1 A. T. Bell, J . Macromol. Sci., Chem. 10, 369 (1976). 42 H. Hasuda, J . Macromol. Sci., Chem. 10, 383 (1976). 43 See papers published inJ. Macromol. Sci., Chem. 10, No. 3 (1976). 44 M. Hudis, in “Techniques and Applications of Plasma Chemistry” (J. R. Hollahan and A . T. Bell, eds.), p. 113. Wiley, New York, 1974. 45 K . Rossman, J . Polym. Sci. 19, 141 (1956). 46 J . L. Weininger, Nature (London) 186, 546 (1960). 47 R. C. Mantell and W. L. Ormand, Ind. Eng. Chern.. Prod. Rrs. Deii. 3, 300 (1964). R. H. Hansen and H. Schonhorn, J . Polym. Sci., Part B 4, 203 (1966). *@J . R. Hollahan and G . L. Carlson, J . Appl. Polym. Sci. 14, 2499 (1970). * J . R. Hall, C. A. Westerdahl, A . J. Devine, and M. J. Bodnar, J . Appl. Polynr. Sci. 13, 2085 ( 1969). a1 P. Blais, D. J. Carrlson, and D. M. Wiles, J . Appl. Polym. Sci. 15, 129 (1971). 38

40

8.

242

CHEMICAL METHODS I N POLYMER PHYSICS

used for etching of polymer surfaces. The reactors have been summarized recently by H ~ d i and s ~ ~are described in detail in the original reference^.^^-^^ A typical apparatus consists of a gas cylinder, a flow meter, a vacuum gauge, a plasma source, a sample chamber, and a vacuum pump in series. Oxygen or other gaseous ions are produced by electrode corona discharge, radio frequency (rf) corona discharge, electric glow discharge, or rf glow discharge, and are used on impingement on polymers to remove layers of material. The mechanism of plasma etching is very complicated and has not been studied in detail. It is believed that the etching is due to degradation and oxidation (if oxygen is used) of polymer surfaces. It has been observed that nearly all polymers lose weight when exposed to a plasma of argon, helium, nitrogen, or oxygen.42*52The technique used by Reneker and B o l ~ which , ~ also measures the concentration of atomic oxygen, is described here. The apparatuses used by others are quite similar except in the type of plasma used. Reneker and Bo19 etched surfaces of bulk and solution-grown crystals of polyethylene by oxygen plasma. The procedure is as follows: The apparatus used is shown in Fig. 1 . Oxygen at a pressure of around 133 N/m2 (1

TOIT)was pumped through the glass tube at a measured rate. A 5 to 10 mA direct current discharge was established in the curved part of the tube between a copper electrode and the metal part of the glass-to-metal seal where the gas was admitted to the apparatus. The purpose of the curved segment was to reduce the intensity of the light from the discharge that reached the sample area. The concentration of 0 atoms in the sample area was measured by a NOz titration method described by K a ~ f m a n . ~ - ~ ~ a*

H. Yasuda, C. E. Lamaze, and K. Sakaoku, J . Appl. Polym. Sci. 17, 137 (1973). D. H. Reneker and L. H. Bolz, J . Macromal. Sci.. Chem. 10, 599 (1976). F. Kaufman, in “International Conference on Chemiluminescence” (M.J. Cormier, D.

M. Hercules, and J. Lee, eds.). Plenum, New York, 1973. 55 F. Kaufman. Proc. R. Sac. London, Ser. A 247, 127 (1958).

i

Oxygen

Sample Area

Rotary Vacuum

FIG.1 . Schematic diagram of a glow discharge apparatus used by Reneker and Bolzm to expose polyethylene to atomic oxygen.

8.3.

PLASMA-ETCHING

243

Accordingly, NO, was admitted at an adjustable and measured rate. At low concentrations, NO, reacts very rapidly with 0 to form NO which in turn reacts more slowly with 0 to form an excited state of NO,, which then decays with the emission of a greenish-yellow light. Relative concentrations of 0, and NO2 were adjusted by controlling the flow. Flow rates were measured with flowmeters of the type in which the flowing gas lifts a ball in a vertical variable-area tube. As the concentration of NO, increased, the intensity of the greenish-yellow light first increased and then, at the concentrations at which the fast reaction between NO, and 0 consumed all the 0, the light disappeared. This sharp end-point established the amount of NO, required to equal the concentration of 0. The rate of attack was established by subjecting a small sheet of polymer to the active oxygen and measuring the weight loss a s a function of exposure time. Etching rates were around 0.5 pm/hr, at 0 atom concentrations of 10" to oxygen atoms/cm3. The etching rate was about 5 times higher in the region immediately adjacent to the electrode and decreased to the above rate at a point 5 cm downstream from the electrode.

The etched surfaces were replicated and replicas were examined under electron microscope. Familiar morphological features of polyethylene were revealed in the interior of thick specimen. Figure 2, for example, shows a replica of a ringed spherulite of high-molecular-weight linear polyethylene. Several micrometers of polymer were removed from the surface before the surface was shadowed with a platinum palladium alloy. The metal replica was then pulled off the sample with a dried drop

FIG.2. Metal replica with adhering polymer fragments from a surface revealed by atomic etching of a spherulitic sample of polyethylene (after Reneker and Bolz").

244

8.

CHEMICAL METHODS I N POLYMER PHYSICS

of polyacrylic acid-polyvinyl alcohol mixture, which forms a tight bond to the replica. In high-magnification photographs of this sample, the edge of 10 nm thick lamellae were discernible near the centers of spherulites and in the rings where the lamellae were nearly normal to the surface. In addition to the use of oxygen in plasma-etching, nitrogen, helium, and argon are also frequently u ~ e d . However, ~ ~ ~ ~the~etching ~ ~ would ~ - ~ ~ be slower because it will be only due to degradation of polymer. Gas pressure is varied from 100 pm to several centimeters. Anderson and H ~ l l a n etched d ~ ~ ~surfaces ~~ of nylon 6,6 and poly(ethylene terephthalate) fibers by argon plasma to develop a characteristic etch pattern possibly related to the lamellar crystalline structures. Recently, internal structures of the same polymers have been studied by Pande el ~ 7 1 using . ~ ~ nitrogen plasma obtained by a rf electrodeless glow discharge. In contrast to the very selective oxidation of nitric acid and ozone (see Section 8.5.11, a plasma etches both crystalline and amorphous regions. However, the rate of etching could be different, i.e., higher for amorphous regions. Spits7 has shown that crystalline parts of cellulose are relatively more stable to ion bombardment than the amorphous parts. If the temperature of the plasma is higher, one may observe melting and annealing effects. Nearly all polymers lose weight when they are exposed to plasma, and the rate of weight loss is proportional to the time of exposure. A surface of a polymer etched by the plasmas is always active and the adhesive properties become so strong that the carbon or other replica subsequently applied cannot be separated from the polymer. Lebedev et have described two techniques which increase the probability of separating the replica from the polymer. The following techniques were used when separating the replica with gelatin drops (30% solution in water). As the first technique, the plasma treated surface was passivated by prolonged retention, up to several days in vapors of butyl or isobutyl alcohols. However, this technique takes longer time and can be applied to polymers which are not affected by the solvents. A second highly effective method involved the brief repeated treatment of the sample examined in the plasmas but after applying the replica. This treatment makes

sB

M . R. Pande, N . V. Bhat, and P. K . Mittal, Text. Res. J . 46, 502 (1976).

57

D.J. Spit, Polymer 4, 109 (1963).

P. J . Goodhew, J . Mater. Sci. 8, 581 (1973). Ye. V. Lebedev, I. Gede, L. I. Bezruk, and Yu. S. Lipatov, Vysokomol. Soyed A19, 1171 (1977); for the English translation see Polym. Sci. USSR 19, 1351 (1978). la

8.4.

THE SURFACE MODIFICATION TECHNIQUES

245

the surface of the carbon or platinum-carbon replica lyophilic and markedly increases adhesion to gelatin. The plasma-etching technique is more detailed in revealing structure differences by removing surface layers than the solvent-etching technique but it is limited by the fact that the etched material cannot be analyzed further, except by electron microscopy.

8.4. The Surface Modification Techniques 8.4.1. Halogenation

F2, C1, , and Br, are used for halogenation of polymers. Certain properties like thermal stability, chemical resistivity, insolubility, dyeability, adhesive joint strength, oil phobic character, and flame resistance of many polymers have been modified by fluorination .5g.60 Fluorination of polymers leads to perfluorinated polymer^.^^,^^ Fluorine diffuses readily into the crystalline parts of the polymers and hence is not useful for selective halogenation of the fold surfaces. Chlorine and bromine under mild conditions react only with the fold surfaces and have been used to study the nature of these surfaces.26*27,65-67 A number of techniques have been devised for chlorination of polymers, especially of polyethylene.66 Chlorinated polyethylene has many properties that are superior to unchlorinated polyethylene.68 Commercial chlorination is usually carried out in solution or in a suspension at high temperature resulting in random chlorination. The halogenation of polymers like polyethylene is free radical in nature. The reaction is catalyzed by radiation or free-radical initiators. The halogenation of polymers is also considered to be a chain reaction: Cl, -CHz-C!-12-CHz-CH,-CH-CH,-+

--

+ h u +2CI.

+ CI-

CI2

-CH,-~H-CH,-+ -CH,--CH(Cl)-CH,-+

HCI CIS

H . Schonhom and R. H. Hansen, J . Appl. Polym. Sci. 12, 1231 (1968). S. P. Joffre, U.S. Patent 2,811,468 (1957). '' M. Okada and K. Makuchi, Ind. Eng. Chern., Prod. Res. Dev. 8, 334 (1969). " D. T. Clark, W. J. Feast, W. K. R . Musgrave, and I. Ritchie, J . Polym. Sci., Part A-1 13, 857 (1975). ffl R. J . Lagow and J. L. Margrave, J . Polym. Sci.. Part B 12, 177 (1974). ~4 H. Schinohara, M. Iwasaki, S. Twjimura, K. Watanabe, and S. Okazaki, J . Polyrn. Sci., Purr A - l 10, 2129 (1972). I . R. Harrison and E. Baer, J . Polym. Sci., Purr B 9, 843 (1971). 88 D. C. Bassett, Polymer 5, 457 (1964). '' D. E. Witenhafer and J . L. Koenig,,J. Polym. S c i . , Part A-2 7 , 1279 (1969). 88 P. J . Cantenno, Encyl. Polyrn. Sci. Techno/., 6, 431 (1%7). 58

Bo

8.

246

CHEMICAL METHODS IN POLYMER PHYSICS

8.4.1. l . Toxicity. All halogens, F, , Cl,, Br,, and I, , are highly toxic chemicals. Halogenation experiments should therefore be carried out under a fume hood with a powerful exhaust system. No halogen should be handled with bare hands. Br, and Clz are commonly used for selective halogenation of amorphous surfaces of polymer crystals. Bromine and its vapors are extremely toxic and even at relatively low concentrations are highly irritating to the skin, the eyes, and the respiratory system. The liquid rapidly attacks the skin and other tissues, causing severe and painful burns. Exposure to dangerous concentrations of the vapors results in serious inflammation and edema, which may lead to pneumonia and even death. The maximum concentration considered . ~this ~ level, safe for an exposure period of 8 hours is less than 1 p ~ m At bromine is easily detected by its odor. A 10 ppm concentration can be tolerated for only a few minutes. Exposure to 40-60 ppm for 0.5- 1 hour can be fatal. A very short exposure to a 500- 1000 ppm concentration is rapidly fatal.sB Liquid chlorine is a skin irritant, causing burns on prolonged contact; it can cause severe eye damage. Gaseous chlorine is an irritant affecting the mucous membranes, respiratory system, eyes, and skin if present in sufficient quantities. Chlorine produces no known cumulative effects. Its characteristic odor gives adequate warning of its presence. The physiological effects of chlorine gas are summarized in Table O'.I 8.4.1.2. Halogenation in a Liquid Medium. This method is suitable for halogenation of solution-grown crystals of polymers. Single crystals of polymers can be grown using a method described in Part 10. If the OB

V. A. Stenger, Encyl. Ind. Chem. Anal. 8, 1 (1969). G. Oplinger, Encyl. Ind. Chem. Anal. 9, 304 (1970).

TABLEI. Physiological Response to Various Concentrations of Chlorine Gaslo Concentration of chlorine in air (ppm by vol) 1 3-3.5 4 10- 15 30 40-60

lo00 1800

Effect of exposure slight symptoms after several hours odor detectable maximum allowable for exposures of 0.5-1 hr least amount causing immediate irritation to throat causes coughing dangerous in about 30 rnin fatal in 30 min or less fatal in 10 min

8.4.

THE SURFACE MODIFICATION TECHNIQUES

247

solvent used for the crystallization is not a halocarbon, the solvent can be exchanged with a halocarbon solvent such as carbon tetrachloride or a Freon. To exchange the solvent, the crystals should be filtered and washed repeatedly in the halocarbon. The halocarbon selected should not dissolve or swell the original polymer or the halogenated polymer. Swelling or dissolution of the crystals may cause random halogenation. Chlorination or bromination of the suspended polymer crystals can be carried out in a flask either by bubbling chlorine or bromine gas or, more conveniently, by adding the liquid halogens. The suspension should be stirred for uniform reaction. The reaction is initiated by radiation especially UV radiation. Keller er af.26 brominated crystals of Marlex SO@ high-density polyethylene in tetrachloroethylene. The crystals were grown in tetrachloroethylene at 70°C and the suspension was treated with bromine under incandescent light. Harrison and Baef17brominated crystals of Marlex 6015Q high-density polyethylene in carbon tetrachloride: The crystals were grown at 83°C in xylene. The solvent was changed to distilled CCI, by washing crystals six or seven times. Approximately I g of single crystals in 200 ml of CCI, with 15 ml of bromine was irradiated with UV light while being stirred at room temperature. Zero time was taken as the time the UV lamp was switched on, and samples (25 ml) were taken at appropriate intervals of time. The samples were washed free of bromine by centrifugation and washing with distilled CCl, until the wash was clear. They were further washed with acetone to remove any traces of free bromine before being dried under vacuum at room temperature for several days. Bromine uptake was measured by chemical analysis.

Harrison and Baerz7 have reported that chlorination of polyethylene with chlorine gas in CCl, at room temperature resulted in a too rapid and too complete halogenation of the polymer. It was thought that the extent of reaction was due to swelling of the chlorinated surfaces by CCI, ,causing further chlorination of the sample and its eventual solubility. However, they were successfuls5 in selective chlorination of polyethylene crystals surfaces using Freon- I 1 (trichlorofluoromethane) as a solvent medium: Freon-I 1 (200 ml) was cooled to 0°C and saturated with chlorine gas at this temperature. This solution was then placed in a dry-ice/acetone bath. To the Freon-I I/chlorine solution was added a slurry of l to 2 grams of nascent polyethylene of the Ziegler-Natta type in 50 ml of Freon-I 1 which had been precooled to dry-ice temperature. Reaction was initiated by UV light which was kept on throughout the course of the reaction. Samples were extracted at appropriate time intervals and washed and dried prior to analysis.

8.

248

CHEMICAL METHODS I N POLYMER PHYSICS

have been successful in chlorination of amorRecently, Ogama et phous surfaces of single crystals of polyethylene by bubbling chlorine gas. The single crystals were grown in xylene. Xylene was exchanged for carbon tetrachloride. Chlorination was done by bubbling chlorine gas through the stirring suspension at room temperature under irradiation of a high-pressure mercury lamp. The degree of chlorination was controlled by bubbling and irradiating time and was determined by elemental analysis. A different method of bromination was used by Witenhafer and K ~ e n i g . They ~ ~ grew crystals of polyethylene, Marlex 6050, in tetrachloroethylene. Tetrachloroethylene was exchanged for carbon tetrachloride by repeated washing. An excess of N-bromo-succinimide was added to a portion of the above suspension and the mixture refluxed (77°C) for 16 hours. The mixture was washed with carbon tetrachloride, filtered, and vacuum-dried for 3 days at 50°C. They studied the effect of bromine on vinyl end group unsaturation. 8.4.1.3. Gaseous Halogenation. Dried solution-grown crystals or thin bulk films can be halogenated by exposing them to chlorine or bromine gas under UV light. Many apparatuses have been described in the literature for gaseous chlorination of polymer^.^^.^^.'^ They are similar in design and principle. An apparatus described by Nakagawa and Yamada?' as shown in Fig. 3 can be used. The cell can be modified according to specific need. Keller et chlorinated single crystals and bulk polyethylene by suspending the samples from a balance in a stream of chlorine gas. The chlorine uptake was measured as weight increase and confirmed by chemical analysis. The reaction was promoted by either light or heat. Bassettas exposed poly(4-methyl-pentene-1)films to chlorine gas at 60°C for intervals between 0.5 and 20 hours. Nakagawa and Yamada71chlorinated films of high- and low-density polyethylene, polypropylene, and polystyrene using the apparatus illustrated in Fig. 3. 8.4.1.4. Deduction of Crystal Morphology. A number of physical methods as described below have been employed to deduce the morphology of polymer crystals. If the halogenation is selectively concentrated at the fold surface, the intensity of low-angle X-ray diffraction may change but the low-angle X-ray period and the wide-angle pattern should remain unaltered. Accordingly, Bassetts6 found that chlorination of poly(4-methyl-pentene-I)crystal film at 60°C showed a marked increase Ioa

T. Oyama, K. Shiokawa, and Y. Kawamura, Polym. J . 9, 1 (1977).

'*T. Nakagawa and S. Yamada, J . Appl. Polym. Sci. 16, 1997 (1972).

7p

M. Luttinger, C. W. Cooper, and G . P. Hungerford, J . Polym. Sci., Part C 24, 257

(1968).

8.4.THE

SURFACE MODIFICATION TECHNIQUES

249

D

A

B

J FIG.3. Schematic diagram of photochlorination apparatus of Nakagawa and Yamada," only one cell is shown in this figure. (A) Sulfuric acid trap, (B)calcium chloride trap, (C) phosphorus pentachloride tube, (D) manometer, (E) stop cork, (F) sample film, (G)irradiation cell, (H)quartz window, (I) chlorine gas reservoir, (J) lamp.

in the intensity of the low-angle X-ray maxima while the wide-angle pattern was still unchanged from the control. He concluded that chlorination of poly(4-methyl-pentene-1)occurs only at the fold surface. Keller et u / . chlorinated ~ ~ polyethylene at 60°C and observed that the X-ray pattern gradually became weaker until it disappeared completely. They have concluded that at temperatures of 60°C or higher, reaction of the crystal lattice with chlorine proceeds to completion. For selective chlorination of polyethylene crystals, therefore, the reaction should be carried out at a low temperature. Low-angle X-ray diffraction patterns of brominated single crystals of polyethylene were studied quantitatively by Harrison and Baer.27 They observed a slight increase in the fold period upon bromination. As bromination increased, the intensity of the low-angle X-ray reflection first decreased to a minimum at about 0.5-1 bromine per fold and then increased. This result confirmed that bromination occurs in the amorphous regions. 8.4.1.4.1.THERMAL ANALYSIS.For a given molecular weight of a polymer, melting point and heat of fusion depend upon crystallinity or lamellar thickness. If halogenation occurs only on the surface of,ttie crystals, there may be a change in melting point but the heat of fusion would remain unaltered. Harrison and BaeP' observed that the melting

8.

250

CHEMICAL METHODS I N POLYMER PHYSICS

point of brominated single crystals of polyethylene decreases continuously up to the addition of one bromine per fold and then stays constant up to four bromines per fold. The heat of fusion of the single crystals, as prepared, is apparently unchanged on addition of up to four bromine atoms per fold. In contrast, the heat of fusion of the melt-crystallized brominated material decreases with addition of bromine as shown in for Fig. 4. Similar results have been reported by Oyama et surface-chlorinated single crystals of polyethylene. The interpretation of the constant value for the heat of fusion of the crystals, as prepared, is that the crystalline portion of the single crystals is unchanged upon bromination. The depression of heat of fusion of the melt-crystallized brominated material. is consistent with the inclusion of defects (bromine atoms) within the crystals, thereby lowering the crystallinity. 8.4.1.4.2. INFRARED ANALYSIS. Once the regions of the polymer preferentially attacked by a halogen have been identified by heat of fusion and the position of those regions has been located by small-angle X-ray diffraction, one can study infrared spectra of the original and halogenated polymers to determine selectivity of attack within these regions. Infrared spectra of some polymers like polyethylene have been studied in detai173.74 NO. OF BROMINES PER FOLD

I

2

3

FIG.4. Heat of fusion Hfof polyethylene vs. mole fraction X , of (-CHz-CHBr-)in -(CHz-CHz-)for (a) as-prepared single crystals after bromination and (b) the same materials after melt recrystallization (after Harrison and BaerZ7).

73 74

J. R. Nielsen and R. F. Holland, J . Mol. Specfrosc. 4, 488 (1960). R. G. Snyder, J . Chem. Phys. 47, 1316 (1967).

8.4.

THE SURFACE MODIFICATION TECHNIQUES

25 I

and bands due to different sequences of rrans (T) and gauche (G) conformations of methylene groups in crystalline and amorphous parts have been identified.l From the change in intensities of these bands one can predict the conformation of folds in the amorphous regions. Band assignments have been 'made for polyethylene by a number of author^.^^.^^ The 1368 cm-1 band has been assigned to the symmetric wagging mode of -GTG or -GTG' conformations about the trans bond, more probably -GTG. Also associated with this mode is a band at 1308 cm-' due to antisymmetric wagging. The band at 1352 cm-' has been assigned to the wagging of a methylene isolated by gauche bonds, i.e., the -GG - conformation. Harrison and Baef7 observed that the ratio of A1368/Ar35s7, increases with increase in bromine content. Since it has been demonstrated that 1368 and 1352 cm-l bands are both due in part at least to folds, the decrease in the ratio due to a decrease in the 1352 cm-I band could imply a selective attack of -GG- structures in the fold. The -GG- structure is of a higher energy than the -GTG structure; hence it is quite reasonable that the former is the unit that undergoes preferential attack. If bromination of the surface results in attack on specific conformations, then perhaps specific brominated products can be obtained. If this is the case there should be bands present in the C-Br stretching region from these conformations. Even at very low bromine contents, theyz7 observed two bands at 555 and at 575 cm-', and a third band at 625 cm-l. As the bromine content increases from zero to one bromine per fold, the ratio A5,5/A555 drops rapidly, indicating the steady increase of the 555 cm-l band. Once the bromine content reaches one bromine atom per fold, however, the ratio A575/A555is held, even out to four bromines per fold. From the thermal, X-ray diffraction and IR data, the authors have concluded that the fold model that best fits the bromination scheme is that of a regular adjacent reentry fold. Halogenation can be used for the location of unsaturated end groups in polymer crystals. Some polymers like polyethylene have a vinyl group, -CHz=CH,, at one end of the molecule. Witenhafer and Koenigs7 observed that the bands at 909 and 990 cm-I assigned to vinyl endgroup unsaturation are both reduced to about 10% of their original value upon bromination by N-bromo-succinimide. Similar observations have been reported by Harrison and Baer.27

t See also Chapter 3 . I (this volume, Part A).

8.

252

CHEMICAL METHODS IN POLYMER PHYSICS

8.4.2. Dehydrohalogenation of Poly(viny1ideneChloride)

Harrison and B a e P dehydrochlorinated poly(viny1idene chloride) (PVDC) by pyridine in order to distinguish between regular folds and other possible fold structures. The dehydrohalogenation of PVDC with organic amines and other bases is well k n o ~ n . ~A~possible - ~ ~ mechanism for its reaction with pyridine is shown below: First Stage H C1 H C1 H I I I I I -C-C-C-C-C-CI I I I I H C1 H C1 H

C1 I

H I

C1 H I

--C-C=C-C-C-C

I C1

I

H

I

C1 H

C1

I l C1 H

l C1

I

I

I

+

The allylic chlorine atoms now present in the polymer are many times more reactive than chlorine atoms in a saturated compound.70 Hence, an adjacent double bond should form more quickly than would an isolated bond, as illustrated in the second stage of the reaction: Second Stage H I

C1 I

H C1 H C1 I

I

I

--C-C=C-C-c-CI I l H C1 H

I

l C1

-

H C1 H C1 H I I I I I -C-C=C-C=C-CI

H

C1 I I c1

+

For each double bond produced, a molecule of pyridinium hydrochloride is obtained. Harrison and Baer2*used unbranched PVDC, which was obtained by bulk polymerization initiated by UV light. The polymer was dissolved in cyclohexane, held at 115"C, and crystallized isothermally at 70°C. The slurry of crystals was filtered and washed with distilled cyclohexane at the crystallization temperature. The suspending liquid was then changed L. A. Matheson and R. F. Boyer, Ind. Eng: Chem. 44, 867 (1952). E. Tsuchida, C. N. Shih, I. Shinoharra, and S. Kambara,J . Po/ym. Sci., Parr A 2, 3347 (1964). 77 E. Tsuchida, C. N. Shih, and I. Shinohara, J . Po/ym. Sci., Part B 3, 643 (1965). G. Laurent, J. P. Roth, P. Rempp, and J. Parrod, Bull. SOC. Chim. Fr. 9, 2923 (1966). 70 R. F. Boyer, J . Phys. Colloid Chern. 51, 80 (1947). 75

76

8.4.

THE SURFACE MODIFICATION TECHNIQUES

253

from cyclohexane via acetone to water and the crystals were either freeze-dried or filtered. The dehydrochlorination of the crystals was carried out at 45°C. The crystals were stirred in a nonsolvent benzyl alcohol/isopropyl alcohol mixture. Pyridine was then added and the rate of reaction was determined by following the production rate of pyridinium hydrochloride. A sample of the suspending liquid was added to a water/isopropyl alcohol mixture; this mixture was then titrated directly with N/30 sodium hydroxide or a known excess of sodium hydroxide was added and then back-titrated with N/30 hydrochloric acid. The endpoints were determined conductometrically . The dehydrochlorinated samples were studied by electron microscopy, small-angle X ray, IR, and thermal analysis. The plots of percentage of chlorine removed from the polymer vs. reaction time suggested that a rapid initial attack on the polymer is due to dehydrochlorination of folds followed by a progressively slower reaction due to dehydrochlorination of the crystalline regions. PVDC showed a singlet at 1648 cm-’ due to formation of an isolated alkene after 24 hours of reaction and a doublet at 1642 and 1612 cm-l due to formation of conjugated diene after 500 hours of reaction. From the kinetics, IR, low-angle X ray, and thermal analysis data, Harrison and B a e P have concluded that the preferential dehydrochlorination of PVDC at the fold surfaces of lamellae and the stepwise nature of this reaction indicate that neither the fringed micelle nor the “switchboard” model are the preferred models for this system. Although the regular fold model is preferred, one cannot distinguish between the smooth and rough surface mode1s.t Dehydrohalogenation of PVDC by pyridine is not very selective at the fold surface since it slowly attacks the crystalline parts as well. The technique may be used for other polymers having C-Cl (alkyl halide) bonds. As dehydrohalogenation produces unsaturation in the amorphous regions, the polymers can be selectively oxidized at the unsaturated sites by oxidizing agents like HN03 and O3 and the degraded products can be studied by gel permeation chromatography (GPC). 8.4.3. Methoxymethylation of Polyamides

The surface modification technique has been used by Arakawa et U ~ . ~ to determine the true melting point inherent to the morphology of drawn nylon 6. When a polymer sample is heated in a thermal analysis experit A discussion of fold surface models is given in Chapter 5.4 (this volume, Part A). 8o

T. Arakawa, F. Nagatoshi, and N. Arai, J . Polym. Sci., Part A-2 7, 1461 (1969).

O

254

8.

CHEMICAL METHODS IN POLYMER PHYSICS

ment, unstable defect crystals are usually reorganized into more stable states before melting occurs. As a result a higher melting point is observed. On the other hand, a higher heating rate can cause superheating of the crystals. Thus the melting point inherent to the polymer morphology can be obtained when both reorganization (the annealing effect) and superheating of the crystals are prevented during thermal analysis.? Annealing or reorganization of polymer chains can be prevented by modifying the crystal surfaces by, for example, halogenation of polyethylene crystals26*2T and crosslinking the fold surfaces by ionizing radiation.81 treated nylon 6 with paraformaldehyde and methanol in Arakawa et the presence of catalysts to methoxymethylate amide groups. Methoxymethylated amide groups, -CON(CH,OCH,)-, react with neighboring amide groups, -CONH-, to form crosslinks: -CON(CHZOCH3)-

+ -CONH-

-

-CON -

Ca1alyst

I I

CH,

+ CH,OH

-CON -

Methoxymethylation of drawn nylon 6 was carried out as followse0: A mixture of 75 g of paraformaldehyde, 75 g of methanol, and 0.1 g of potassium hydroxide was stirred at 60°C until it became a clear liquid. Then 6 g of anhydrous oxalic

acid was added as catalyst for the methoxymethylation reaction. Sample filaments were dipped in this reaction mixture at 30°C without stirring for a period of time. They were then washed with distilled water, immersed twice in a large quantity of distilled water for a few hours and were dried over phosphorus pentoxide.

Methoxymethylated nylon 6 was studied by DSC.80 Highly oriented, annealed nylon 6 sample had a well-defined melting peak at 220°C. With progressive methoxymethylation of amide groups, the single peak at 220°C divided into a sharp low-temperature peak and a broad hightemperature peak. The low-temperature peak shifted continuously to lower temperatures with increasing reaction time and finally converged to 195°C after 30 hr of reaction. Further methoxymethylation did not influence the peak temperature appreciably. Chemical analysis showed that methoxymethylation of about 8% of total amide groups is enough for the low-temperature peak to converge to a fixed point. Methoxymethylated amide groups would not crystallize or reorganize during heating and hence the melting point observed would be that of crystalline core only. This technique needs to be studied further in order to confirm that methoxymethylation occurs only in amorphous regions. t See also Part 9. H. E. Bair, R. Salovey, and T. W. Huseby, Polymer 8, 9 (1967).

8.5.

THE SURFACE DEGRADATION TECHNIQUES

255

8.4.4. Acylation of Polystyrene

Lemstra et ~ 1have . modified ~ ~ amorphous surfaces of single crystals of isotactic polystyrene by the Friedel-Crafts reaction. The Friedel-Crafts reaction is known to introduce benzoyl groups on the phenyl groups of polystyrene. Single crystals of isotactic polystyrene were grown from dilute solutions in dioctylphthalate at 125°C. These crystals were chemically modified by a Friedel-Craft acylationE2: To 100 ml of nitrobenzene at 50°C were added 0.5 g of the single crystals, about 0.25 g AICl,, and 1 ml of benzoyl chloride. AICI, dissolved in nitrobenzene, but the single crystals did not, since the reaction temperature of 50°C is far below their dissolution temperature of about 100°C. After two hours, the reaction mixture was poured into a mixture of methanol/HCI (0.1 N). After filtration the collected crystals were washed thoroughly with methanol and ether, and dried under vacuum at 50°C.

Both the original and acylated crystals were characterized by IR, X ray, and DSC.82 Benzoylation of polystyrene crystals yielded an extra IR band at 1660 cm-l, which is characteristic for diary1 ketones. X-ray diffraction patterns demonstrated that crystalline structure of the single crystals remained intact during acylation, i.e., acylation occurs in the amorphous regions.

8.5. The Surface Degradation Techniques 8.5.1. Selective Degradation Reagents

Selective degradation techniques are based on greater reactivity of less perfect, amorphous, parts of polymer crystals. They were first developed for analysis of c e l l u l o ~ e .Chemically, ~ ~ ~ ~ ~ the amorphous content of cellulose is selectively degraded by acid hydrolysis, HCI or H2S0, in water or ethanol. Polyesters and polyamides have also been degraded by hydrolysis .83-85 Palmer and Cobbold31showed that amorphous surfaces of polyethylene crystals can be selectively etched by fuming nitric acid. Another selective oxidizing agent proposed by them is ozone. Priest32 used ozone for freeze-dried single crystals of polyethylene. Since then nitric acid and ozone have been widely used for a large number of polymers and copolymers to study the amorphous surfaces. Other P. J. Lemstra, A. J. Schouten, and G. Challa,J. P d y m . Sci., Phys. Ed. 10,2301 (1972). D. A. S. Ravens and I. M. Ward, Trans. Faraday SOC. 57, 150 (1961). R. E. Mehta and J. P. Bell, J. Polym. Sci.,Part A-2 11, 1793 (1973). B. Wunderlich, “Macromolecular Physics,” Vol. 1, p. 408 Academic Press, New York, 1973.

8.

256

CHEMICAL METHODS IN POLYMER PHYSICS

reagents, like oxygen, sulfur dioxide, and nitrogen dioxide have also been found to degrade amorphous parts of polymers.86-8B Nitric acid and ozone degradation techniques will be discussed in detail, as these techniques have been used for a large number of polymers to elucidate fold structures.

8.5.2.Nitric Acid Degradation No special apparatus is required for this technique. The degradation reaction can be carried out in any convenient glass vessel, for example, a flask or a test tube. Polymers in any physical form such as films, fibers, pellets, powders, sedimented mats, or freeze-dried solution-grown crystals can be oxidized.Bo-loBConcentrated (above 60%) or fuming nitric acid should be used.B0 If the density of polymer is lower than that of the S. D. Razumovskii, A. A. Kefeli, and G . E. Zaikov, Eur. Polym. J . 7 , 275 (1971). H. H. G. Jellinek and A. K. Chaudhuri, J. Polym. Sci.,Part A-1 10, 1773 (1972). aa P. Hrdlovic, J. Pavlinec, and H. H. G . Jellinek, J . Polym. Sci., Part A-1 9, 1235 (1971). H. H. G. JeUinek, Text. Res. J . 43, 557 (1973). QO K. H. Illers, Makrornol. Chem. 118, 88 (1968). O1 F. M. Rugg, J. J. Smith, and R. C. Bacon, J. Polym. Sci. 13, 535 (1954). E. G . Bobalek, J. N. Henderson, T. T. Serafiny, and J. R. Shelton,J. Appl. Polym. Sci. a6 87

2, 210 (1959).

H. C. Beacheli and G . W. Tarbet, J . Polym. Sci. 45, 451 (1960). T. H. Meltzer, R. F. Hoback, and R. N. Goldey, J . Appl. Polym. Sci. 7 , 71 (1963). O5 A. Keller and S. Sawada, Makromol. Chem. 14, 190 (1964). 88 C. W. Hock, J . Polym. Sci., Part A-2 4, 227 (1966). O7 D. J. Blundell, A. Keller, and I. M. Ward, J. Polym. Sci., Part B 4, 481 ;1966). * A. Peterlin, G. Meinel, and H. G . Olf, J. Polym. Sci., Part B 4, 399 (1966). F. H. Winslow, M. Y. Hellman, W. Matreyek, and R. Salovey, J . Polym. Sci.,Part B 5, 89 (1967).

D. J. Blundell, A. Keller, and T. M. Connor, J. Polym. Sci., Part A-2 5, 991 (1967). T. Williams, A. Keller, and I. M. Ward, J . Polym. Sci., Part A-2 6 , 1621 (1968). lo* F. M. Willmouth, A. Keller, I. M. Ward, and T. Williams, J . Polym. Sci., Part A-2 6 , loo lol

1627 (1968). lO3

D. M. Sadler, T. Williams, A. Keller, and I. M. Ward, J . Polym. Sci.,Part A-2 7 , 1819

(1969). lo4

T. Williams, Y. Ydagawa, A. Keller, and I. M. Ward, J. Polym. Sci., Part A-2 8, 35

(1970).

T. Williams, D. J. Blundell, A. Keller, and I. M. Ward, J. Polym. Sci., Part A-2 6 , 1613 (1968). lo6

N . E. Weeks, S. Mori, and R. S. Porter, J. Polym. Sci., Polym. Phys. Ed. 13, 2031

(1 975). lo'

A. Keller and E. Martuscelli, Makromol. Chem. 151, 169 (1972). P. J. Holdsworth, A. Keller, I. M. Ward, and T. Williams, Makromol. Chem. 125,70

(1 969). loo

P. J. Holdsworth and A. Keller, Makromol. Chem. 125, 82 and 94 (1969).

8.5.

THE SURFACE DEGRADATION TECHNIQUES

257

acid at the reaction temperature, the sample should be held at the bottom of the vessel by a glass rod or by glass wool. An excess of nitric acid is used, about 10 g of the acid for 1 gm of polymer. The reaction is usually carried out between 60 and 90"Cgobecause the oxidation is rapid at these temperatures. The polymer is placed in concentrated or fuming nitric acid at room temperature (RT) and is left for some hours to allow diffusion of the acid into the polymer. Oxidation of polymers is very slow at RT. The reaction is then accelerated by putting the tube in a constant temperature bath at 60°C or higher. Alternatively, a sample can be placed in hot nitric acid. Polyalkenamers have been degraded at RPo7and polypropylene has been degraded at l20"Cg6 After the desired treatment time, the contents of the tube are poured into an excess of water or a piece of polymer is removed from the acid and washed repeatedly with distilled water until no acid remains in the wash. After washing, the sample is extracted with a volatile organic nonsolvent like acetone or methanol and dried under vacuum. 8.5.3. Ozone Degradation 8.5.3.1. Properties of Ozone. Ozone is an unstable gas with a pungent odor. At ambient temperature, ozone is a blue gas but the color is not noticeable at normal production concentrations, unless the gas is viewed through considerable depth. At - 112"C, ozone condenses to a dark blue liquid. Liquid ozone explodes easily, as do concentrated ozone-oxygen mixtures (above about 30% ozone) in either the liquid or the vapor state. Explosions may be initiated by minute amounts of catalysts or organic matter, shocks, electric sparks, a sudden change in temperature or pressure, etc. The structure of the ozone molecule is that of an obtuse angle in which a central oxygen atom is attached to two equidistant oxygen atoms with an included angle of about 127". Ozone is a potent germicide and a powerful oxidizing agent. It is unstable and slowly decomposes to ordinary oxygen. The decomposition is very slow at room temperature, but is catalyzed by heat and moisture, by some metals and metal oxides, by sodium hydroxide, by soda lime, and by chlorine and bromine, etc. Saturated hydrocarbons are oxidized slowly by ozone. It reacts readily with unsaturated organic compounds by adding all three oxygen atoms at a double or triple bond. The resulting compounds are called ozonides. Many ozonides are explosive. Decomposition of ozonides results in rupture at the position of the double or triple bond and formation of aldehydes, ketones, and acids. Typical groups that are readily oxidized by ozone are -SH, =S, -NH2, =NH, -OH (phenolic) and -CHO.

258

8.

CHEMICAL METHODS I N POLYMER PHYSICS

The pungent odor of ozone can be detected in very low concentrations; approximately 0.02-0.05 ppm by volume.11oJ11 Continued exposure 8 hr) at somewhat higher concentrations induces severe coughing. Still stronger concentrations produce this effect immediately. Other effects sometimes observed are depression, cyanosis, and nausea. Ozone could be lethal in a few minutes in concentrations over 1700 ppm.llO.lll All experiments on oxidation of polymers should be performed in a hood with a very powerful exhaust system. 8.5.3.2.Ozone Generation. Ozone can be produced photochemically112by the action of UV light of wavelength shorter than 2200 8, on oxygen or air. The concentration of ozone produced by this photochemical method is very low. In practically all laboratory or industrial applications ozone is produced by the action of an electrical discharge on air or oxygen. A number of ozone generators have been reviewed by Hann and Manley.l13 A small compact ozonizer for laboratory use has been described by Whaley.l14 This apparatus is shown in Fig. 5 . The generator was made of three individual ozonizer tubes constructed of soft glass. Ozone production in this type of silent-electrical discharge ozonizer is greatly affected by the type of glass which comes in contact with the oxygen (soft glass is the best). The inner tube was 12 mm O.D., the outer 22 mm O.D., and the outlets were 8 mm O.D. A somewhat larger size tubing must be used for the inner tube if the secondary voltage of the transformer employed is less than 8000 V. The seal of inner to outer tubes should be as close as possible to the inlet in order that maximum electrode area may be obtained. The container which holds the ozonizer tubes is constructed of 74 mm O.D. Pyrex glass tubing, 21 cm in length and slightly flared at the top. A 600 ml beaker is adequate for this purpose. The three ozonizer tubes are held in place by a support cut from a clay plate on the bottom of the container. If anoncorrosive material is to be used as an electrode this may be conveniently constructed from a large rubber stopper. The support at the top was made of&inch plywood, shaped to fit the flared top of the container, designed to maintain the tubes in a vertical position. The ozonizer tubes were connected in series by means of TygonQ tubing; both Pyrex glass container and inner types were filled with 7.7 M zinc chloride solution; and copper wires leading from the secondary of the transformer were permitted to dip into the electrode solution. The inner electrode may be mercury or any type of electrolytic conductor which will not freeze at the temperature of the cooling bath. The ozonizer was then placed in a 2000 ml Dewar vessel which contained a suitable cooling bath supported at the top by a cork ring. The cork ring was fitted with an opening to permit ad“Ozone Chemical Technology.” Am. Chem. SOC.,Washington, D.C., 1959. “Ozone Reactions with Organic Compounds,” Adv. Chem. Ser. 112. Am. Chem. SOC.,Washington, D.C., 1972. 11* N. I. Sax, “Dangerous Properties of Industrial Materials,” p. 989. Van NostrandReinhold, Princeton, New Jersey, 1968. V. A. Hann, and T. C. Manley, Encyl. Ind. Chem. Anal. 16, 538 (1972). 114 T. P. Whaley, J . Chem. Educ. 34, 94 (1957). 111

8.5.

THE SURFACE DEGRADATION TECHNIQUES

259

To Troniforrner

\

FIG.5 . Laboratory ozonizer of Whaley.114

dition of solid carbon dioxide. The transformer used was capable of 10,800 volts secondary voltage. Discharge between electrodes did not take place at voltage less than 8000 volts. This type of ozonizer produced between 7.5% and 8% (by weight) ozone at a temperature of -40°C and a 60-cycle voltage of 10,500volts. The conversion may be increased by maintaining the temperature somewhat lower than -40°C and also by using voltage of greater frequency.

8.5.3.3.Freeze-Drying Technique. For the reaction to proceed at its maximum rate, the surfaces of the crystals should be freely accessible to contact with the reactant gas, a situation that minimizes complications of interpertation due to long times of diffusion of gases to the reaction sites. The best way of achieving this is to freeze-dry solution-grown crystals. The solvent used for crystallization is removed by sublimation at a tem-

260

8.

CHEMICAL METHODS I N POLYMER PHYSICS

perature below its freezing point, thus eliminating the surface tension effects that draw the crystals into close contact when liquid is evaporated from suspension. The solvent used for crystallization should be such that it can be solidified at moderately low temperature, around 0°C and can be sublimed rapidly under vacuum. If the solvent used for crystallization has a very low freezing point or takes longer to sublime, it can be exchanged for an appropriate solvent. Para-xylene (m.p. 13.2"C)is a very suitable solvent for crystallization of many polymers, which later can be removed by sublimation at about 0°C. Two main limitations of the ozone degradation technique are (1) it can only be applied to freeze-dried polymer crystals, and (2) the single crystals should be mainly monolayer. Films of polyethylene that were exposed to ozonized oxygen showed that the rate of reaction was negligible compared with the rate of reaction with freeze-dried Monolayer crystals can be obtained by self-seeding te~hnique."~."~Freeze-dried monolayer single crystals of polyethylene can be obtained by a procedure described below. The procedure can be used for other polymers by selecting proper solvent, and proper seeding and crystallization temperatures. Polyethylene is dissolved in hot p-xylene to prepare about 0.1% solution. The solution is then cooled to 70°C for crystallization. The crystal suspension is gradually heated from 70°C to a seeding temperature (101 & 0.YC for linear polyethylene) in about 3 hr. At this temperature very small fragments of the crystals are left undissolved. The solution is then rapidly quenched to a desired crystallization temperature. The major part of the solvent (about 90%) is removed either by decantation or by slow filtration under gravity. The resultant crystal slurry is cooled to liquid nitrogen for quick freezing of p-xylene. The p-xylene is sublimed at about 0°C under vacuum. 8.5.3.4. Degradation of Polymer Crystals. Priest32was the first to use ozone for selective degradation of polyethylene crystals. The ozonizer used by Priest32for selective oxidation of polyethylene single crystals was a modification of the ozonizer described by Whaley."* It had two ozonizer tubes instead of three and, for most of the work, was cooled by water instead of solid carbon dioxide. This method was more convenient for prolonged experiments even at the expense of the amount of ozone produced. The transformer used gave 10,000 V output at 50 Hz. The whole system as shown in Fig. 611'-'30 consisted of an oxygen cylinder fitted with a pressure-reducing valve, a drying tower containing silica gel, a neella

lle

D. J. Blundell, A . Keller, and A . J. Kovacs, J . P d y m . Sci., Purr B 4, 481 (1966). D. J. Blundell and A. Keller, J . Macrumol. Sci., Part B 2, 337 (1968).

8.5.

26 1

THE SURFACE DEGRADATION TECHNIQUES

TRANSFORMER

-K

FIG.6. Schematic diagram of an apparatus used by PriestS*and o t h e r ~ l ~for ~ - degradal~~ tion of freeze-dried crystals of polymers by ozone. (A) Oxygen cylinder, (B) pressure regular, (C) needle valve, (D) flow meter, (E) silicagel drying tower, (F) glass wool, (G)two-tube ozonizer, (H) sintered disk degradation cell, (I) freeze-dried crystals, (J) KI solution, (K) to exhaust system.

dle valve, a flow meter, the ozonizer, the sample holder and two wash bottles containing aqueous potassium iodide solution (about 10% KI) to remove unused ozone. The amount of ozone produced was measured occasionally by first passing the gas into a solution of about 4 g of potassium iodide in 200 ml of water with a little boric acid at a flow rate of 100 cm3/min for 10 min. The iodine produced was titrated with 0.1 N sodium thiosulfate solution (1 ml 0.1 N N*S2O3 = 2.4 mg ozone). Usually the oxygen contained 1-1.5 wt% of ozone. Ozone offers the following advantages32over nitric acid: (a) It works at room temperature, well below temperatures that might affect the structures to be studied. (b) The chemical reaction is simpler, since only oxygen atoms can be added to the polymers and because reaction would be carried out without a liquid phase being present. (c) Characteristically, treatment with concentrated nitric acid causes loss in weight as degraded fragments are dispersed in the liquid. With ozone, the weight increases, which means products of degradation are retained and are available for study. 8.5.4. Deduction of Crystal Morphology

Both HN03 and O3 are strong oxidizing agents. They diffuse into amorphous regions but not into the crystalline ones. They oxidize

8.

262

CHEMICAL METHODS IN POLYMER PHYSICS

polymer chains in the amorphous parts including folds resulting in chain scission. The end groups of the cut chains become -COOH, -CHO, or -NOz. The reaction mechanism of oxidative degradation of polymers by nitric acid has not been studied in detail. Ozone reacts with saturated aliphatic polymer chains to form intermediates like peroxides and hydroperoxides or with unsaturated aliphatic chains to form ozonides. Peroxides, hydroperoxides, and ozonides decompose and, through chaincleavage, yield polymer chains having -COOH or -CHO end groups .86J31 There is evidence"' that loose folds and loose ends are attacked first and sharp folds are attacked later on. Once a fold is cut the carboncarbon bond adjacent to -COOH or -NOz groups is attacked. This shortens the cut ends until they are level with the crystalline core. As the reaction proceeds, more folds are cut and the cut ends are shortened until all folds are cut. The oxidation reaction can be followed by measuring changes in the following parameter^^^-'^^^^^^: weight, density and crystallinity, low- and wide-angle X-ray diffraction, melting point and heat of fusion, mobile fraction by NMR, and molecular weight and molecularweight distribution. Removal of amorphous content is indicated by increase in weight ~ ~ ~ ~ , 3 1 ~ 3 2 ~ B 0 ~ 9 5 ~ B 6 ~ 1 0 0 ~ 1 0 6density ~ 1 0 8 ~ 1 0 and B ~ 1 Zcrystallinity 0 31~B0,05.B6.B8.103 heat of mobile fracfusion, and decrease in melting point,31~Bo~B6~Bs~103~*07~10n~1zo tion,98*100 X-ray diffr~ction,BO~95~100~103~105-107~10B~117 and molecular weight .B0~98~BB~101-108~117-120 The rates of these changes decrease rapidly with increase in degradation time. Upon prolonged degradation, no appreciable change in the above property is observed. Figure 7 shows the change in weight, crystallinity, melting point, and molecular weight of polypropylene treated with boiling (120°C) HN03. All the follow-up techniques mentioned above can be used to (1) confirm that polymer crystals are composed of crystalline parts and amorphous surface layers, and (2) measure amount of amorphous material in polymers. However, they cannot be used to (1) differentiate between fringe micelle and chain-folded models, (2) confirm whether all folds are cut, (3) differentiate between sharp and loose folds, (4) determine distribution of folds, ( 5 ) determine attack of oxidizing agents on crystalline core.

11' 118

G . N. Patel and A. Keller, J . Polym. Sci., Phys. Ed. 13, 2259 (1975). G. N. Patel and A. Keller, J . Polym. Sci., Phys. Ed. 13, 2275 (1975). G. N. Patel, A. Keller, ND E. Martuscelli, J . Polym. Sci., Polym. Phys. Ed. 13, 2281

(1975).

A. Keller and E. Martuscelli, Mukromol.

Chem. 151, 189 (1972).

8.5.

THE SURFACE DEGRADATION TECHNIQUES

263

ACID TREATMENT, HOURS

FIG.7. Plots of change in molecular weight, weight loss, crystallinity, and melting point of polypropylene against nitric acid treatment time at 120°C (after Hockw).

For the above purposes, gel permeation chromatography (GPC) is used. GPC is used to determine molecular weight and molecular-weight distribution of polymers has been described in detail in Part 2 (this volume, Part A). Oxidizing agents remove amorphous material including folds. If polymer chains are folded, as shown in Fig. 8a, one can expect formation of single, double, triple, etc. traverses in the crystals as shown in Fig. 8b. When all folds are cut, Fig. 8c, the crystals will consist of single traverses only. GPC of degraded crystals therefore should show peaks for single, double, triple, etc. traverses. Figure 9 shows GPC traces at different stages of degradation of polyethylene single crystals. Discrete peak for single and double traverses can be seen. The molecular weight of single and double traverses is calculated from the position of the peaks using the calibration curve. The chain length of a single traverse L, is almost equal to half the chain length of a double traverse L2.101-108,117-120 L , and 0.5L2 agreed well with the fold period105-107,117 L, measured by X-ray diffraction. This confirms that polymer chains are folded back and forth in the lamellae. If polymer chains are crystallized in fringed micelle form, one cannot expect discrete peaks in GPC; instead there should be only one broad peak. This is explained in Section 8.5.8.1. In order to determine the distribution of fold lengths, precise measurements on molecular weight of single and double traverses are required. Therefore, single- and double-traverse peaks should be well resolved. Molecular weight of single and double traverses ranges from 1000 to 15,000 depending upon the fold period. Better resolution of the peaks

264

8.

CHEMICAL METHODS IN POLYMER PHYSICS

Triple

Double ;e

Single Traverse

Single

4

lo1 Ibl Icl FIG. 8. Schematic presentation of different stages of degradation of polymer single crystal. (a) Undegraded single crystal (the possible additional presence of cilia and nonadjacently reentrant loops are not shown). (b) Partially degraded single crystal. (c) Fully degraded single crystal ("crystal core"). The dots terminating the chains represent COOH groups (after Patel and Keller12s).

can be obtained by using two to four columns in the 500-20,000 8, porosity range.ll7s12OJ23 Errors in molecular weight due to variation in experimental conditions like flow rate, temperature of the siphon, and temperature of the columns can be minimized by using an internal ~ t a n d a r d . ~ ~ Use ~ J ~of~ an J~~ oligomer, possibly having a chemical composition similar to that of the samples is recommended as an internal standard in GPC experiments for better reproducibility and precision. Adsorption of the end groups, -COOH and -NO2 of single and double traverses, affects the peak position of the Such effects should be corrected by using calibration samples having the same end groups. Preparation of such calibration samples is described in Section 8.5.7.

If only the folds are cut and the cut ends are not shortened, (1) L, should be equal to 0.5& at any stage of degradation, and (2) there should not be decrease in L , or L2 with increase in P,,where P, is the weight fraction of molecules in a single traverse peak and is a measure of the extent of cut folds (see Section 8.5.6 for further detail). Precise measurements of L , and L2 by Patel and Keller"' on ozone-degraded single crystals of polyethylene showed that 0.5& is higher than L, and 0.5L2decreases with increase in PI,as shown in Fig. 10. This indicates that the M. Chang, T. G. Pound, and R. St. John Manley, J. Polytn. Sci.,Polym. Phys. Ed. 11, 399 (1973). IZ2 G. N. Patel and J. Stejny, J . Appl. Polym. Sci. 18, 2069 (1974). lZs G. N. Patel and A. Keller, J . Polym. Sci., Polym. Phys. Ed. 13, 332 (1975).

8.5. THE SURFACE DEGRADATION TECHNIQUES

h

Tc :70' C 470hr

Tc 90' C 350 hr

LI, I I , I I , I I 20 25

I

-

12.0 hr

v)

c .E

a

E

=

n

L

Q

265

,j il,

5.5 hr

,;oll,J

W V

c W

W L

c

c

0

x

2.5 hr

W -0 c

W

5 .c

u

e

c

oz 0)

16 min

15 min

Elution Volume (Pulse) FIG.9. GPC traces of single crystals of polyethylene grown at 90°C and degraded by O3at 60°C for increasing length of time. The peak with the asterisk is due to n-decane used as an internal standard (after Patel and Kellerl*o).

cut ends are shortened rapidly. Though there is a considerable decrease in L, and 0.5Lz at the early stages of degradation, there is no appreciable change in L,, the fold period measured by X ray. This suggests that loose folds are cut first. At the intermediate stages of degradation, there is little decrease in L , , 0.5Lz, and L,. This can be expected if relatively

266

8.

CHEMICAL METHODS I N POLYMER PHYSICS 1, ot RT 1, at RT o 1, at SO'C 0

90°C Crystols Woshed

8I/z

Lz 01

.I/*

wc

- -l/z~l,*~-121

------------1LO 130

I

\

-

llO\

x\

I

I

I

I

I

0.2

04

0.6

0.8

1

! FIG.10. Plots of traverse length against P, for polyethylene crystals grown at 90°C and degraded by ozone. The degraded crystals were washed with water before the GPC test. B, L, degraded at room temperatufe; . , O X 2 degraded at room temperature; 0, L , degraded at 60°C; 0 ,0.5L2degraded at 60°C. Lt0 = extrapolation of L , to P1 = 0; =, low-angle X-ray . , curve for spacings (L,); L,O = L, at P = 0; L,O corrected for chain obliquity; 0.5(L2 + Llo - L J , which is a correction (upper limit) for continued cutting of severed chains from their cut ends (after Patel and Keller"').

-

sharper folds at the level of the crystalline core are cut. The decrease in L , , 0.5L2, and L, with further increase in P , indicates that sharp fold buried within the crystals are cut. Normally, a plot of 0.5L2 against P , should represent the distribution of folds, but value of o.sL2 should be corrected for the decrease in length from both ends. Single traverses also have two ends and would suffer the same loss in length as double traverses. So as to set an upper limit of 0.5Lz,one has to add O.5(Ll0- Ll), in OSL,, where LIois single-traverse length at P , = 0, which can be obtained by extrapolating the plot of L , against P , to P , = 0. Ll0 - L, measures shortening of single traverse from both the ends and is equal to the shortening of double traverses from the ends at any given stage of degradation. Thus a plot of 0 . 5 L ~-+ o.5(Ll0 - Ll) or 0.5(Lz + Ll0 - L,) against P , represents distribution of folds in polymer crystals, see the uppermost curve in Fig. 10. 8.5.5. Location of Unsaturation and Branches

Many homopolymers and copolymers contain carbon-carbon double bonds and methyl and ethyl branches. It is important to determine the location and distribution of these groups, as many physical and chemical

8.5.

THE SURFACE DEGRADATION TECHNIQUES

267

properties will be affected according to whether they are distributed uniformly or excluded from the crystalline parts. IR spectroscopy is used to determine the location of these groups, as they have characteristic absorption frequencies in the IR region. IR spectra are recorded for samples at different stages of degradation. About 1-20 mg of degraded polymer is thoroughly mixed with about 0.5 g of dry KBr and pressed into a pellet under vacuum. The IR band intensity is measured for samples degraded for different periods of time. To avoid any error in weighing the sample, the ratio of the intensity of the group of interest to that of an internal standard is measured. The ratio is plotted against the degradation time or against the fraction of uncut folds. Using this technique, Priest and Ke11er124J25 observed that about 90%of the vinyl end groups in polyethylene are located outside the crystalline and Patel et al. llQhave determined the core. Holdsworth and Keller108Jog distribution of methyl and ethyl branches in ethylene-propylene and ethylene -butene copolymers. Keller and M a r t u ~ c e l have l ~ ~studied ~~~~~ degradation of guttapercha (trans 1 ,Cpolyisoprene) and polyalkenamers by HNO, and 0,in order to determine the distribution of trans vinylene double bonds. For polymers having high concentration of unsaturation, either very low concentration of ozone, less than 0.5% should be used or the degradation reaction should be carried out at a low temperature, below 0°C.107 The results reveal that while there is a certain amount of preference for the branches and the double bonds to accumulate near the fold surface, a substantial portion of them can still be located in the crystal interior. 107-109~11Q~120 8.5.6. A Test for Random Attack

Assuming a random attack of oxidizing agents on folds of polymer crystals, Williams and Ward133derived an equation that related p , the fraction of uncut sites (folds), to P , and P 2 , the weight fraction of molecules in the single- and double-traverse peaks, respectively: p = 1 - ( P p = P2/2P,. lz4

(8.5.1)

A. Keller and D. J. Priest, J . Polym. Sci., Purl B 8, 13 (1970).

'" D. J. Priest and A. Keller, J . Mucrurnol. Sci.,Phys. 2, 479 (1968). G . N. Patel, L. D'Ilario, A. Keller, and E. Martuscelli, Mukromol. Chem. 175, 983 ( 1974).

G . N. Patel and A. Keller, J . Polym. Sci.,Pulym. Lett. Ed. 11, 737 (1973). G . N . Patel and A. Keller, J . Polym. Sci., Polym. Phys. Ed. 13, 303 (1975). G . N . Patel and A. Keller, J . Polym. Sci., Polym. Phys. Ed. 13, 323 (1975). 130 G . N. Patel, J . Polym. Sci., Polym. Phys. Ed. 13, 351 (1975). 131 A . A. Kefeli, S . D. Razumovskii, and G . E. Zaikov, Vysokomol. Soldin.. Ser. A 13,803 (1971). 13' G . N. Patel, J . Appl. Polym. Sci. 18, 3537 (1974). lS I. M. Ward and T. Williams, J . Pulym. Sci., Purr A-2 7, 1585 (1969). 12*

268

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CHEMICAL METHODS IN POLYMER PHYSICS

Usually the GPC peaks of single and double traverses are not clearly resolved, but can be separated by assuming that they are ~ymmetrica1.l~~ The ratio of the area under the single- and double-traverse peaks to that under the whole chromatogram will give P, and P2,respectively. If the attack is random, 1 - P1112 = P2/2P1 or P2 = 2(1 - P11/2)P1. Williams for polyethylene degraded by niand Ward133found that 1 - P, = P2/2P1 tric acid. Pate1 and Keller118confirmed random attack of ozone on folds of single crystals of polyethylene by plotting P2against P,. The observed plot of P2against P,coincided with that calculated from Eq. (8.5.1) for p ranging from 1 to 0. 8.5.7. Preparation of Low-Molecular-Weight Fractions for GPC

As mentioned earlier, low-molecular-weight fractions of the same polymer with the same end groups are required for calibration of GPC. The calibration samples can be obtained by the following steps: (1) Single crystals of polymer are grown at different temperatures or annealed at different temperatures in order to get crystals of different thicknesses. (2) The single crystals are selectively degraded until all folds are cut; only a single traverse peak showed in GPC. (3) Measure molecular weight of the degraded samples by light scattering, osmometry, etc.

8.5.7.1. Preparation of Long-Chain Paraffins. Keller and Udagawa13* developed a method of preparation of long-chain paraffins from fully degraded polyethylene crystals. They grew single crystals of polyethylene and degraded them with 95% fuming nitric acid at 60°C for about 3 weeks. This treatment removed all folds and showed a single traverse peak in GPC. Nitric acid degradation introduced -NO2 and -COOH end groups which then have to be eliminated. They first eliminated -NO2 groups: For this purpose 200 mg samples were treated with 10 ml98% H,SO, at 60°C for 2 days. The resulting material was dissolved in xylene and cleared with activated charcoal. The absence of the 1557 cm-l band in the infrared spectrum reveals that the nitro group has been removed. The method used for short carboxylic acids by Barton ei a / .lsa was adopted to eliminate -COOH groups. This consists of the substitution of the carboxyl group with an iodine atom through the intermediary of oxyiodide. The iodine is then replaced by a hydrogen leading to a methyl terminated paraffin. Iu

A. Keller and Y.Udagawa, J. Polym. Sci., Purr A-2 8, 19 (1970).

8.5.

THE SURFACE DEGRADATION TECHNIQUES

269

In detail, 100 mg of dicarboxylic acid was dissolved in carbon tetrachloride; 2.3 g of lead tetraacetate and 1.5 g of iodine were added and the photochemical reaction described by Barton er al.135was performed under reflux in an atmosphere of nitrogen with the aid of a 150 W tungsten lamp. At this stage the 1712 cm-l carboxyl band has disappeared in the infrared spectrum while the strong iodide absorption peak at 1169 cm-' becomes apparent. The reduction of the iodide was carried out by means of the Clemensen reduction method with zinc amalgam. The infrared spectrum reveals that the 1169 cm-I iodide peak is now absent. However, two small bands at 1746 cm-l and 1240 cm-l not contained by the pure paraffin spectrum are still visible. These two small bands could be removed by treatment with hydroiodic acid in the presence of red phosphorus in a sealed tube at 150°C for 6 days. At this last stage however, a small carboxylic acid peak at 1712 cm-1 reappeared again. Potentially, the 3% carboxyl should be removable by repetition of the whole procedure. A single repetition should give 99.9% pure para%n, and so on.

Ballard and D a w k i n ~ have ' ~ ~ prepared block copolymers of the dicarboxylic acid (prepared by the above procedure) and 1,lO-decane diol, hexamethylene diamine, poly(propy1ene glycol), diphenylmethane-4,4'diisocyanate, and 2,4-toluene diisocyanate. 8.5.8. Hydrolysis

Hydrolysis is a chemical reaction in which water reacts with another compound so that both water and the compound are cleaved or split; the water decomposes to its ions, each of which reacts with a portion of the cleaved compound. Addition of an acid or base, thereby increasing the concentration of hydrogen or hydroxyl ions, generally accelerates the hydrolysis reaction. Many types of hydrolysis are important. Among them are (1) formation of an acid and an alcohol by hydrolysis of an ester, R-COO-R' + H20 + R-COOH + HOR', (2) conversion of starch and cellulose to glucose or sugar, and (3) hydrolytic cleavage of proteins to amino acids. Many synthetic and natural polymers like polyesters, polyamides, and cellulose have been selectively degraded by hydrolysis. 8.5.8.1. Cellulose. Amorphous parts of cellulose are selectively degraded by acid hydrolysis in water or ethanol at 80-100°C. Typical concentrations are 1-3 N HCl or H,S04.2930J21 The end groups of hydrolyzed cellulose are -CHO, due to cleavage of the (1-4)-p-glycosidic have studied hydrolyzed cellulose by X-ray difbonds. Baudisch et 135 D. H. R. Barton, H. P. Faro, E. P. Serebryakov, and N. F. Woolsey, J. Chem. Soc. p. 2438 (1965). 138 D. G. H. Ballard and J. V. Dawkins, Eur. Polym. J. 9, 21 1 (1973). 13Ba S.Wantabe, J. Hayashi, and T. Akahori,J. Polym. Sci., Chem. Educ. 12,1065 (1974). 13' J. Baudisch, J. Dechant, D. Nghi, B. Phillipp, and C. Ruscher, Faserforsch. Textiltech. 19, 62 (1968).

270

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fraction, deuterium exchange, formylation, water vapor absorption, weight loss, and molecular-weight determination. Their results suggest that amorphous cellulose is removed by hydrolysis. However, the results do not differentiate between the two proposed models for crystals of cellulose, One model represents the protofibril as a parallel assembly of several fully extended chain^.'^^-'^^ In the other the chains may have a folded conformation similar to that found in lamellar single crystals of synthetic polymers. M a n l e ~ has ' ~ ~proposed a compromised model in which the protofibril is thought to be composed of a single chain folded on itself with a periodicity of about eight glucose units. Chang et u1.lz1 studied hydrolyzed cellulose by GPC in order to differentiate between the different models proposed for crystals of cellulose.138-143If the chain molecules in cellulose are folded, one can expect discrete peaks in the gel permeation chromatogram of hydrolyzed cellulose, as observed in the chromatogram of selectively oxidized polyethylene crystals. The hydrolysis method of Chang et u1.121 and the GPC study of degraded cellulose is described below: Sample Residue afrer Hydrolysis. A sample of cotton linters cellulose was hydrolyzed for various periods of time with 1 N hydrochloric acid at 80°C. After hydrolysis, the residues were washed to neutrality with distilled water, dried in vacuo at S O T , and nitrated with a mixture of fuming nitric acid and phosphorus pentoxide following a procedure described by Alexander and Miche11144slightly modified as follows: After an appropriate time the reaction was terminated by adding an excess of iced water to the reaction mixture which was then stirred until colorless. Finally, the sample was separated by filtration and washed to neutrality. Acid-Soluble Hydrolysis Products. During hydrolysis, low molecular weight fragments are dissolved by the hydrolyzing acid. To obtain an adequate quantity of this material for study, a high rate of degradation was required, and this necessitated fairly drastic conditions of hydrolysis. In addition, it was necessary to minimize the contact time between the acid and the dissolved hydrolysis products. In order to realize these objectives, the flow hydrolysis apparatus described by Millett et a/.145was used with appropriate modifications to facilitate rapid collection of the hydrolyzate (see Fig. 11). The sample is contained in a sintered glass filter funnel of coarse porosity. Constant-boiling hydrochloric acid (6.5 N) is distilled into the column and maintains the sample at a constant temperature of 108'C. The condensate runs back from the condenser along a glass rod and onto the sample so that splashing is prevented. The acid percolates through the sample, carrying the dissolved hydrolysis products with it, and is

A. Frey-Wyssling and K. Muhlethaler, Makromol. Chem. 62, 25 (1963). Muggli, Cellul. Chem. Technol. 2, 549 (1968). 140 R. Muggli, H. G. Elias, and K. Muhlethaler, Makromol. Chem. 121, 290 (1969). 141 H. Dolmetsch, and H. Dolemetsch, Kolloid-Z. 185, 106 (1962). l4* M. Chang, J . Polym. Sci., Part C 38, 343 (1971). R. St. John Manley, J . Polym. Sci., Purr A-2 9, 1025 (1971). W. J. Alexander and R. L. Mitchell, Anal. Chem. 21, 1497 (1949). l4& M. A. Millett, W. E. Moore, and J. F. Sueman, Ind. Eng. Chem. 46, 1493 (1954). lsB

'WJ R.

8.5.

THE SURFACE DEGRADATION TECHNIQUES

27 1

FIG. 11. Schematic drawing of the apparatus used for the acid hydrolysis of cellulose. The apparatus facilitates the collection of the soluble hydrolysis products with minimum homogeneous degradation. (A) Flask containing constant boiling hydrochloric acid, (B) water-cooled condenser, (C) coarse sintered-glass filter funnel containing the cellulose sample, (D) dry ice-acetone bath (after Chang et directly discharged into a flask maintained at about -70°C by a dry ice-acetone bath. In this way homogeneous hydrolysis of the dissolved hydrolysis products is effectively minimized, if not altogether prevented. It is essential to prevent the accumulation of acid above the sample in the filter funnel. This can be accomplished by appropriate adjustment of the distillation rate and by application of very gentle suction at the outset of the experiment. By using a constant reflux rate of about 6 ml/min, aliquots of the hydrolyzate were collected and reduced to dryness by low-temperature vacuum evaporation (a freezedrying apparatus was used for this purpose), giving a thin clear film on the walls of the containing flask. The samples were then nitrated by the method mentioned above and stored in a desiccator until required. Whole Degraded Sumples. Whole degraded samples, comprising the product soluble in the hydrochloric acid plus the residue, were prepared with the aid of the flow hydrolysis apparatus mentioned above. A weighed sample of cellulose was placed in the apparatus and, by using the constant-boiling (6.5 N) hydrochloric acid at about 1Os"C, the degradation was allowed to proceed for a period long enough to produce weight losses in the range 6-70%. The hydrolysis products dissolved in the acid were recovered by the method described earlier. The residual hydrocellulose was also recovered

8.

272

CHEMICAL METHODS I N POLYMER PHYSICS

from the apparatus and rinsed with distilled water until the washings were neutral. The residues were then vacuum-dried at 50°C and weighed in order to determine the weight loss of the original cellulose. Finally, the residue was mixed with the dissolved hydrolysis products to give a whole degraded sample, which was nitrated as described earlier and stored until ready for use.

8.5.8.1.1. GEL PERMEATION CHROMATOGRAPHY. Molecular weight distributions of degraded cellulose were determined by GPC. Tetrahydrofuran was used as a solvent. High-porosity columns, mainly over lo3 8, porosity were used for the residue after hydrolysis, low-porosity columns, mainly below 2 x lo3 8, porosity, were used for acid-soluble hydrolysis products, while both high- and low- porosity, from lo5 to 250 A columns were used for whole degraded sample. The purpose of using sets of different porosity columns for different products was to obtain better resolution of the degraded products. The most dramatic feature of the chromatograms, as shown in Fig. 12, is that the distribution curves do not show the development of any specific peak corresponding to a progressive accumulation of any particular molecular-weight species as a re~

0

3

!0

,

1

1

7

z w

5

LL

&

0 X

I

1

,

1

1

1

,

r

--- 265%% WWTT LOSS LOSS -.-.40% WT. LOSS ----- 70% WT LOSS

I

8.5.

THE SURFACE DEGRADATION TECHNIQUES

273

sult of hydrolysis. The results show that the total number of crystallites decreases during hydrolysis while the chain length distribution remains constant. These results are in sharp contrast to the peak pattern observed in chain-folded crystals. The results suggest that cellulose chains may not be folded. 8.5.8.2. Cellulose Triacetate. M a n l e ~ has ' ~ ~investigated the degradation of lamellar chain-folded single crystals of cellulose triacetate (CTA) in order to ascertain whether the gel permeation chromatogram of the degradation products exhibits multiple peaks consistent with chain folding. Lamellar single crystals were prepared by dissolving CTA in boiling nitromethane and subsequently cooling the solution to a desired crystallization temperat~re.'~'The crystals were centrifuged from solution, and washed with aliquots of nitromethane and then with ethanol. To compare the degradation results, completely amorphous CTA was prepared by dissolving it in a mixture of methylene chloride and methanol (90: 10, v/v), and precipitating the polymer by adding more methanol. The amorphous and lamellar crystals were degraded with a mixture of 10% by weight of sulfuric acid in methanol at a refluxing temperature of about 67°C for various periods of time. Sample weights before and after degradation revealed that there was some weight loss. The samples were then washed with methanol, suponified with 2% sodium methoxide in methanol, and then nitrated with a mixture of phosphorus pentoxide and fuming nitric The nitrated samples were finally dissolved in tetrahydrofuran. The chain length distributions were then determined by GPC. For the amorphous material, the distribution profile of the undegraded sample showed a continuous shift to lower molecular weights. After extensive degradation, approximately 2 hr a new peak appeared at an elution volume correspondhg to a degree of polymerization (DP) of about 15. The area under this peak increased and the peak position shifted continuously to a lower DP of about 8. In the lamellar crystals after relatively short period of treatment time, 0.5-1 hr, the original single peak split into two overlapping maxima with peak position at elution volumes corresponding to DP of 30 and 60. As the treatment time increased, the peak at a DP of 30 increased in magnitude at the expense of the peak at a DP of 60. Simultaneously, the peak at a DP of 30 exhibited a slow downward shift during prolonged treatment. Finally, the sample exhibited the same low DP (8-15) peak observed in the amorphous material. 146

'41 14*

R. St. J. Manley, J . Polym. Sci., Polym. Phys. Ed. 11, 2303 (1973). R. St. J. Manley, J . Polym. Sci., Purr A - I . 1875 (1963). W. J. Alexander and R. L. Mitchell, Anul. Chem. 21, 1497 (1949).

274

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CHEMICAL METHODS IN POLYMER PHYSICS

From the results, M a n l e ~ has l ~ ~concluded that CTA chains are folded in the lamellar crystals and the molecules contain some kind of periodic structural weakness repeating approximately every 15 glucose units and susceptible to selective degradation by methanolysis. 8.5.8.3.Polyesters. Three methods have been developed for selective degradation of (1) hydrolysis catalyzed by an acid or a base at moderate temperatures ,150-152 (2) hydrolysis at elevated temperatures without catalysts,l= and (3) aminolysis by methylamine.154-158 HYDROLYSIS. A 0.1 -10% aqueous solution of 8.5.8.3.1. CATALYTIC hydrochloric acid, potassium hydroxide, or sodium hydroxide is used for the selective hydrolysis of polyesters. The reaction is usually carried out between room temperature and 70°C. Polymer film, fiber, or solutiongrown crystals are immersed in the solution and the solution is stirred continuously. The solvent used for crystallization of polyester is exchanged with water before hydrolysis. After the desired treatment time, the samples are washed thoroughly with water to remove the catalysts and are dried under vacuum. The degraded samples are characterized as usual by weight loss, X ray, DSC, IR, density, and GPC. Ravens150 hydrolyzed the amorphous regions of poly(ethy1ene terephthalate) (PET) fibers by 0.5 N hydrochloric solution at 70°C. Amorphous surfaces of solution-grown crystals of PET and some copolyesters were hydrolyzed by Hachiboshi et a1.151using 1-3% solution of potasdegraded the disordered sium hydroxide at 50°C. Fischer et regions of single crystals of a copolymer of D( -)-lactide and DL-lactide CH, 0 H-0[-C-C-0-],,H I II

I

H

by sodium hydroxide at 20°C. described below:

The method of Fischer et

~ 1 . l is~ ~

Single crystals of the copolymer were grown in xylene. Before hydrolysis, xylene was exchanged by methanol. After the exchange, water was added to obtain 1 :2 waterWegner, Angew. Makromol. Chem. 58/59, 37 (1977). D. A. S. Ravens, Polymer 1, 375 (1960). ls1 M. Hachiboshi, T. Fukuda, and S. Kobayashi, J . Macromol. Chem., Phys. 3, 525 (1969). la E. W. Fischer, H. J. Sterzel, and G. Wegner, Kolloid-Z. Z . Polym. 251, 980 (1973). lP A. Miyagi and B. Wunderlich, J . Polym. Sci., Polym. Phys. Ed. 10, 2073 (1972). lM G . Farrow, D. A. S. Ravens, and I. M. Ward, Polymer 3, 17 (1962). mS J. L. Koenig and M. J. Hannon, J . Macromd. Sci., Phys. 1, 119 (1967). Ise J. R. Overton and S. K. Haynes, J . Polym. Sci., Part C 43, 9 (1973). Is' K. K. Mocherla and J. P. Bell, J . Polym. Sci., Polym. Phys. Ed. 11, 1779 (1973). le D. T. Duong and J. P. Bell, J . Polym. Sci., Polym. Phys. Ed. 13, 765 (1975). 14B G . lao

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THE SURFACE DEGRADATION TECHNIQUES

275

methanol mixture. The mixture was slightly stirred and held at 20°C. For hydrolysis a 1 :2 water-methanol mixture, containing 0.02 to 0.04 moles per liter of sodium hydroxide was added at the same temperature. During the hydrolysis, ester bonds are cleaved and lactic acid anions are obtained. The reaction was followed by measuring electrical conductivity of the suspension, because substitution of the hydroxide ions by the lactic acid anions, possessing a lower mobility, gives rise to a decrease in conductivity.

Electron microscopy, GPC, and small-angle X-ray scattering were applied for proving the selectivity of the hydrolysis. GPC traces of the degraded polyester showed peaks for single, double, and triple traverses. 8.5.8.3.2. HYDROLYSIS WITHOUT CATALYSTS. Miyagi and Wunderlich153used water for the hydrolysis of PET: PET was crystallized at 250°C and 140°C from the melt. After 48 hours at the crystallization temperatures, the samples were quenched in ice-water. The crystallized PET was crushed into particles less than 2 mm in diameter. About 2.5 g of the sample was placed in a glass tube with 0.5 ml of water. The glass tubes were 8 mm in diameter, 20 mm in length, and 1 mm in wall thickness. After evacuating until about 5% of the water was lost, the glass tubes were sealed and annealed around the seal to remove strain. Hydrolysis was carried out at temperatures of 180 and 130°C (9.9 and 2.7 atm water vapor pressure) for various times. The inner volume of the glass tube was about 10 ml, small enough that saturation with water vapor was maintained throughout the hydrolysis.

Hydrolysis was carried out at 130, 180, 190, and 210°C for different periods of time. The hydrolyzed samples were extracted three times in about 300 ml of ethanol at 74°C for 16 hours, sufficient to dissolve terephthalic acid and ethylene glycol, the low-molecular-weight byproducts. After extraction, the samples were dried in vucuo at 60°C for 24 hours. The hydrolyzed PET was analyzed by weight loss, density, viscosity, molecular weight, end group determination, thermal analysis, low- and wide-angle X-ray analysis, and electron microscopy. On hydrolysis of a 67% crystalline polymer at 180°C for about 300 minutes, almost fully crystalline extended chain oligomers can be obtained with about 65% yield. This technique gives very reproducible results and the experimental conditions are controllable. However, if the technique is applied to solution-grown single crystals of PET, there may be some annealing effects at higher hydrolysis temperatures. 8.5.8.3.3. AMINOLYSIS. Aminolysis of polyesters takes place according to the following chemical reaction: R’-CO-0-R”

+ H,N-CH3

-

R’-CO-NH-CHI

+ HO-R”

The technique was first applied by Farrow et u1.,IM who degraded PET in 20% aqueous methylamine at RT. They concluded that although there is a rapid initial degradation of the amorphous regions, extensive degrada-

276

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CHEMICAL METHODS IN POLYMER PHYSICS

tion leads to attack of both crystalline and amorphous regions. Overton and H a y n e ~ reacted '~~ PET with 40% aqueous methylamine at 30°C and the reaction products were examined by GPC. 40% aqueous methylamine was apparently selective enough to separate amorphous and crystalline regions.lJB Duong and Bell158degraded PET by 20 and 40% methylamine. They have concluded that the 40% aqueous methylamine is effective in separating the two morphological phases in semicrystalline PET films. The procedure they used is described below: Degradation experiments were carried out at room temperature, ca. 18°C with mild agitation for various lengths of time up to 24 hr. About 3.0 g of the PET film was first broken into small particles of approximately 4 mm diameter before being placed in a sealed bottle containing 200 ml of 40% aqueous methylamine. Weight losses of the samples were recorded during the course of degradation. Before weighing, the PET samples were filtered, carefully washed with distilled water, and vacuum dried for 3 hr at 30°C. It was observed that after 12 hr of contact with methylamine the PET films disintegrated into very fine particles.

The GPC results showed that the amorphous and the chain-folded regions were completely removed by 40% aqueous methylamine after a period of 24 hours, while the 20% solution under the identical conditions showed a broad distribution of molecular weights indicating little selective degradation. The results by weight loss, crystallinity, X ray, IR, and GPC by Duong and B e l P and by other^'^^-'^^ show that 40% methylamine is selective enough to degrade amorphous regions of polyesters. 8.5.8.4. Polyamides. Polyamides can be hydrolized either by an aqueous acid or base solution. Battista150degraded nylon 6 and nylon 6,6 by 1.5 N aqueous HCl solution at 72°C. The melt-crystallized polyamide fragments so hydrolyzed had dimensions of 30-80 A, the order of magnitude of the folded-chain lamellae. The molecular weight was 3000-4000, indicating the etching was only sufficient to cut all tie molecules and separate the crystals. Many folds were still intact. Also, the hydrolyzed and water soluble portion was of reasonably high molecular weight, -1500, which shows that for complete degradation a poorer solvent must be used. Koenig and AgboatwalalB0used a dilute solution of sodium hydroxide, 2 g/liter at 98°C to hydrolyze single-crystal mats of nylon 6,6. The degraded crystals were thoroughly washed with water, then acetone, and were dried in a vacuum oven. A similar degradation of nylon 6,6 filaments has been carried out by Rochas and Pierret.161 Hydrolysis of 0. A. Battista, U.S.Patent 3,299,011 (1967). J . L. Koenig and M. C. Agboatwala, J . Macromol. Sci., Phys. 2, 391 (1%8). P. Rochas and S. Pierret, Bull. Inst. Texr. Fr. 93, 81 (1961); Chem. Absrr. 25, 1811a (1961). ls0

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277

single crystals of nylon 6,6 resulted in a decrease in molecular weight of the polymer. The hydrolysis left the crystalline region unaffected, as exhibited by the insignificant change in intensity of the 936 cm-l IR band. The 1329 and 1224 cm-l bands,160due to fold-regions, decreased considerably in intensity on hydrolysis. The results suggest that sodium hydroxide attack is concentrated primarily on the surface of the crystals. However, a detailed study using GPC is required to confirm the selectivity of hydrolysis with HCl and NaOH. 8.5.9. Degradation by a Mixture of Potassium Permanganate and Sulfuric Acid

Krsova161a reported that linear polyethylene can also be etched by a freshly prepared mixture of potassium permanganate and concentrated sulfuric acid in the ratio of 1 : 1. The. samples *ere etched for 60 minutes at room temperature. After the etching, the specimens were first washed with several gradually more diluted sulfuric acid solutions ( 1 : 1, 1 : 3, 1 : 7) and finally with distilled water. The electron microscopic studies show that polyethylene is etched; however, it is not clear from the author's results that the mixture selectively removes the amorphous surfaces only. 8.5.10. Hydratinolysis of Polyamides

Hughes and Belllslb degraded films and single crystal mats of nylon 6,6 by refluxing them in 95% hydrazine. Hydrazine undertakes a nucleophilic attack at the carbonyl of the amide linkage of the nylon yielding a hydrazide and a primary amine residue, R.CO NH*R' + NH2.NH, +RCONH.NH2 + R'NHz.

Weight loss, density, GPC, and microscopy results suggest that the disorder and noncrystalline regions of the material are removed upon the treatment.

lala A. Krsova, Plusfy u Kuucuk 14, 41 (1977); for the English translation, see Intern. Polym. Sci. and Tech. 4(6), TI34 (1977). lalb A. J. Huges and J. P. Bell, J . Polym. Sci., Polym. Phys. Ed., 16, 201 (1978).

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CHEMICAL METHODS IN POLYMER PHYSICS

8.6. Irradiation and Selective Degradation 8.6.1. Radiation-Induced Chemical Changes

When a polymer is subjected to ionizing radiation, positive ions, free electrons, and excited molecules are formed in the polymer. 162-168 Reaction of these primary products leads to formation of a second generation of active intermediates like positive ions, negative ions, and free radicals. These active intermediates have been detected in polymers irradiated at very low temperatures.168 The ions have very short life, while radicals are relatively stable at low temperatures. Reaction of primary and secondary intermediates introduces some permanent changes like crosslinks, degradation, and unsaturation162-168(trans vinylene double bonds, -CH=CH-, are formed while vinyl and vinylidene double bonds, -CH=CH2 and -C=CH,, decay during irradiation). The physical and chemical properties of polymers are affected by these changes. Crosslinking converts a soluble polymer into an insoluble and infusible network, which retains its shape beyond its crystalline melting point. Crosslinking in soft and flexible polymers gives an increase in elastic modulus, a marked increase in hardness, and a reduction in ultimate elongation.169*170Long term time-dependent properties like creep and stress cracking are dramatically improved by crosslinking.169,170Degradation produces the opposite effects in polymers. Crosslinking and degradation are simultaneous and competitive phenomena. The ratio of the two determines the net effect. Many polymers used for electrical insulation and heat-shrinkable packaging are crosslinked by radiation.171 Methods of measuring crosslinks and scission have been described by Charlesby and others. 162-167 A. Charlesby, “Atomic Radiation and Polymers.” Pergamon, Oxford, 1960. A. Chapiro, “Radiation Chemistry of Polymeric Systems.” Wiley (Interscience), New York, 1962. lM M. Dole, ed., “The Radiation Chemistry of Macromolecules,” Vols. I and 11. Academic Press, New York, 1972. Ie5 F. A. Makhlis, “Radiation Physics and Chemistry of Polymers.” Wiley, New York, 1975. J. E. Wilson, “Radiation Chemistry of Monomers, Polymers, and Plastics.” Dekker, New York, 1974. lE7 M. Dole, Adv. Radiat. Chem. 4, (1974). D. T. Turner, J. Polym. Sci., Purr D 5 , 229 (1971). IeB B. J. Lyons and F. E. Weir, in “Radiation Chemistry of Macromolecules” (M. Dole, ed.), Vol. 2, p. 282. Academic Press, New York, 1972. 170 H. Jenkins and A. Keller, J. Macromol. Sci., Phys. 11, 301 (1975). l7I 1st International Meeting on Radiation Processing, Dorado Beach, Puerto Rico, May 10-13. 1976.

8.6.

279

IRRADIATION A N D SELECTIVE DEGRADATION

0 0

250 mrad 400rnrad Unirradiated

00 3 0

io 6 I0

!O

Degradation Time (Min)

FIG.13. Percentage soluble fraction and percentage of material in form of single traverses ( P I )as a function of degradation time at 60°C (after Patel and Keller128).

If polymers are irradiated at low temperatures, radicals can be detected. Alkyl radicals, -eH,-CH-CH2-CH,--, are produced when polyethylene is irradiated at liquid nitrogen temperature '(77 K). Upon thermal annealing, alkyl radicals migrate to the surfaces of crystals to form crosslinks or migrate up to a double bond to produce ally1 radicals, -CH,-CH=CH-eH-CH,-. Allyl radicals are stable at room temperature. Allyl radicals and double bonds are very susceptible to oxidation and cause post-irradiation degradation. Allyl radicals can be decayed by annealing polyethylene above 100°C. However, the production of double bonds remains an undesirable change. Concentration of radiand that of doucals is measured by electron spin resonancet (ESR)1s7*1as ble bonds is measured by IR and UV ~ p e c t r o ~ c o p y . ' ~ ~ - ' ~ ~

t See also Chapter 5.3 and Part 14 (this volume, Parts A and C, respectively). M. Dole, D. C. Miller, and T. F. Williams J . A m . Chem. SOC.80, 1580 (1958). E. J. Lawton, J. S. Balwit, and R. S. Powell, J. Polym. Sci. 32, 257 (1958). 174 W. C. Sears, J. Polym. Sci., Part A 2, 2455 (1964). 175 M. B. Fallgatter and M. Dole, J. Phys. Chem. 68, 1988 (1964). 176 A. Charlesby, A. R. Gould, and K. J. Ledbury, Proc. R. SOC:London, Ser. A 277,348 (1964). D. M. Bodily and M. Dole, J. Chem. Phys. 45, 1433 (1966). 178 M. Budzol and M. Dole, J. Phys. Chem. 75, 1671 (1971). 179 N . A. Slovokhotova, A. T. Koritskii, V. A. Kargin, N. Ya. Buben, V. V. Bibikov, Z. F. Ilicheva and G. V. Rudnaya, Polym. Sci. USSR (Engl. Trans.) 4, 1244 (1963). 180 N. A. Slovokhotova, A. T. Koritskii, V. A. Kargin, N. Ya. Buben, and Z. F. Ilicheva, Polym. Sci. USSR (Engl. Trans/.)4, 1251 (1963). 172

ITJ

280

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CHEMICAL METHODS I N POLYMER PHYSICS

When the effects of ionizing radiation on polymers were discovered, it was assumed that all radiation-induced chemical changes occurred randomly in the polymer matrix.162Ja2 Measurements on concen, ~ ~since ~ ESR spectra of selectribution of radicals in poly p r ~ p y l e n ebut, of different crystallinity started casting doubt about the randomness of the changes. The results showed that concentration of radical^,^^^-^^^ crossl i n k ~ , ~ and ~ ~scission1w-200 ~ ~ ~ * - ~increased ~ ~ with increase in amorphous content in polymers. The selective oxidation technique has been used to determine distribution of crosslinks, scission, and trans vinylene double bonds in polyethylene.12s-130~201~202 The technique has also been used to determine the distribution of radicals in polypropylene,2mbut, since ESR spectra of selectively oxidized polypropylene are complicated due to -COOH and -NO, groups, it is not discussed here. 162~181

lel M. Dole, “Effect of Radiation on Colloid and High Polymeric Substances,” Report of Symp. IV. Tech. Command, U.S. Army Chem. Cent., Maryland, 1950. lS2A. Charlesby and S. H. Pinner, Proc. R . Soc. London, Ser. A 249, 367 (1959). le3 S. Nara, S. Shimada, H. Kashiwabara, and J. Sohma, J . Polym. Sci.,Part A-2 6, 1435 (1968). le4 N. Kusumoto, T . Tamamoto, and M. Takayangi, J. Polym. Sci., Part A-2 9, 1173 (1971). lS6 K. Tsuji and H. Nagata, Rep. Prog. Polym. Phys. Jpn. 15, 567 (1972). lee D. R. Johnson, W. Y.Wen, and M. Dole, J . Phys. Chem. 77, 2174 (1973). la’s.Schimada, M. Maeda, Y. Hari, and H. Kashiwabara, Polymer 18, 19 (1977). R. Salovey and A. Keller, Bell Syst. Tech. J . 40, 1397 and 1409 (1961). la8R. Salovey, J. Polym. Sci. 61, 163 and 463 (1962). loo T. Kawai, A. Keller, A. Charlesby, and M. G . Ormerod, Philos. May., [8] 10, 779 (1964). lel T. Kawai, A. Keller, A. Charlesby, and M. G. Ormerod, Philos. Mag., [8] 12, 657 (1965). lrn B. J. Lyons, Polym. Prepr., Am. Chem. Soc., Div. Polym. Chem. 8, 672 (1967). 183 B. J. Lyons, J. Polym. Sci., Part A 3, 777 (1965). A. M. Rijke and L. Mandelkern, Macromolecules 4, 594 (1971). le5 S. Rafi Ahrnad and A. Charlesby, Int. J . Radiar. Phys. Chem. 8, 497 (1976). l W M. Tutiya, J. Phys. SOC.Jpn. 25, 1518 (1968). lo’ R. P. Kusy and D. T. Turner, J . Polym. Sci., Part A-I 10, 1745 (1972). lea D. M. Pinkerton and K. R. L. Thompson, J . Polym. Sci.,Part A-2 10, 473 (1972). log T . Ishibashi, M. Takeuchi, and A. Odajima, Rep. Prog. Polym. Phys. Jpn. 15, 233 (1 972). M. Tukiya, Soc. Polym. Sci., Jpn. 6, 39 (1974). ‘01 R. Salovey, M. Y. Hellman, W. Matreyek, and F. H . Winslow, J . Polym. Sci., Part B 5, 1131 (1967). 2 M D. M. Bodily, J . Polym. Sci., Part A-2 10, 1709 (1972). zos N. Kusumoto, K. Matsumoto, and M. Takayangi, J . Polym. Sci., Part A-1 7, 1773 ( 1969).

8.6.

IRRADIATION AND SELECTIVE DEGRADATION

28 1

8.6.2. Location of the Chemical Changes

Two procedures have been adopted in order to determine the concentration of chemical changes in amorphous and crystalline . (1) irradiation of polymer crystals followed by selecregion~l26-130,201.202. tive oxidation of amorphous regions, and (2) selective oxidation of all amorphous material followed by irradiation. Both procedures are used to measure the concentration of chemical changes in crystalline regions. Hence, the concentration of a chemical change can be obtained by subtracting the concentration of the change in crystalline regions from that of whole crystals. Randomness of the chemical changes in crystalline regions can be studied by measuring their concentrations in crystalline regions of different thicknesses. Crystalline regions of different thicknesses can be obtained by crystallizing polymer at different temperatures followed by selective oxidation of amorphous regions. Bodilyzo2irradiated high-density polyethylene at 25°C with X rays from 1 MeV linear electron accelerator equipped with a gold target. Samples were sealed in glass tubes under vacuum. After irradiation, the tubes were placed in an oil bath at 90°C for 2 hr in order to decay all radicals before opening to the atmosphere. Irradiated samples were oxidized with fuming nitric acid for variable periods of time at 80°C. Oxidized samples were washed with water and extracted with boiling acetone in a Soxhlet extractor for over 4 hr. After extraction, the samples were dried in V L I C K O . Concentration of rrans vinylene double bonds was measured from the intensity of the IR band at 966 cm-l. Bodily202observed an initial rapid decrease in the absorbance of about 35% in a sample of 75% crystallinity followed by a slow decrease at longer oxidation times. The irridiation process introduces defects into the polymer chains and may alter the amount of polymer subjected to rapid oxidation. This may account for the observation that the fractional decrease in the vinylene groups is greater than that of the amorphous fraction of the sample. The vinylene group appears to be very nearly randomly distributed throughout both amorphous and crystalline regions. Bodily202also irradiated partially oxidized poiyethylene samples. He found no significant difference in concentration of the vinylene groups in oxidized and unoxidized samples; that is, they are randomly distributed in crystals . In order to determine distribution of crosslinks, scission, and double bonds, Pate1 and Keller128-130also adopted the same two procedures. They grew freeze-dried single crystals of different thicknesses by crystallizing linear polyethylene at different temperatures, namely, 70, 80, and

282

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90°C. The crystals grown at 70°C were further annealed at 132.2"C for 72 hours to produce crystals with fold periods greater than can be obtained by isothermal crystallization alone. The main purpose of growing crystals of different thicknesses was to study randomness of the chemical changes in crystalline regions. These samples were used for both the procedures and are described in Table 11. 8.6.2.1. Procedure 1. The freeze-dried crystals grown at 90°C were irradiated with y rays from a s°Co source at room temperature with doses of 250 and 400 Mrad. t Both the irradiated crystals and unirradiated controls were annealed at 60°C for 4 days (to decay the radicals from the irradiated samples) and subsequently degraded with ozone both at room temperature and at 60°C. Complete degradation to single traverses can only be achieved at 60°C. After degradation, the soluble fractions were extracted in a Soxhlet apparatus appropriately modified to allow a steady stream of oxygen-free nitrogen to be passed through it and to maintain the extraction zone at the boiling temperature of the liquid. The amount of insoluble residue was then determined by weighing and the molecular weight distribution of the soluble fraction was measured by GPC. The solvent used for extraction of the soluble fraction should be the same as that used in GPC so that the soluble fraction can be directly injected into the GPC column. High dose is preferred, so that a high concentration of the radiation-induced chemical changes can be introduced. With increasing degradation times, the solubility increased rapidly and, after an hour at 60°C practically all material became soluble (Fig. 13). If the crosslinks were randomly distributed in the crystals, one might expect insoluble gel even for the disconnected stems of the core material alone. The solubility data suggest that the crosslinks responsible for the original t Rad is defined as the quantity of ionizing radiation that results in absorption of 100 ergs of energy per gram of irradiated material. TABLE11. Characterization of Starting Materials Used for Irradiation" Crystallization temperature, "C (or treatment conditions)

Original fold period (by X ray), 8,

70

112 127

75 95

150

110

225

180

85 90

Crystals grown at 70" and annealed at 123-122" for 72 hr After Patel and Keller.'28

Thickness of the crystal core (by GPC), A

8.6.

IRRADIATION A N D SELECTIVE DEGRADATION

283

insolubility cannot be present in the lattice. However, the solubility data are not sufficient to determine concentration of crosslinks in crystalline region. Concentration of crosslinks in crystalline regions can be measured by GPC. Figure 14 shows the development of GPC traces at consecutive stages of degradation for a sample irradiated at 400 Mrad. Comla1

r

22 23 24 25 26

Elution Volume [Pulse1 GPC traces of single crystals grown at 90"C, irradiated with 400 Mrad, and degraded with ozone at 60°C for increasing lengths of time. (a) 15 min, (b) 25 min, (c) 1 hr, 50 min, (d) 220 hr, (e) crystal core material obtained by degrading crystals grown at 90°C to PI = 100% was irradiated with 400 Mrad and then degraded at 60°C for 200 hr. The peak with the asterisk is due to decane used for calibration. It should be noted that the low molecular-weight tail in (d) and (e) are almost equal in magnitude (after Patel and Keller '9. FIG. 14.

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CHEMICAL METHODS IN POLYMER PHYSICS

parison of Fig. 14 with GPC traces of unirradiated polyethylene, Fig. 9, shows that the pattern is essentially similar, i.e., single- and doubletraverse peaks appear in appropriately varying radios until only one peak, the single-traverse peak, is left. There are nevertheless certain differences between irradiated and unirradiated samples. In the late stage of degradation of irradiated sample, the single-traverse peak has a lowmolecular-weight tail. The appearrance of a low-molecular-weight tail raises the possibility that crosslinks might have been present initially in the crystalline regions but that the chains were cut at the tertiary carbon atoms. Patel and Keller12Q have shown that the low-molecular-weight tail is due to cutting the chains at trans vinylene double bonds in the crystalline cores. Ozone was found to penetrate into crystalline region and degrade the double bonds.128 At 60°C all double bonds were cut within about 40 min. About Q of the double bonds remained uncut by ozone even after prolonged degradation of irradiated samples at room temperature. Hence, for determination of the distribution of double bonds one has to degrade polyethylene at such a low temperature that ozone does not penetrate into the crystalline core. At low temperatures the reaction rate was slow and hence Patel and Keller129used the second procedure to determine the distribution. 8.6.2.2. Procedure 2. All preparations described in Table I1 were degraded by ozone at 60°C to the stage where all folds are cut and only -COOH capped single traverses remain. This was verified by GPC from which the chain length corresponding to the peak was determined. All fully degraded samples, i.e., crystalline cores, were irradiated at room temperature with 6oCoy rays while in vacuo. The doses ranged from 6 to 600 Mrad. The irradiated samples were examined for changes in molecular weight by GPC. The samples were also examined for trans vinylene double bonds by IR spectroscopy. All samples remained fully soluble on irradiation and hence were studied by GPC. A series of the traces is given in Fig. 15 for crystalline core of 95 A. As can be seen from the figure, a multiple-peak structure evolves at the higher molecular weight side of the original peak, with a second peak clearly and a third peak poorly resolved. It is remarkable that, even for the highest dose, the area of the trace beyond the original peak, i.e., the total amount of crosslinked material, represents only afraction of the main peak. The presence of this small fraction of crosslink material apparently suggest that there is a small fraction of crosslinks in crystalline cores. But comparison of this small fraction of crosslinked material in crystalline cores of different thicknesses, showed that for a given dose, the fraction of all material that has become crosslinked is es-

8.6.

IRRADIATION A N D SELECTIVE DEGRADATION

I

-c

-E a l

._

0 Mrad

LOO Mrad

mo Mrad

600 Mrad

$bm%hkk’ 2b 2; 2; 2k h o;

2;

285

i ;6

Elution Volume (Pulse)

FIG. 15. GPC traces of unirradiated and irradiated crystalline core of thickness 95 A. The crystalline core was obtained by complete degradation of amorphous surfaces of single crystals of polyethylene grown at 90°C by ozone. The peak with an asterisk is that of ndecane used as an internal standard (after Patel and Keller’**).

sentially constant. The result indicates that crosslinks are located at the ends of crystalline cores and not within them, i.e., there are no crosslinks in crystalline regions. As the primary energy deposition must be uniform throughout the crystal, excitations and subsequent formation of radicals is expected throughout the crystal. That such radicals do not lead to crosslinks within the lattice must be due to the fact that the carbon atoms within the lattice are too far apart, and their position is too rigidly fixed to allow a C-C bond to arise when two radicals in neighboring chains become adjacent. The length of a C-C bond is 1.54 A, while the closest distance between carbon atoms of adjacent chains, which is along (110) planes, is 4.1 A. Vibrations needed to bring such chains close enough to permit the formation of a C-C bond are likely to be too large for the lattice to sustain without involving melting. Had there been any degradation during irradiation of crystalline cores, it would have appeared as a low-molecular-weight tail of single-traverse peak in GPC. The single-traverse peak remained unaltered both in position and in width for all doses (see Fig. 15). In particular, there was no

286

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CHEMICAL METHODS IN POLYMER PHYSICS

0.10

v , 0 x

0

95% 758

lld

1

Hale grn”xl0‘

2

A is08

Dose (Mrod)

FIG. 16. Plot of ratio of absorbance at 966 cm-l to that of 1470 cm-I against dose for polyethylene crystalline cores of different thicknesses. The corresponding concentration of truns vinylene groups in rnoles/gram is given on the right-hand ordinate (after Patellso).

detectable addition of material at the low-molecular-weight end of the peak. This indicates that there is no chain scission in the crystalline regions. It could be argued that scission does take place, but, as the severed ends are rigidly held in the lattice, they will recombine (case effect). If this happened it is unobservable. Concentration of trans vinylene double bonds and trans, trans conjugated diene was measured from the absorbance of bands at 966 and at 988 cm-’, re~pectively.’~~ The band at 1470 cm-I was used as an internal standard. Figure 16 shows a plot of concentration of trans vinylene groups against dose for crystalline cores of different thicknesses. It is evident from the figure that the concentration of trans vinylene groups is independent of the thickness of the crystalline core; i.e., the double bonds are randomly distributed in the crystalline regions. Similarly, concentration of trans, trans conjugated diene was also independent of thickness of the crystalline core, i.e., the dienes are also randomly distributed in the crystalline regions. For a given dose, the concentration of trans vinylene and diene groups was slightly lower in undegraded chain-folded single crystals than in the crystalline regions. The independence of concentration of trans vinylene and trans, trans conjugated diene groups on the thickness of crystalline cores suggests that both groups are formed randomly throughout the crystalline regions. The lower concentration of both the groups in chain-folded crystals suggests that the concentration of both the groups is lower in the amorphous regions than that in crystalline ones.

9. THERMAL ANALYSIS OF POLYMERS?

By James Runt and Ian R. Harrison 9.1. Introduction Modern thermal analysis can trace its origin to the pioneering work of Le Chatelier in differential thermal analysis.’ However, extensive application of thermoanalytical techniques to polymers has taken place only over the past 15 years. The term “thermal analysis” applies to a family of techniques, all of which monitor primarily physical properties as a function of temperature or time at a fixed temperature. In this chapter we concentrate on two dynamic techniques that have been employed with great success with polymer systems: differential thermal analysis (DTA) and differential scanning calorimetry (DSC). An abundance of experimental information on polymers has become available through the use of DTA/DSC and the following sections are largely devoted to reviewing this work.

9.2. Instrumentation and Method Many thermal analysis techniques are available to the polymer scientist and recent publications have described the instrumentation and use of these systems in detail.23 This section briefly describes two of these: DTA and DSC. Both DTA and DSC are techniques that monitor enthalpy changes of a material as a function of temperature or time. The versatility of DTA/DSC stems from the fact that most chemical reactions and physical transformations result in an energy change. DTA involves measurement of the temperature difference (AT) between a sample and an inert reference material as the two are being t See also Chapters 6.1 and 6.2 H. Le Chatelier, Z . Phys. Chem. 1, 396 (1887). Blazek, “Thermal Analysis.” Van Nostrand-Reinhold, Princeton, New Jersey, 1973. P. E. Slade and L. T. Jenkins, “Techniques and Methods of Polymer Evaluation,” Vols, 1 and 2. Dekker, New York, 1966 and 1970.

* A.

287 METHODS OF EXPERIMENTAL PHYSICS, VOL. 16B

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

288

9.

THERMAL ANALYSIS OF POLYMERS

heated or cooled at a known (linear) rate. Experimentally, this can be accomplished by employing a heating block that contains two symmetrically located and identical chambers (Fig. l).4 The sample is placed in one chamber while an inert reference material of similar heat capacity is placed in the other. Thermocouples or other suitable temperaturesensing devices are then placed into the centers of the sample and reference materials. This system is then heated or cooled at a uniform rate. The resulting AT signal can then be recorded vs. time or sample temperature. Providing the sample undergoes reaction or transition within the temperature or time period monitored, the recorded trace shows either increases or decreases in AT over certain ranges. Whether a peak or trough is obtained depends on the recording system and whether the change in the sample is accompanied by absorption or liberation of heat. Prior to transition, AT should be effectively zero since the sample and reference are heated at identical rates (region A in Fig. 2). As the sample undergoes a transition, for example, melting, its temperature will remain B . Wunderlich, “Thermal Analysis of Linear Macromolecules,” p. D-1. Rensselaer Polytechnic Institute, Troy, N.Y., 1975.

PROGRAMMER FURNACE CONTROL THERMOCOUPLE

w OTA.FURNACE

FIG.I . Schematic diagram of a differential thermal analysis apparatus. R and S refer to the reference and sample chambers, respectively (from Wunderlich4).

9.2.

INSTRUMENTATION AND METHOD I

t

A -I-

I I

ATw dh

t

B

I

C

-I-

I

289

-

iAi

I I

constant until the process is completed. Meanwhile, the temperature of the reference continues to increase. Thus a difference between the sample and reference temperatures exists and a signal characteristic of this difference is monitored (region B). At the conclusion of the transition the sample temperature catches up with the reference and again AT = 0 (region C). DTA is at best a semiquantitative technique since the sample forms the major part of the thermal conduction path. The sample thermal conductivity varies in a manner that is generally unknown during the transition; hence the proportionality between AT and energy changes is also unknown . DSC, however, provides a more convenient technique with which to study heats of transition. In DSC, heat flow into or out of a sample and reference is measured as a function of time or temperature. A block diagram of a typical system is shown in Fig. 3.5 Two symmetrically located sample holders are fixed inside a metal block. However, unlike DTA, the temperature-sensing devices and heating units are built into the holders to allow individual heating of the sample and reference materials. In DSC the requirement is to maintain equivalence of sample and reference temE. S. Watson, M. J. O’Neill, J. Justin, and N . Brenner, Anal. Chem. 36, 1233 (1964).

290

9.

THERMAL ANALYSIS OF POLYMERS

DiFFEREmik TEMP CDN~ROL LM)p

I

AVERAGE TEMP CoNmoL Loop

SAMPLETEMPERATURE RECORDER (TEMPERATURE MARKING)

RECORDER (DIFFERENTIAL POWER)

REFERENCETEMPERATURE

REFERENCE TEMPERATURE

FIG.3. A schematic of a typical differential scanning calorimetry system (from Watson et a1.5).

peratures at all times. The energy difference between sample and reference heaters as the sample and reference experience a linear temperature or time change is measured. Prior to transition, the energy difference is effectively constant (region A in Fig. 2). As a sample undergoes transition, the sample heaters must supply additional energy to the material in order to keep the sample and reference temperatures the same. As differential energy is recorded as a function of temperature or time, one detects a deflection from the baseline (region B). At the conclusion of the transition, the energy supplied to the sample to maint.ain temperature equivalence drops back to its pretransition level (region C). In contrast to DTA, DSC is a quantitative technique in that the area under a curve can be directly related to the heat of transition. A minor practical disadvantage to DSC is that many commercial instruments only allow heating up to 1000 K. However, this is not normally a restriction in the usual polymeric systems. DSC and DTA also differ with respect to thermocouple placement. In DTA the sample and reference thermocouples sit directly in the sample and reference materials and directly record temperature differences. This results in transition temperatures being accurately recorded at most heating rates. In DSC the thermo-

9.3.

THEORY

29 1

couples record the temperatures of the metal holders in which the sample and reference sit. As a result the sample temperatures must be calibrated and the calibration is usually different for different heating rates.

9.3.Theory There have been a variety of theoretical treatments of DTA/DSC curves reported in the literature.6-16 These differ in complexity according to the simplifying assumptions made. Regardless of the assumptions employed, one feature common to all is that the area under an exotherm or endotherm can be related to the heat of reaction. A simplified treatment will be 0ut1ined.l~ It is assumed that a AT vs. time recording system is employed. This was originally derived for a well-stirred liquid reaction system, but has since found application to s01ids.I~ Consider two cells immersed in a bath at temperature TB (Fig. 4), one cell containing reactant solution (sample) at temperature Ts and the other containing reference solution at TR . The bath temperature is then increased at a uniform rate. The heat balance for the sample cell can then be written CsdTs

=

dH

+ Ks(TB - Ts) d t ,

(9.3.1)

where Cs and K , are the total heat capacity and the heat transfer coefficient of the sample cell, respectively, and dH the heat generated by the sample in time dt. Similarly, for the reference cell, CRdTR

=

KR(TB- TR) d t ,

(9.3.2)

where R refers to the reference cell. If identical cells are used, K s = KR = K . Assume that Cs = CR = C; that is, that the heat capacities of the reference and sample are the same. Subtracting (9.3.2) from S. Speil, L. H. Berkelhamer, J. A. Pask, and B. Davies, US.,Bur. Mines, Tech. Pap. 664, 81 (1945).

’ M. J. Vold, Anal. Chem. 21, 683 (1949). * J. L. Soule, J . Phys. Radium

13, 516 (1952). H. J. Borchardt and F. Daniels, 3. Am. Chem. Soc. 79, 41 (1957). lo R. L. Reed, L. Weber, and B. S. Gottfried, Ind. Eng. Chem., Fundam. 4, 38 (1965). l1 S. L. Boersma, J . Am. Ceram. SOC. 38, 281 (1955). l2 J. J . Kessis, C . R . Hebd. Seances Acad. Sci., Ser. C 270, 1 (1970). l3 J. J. Kessis, C . R . Hebd. Seances Acad. Sci. Ser. C 270, 1201(1970). l4 L. Reich and S. S. Stivala, “Elements of Polymer Degradation,” p. 116. McGraw-Hill, New York, 1971. l5 M. J. O’Neill, Anal. Chem. 36, 1238 (1964). l6 R. N. Goldberg and E. J . Prosen, Thermochim. Acta 6, 1 (1973).

9.

292

THERMAL ANALYSIS OF POLYMERS DTC

I

(I

If I

i

SAMPLE

REFERENCE

'61,

W

FIG.4. Simplified DTA apparatus refered to in theoretical analysis section (from reference 9). Ts , TR,and Ts are the sample, reference and bath temperatures, respectively. The differential thermocouple is denoted by DTC.

(9.3.1) yields d H = C d ( A T ) + K AT d t , where AT = Ts

-

(9.3.3)

TR. Upon integration of (9.3.3), AH = K

lom AT d t .

(9.3.4)

It is assumed that sample and reference are at the same temperature before and after reaction; therefore, AT is equal to zero initially and at infinite time. According to the assumptions previously made, the integral in (9.3.4) is zero before and after transition. Hence the integral in (9.3.4) represents the area ( A ) under the DTA peak. We can then write AH = K A ,

(9.3.5)

showing that the area under the peak is directly proportional to the heat of reaction. Treatments that yield a similar result can be derived for systems where the difference in energy supplied to individual sample and reference heaters is monitored vs. TR.I5J6The shapes of the DTA/DSC curves do not represent the actual course of the endothermic or exothermic reaction. There are built-in delays due to the heat capacities of the sample, reference, and thermocouples. Similarly there are delays due to thermal conductivities between the sample, thermocouple, sample

9.4.

FACTORS AFFECTING THE DTA/DSC CURVE

293

container, and heater. Theoretical derivations of true peak shapes have been reported by K e ~ s i s ~ and ~ J~thers.~JO ~

9.4. Basic Factors Affecting the DTA/DSC Curve The main quantities that are determined from a DTA/DSC curve are (1) the temperature of the beginning and end of the thermal event, (2) the temperature of the peak maximum, and (3) the amount of material involved in the transition or the heat of transformation. Great attention has been devoted to standardization of the method itself, and determination of those factors which might possibly affect the curve. It has been found that the results of DTA/DSC are affected by factors connected with the type of apparatus, physical and chemical nature of the samples, and also the technique employed. These factors can be summarized as follows. 9.4.1. Instrumental Factors 9.4.1.1. Heating Rate. DTA measurements have shown that a higher rate of heating can lead to a shift to higher temperature in the observed melting point (peak maximum) and an increase in the area under the curve." Strella and Erhardtl* have shown that, depending on the rate, differences as large as 15°C in polymer transition temperatures can be observed in DSC experiments (see Section 9.5.4). 9.4.1.2. Sample Holder Asembly (Block). Materials such as copper, silver, aluminum, and ceramic have been used for construction of sample holders. Since these materials have different thermal conductivities and heat capacities, they affect the shape of the resulting curve and the magnitude of the thermal events. The geometry of the holder also has a large effect on the magnitude of the peaks and the peak areas obtained. 9.4.1.3. Temperature Sensor. DTA curves are affected by the type and size of the thermocouples employed and thermocouple placement with respect to the block and the sample. In DSC, however, the temperature sensors are permanently embedded in the sample and reference holders. 9.4.1.4. Recording of the Thermal Event. The shape of the DTA curve will be different for different recording schemes: AT vs. sample temperature, reference temperature, or time. 9.4.2. Sample Factors 9.4.2.1. Particle Size. The effect of particle size is difficult to evaluate because other factors usually operate simultaneously. However, in genI'

W. W. Wendlandt, Chem. Anal. 19, 152 (1964).

S. Strella and P. F. Erhardt, J . Appl. Polym. Sci.

13, 1373 (1969).

294

9.

THERMAL ANALYSIS OF POLYMERS

eral, it can be said that the particle size and particle size distribution affect both peak shape and peak temperature. 9.4.2.2. Sample Packing. Sample packing affects heat transfer to the temperature sensor and in decompositions can affect the diffusion of gases in or out of the sample. Close packing can improve heat transfer to the temperature sensor. 9.4.2.3. Sample Size. Small samples yield smaller, but narrower, peaks and also reduce the probability of overlap between neighboring effects. However, the probability that the small sample is not representative of the whole is increased. 9.4.2.4. Atmospheric Control. Replacement of air by an inert gas or vacuum can suppress unwanted reactions that may interfere with the effect being monitored. If evolution or absorption of a gaseous component occurs, the peak temperature and curve shape can be affected by the gas pressure in the system. 9.4.2.5. Sample Dilution. Sample dilution is usually employed (1) to reduce baseline drift, (2) to avoid a differential signal that cannot be kept on the recorder scale, and (3) to try to match the heat transfer characteristics of the sample and reference. As long as a change in state rather than a decomposition is being examined, the effect of a diluent should only be the reduction of the peak height and area; the peak temperature should remain unaffected. Care must be taken, however, to ensure that the diluent does not react with the sample. 9.4.2.6. Sample Pretreatment. Sample pretreatment may be used to eliminate some material from a sample or to bring a specimen to a certain state. For example, one may wish to premelt or anneal a sample to remove solvent or to destroy its previous thermal history. 9.4.3. Reference Material

A reference material for DTA studies should be inert over the temperature range of interest. It is also desirable to use a reference with thermal characteristics and a particle size range close to that of the sample. No reference material need be used in DSC studies.

9.5. Melting Behavior of Polymers 9.5.1. Introduction

The melting of crystalline polymers is a thermodynamic phenomenon directly relatable to the melting of low-molecular-weight substances. For most crystalline polymers this fact is somewhat obscured by experimental

9.5. MELTING

295

BEHAVIOR OF POLYMERS

I

I

I

100

105

110

I

115 TEMPERATURE (“C)

I

I

I

120

125

130

FIG. 5. DSC melting curve for polyethylene single crystals (molecular weight 3 grown at 90°C. The heating rate was 10”C/min (from Mandelkern and Allowsz).

X

lo5)

observations such as broad melting ranges and seemingly nonequilibrium melting temperatures. However, evidence in several forms supports the notion that this transition in polymers is a true melting phenomenon.lg A typical DSC melting curve is shown in Fig. 5 . The melting “point” (T,) is usually defined to be the temperature at the peak maximum, but care must be taken since it is well known that the peak temperature is a function of sample size. If the peak is flat or irregularly shaped, a T, determination based on the construction of a geometrical median line has been recommended.20 Several other methods of defining T, have also been proposed .20 Reviews on polymer melting have been published by Zachmann,21 Wunderlich,22and other^.^^.^^ It has been seen that polymer melting can be complicated by a number of phenomena, the most important of which are superheating, and melting and reorganization during heating. Superheating is a process whereby a sample is heated faster than the meltcrystal boundary can progress through the sample. The unmelted inner portion may then temporarily superheat and melt at unusually high teml8 u,

22

24

L. Mandelkern, “Crystallization of Polymers,” p. 20. McGraw-Hill, New York, 1964. E. Pella and M. Nebuloni, J . Therm. Anal. 3, 229 (1971). H. G . Zachmann, Fortschr. Hochpo1ym.-Forsch. 3, 581 (1964). B. Wunderlich, Thermochim. Acta 4, 175 (1972). W. Wrasidlo, Fortschr. Hochpo1ym.-Forsch. 13, 55 (1974). B. Ke, Polym. Rev. 6, 347 (1964).

296

9 . THERMAL ANALYSIS OF POLYMERS

peratures. Upon heating, it is also possible for a polymer sample to partially or completely melt, to recrystallize in a more stable form, and to melt again. If either of the above processes occurs, the melting point may be shifted to higher temperatures. In such cases the observed tnelting point is not characteristic of the original sample but is dependent on the process taking place. The equilibrium melting point (Tm0)of a polymer is defined as the melting point of a group of large crystals, each in equilibrium with the polymer melt. A further provision is that the crystals at Tm0have the equilibrium degree of crystalline perfection consistent with the minimum free energy at T,0.z5 Polymers crystallized by the usual procedures tend to have melting points well below Tm0(in the absence of superheating or melting followed by recrystallization). This is due to the fact that these chain-folded crystals are small (thin platelets) and somewhat imperfect. Due to the recent development of high-pressure crystallization techniques, extended-chain crystals can be readily formed. These crystals are close enough to thermodynamic equilibrium that, on slow heating, a reasonably close approximation to Tm0 is obtained. Hoffman and Weeksz5have proposed a novel way of obtaining Tm0. A plot of the observed melting point vs. crystallization temperature (T,) is extrapolated until it intersects a line given by T, = T,. Only at the equilibrium melting point will T, = T, . Thus the value of T, at the point of intersection is Tm0. Several equations have been developed to account for the observed melting point (T,).2s,z7 For instance, the melting temperature of polymer single crystals can be related to the lamellar thickness by the following equationz5 T, = Tm0[1 - (2ue/1AH,)]

(9.5.1)

where AHf is the heat of fusion, 1 the lamellar thickness, and T,,, and uethe melting point and specific surface free energy, respectively. Any variable that changes lamellar thickness or surface energy will change the melting temperature of the polymer. Tm0and AH, are usually considered constants in the above equation, for a given homopolymer. Several factors have been shown to affect the melting of polymers. The following sections are devoted to the analysis of these.

J. D. Hoffman and J. J. Weeks, J. Res. Nail. Bur. Srand.. Sect. A 66, 13 (1962). L. Bohlin and J. Kubat, J . Polym. Sci, Polym. Phys. Ed. 14, 1169 (1976). " J. N. Hay, J. Polym. Sci., Polyrn. Chem. Ed. 14, 2845 (1976). *6

9.5. MELTING BEHAVIOR OF POLYMERS

9.5.2. Dependence of Melting Conditions

297

on Crystallization

It has been shown that the thickness of polymer lamellae is determined by the crystallization temperature. In the case of single crystals it was reported that the thickness increased roughly exponentially with increasing crystallization temperature.28 However, subsequent studies using different solvents have revealed that crystallization temperature alone is not a consistent variable but rather one should determine the degree of super~ o o l i n g . ~For ~ *crystallization ~~ from solution the degree of supercooling is defined as the difference between the dissolution temperature of an infinite extended-chain crystal and the crystallization temperature. The smaller the supercooling, the thicker the crystal and the higher the melting point. Lamellar thickness seems to be unaffected by the concentration of the solution from which the crystals were f ~ r m e d ~and l . ~by~ variations in molecular However, this is not the case for extremely lowmolecular-weight polymers (molecular weight less than 15,000) where both dissolution temperature and therefore supercooling are markedly changed. An important characteristic of lamellar thickness is that it is unaffected by the thickness of the crystal at previous stages of growth. Thus, if the supercooling is altered during growth, the lamellar thickness will change and correspond to that due to the new s u p e r ~ o o l i n g . ~ ~ The pronounced effect that crystallization conditions can have on melting behavior was illustrated by Wunderlich.22 Figure 6 shows a plot of the observed melting peak temperature vs. heating rate for chemically identical polyethylene samples crystallized under different conditions. Note that the melting points can differ by as much as 25°C depending on the heating rate. For example, at a heating rate of 25"C/min, the melting peak temperatures of the slowly cooled melt-crystallized sample (Fig. 6, curve 2) and the solution-grown crystals (Fig. 6, curve 4) differ by about 12°C.

28 29

30

A. Keller and A . O'Connor, Discuss. Faruday SOC. 25, 114 (1958). T. Kawai and A. Keller, Philos. Mag. [8] 11, 1165 (1965). A . Nakajima, S. Hayashi, T. Korenaga, and T. Sumida, Kolloid-Z. & Z . Polym. 222,

124 (1968). 31

32

D. D. A. A.

C. Bassett and A. Keller, Philos. M u g . [8] 6 , 344 (1961). C. Bassett and A. Keller, Philos. M a g . [8] 7 , 1553 (1962). Keller and A . O'Connor, Polymer 1, 163 (1960). Keller, Polymer 3, 393 (1962).

298

9.

THERMAL ANALYSIS OF POLYMERS

4 I

0

I

25 50 HEATING RATE (OCIMIN)

i

FIG.6. DTA melting peak temperatures of chemically identical polyethylenes of different morphology as a function of heating rate. Curve 1 refers to extended-chain crystals, curves 2 and 3 to slow- and fast-cooled melt-crystallized spherulites, respectively, and curve 4 to solution-grown single crystals (from WunderlichZ2).

9.5.3. Dependence of Melting on Molecular Weight and Molecular-Weight Distribution

Melting of crystalline polymers usually occurs over a rather broad temperature range. There are several factors that may influence the breadth of this temperature range. A comparison of a whole polymer and a molecular-weight fraction shows that the fraction has a much sharper melting range.35 Part of the effect of molecular weight may be due to the chain ends, which may or may not be able to fit into the crystal lattice. More recently, Harland et showed that the annealing temperature 36

se

R. Chiang and P. J. Flory, J . Am. Chem. SOC.83, 2857 (1961). W. G. Harland, M. M. Khadr, and R. H. Peters, Polymer 13, 13 (1972).

9.5.

MELTING BEHAVIOR OF POLYMERS

299

and the annealing temperature range, within which double melting peaks may be obtained on subsequent melting, is dependent on molecular weight and molecular-weight distribution. 9.5.4. Annealing

When a polymer is held at or above its crystallization temperature but below its melting point, structural rearrangements can occur. The outstanding feature of this process is an increase in lamellar thickness. Annealing of polyethylene single crystals has been studied extensively but there still exists uncertainty about the actual thickening mechanism involved. The main features of the annealing process may be summarized as follows: 1. The thickening process is i r r e ~ e r s i b l e . However, ~~ BurmesteI3* has shown that subsequent heating and cooling of solution-grown crystals of polyoxymethylene above 128°C produced reversible fold period changes. 2. Thickening takes place in a temperature range close to the melting point .39 3. On holding at constant temperature, the fold period increases linearly with the logarithm of annealing time but only after a large initial inc r e a ~ e . ~ ~ However, *~O for very high-molecular-weight polymers practically no change in long spacing occurs beyond the first sudden increase.41 4. Molecular orientation and alignment are generally retained, although reorientation by rotation around a specific crystallographic axis can take place at high temperature^.^^,^^,^^ 5 . The onset of thickening occurs at a lower temperature for crystals with a smaller initial fold p e r i ~ d . ~ ~ , ~ ~ 6 . Thickening occurs with increasing rate the lower the initial fold p e r i ~ dand ~ ~ molecular ,~~ weight .44 The rate also increases with increasing annealing t e m p e r a t ~ r eand , ~ ~is higher in liquids (potential solvent and nonsolvent) than in ai137,45at the same temperature. T. Kawai, Kolloid-Z. & Z . Polym. 201, 104 (1965). A . F. Burmester, Ph.D. Thesis, Case Western Reserve University, Cleveland, Ohio (1 970). 38 E. W. Fischer and G . F. Schmidt, Angew. Makromol. Chem. 74, 551 (1962). H. Nagai and N. Kajikawa, Polymer 9, 177 (1968). 4 1 Takayanagi and F. Nagatoshi, Mem. Fac. Eng., Kyushu Univ. 24, 33 (1965). 42 F. J . Balta Calleja, D. C. Bassett, and A. Keller, Polymer 4, 269 (1963). 43 D. C. Bassett and A. Keller, J . Polym. Sci. 40, 565 (1959). 44 A. Peterlin, Makromol. Chem. 74, 107 (1964). 45 N. Hirai, Y. Yarnashita, T. Mitsuhata, and Y. Tamura, Chem. High Polym. 18, 33 (1961). 37

38

300

9.

THERMAL ANALYSIS OF POLYMERS

7. Nakajimaet found that on annealing mats of polyethylene single crystals the lamellar thickness in the early stages of annealing decreases or stays constant if the annealing is performed at a supercooling greater than that experienced by the crystals during crystallization. To account for the observed features, various thickening mechanisms have been proposed. A sliding diffusion mechanism was postulated that involved molecular motion along the backbone within the ~ r y s t a l . ~ ~ , ~ ~ This could involve either the migration of point defects48 or collective jumps of entire segments of a m 0 1 e c u l e . ~Alternatively, ~~~~ Dreyfuss and Keller51have postulated that chains rearrange locally by pulling out a limited number of chain folds at a time. In contrast, others argue that partia137,30,52-55 or complete56melting followed by recrystallization to a longer fold period occurs on annealing. Bair et a / .57 have proposed that different annealing mechanisms are operative in different temperature ranges. Lamellar thickening without true melting was believed to occur at the lower end of the annealing temperature range, while melting followed by recrystallization occurs at higher temperatures. A similar explanation was also proposed by Takayanagi and NagatoshL41 Nakajima et proposed that isothermal annealing takes place in the initial stages by partial melting and recrystallization and later by “thickening nucleus formation”4s and chain translation. The main point here is that during a simple melting experiment the crystal is exposed to annealing conditions. These could well change the fold period of the original crystal and, hence, its apparent melting point. A variety of approaches have been suggested to avoid this problem, one of which is to observe the effect of heating rate on melting behavior. 9.5.5. Effect of Heating Rate on Polymer Melting

If annealing occurs upon heating a polymer sample to its melting point, fold period changes can occur. The amount of change will depend on the A. Nakajima, S. Hayashi, and H. Nishimura, Kolloid-Z. & 2. Polym. 229, 107 (1968). J. D. Hoffman, SPE Trans. 4, 315 (1964). 48 D. H. Reneker, J . Polym. Sci. 59, 539 (1962). 48 A. Peterlin, J . Polym. Sci., Part B 1, 279 (1963). A. Peterlin, Polymer 6, 25 (1965). 51 P. Dreyfuss and A. Keller, J . Polym. Sci., Part B 8, 253 (1970). 52 L. Mandelkern and A. L. Allou, Jr., J . Polym. Sci., Part B 4, 453 (1966). L. Mandelkern, A. S . Posner, A. F. Dioro, and D. E. RobertsJ. Appl. Phys. 32, 1509 ( 1961). J. B. Jackson, P. J. Flory, and R. Chiang, Trans. Faraday Soc. 59, 1906 (1963). 55 L. Mandelkern, R. K. Sharma, and J. B. Jackson, Macromolecules 2, 644 (1969). X I A. Keller, J . Macrornol. Sci. 8, 105 (1973). ” H. E. Bair, R. Salovey, and T. W. Huseby, Polymer 8, 9 (1967). 46

47

9.5.

30 I

MELTING BEHAVIOR OF POLYMERS

A

dt

B

TEMPERATURE

FIG.7. Melting behavior of a polymer sample at different heating rates. A, B, and C illustrate slow, fast, and intermediate heating rates, respectively.

time the sample spends in the region 0-40°C below its T, (annealing region). On slow heating, the sample will spend enough time in this region so that fold period changes can occur (Fig. 7A). The observed T,,, will correspond to the melting point of the annealed sample. However, on fast heating the sample will spend a relatively short time in this region. One would, therefore, expect little or no change in the fold period. In this case the observed melting point will be that of the original material (Fig. 7B). At intermediate rates two melting endotherms may be seen (Fig. 7C) where the low- and high-temperature peaks correspond to the melting of the original and annealed material, respectively. Note also that heating too quickly can result in superheating; the temperature of the solid sample may rise faster than it can melt. Many authors have detailed the effect of heating rate on a wide variety of polymer system^.^**^^^^-^^ For instance, Miyagi and WunderlichWdetermined the melting points of several poly(ethy1ene terephalate) samples JB 59

Bo

E. Hellmuth and B. Wunderlich, J . Appl. Phys. 36, 3039 (1965). N. Overbergh, H. Berghmans, and G . Smets, J . Polym. Sci., Part C 38, 237 (1972). K. Kamide and K. Yamaguchi, Makromol. Chem. 162, 205 (1972). I . R. Harrison, J . Polym. Sci., Polym. Phys. Ed. 11, 991 (1973). H . W. Holden,J. Po/ym. Sci., Part C 6, 53 (1964). Z. Pelzbauer and R. St. John Manley, J . Polym. Sci., Part A-2 8, 649 (1970). A . Miyagi and B. Wunderlich, J . Polym. Sci., Par? A-2 10, 1401 (1972).

302

9.

THERMAL ANALYSIS OF POLYMERS

at different heating rates (Fig. 8). The variation of the observed T, with heating rate for samples A, B, and C was explained on the basis of superheating, the behavior of D on the basis of reorganization. Superheating is revealed by an increase in the observed T, with increasing heating rates; reorganization by a decrease in T, with increasing rate.s4 Slow-heating experiments have been used to obtain high melting points, which are supposedly close to the equilibrium value needed for thermodynamic and kinetic calculations.60 These “equilibrium’’ melting experiments assume that if a material is given a long time to stabilize after being heated to a higher temperature, then the crystallites have a chance to reorganize or perhaps even partially melt and recrystallize to a more perfect form. Thus, an originally imperfect folded-chain crystalline polymer would anneal toward its most perfect state; the extended chain crystal.

tI

o

I

20

I

1

1

40 60 m HEATING RATE (“C/MlN)

I

100

1 0

FIG.8. DTA melting peak temperatures of poly(ethy1ene terephthalate) as a function of heating rate. (C) Sample crystallized at 250°C for 2 days; (A) sample C annealed at 260°C for 2 days; (D) sample C hydrolyzed at 210°C for I hour; (B) sample D annealed at 250°C for 1 day (from Miyagi and Wunderlichs4).

9.5.

MELTING BEHAVIOR OF POLYMERS

303

9.5.6. Multiple Melting Peaks

The phenomenon of multiple melting endotherms seems to exist with most polymers. For example, polyethylene gives rise to two endotherms after it has been isothermally crystallized or annealed, subsequently quench-cooled, and remelted. For a given polymer sample the phenomenon has a number of well-defined characteristics. First, the area of the peaks and the peak temperatures are dependent on the crystallization conditions. Second, prolonged annealing causes the high-temperature peak to increase in area and the peak may move to higher temperatures. Third, the low-temperature peak either decreases in area and moves to lower temperatures and eventually disappears or increases in area and joins the high-temperature peak. The direction of movement depends on the temperature of crystallization and any subsequent annealing. Various explanations have been proposed for the phenomenon of the multiple-peak endotherms. 9.5.6.1. Secondary Crystallization. Mandelkern et ~ 1 investigated . ~ ~ the fusion of linear polyethylene by differential scanning calorimetry. Their samples were isothermally crystallized at 130°C and then cooled to room temperature over 24 hours. Two endotherms were obtained on melting these samples. A minor endotherm at lower temperature was believed to be due to the melting of crystals formed during the cooling of the samples from 130°C to room temperature. Lemstra et ul.ss,s7observed three endotherms for isotactic polystyrene melt-crystallized at larger supercoolings (Fig. 9). The lower temperature peak was thought to originate from secondary crystallization of intercrystalline links that occurred on cooling. However, other^^^*^^^* have suggested that this lowtemperature peak is due to melting of “impurities,” i.e., sterically inhomogeneous species. Hybart and PlatP found two melting endotherms for drawn, annealed, and precipitated nylon 6,6. The lower temperature peak was likewise thought to be a result of secondary crystallization. 9.5.6.2. Different Mechanism of Nucleation. Holden investigated the effect of thermal conditioning on the melting behavior of polyethylene by differential thermal analysis.62 The multiple peaks observed in the melting curves of nonisothermally crystallized samples were assumed to be caused by the melting of crystals formed from heterogeneous and e5 L. Mandelkem, J. G . Fatou, R. Denison, and J. Justin, J . Polym. Sci., Purr B 3, 803 (1965). a~P. J. Lemstra, T. Kooistra, and G . Challa, J . Polym. Sci., Purr A-2 10, 823 (1972). 6’ P. J . Lemstra, A. J. Schouten, and G . Challa,J. Polym. Sci., Polym. Phys. Ed. 12, 1565 (1 974). 88 A. Lety and C. Noel, J . Chim. Phys. Phys.-Chim. Biol. 69, 874 (1972). e9 F. J. Hybart and J. D. Platt, J . Appl. Po/ym. Sci. 11, 1449 (1967).

3 04

9.

THERMAL ANALYSIS OF POLYMERS

0 TEMPERATURE

(OC)

FIG.9. DSC melting curve of isotactic polystyrene crystallized from the melt at 233°C. The heating rate was 8”C/min (from Lemstra el ~ l . ~ ~ ) .

homogeneous nucleation. Two endotherms were found by Cooper et al. 70 in slowly crystallized isotactic poly(propy1ene oxide). They suggested that these result from different modifications of a single, crystalline structure produced by different nucleation mechanisms during the crystallization process. 9.5.6.3. Crystallization in Two Different Lattice Structures. Four melting peaks were found by DSC studies of extruded isotactic polypropylene at low heating rates’l (Fig. 10). For small samples two exotherms were also observed in the melting region. It was postulated that the first endotherm was the melting of the p crystalline form. The p form subsequently recrystallized and melted again, giving rise to the second endotherm. The remaining two endotherms were then assigned to the melting of the CY crystalline modification. other^^^,^^ have also attributed the multiple-melting behavior of isotactic polypropylene to the existence of two different crystal forms. The multiple endotherms observed for W. Cooper, D. E. Eaves, and G . Vaughan, Polymer 8, 273 (1967). A. A. Duswalt and W. W. Cox, in “Polymer Characterization” (C. D. Craver, ed.), p. 147. Plenum, New York, 1971. 72 J. L. Kardos, A. W. Christiansen, and E. Baer, J . Polym. Sci.,Part A-2 4,447 (1966). 73 K. D. Pal, J . Polym. Sci., Part A-2 6, 657 (1968). 70

71

9.5. MELTING BEHAVIOR OF POLYMERS

305

polybutene- 1,74 natural and synthetic trans-polyisoprene~,~~ and oriented fibers of polypivalolactone7s were also thought t o be due to different polymorphic crystal forms. 9.5.6.4. Partial Melting Followed by Recrystallization Followed by Complete Melting. The melting and annealing of irradiated and unirra-

diated polyethylene single crystals was studied using differential scanning c a l o r i m e t r ~ . Samples ~~ were irradiated in hopes of suppressing lamellae reorganization during heating. Unirradiated crystals grown at 85°C from a dilute solution of the polymer in xylene had a pronounced multiplepeaked melting pattern that was attributed to a lamellar thickening process. A single endotherm resulted on irradiation of these crystals to 26 Mrad. Irradiation apparently suppresses the lamellar thickening process so that the crystals melt without reorganization during heating. It was suggested that the annealing process of the single crystals between 118 and 126°C was due to lamellar thickening without true melting, while annealing above 126°C was attributed to the recrystallization of melted lamellae. Differential calorimetric studies of the fusion of single crystals of linear polyethylene by Mandelkern and AllouS2indicated that a definite exothermic process was observed between two endothermic peaks and it was concluded that partial melting followed by recrystallization is the major process that is occurring (Fig. 11). Partial melting followed by B. H. Clampitt and R. H. Hughes, J . Polym. Sci., Part C 6, 43 (1964). Cooper and R. K. Smith, J . Polym. Sci., Part A 1, 159 (1963). '@ R. E. Prudhomme and R. H. Marchessault, Makrornol. Chem. 175, 2705 (1974).

74

" W.

I

I

120

130

I

I

140 150 TEMPERATURE ("C)

I

I

160

170

FIG. 10. DSC melting curve of isotactic polypropylene with 6-10% quinacridone nucleator at 5"C/min (from reference Duswalt and Cox'l).

9.

306

THERMAL ANALYSIS OF POLYMERS

I

1

1

I

1

I

95

loo

105

110

115

120

I

I

125

130

TEMPERATURE ("C)

FIG.1 1 . Melting curve of polyethylene single crystals grown at 90°C (molecular weight 2.8 x lW). The heating rate was 2.5"C/min (from Mandelkern and AllouS2).

reorganization has also been evoked to explain the multiple-melting meltbehavior of annealed or drawn poly(ethy1ene tere~hthalate),~'-~l crystallizeds2and single crystalse3of isotactic polystyrene, poly(N-methyl laurolactam),84 some p0lyamides,8~ poly(cis-l,4 butadiene),es meltcrystallized polycaprolactam heated slower than 20°C/min,87and polyoxymethylene single crystals heated below 10"C/min.88 9.5.6.5. Differences in Morphology and Degree of Crystalline Perfection. Linear polyethylene crystallized from the melt or heat-treated at a temperature above 110°C showed an additional melting endotherm at a lower temperature (10-20°C) than the major peak.s0 The appearance of the lower peak depended both on the crystallization temperature and the time and temperature of heat-treatment. If the sample contained two 77

P. J. Holdsworth and A. Turner-Jones, Polymer 12, 195 (1971).

R. C. Roberts, J . Polym. Sci., Part B 8, 381 (1970). M. Ikeda, Chem. High Polym. 25, 87 (1968). G. W. Miller, Thermochirn. Acra 8, 129 (1974). G. E. Sweet and J. P. Bell, J . Polym. Sci., Part A-2 10, 1273 (1972). J. Boon, G. Challa, and D. W. VanKrevelen, J . Polym. Sci., Part A-2 6 , 1791 (1968). P. J. Lemstra, A. J. Schouten, and G. Challa, J . Polym. Sci., Polym. Phys. Ed. 10,2301 (1972). S. W. Shalby, R. J. Fredericks, E. M. Pierce, and W. M. Wenner, J. Polym. Sci., Polyrn. Phys. Ed. 12, 223 (1974). N. Lanzetta, G. Maglio, C. Marchetta, and P. Palumbo, J. Polym. Sci., Polym. Chem. Ed. 11, 913 (1973). E. A. Collins and L. A. Chandler, Rubber Chem. Technol. 39, 193 (1966). 87 F. N. Liberti and B. Wunderlich, J . Polym. Sci., Part A-2 6, 833 (1968). M. Jaf€e and B. Wunderlich, Kolloid-Z. & Z . Polym. 216-217, 203 (1967). K. Hoashi and T. Machizuki, Makromol. Chem. 100, 78 (1967). 7@

9.5.

MELTING BEHAVIOR OF POLYMERS

3 07

endothermic peaks, then further heat-treatment at a temperature near that of the lower peak caused additional changes. The lower endotherm split into two peaks that are higher and lower in temperature than the original lower peak. This effect was interpreted by assuming that the crystalline regions that correspond to the initial lower peak maximum form more stable crystals of greater thickness and higher melting point. In addition, they partially melt and on cooling form smaller crystallites with a lower melting point. EdwardsQostudied the melting behavior of 1,4-cis-polyisoprene by differential scanning calorimetry and high-resolution transmission electron microscopy. Two melting endotherms were observed and assigned to two types of spherulitic lamellar crystals having the same crystal structure but different growth rates, lamellar thicknesses, and growth planes. Crystals differing in lamellar thickness were also used to explain the multiple-melting behavior of an annealed poly(ethy1ene-m-methacrylic acid) ionomerQ1and molecular-weight fractions of partially isotactic polypropylene The dual melting peaks found for certain poly(4methyl pentene-1) single annealed and drawn nylon 6,6,94and stress-crystallized high-density polyethyleneQ5were also explained on the basis of two different morphological species. Two melting endotherms were also observed for solution-grown single crystals of syndiotactic POlypropylene.m In this case the lower temperature peak was ascribed to the melting of crystals or parts of crystals with incorporated chain defects and the high-temperature endotherm to more regular crystal structures. Differences in degree of crystalline perfection were also evoked to explain the multiple-melting behavior of annealed poly(ethy1ene tere~hthalate),~' cold-drawn high-density p ~ l y e t h y l e n eisothermally ,~~ crystallized natural rubber,0Qand segmented polyurethane systems.loo~lol B. C. Edwards, J . Polym. Sci., Polym. Phys. Ed. 13, 1387 (1975). C. L. M a n and S. L. Cooper, Makromol. Chem. 168, 339 (1973). 82 C. Booth, C J. Devoy, D. V. Dodgsen, and I. H. Hillier,J. Polym. Sci., Purr A-2 8,519 (1 970). D. R. Morrow, Ci. C. Richardson, L. Kleinman, and A. E. Woodward, J . Polym. Sci., Part A-2 5, 493 (1967). J. P. Bell, P. E. Slade, and J. H. Dumbleton, J . Polym. Sci., Part A-2 6, 1773 (1968). O5 D. J. Blundell, F. N. Cogswell, P. J. Holdsworth, and F. M. Willmouth, Polymer 18, 204 (1977). 88 A. Marchetti and E. Martuscelli, J . Polym. Sci., Polym. Phys. Ed. 12, 1649 (1974). O7 R. C. Roberts, Polymer 10, 117 (1969). W. G. Harland, M. M. Khadr, and R. H. Peters, Polymer 15, 81 (1974). H. G. Kim and L. Mandelkem, J . Polym. Sci., Part A-2 10, 1125 (1972). loo S. L. Samuels and G. L. Wilkes, J . Polym. Sci., Polym. Phys. Ed. 11, 807 (1973). R. W. Seymour and S. L. Cooper, Macromolecules 6, 48 (1973).

308

9. THERMAL

ANALYSIS OF POLYMERS

9.5.6.6. Chain-Folded and Extended-Chain Crystallites. The phenomenon of double melting was studied in nylon 6,6 and isotactic polystyrene as a function of sample treatment by annealing or drawing.Io2 One endotherm was thought to be caused by the melting of folded-chain crystals, while the other was due to the melting of extended-chain structures obtained by conversion of folded-chain crystals on annealing. Pals et al. lo3 found that the two melting endotherms of stretched, oriented polypropylene films could be explained by assigning one endotherm to “stretched nuclei material”103and the other to less oriented crystallites. Similarly, it was postulated that the melting behavior of poly(ethy1ene oxide) fractions ,lO4 polyethylene crystallized from the melt under elevated pressure,1o5 and annealed isothermally crystallized poly(ethy1ene terephthalate)Ios among otherslo7J08arises from the melting of chainfolded and extended-chain structures. Kardos et al.109observed four endotherms for polyethylene crystallized under pressure. Two were found to be due to the melting of chain-folded and extended-chain structures, while the others were thought to arise from the onset of segmental motion in the crystals. 9.5.6.7. Molecular Weight Fractionation. The melting characteristics of high-density polyethylene were found to depend on molecular weight and its distribution, and on annealing and crystallization history.36 Double-melting endotherms were suggested to have arisen from molecular-weight fractionation that occurred during annealing. There is assumed to be a temperature-dependent preferential crystallization of lower or higher molecular-weight species.

None of the above phenomena can be the sole explanation for the multiple-peaked endotherms. More than one phenomenon can occur at the same time and undoubtedly differences exist between different polymer types. 9.5.7. Dried and Suspension Crystals

Sample sizes for DSC of single crystals of polyethylene are typically 2-6 mg, although smaller samples have been used. DTA samples are J. P. Bell and J . H. Dumbleton, J . Polym. Sci., Part A-2 7 , 1033 (1969). D. T. F. Pals, P. Van Der Zee, and J. H. M. Albers, J . Macrornol. Sci. B6,739 (1972). D. R. Beech, C. Booth, D. V. Dodgsen, R. D. Shape, and J. R. S. Waring, Polymer

lop

laS

IM

13, 73 (1973).

B. Wunderlich and T . Arawawa, J . Polym. Sci., Part A 2, 3697 (1964). D. L. Nealy, T. G . Davis, and C. J. Kibler, J . Polym. Sci.. Part A-2 8, 2141 (1970). lo’ K. D. Pae and J. A. Sauer, J . Appl. Polym. Sci. 12, 1901 (1968). N. Okui and T. Kawai, Makromol. Chem. 154, 161 (1972). looJ. L. Kardos, E. Baer, P. H. Geil, and J. L. Koenig, Kolloid-Z. & Z. Polym. 204, 1 lo5

lO8

(196.5).

9.5.

309

MELTING BEHAVIOR OF POLYMERS

usually 1-2 mg. Using these sample sizes it has been reported that for high-molecular-weight polyethylene ,molecular weight greater than lo5) simple fusion curves are obtained for all crystallization temperatures .l10 Further, the fusion process is characteristically a very broad one (Fig. 5 ) . However, HarrisonelJll has shown that the DTA curves obtained for polyethylene crystals are dependent on sample weight. Even highmolecular-weight polyethylene gives multiple-peaked melting curves when small sample weights are used, e.g., 0.07 vs. 2.70 mg (Fig. 12). When single-crystal samples have been melted and then recrystallized, a broad peak is obtained for all sample weights on remelting. If identical melt crystallization conditions are employed, the shape and position are essentially the same regardless of sample weight. This latter observation suggests that the sample weight differences shown may be due to the poor thermal conductivity of the powdered solid sample. This same effect is not observed in low-molecular-weight organic compounds. Presumably this arises from the fact that low-molecular-weight organic compounds melt to yield low-viscosity liquids, which easily flow and allow sample collapse to the hot container walls. In an effort to resolve this problem with polymers (which have a high melt viscosity), the air spaces within the original powdered sample were filled with silicone oil. This oil is a nonsolvent for the polymer and has a thermal conductivity of the same order as solid polyethylene. In all cases the oil tends to narrow the peaks of the melting curves61*111 (Fig. 13). At a ll1

L. Mandelkern, Prog. Polym. Sci. 2, 165 (1970). I. R. Harrison, J . Macromol. S c i . , Chem. 8, 43 (1974).

.07 0 x W

.66

4

c -4

2.70

f 0

n

z

W

SOLID I

120

1

130 TEMPERATURE (OC)

1

140

FIG.12. DTA melting curves of dried polyethylene (Marlex 6001) single crystals grown at 86°C. All runs were conducted at 2O0C/min and the sample weight given is in milligrams (from Harrisona1).

3 10

9. I

THERMAL ANALYSIS OF POLYMERS I

I

I

I

I SOLID IN OIL

0.06

SOLID 0.67

IN OIL

SOLID

2.02

IN OIL

I 120

I

I I 130 TEMPERATURE ("C)

I

I

140

FIG.13. DTA melting curves for three different weight ranges of dried (solid) and in-oil polyethylene single crystals grown at 86°C. The sample weight shown is in milligrams. All samples were heated at ZO"C/min (from Harrisons*).

sample weight of 2 mg, the in-oil sample shows two distinct peaks and a possible shoulder, none of which are resolvable in the solid sample.61 Presumably, oil keeps the powder particles in better thermal equilibrium with each other. This leads to a much narrower distribution of thermal states within the in-oil sample compared to that which exists in the solid sample. As a result the thermocouple in-oil senses the individual components of the melting process rather than presenting an average of the broad distribution of thermal states present in the solid sample.

9.5.

MELTING BEHAVIOR OF POLYMERS

31 I

An even more thermally homogeneous system is obtained by exchanging a suspension of the lamellae to oil without ever drying them down. Comparative melting curves for the three types of samples are shown in Fig. 14. A stylized drawing of the different samples is also shown (Fig. 15). The exchanged-to-oil sample has narrower peaks, i.e., approximately 15°C at half-peak height for the largest peak in the DTA trace. Endotherms this narrow have not previously been reported, eveR for low-molecular-weight polyethylene, as it is customarily run as a solid sample. The positions of the three peaks in the exchanged-to-oil samples are essentially independent of heating rate over the range 5-8O0C/min. In the light of previous work on annealing and recrystallization of polymers, it was suggested that the highest broad endotherm represents the melting behavior of single crystals that have already undergone some transformation, such as annealing or recrystallization. With regard to the two lower melting peaks, several alternatives present themselves. However, if initial assumptions regarding purity and isothermal crystallization are correct, the implication is that isolated lamellae of polyethylene can exhibit two independent melting transitions. This latter observation is not without precedence. Annealing studies and heat treatments on truncated polyethylene single crystals have demonstrated that material contained in the 100 fold sectors showed signs of having reorganized at lower temperatures than material in the 110 fold

I

1

I

120

I

130 TEMPERATURE (OC)

I

140

FIG.14. DTA melting curves for solid, in-oil, and exchanged to oil polyethylene single crystals all run at 20"C/min. Each sample contained approximately 0.07 mg of polymer (from Harrisons1).

312

9.

THERMAL ANALYSIS OF POLYMERS

SOLID

FIG.15. Idealized states for the three samples in Fig. 14. “Solid” sample contains particles made up of collapsed lamellae. Both particles and lamellae may be in poor contact with appropriate neighbors. In-oil sample shows that particle contact has been improved by presence of oil. Voids and poor lamellae contact may still exist within the particle. Exchanged-to-oil samples show isolated lamellae in a matrix as thermally conductive as they are themselves (from Harrisone1).

It has been suggested that a definite difference in transformation temperature of the two sectors should exist. Harrisons1 coupled thermal studies with electron microscopy and showed that when samples were heated to the first peak, the 100 fold sectors showed signs of having undergone some sort of transformation. This study was extended to single crystals grown at a variety of temperature^."^ It was shown that the peak area ratios as measured by DTA were approximately the same as the area ratios measured for 110/100 sectors by microscopy. Hence the impliczition is that the DTA endotherms seen represent the melting of 100 and 110 sectors, respectively. This work leads one to question the reliability of much of the previous work where relatively large amounts of dry crystals were used. D. C. Bassett, F. C. Frank, and A. Keller, Nature (London) 184, 810 (1959). A. Keller and D. C. Bassett, J . R. Micros-. SOC. 79, 243 (1960). I. R. Harrison and G. L. Stutzman, Anal. Calorim., Proc. Third S y m p . , 1974 Vol. 3, p. 579 (1974). 112

9.5.

MELTING BEHAVIOR OF POLYMERS

313

9.5.8. The "True" Melting Point

The annealing processes that occur during the heating of folded-chain lamellae crystallites present problems when the objective is to measure the true melting point. In other words, when thickening occurs there will be no simple correlation between the observed melting point and the properties of the original material. Thus it is desirable to find a heating path such that these reorganizational processes are minimized. Such a heating program has been defined as a zero entropy production path.21,115 This entails heating at a rate fast enough to prevent reorganization, but not so fast as to cause superheating. Using DTA and hot stage microscopy, Hellmuth and WunderlichJ8 studied the time dependence of the melting of dendrites and single crystals of polyethylene. It was reported that when the single crystals were heated from 0.5 to 15"C/min, the observed T, decreased with increasing heating rate (Fig. 16A). This was thought to be a result of thickening (stabilization) prior to melting. At rates higher than lS"C/min, no additional change in T, was detected. From this it was concluded that no reorganizational processes were taking place at these rates. This implies that above 15"C/min the melting process follows a zero entropy pro-

FIG.16. Melting points of polyethylene crystals as a function of heating rate observed under an optical interference microscope. (B) Dendrites; (A) single crystals. The bars represent the limits of a large amount of data collected at a particular rate (from Hellmuth and Wunderlich?

B. Wunderlich, Polymer 5, 611 (1964).

3 14

9.

THERMAL ANALYSIS OF POLYMERS

duction path. However, Nakajima er ~ 1 have . reported ~ ~ that the observed T, for polyethylene crystals decreased continuously with increasing heating rate up to 45"C/min. This suggests that reorganization can take place even at high rates for single crystals. For dendritic crystals only a small decrease in the measured T,,, was found over a wide range of heating rates (Fig. 16B). The dendrites were crystallized at a temperature below that of the single crystals and are thinner and less perfect. As a consequence, the dendrites should have a lower T, than the single crystals. In practice, the reverse was seen to be the case at all heating rates. It was proposed that this unexpected melting behavior was due to the fact that dendrites can reorganize even at very high heating rates (up to 3000"C/min). As noted previously, polyethylene single crystals, as grown, can exhibit multiple endotherms (Fig. 17A). This behavior is probably a consequence of reorganization prior to melting. As a result it is difficult to estimate the melting point of the original material. A method for determinining the true T, for single crystals has been proposed by Bair et a1.57*11S-118 It was postulated that irradiation of a polymer sample with

im

125 TEMPERATURE (OC)

l30

135

FIG. 17. Comparative melting behavior of polyethylene crystals grown at 85°C. (A) Original preparation; (B) sample A irradiated at 10 Mrad; (C) sample A irradiated at 26 Mrad (from Bair et a/.57).

H. E. Bair and R. Salovey, J . Macromol. Sci., Phys. 3, 3 (1969). H. E. Bair and R. Salovey, J . Polym. Sci.>Purr B 5 , 429 (1967). l" H. E.Bair, T.W.Huseby, and R. Salovey, Anal. Calorim., Proc. A m . Chem. Symp., 1968 Vol. 1 , g. 31 (1968). *IE

3 15

9.5. MELTING BEHAVIOR OF POLYMERS

high-energy electrons causes the formation of crosslinks predominantly in the Since fold surface crosslinks had been shown to retard thickening processes,122Bairet al. irradiated single crystals in hopes of inhibiting reorganization during heating. Figure 17 illustrates the effect of increasing radiation dosage on polyethylene crystals grown at 85°C. Note that the high-temperature endotherm (generally thought to be the melting of thickened crystals) has been suppressed. Heat of fusion measurements by Bair et ~ 1 . 5 ' have supported the contention that irradiation at low dose rates does not affect the thermodynamic properties of the crystals. However, Rijke and Mandelkerr~'~~ found that the heat of fusion of polyethylene crystals, irradiated at a number of low dose rates, first decreases by about 15% then increases to a constant value about 94% of original heat of fusion (Fig. 18). It was proposed that the initial drop could be explained by radiation-induced defects within the crystalline regions and the subsequent increase by melt entropy considerations.

t 40

:

I 0

I 20

I

I

I

40

60

80

0

R

FIG.18. Plot of the measured heat of fusion ( A H , ) vs. radiation dose ( R ) in megarads for linear polyethylene (M,= 122,000). The samples was crystallized from dilute xylene solution at 85°C and the irradiationwas conducted at room temperature (from Rijke and Mandelkern19

R. Salovey, J . Polym. Sci. 61, 463 (1962). R. Salovey and D. C. Bassett, J . Appl. Phys. 35, 3216 (1964). Iz1 G. N . Patel, L. D'Illario, A. Keller, and E. Martuscelli, Makromol. Chem. 175, 983 (1974). Iz2 T. Kawai and A. Keller, Philos. Mag. [8] 12, 287 (1965). Iz3 A. M. Rijke and L. Mandelkern, J . Polym. Sci., Purr B 7 , 651 (1969). Ii0

lZo

316

9.

THERMAL ANALYSIS OF POLYMERS

Thus, it appears that the significance of melting points obtain by irradiation techniques can be questioned. DTA/DSC experiments on suspensions of single crystals may provide an alternative method for measuring the true melting point. As discussed previously, Harrison and c o l l a b o r a t ~ r s ~ ~ have * ~ ~performed ~J~~ DTA melting experiments on polyethylene single crystals in silicone oil. Upon heating, three melting endotherms were obtained and the two at lower temperatures assigned to the melting of the 100 and 110 sectors of these crystals. It was found that the peak positions for these two transitions are essentially independent of heating rate in the range 5-8O0C/min. This implies that these endotherms are not influenced by reorganization processes or superheating and hence represent the true melting points of the original material. However, little DTA/DSC research to support this conclusion has been performed on other polymers in suspension. Thus added support for this conclusion is lacking.

9.6. Quantitative Methods From a determination of the area under a DTA/DSC peak associated with a thermal event such as decomposition or curing, one can calculate the heat involved in the reaction or transition. In the following section, general methods for finding heats of transformation are outlined. The determination of the heat of fusion to find weight fraction crystallinity serves as an illustration. However, it should be emphasized that these methods may be applied to other transitions and reactions. Of primary importance in the characterization of semicrystalline polymers is the determination of the weight fraction crystallinity (x,)at a particular temperature T , . This parameter influences a variety of polymer properties and hence its determination is fundamental in understanding and correlating polymer properties. One method of finding x1is based on heat of fusion measurements by DSC. Like most methods of crystallinity determination for polymers, a two-phase structural model is employed (amorphous-crystalline). This model necessitates that certain assumptions be made. These include124:

1. that the polymer consists of distinct amorphous and crystalline regions and that each may be assigned a particular heat capacity at any particular temperature; 2. that the polymer exists in a stress-free state; A. P. Gray, Thrrmochim. Acia 1, 563 (1970).

9.6.

QUANTITATIVE METHODS

317

3. that the heat capacity of the amorphous regions can be extrapolated from or is identical to that of the melt; 4. that the amorphous and crystalline heat capacities are additive.

(9.6.1) where AHfl is the measured heat of fusion of the polymer at temperature T1and AHtl is the heat of fusion of a 100% crystalline polymer determined at T1. Practically, A H f l is found as follows. First, the melting curve of a known weight (Y,) of a pure, well-characterized material such as indium (T, = 156.6”C; AHfr = 6.80 cal/g) is determined. The area under the melting endotherm (A,) can be determined and the heat of fusion (AH,) found in the literature. It can be seen that

Yr(AHfr)= KAr,

(9.6.2)

where K is an instrumental constant that relates energy to the area of recorded output. One can then solve for K. A polymer sample of known weight (Y,) is then scanned through its melting region. The area under the endotherm ( A , ) is again determined. By analogy to (9.6.2),

Y p ( W p ) = KAp,

(9.6.3)

where A H f p is the heat of fusion of the polymer. Since K is an instrumental calibration constant and Y, and A , are known, one can solve for A H f p . A problem that is frequently encountered with polymers is how one defines A,. In other words, in what manner should one connect the pre- and post-transition baselines? Consider the general equation relating measured heat capacity to thermal (9.6.4) where H is the measured heat capacity, T the temperature, and H, and Ha the heat capacities of the crystalline and amorphous material, respectively. In the melting region the last term in (9.6.4) becomes dominant, but, at least in the case of polyethylene, is a minor contribution to H up to 30°C below the melting temperature.125 Hence one customarily attempts to isolate the last term in (9.6.4)from the heat capacity contributions by drawing an appropriate flat baseline tangent to the pre- and postmelting baselines of the melting curve (Fig. 19). However, the construction of lZ5 B . Wunderlich, “Macromolecular Physics,” Vol. 1, p. 401. Academic Press, New York, 1973.

9.

318

~

THERMAL ANALYSIS OF POLYMERS

~~

TEMPERATURE

*

FIG.19. Typical DSC melting curve. The dashed line connects the pre- and postmelting baselines.

this baseline is a very subjective procedure. For sharp-melting materials, the error incurred in this estimation is usually small. Some polymers, on the other hand, may melt over wide temperature ranges and it may be quite difficult to determine where melting actually begins. There may also be appreciable baseline curvature due to instrumental factors or a nonlinear change in the heat capacity with temperature. An added complication is that (9.6.4) is not strictly valid in the melting region, but a term describing the change in AH, with temperature should be added.125 The problem of thermal lag (i.e., that the temperature of the sample deviates from the reference or programmed temperature in the transition region) is also ignored. It is apparent that an accurate determination of the baseline is needed before reliable AH, values are obtained. To this end, a variety of methods have been put forth in order to appropriately construct baselines. One procedure, developed by Brennan et al.,128*127 is illustrated in Figs. 20-23. They attempted to solve the problem of baseline shift by taking into consideration heat capacity changes throughout the temperature Iz6 W. P. Brennan, B. Miller, and J. C. Whitwell, Ind. Eng. Chem., Fundam. 8,314(1969). n7 W. P. Brennan, B. Miller, and J . C. Whitwell, Anal. Calorim., Proc. Symp., 2nd, 1970

Vol. 2, p. 441 (1970).

319 ISOTHERMAL

PROGRAMMED

i

1

ISOTHERMAL

dh dt

2

Xi

\I

FIG.20. Typical DSC traces of (1) empty sample pans, (2) sample, and ( 3 ) “products” of scan 2 (from Brennan er u / . ~ * ~ ) ,

, ISOTHERMAL

c

PROGRAMMED

b

.

ISOTHERMAL

dh dl

0

TEMPERATURE

----f

FIG.21. Construction for zero reference level (from Brennan er

~ 1 . l ~ ~ ) .

320

9. , ISOTHERMAL

,

THERMAL ANALYSIS OF POLYMERS

I

PROGRAMMED

.-

ISOTHERMAL

\I

dh -

L

dt

FIG.22. Thermal event trace with “true” baseline (from Brennan et ~ 1 . l ~ ~ ) .

dh dt

0

TEMPERATURE

----f

FIG.23. Heat contribution of thermal event only (from Brennan et U I . ~ ~ ~ ) .

9.6.

32 I

QUANTITATIVE METHODS

region of interest. This method is based on Brenner and O’Neill’s12* method of determining heat capacities by DSC. First a reference baseline is established on the output chart using empty sample pans (Fig. 20, curve 1). Notice that this trace has three parts: the first, an isothermal at a temperature T I ; the second, the DSC curve over the desired programmed temperature range; and a final isotherm at T 2 . The three-part sequence is repeated with the sample in one of the pans (curve 2). The sequence is again repeated utilizing the “product” of scan 2, the polymer melt (curve 3). Note that this would be impossible in the case of most polymers since some recrystallization upon cooling the melt is inevitable. In practice this technique would probably only be used for heats of reaction or curing. The three curves are then superimposed. As the first step in placing curve 2 on a base from which heat capacity effects are removed, curves 2 and 3 are redrawn with reference to a straight line representing dh/dt = 0, where dh/dt is the enthalpy per unit time of curve 1 starting beyond the point where instrumental lag has been overcome (Fig. 21). Next, the “sensible” heat contributions must be removed from curve so that its final form represents the heat contribution of the transition only (Fig. 22). As a first step, the baseline is obtained according to an iterative procedure outlined by Brennan et al. 126 Assuming no thermal lag this baseline is of the form12‘ (9.6.5) dh/dt = [HAL+ HREAC(~ - f) + H p ~ ~ ~ ( f ) ] [ d T = p /dTJdt] dt

where dTp/dfand dT,/dr are equal and represent the rate of change of the sample and programmed temperature, respectively, HAL the heat capacity of the sample container,fthe fraction of the reactants (solid polymer) that has undergone transition at a certain temperature, and HpRoD and HREAC the total heat capacities of products (melt) and reactants, respectively. Next, the heights Xi of the curve above this baseline are plotted vs. temperature (Fig. 23). Note that the shape of the curve may well be changed. This curve is free of heat capacity effects and thought to be a true picture of the transition. The area under the curve is now solely related to the heat Contribution of the transition. A somewhat different baseline was proposed by Heuval and The final form of their equation is127 dh/dt = [HAL

+ HREAC(1 - f) +

HpRODCf)]

dT,/dt.

(9.6.6)

Note that (9.6.6) differs from Brennan’s equation (9.6.5) in that Tp is replaced by T, . This is due to Heuval and Lind’s assumption that thermal 12*

12*

N . Brenner and M. J. O’Neill, lnsirum. News 16(2), 3 (1965). H. M. Heuval and K. C. J. B. Lind, Anal. Chem. 42, 1042 (1970).

9.

3 22

THERMAL ANALYSIS OF POLYMERS

lag is important. Brennan et ~ 1 . argue l ~ ~that thermal lag is not important for polymers and hence ignore this correction in their treatment. Heuval and Lind120disagree and believe that the Brennan treatment is only applicable to transitions over a “wide” temperature range. Another approach that compensates for heat capacity changes on going from solid to melt was proposed by Guttman and Flynn.130i131 They present arguments to suggest that the “correct” baseline can be obtained by a linear extrapolation of the posttransition baseline to the thermodynamic transition temperature. This is shown in Fig. 24. However, their treatment is only valid for transitions that occur instantaneously at a single temperature. Thus it does not appear applicable to the usual polymeric systems. A method for calculating the weight fraction crystallinity that circumvents the construction of a baseline altogether was proposed by Gray.124 At some initial temperature Tl below the transition region,

H1

=

(9.6.7)

xIHc1 + (1 - x1)Ha1,

where the subscript 1 denotes temperature Tl , At some final temperature ( T z ) where the polymer is completely molten,

H2 = Ha2.

(9.6.8)

By difference,

AH2.1 = H2

-

H I = AHa(2.1) + &XI

3

(9.6.9)

where AHa(2,1)= Ha2 - Hal and A e 1 = Hal - H c l , the energy required to transfer 1 g of purely crystalline material to 1 g of purely amorphous material at Tl . Solving for xl,

x1 = m

AH2,1- b,

(9.6.10)

where m = l / A H t l and b = AHa(2,1)/AHfOl.Equation (9.6.10) states that x1 and the total energy absorbed between Tl and T2 are related. Thus, by measuring AH2,1from a DSC trace and knowing A m at Tl and AHa(2,1), the weight fraction crystallinity may be determined. An important advantage of this method is that practically all polymers will undergo some reorganization while they are being heated through their melting region. This behavior complicates baseline methods, but has no effect on Gray’s approach. AH2,1is a function of the initial and final states only and since the final state is the melt, whose properties are asI3O

C. M. Guttrnan and J. H. Flynn, Anal. Chem. 45, 408 (1973). J. H. Flynn, Thermochim. Acta 8, 69 (1974).

9.6.

QUANTITATIVE METHODS

TEMPERATURE

323

---t

FIG.24. The dashed line represents the correct baseline according to Guttman and Flynn. The heat of transition is defined as the cross-hatched region corrected for the circled area (from Guttman and Flynnlso).

sumed to be independent of thermal history, AH,,, should give the correct crystallinity at T, regardless of the path the polymer takes to T,. The major disadvantage here is that accurate values of A Ha,,,,, and A HFl must be known to find x , . An alternative method for determining x1 makes use of the fact that AH,,, should be independent of the path from T, to T, .124 First, imagine that the polymer is heated from T , and T k , the perfect crystal melting point, keeping x constant. Then

AHo,I

=

Xl(Hc0

-

Hc1) + (1 - xl)(Hao - Hal>,

(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
DTA/DSC has also been used increasing for the determination of the compatibility or incompatibility of polymer blends .227-229 These polyblends are physical mixtures of structurally different homo- or copolymers. Compatibility is a representation of how close a blend can approach the ultimate state of mixing. The lack of compatability leads to phase separation of the components and, ultimately, poor mechanical properties due to the presence of large domains, which have poor interfacial bonding.227 In general, compatible systems exhibit one Tg that falls between the T, of the blend constitutents. Incompatable mixtures will show the Tg of each component.

9.7.5.Polymer Reactions Analysis of polymer degradation and oxidation as well as the effect of stabilizers on these reactions is important for both product processing and performance. DSC/DTA has provided a convenient way of studying these reaction^.^^^-^^^ The quantities that can be derived from these studies include heats of reaction, kinetic parameters, reaction onset tem227 228

D. S. Hubbell and S . L. Cooper, J. A p p l . Polym. Sci. 21, 3035 (1977). J. S. Nolands, N. N.-C. Hsu, R . Saxon, and J . M. Schmitt, Adv. Chem. Ser. 99, 15

(1971) 2z0 W. J. MacKnight, J. Stoelting, and F. E. Karasz, Adv. Chem. Ser. 99, 29 (1971) zso L. Goldfarb, C. R. Foltz, and D. C. Messersmith, J . Polym. Sci., Polym. Chem. Ed. 10, 3289 (1972). P. Dunn and C. Erinis, J. Appl. Polym. Sci. 14, 355 (1970). 232 A. Rudin, H . R . Schreiber, and M. H. Waldman, Ind. Eng. Chem. 53, 137 (1961). * B . Ke, J. Polym. Sci., Part A 1, 1453 (1963).

336

9.

THERMAL ANALYSIS OF POLYMERS

perature, and induction times. Often, DSC/DTA is coupled with mass - ~ ~ ~ to obtain the maxand t h e r m o g r a ~ i r n e t r y * ~in~order imum information from a given amount of sample. Thermogravimetry is primarily employed to follow changes in sample weight as a material undergoes reaction. Effluent gas analysis systems are also available that couple DSC/DTA with a gas chromatograph, which in turn may be coupled with a mass spectrometer. This system is capable of measuring both the quantity and composition of an evolved gas.239 9.7.6. Crystal Iization

The rate of crystallization is important in both a theoretical and practical sense. The mode of crystallization affects the degree and nature of crystallinity and hence a variety of polymer properties. In the past, dilatometry, microscopy, and infrared spectroscopy have been employed to study the crystallization process. Recently, however, DSC has been used increasingly for this p u r p o ~ e . ~ ~ ~Unlike - ~ * ' methods such as dilatometry, DSC requires small sample sizes and quantitative kinetic isotherms can be rapidly obtained over a range of temperatures. Isothermal crystallization curves are normally analyzed according to the Avrami equation248 1 - xt = exp(-Z,tn),

(9.7.5)

where 2, is the crystallization rate constant, t the time, and n the Avrami exponent. The numerical value of n is characteristic of certain nucleation and growth mechanisms. The weight fraction crystallinity at time t (xt) is

J. H. Johnston and C. A. Gaulin, J . Macromol. Sci.,Port A 3, 1161 (1969). K. Vo Van, S. C. Malhotra, and L. P. Blanchard, J . Macromol. Sci., Chem. 8, 843 ( I 974). 236 T. Kotoyon, Thermochim. Acta 5, 51 (1972). 237 S . Igarashi and H. Kambe, Bull. Cham. Soc. Jpn. 37, 176 (1964). 23* L. Reich, Macromol. Rev. 3, 49 (1968). 23@ A . S . Kenyon, Tech. Methods Polym. Eval. 1, 217 (1966). 240 M. Gilbert and F. J. Hybart, Polymer 13, 327 (1972). A. Booth and J. N. Hay, Polymer 10, 95 (1969). 242 G . S. Fielding-Russell and P. S . Pillai, Kolloid-Z. & Z . Polym. 250, 2 (1972). 243 J. N. Hay and M. Sabir, Polymer 10, 203 (1969). 244 V. Era and H. Venalainen, J . Polym. Sci., Part C 42, 879 (1973). 245 A. Savolainen, J . Polym. Sci., Part C 42, 885 (1973). 146 G . Ceccorulli and F. Manescalchi, Makromol. Chem. 168, 303 (1973). %' G . S . Fielding-Russell and P. S. Pillai, Makromol. Chem. 135, 263 (1970). B. Wunderlich, "Macromolecular Physics," Vol. 2, p. 132. Academic Press, New York, 1976. 254

235

9.8.

defined as

SUMMARY

IOt 1:

337

(dHt/dt)dt

Xt =

(9.7.6)

(dH,/dt) dr’

where Ht is the heat evolved up to time t. The numerator in (9.7.6) is the area a in Fig. 29, while the denominator is the total area under the curve. From knowledge of xt at various times, both n and Z1can be obtained by plotting log[ - ln(1 - xt)] vs. log t , the slope being n and the intercept log Z , . 2, can also be determined from knowledge of the time required for 50% crystallization ( t I l z )and the average value of n: 2,

=

In 2/t1,zn.

(9 * 7.7)

It has been found that the crystallization parameters depend on molecular weight, molecular-weight distribution, degree of tacticity, the presence of impurities or nucleation agents, and structural irregularities. For instance, Fatou and B a r r a l e ~ - R i e n d afound ~ ~ ~ that molecular weight strongly influences the isothermal crystallization rate of polyethylene fractions. This effect was attributed to the viscosity of the crystallizing medium, which may influence chain transport or the crystal growth rate. Howeve the authors found that as the crystallization temperature decreases, the influence of molecular weight is not as strong. In a DSC study of ethylene-vinyl acetate copolymers, Johnsen el ul.z50also found that molecular weight affects the kinetics of crystallization as well as the maximum obtainable degree of crystallinity. In addition, they also reported the effects of copolymer composition on crystdlization kinetics. A more detailed treatment of crystallization is given in Part 10.

9.8. Summary From the preceding discussion it is evident that DTA and DSC are powerful techniques that can yield a large body of information concerning the thermal properties of polymers. DTA/DSC along with other thermal analysis techniques not elaborated on here (e.g., thermogravimetric analysis) have contributed greatly to our knowledge of polymer physical properties. The enormous development in the field of polymer thermal analysis in the last 15 years seems likely to continue in the future and lead to an even better understanding of the structure, organization, and properties of polymer systems. J . M . G. Fatou and J. M. Barrales-Rienda, J . Polym. Sci., Port A-2 7, 1755 (1969). U. Johnsen, G. Nachtrab, and H. G. Zachmann, Kolloid-Z. & Z . Polym. 240, 756 (1970). ~ 4 8 2*

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10. NUCLEATION AND CRYSTALLIZATION

By Gaylon S. Ross and Lois J. Frolen 10.1. Introduction 10.1.1. Aims and Objectives

It is well known that the physical, mechanical, and chemical properties of crystalline polymers depend on the morphology of the crystals formed as well as the degree of crystallinity of the sample. Although the study of the morphology of polymer crystals is both interesting and complex, this subject is beyond the scope of this part. Khoury and Passaglia’ have recently published a chapter on the morphology of crystalline synthetic polymers that is written for the nonspecialist and covers many details concerning morphological studies from the mid-1950s to the present. For more details concerning this fascinating subject the reader is referred to other excellent book^^-^ and review^.^-'^ This chapter is written for the nonspecialist. It includes a survey of some of the better experimental methods that are available for the study of polymer nucleation and crystallization. We are primarily concerned with the description and evaluation of the experimental methods rather than surveying the results obtained for individual polymers by different techniques. F. Khoury and E. Passaglia, Treatise Solid State Chem. 3, 335 (1976). H. Geil, “Polymer Single Crystals.” Wiley (Interscience), New York, 1963. B. Wunderlich, “Macromolecular Physics,” Vol. 1 . Academic Press, New York, 1973. * B. Wunderlich, “Macromolecular Physics,” Vol. 2. Academic Press, New York, 1976. A. Keller, in “Growth and Perfection of Crystals” (R. H. Doremus, B. W. Roberts, and D. Turnbull, eds.), p. 499. Wiley, New York, 1958. A. Keller, Rep. Prog. Phys. 31, Part 2, 623 (1968). ’ A. Keller, Kolloid-Z. & Z . Poly. 231, 386 (1969). A. Keller, Phys. Chem., Ser. One 8, 105 (1972). H . D. Keith, in “Physics and Chemistry of the Organic Solid State” (D. Fox, M. M. Labes, and A. Weissberger, eds.), p. 561. Wiley (Interscience), New York, 1963. lo H. D. Keith, Kolloid-Z. & Z. Polym., 231, 421 (1969). l1 P. Ingram and A. Peterlin, Encycl. Polym. Sci. Techno/.. 9, 204 (1968). l2 D. V. Rees and D. C. Bassett, J . Muter. Sci. 6, 1021 (1971). Is R. A. Fava, J. Polym. Sci., Part D 5 , l(1971).

* P.

339 METHODS O F EXPERIMENTAL PHYSICS, VOL. 16B

Contribution of the National Bureau of Standards. ISBN 0-12-415951-2

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NUCLEATION A N D CRYSTALLIZATION

Although we emphasize the study of crystallization in bulk samples, we discuss crystallization from solution to some extent. Information concerning the underlying theory will be provided along with the method for analyzing the results of each type of experiment.

10.2. General Background on Semicrystalline Polymers 10.2.1. Requirements for Crystallization

A polymer is produced synthetically by the systematic adding or linking of small low-molecular-weight units, monomers, producing a distribution of differing molecular-weight chains. The number of monomer units in each chain is referred to as the degree of polymerization. Such materials will crystallize if the resulting structures are such as to permit the chain backbones to be packed in an ordered parallel configuration. This requirement can only be met if nearly complete chemical and stereo regularity exists. If, in the polymerization, all the monomer units attach themselves in an orderly head-to-tail fashion, chemical regularity will exist. Occasionally a head-to-head (or tail-to-tail) attachment will occur that will present a flaw in the repeat regularity of the polymer backbone configuration. Other chemical imperfections can produce branching or side chains in the otherwise linear chain. When the monomer has asymmetrical side groups, a mere head-to-tail addition is not sufficient to produce a polymer chain capable of crystallizing. It must also be stereo regular. The side groups must be either (1) isotactic, i.e., all on the same side of the backbone, or (2) syndiotactic, i.e., an alternation, left-right-left along the backbone. If such side groups are randomly placed left or right along the polymer backbone, crystallization does not occur. For the purposes of this discussion, we only consider crystallizable polymers, namely, those possessing linear chains with few branch points. When the asymmetric monomer side groups do exist, only those polymers that exhibit syndiotactic or isotactic character are considered. There is still another type of polymer system, the block copolymer, which can exhibit limited crystallization. This is a polymer whose monomer units A and B are jointed together in such a fashion as to pro- * .. If the blocks of duce a polymer of the type A-A-A-B-B-B-AAs are the same size and if the same is true of the blocks of Bs, and if the repetition of A and B blocks occurs in a regular fashion, then crystallization can occur in the system. If there are long runs of A or B blocks, one of which is crystallizable, crystallization can also occur. In actual fact, for a large number of reasons, no polymer system is completely crystallizable. At best, a conglomerate of crystalline and amor-

10.2.

GENERAL BACKGROUND ON SEMICRYSTALLINE POLYMERS

34 1

phous regions are produced. It is this semicrystalline polymer system that we will discuss in this part. For a more extensive description of polymer crystallization requirements, the reader is referred to Khoury and Passagha,' Geil,2 and W ~ n d e r l i c h . ~ 10.2.2. Historical Development 10.2.2.1. Early Experiments and Resulting Models. In the case of polymers crystallized from the bulk (i.e., from their own melts), the predominant morphology that is observed using a polarizing microscope is spherulitic (Fig. 1) in nature. This type of structure has also been observed in simple compounds and minerals. The extinction patterns produced, i.e., the familiar maltese cross or the alternation of light and dark concentric rings, remained unchanged as the sample was rotated between

FIG. 1. Transition from spherulitic to axialitic morphology in linear polyethylene ( M , = 30,600, M , / M , = 1.19) with decreasing undercooling (optical micrographs, crossed nicol prisms).

342

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NUCLEATION A N D CRYSTALLIZATION

FIG. 2. (A) Polyethylene spherulite with a schematic of the radiating chain-folded lamellae. Twist of lamellae causes the characteristic dark rings and patches, depending on the regularity of twist. (B) Schematic of the fringed micelle showing crystalline and amorphous regions. (C) Model for surface nucleation and growth of a regularly chain-folded crystal with adjacent reentry. The crystal grows in the G direction by the addition of chains having thickness b and width a . The thickness of the crystal itself is denoted as 1 (the initial thickness prior to annealing is called / g * ) . The lateral and fold-surface free energies are u and re, respectively. (D) Schematic showing possible departures from strictly regular folding as well as possible causes for defects in the body of the crystal (Fig. 2B-D from Hoffman et a/.").

the polarizing and analyzing elements of the microscope, indicating the symmetry of the spherulitic structures. The nature of the long but narrow fibrillar crystals making up the spherulite (Fig. 2A) was not known for quite some time. Indeed the precise nature of such structures still remains a source of controversy and a reason for continued research. Bunn and c o - w ~ r k e rwere s ~ ~responsible ~~~ for several important observations concerning polymer crystals grown from the melt, particularly the crystals found in linear polyethylene. It was known from X-ray line

la

C. W. Bunn, Trans. Faraday SOC. 35, 482 (1939). C. W. Bunn and T. C. Alcock, Trans. Faraday SOC. 41, 317 (1945).

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GENERAL BACKGROUND ON SEMICRYSTALLINE POLYMERS

343

broadening measurements that the polymer crystals were quite small, and also from Bunn it was learned that the polymer chains of the crystallized molecules were perpendicular to the radial direction in the spherulites. In the same studies Bunn demonstrated that the unit cell of crystallized polyethylene did not change as a function of molecular-weight distribution. Finally, he showed that the polymer crystals exhibited a high degree of perfection and, in particular, that the unit cell of crystallized bulk polyethylene was similar to that of the orthorhombic form of the normal alkanes. These observations, coupled with the knowledge that the crystals were of the order of a few hundred Angstroms thick, led to the postulation of the fringed micelle model (Fig. 2B). In this model the polymeric system was viewed as an amorphous, random-coil matrix, with bundles of extended chains of adjacent molecules forming crystalline domains. In such a model any given molecule could contribute to one or more of the crystalline domains, remain completely in the amorphous region, or in fact contribute to both the crystalline and amorphous regions. The amorphous matrix was visualized as being noncrystalline by virtue of atacticity , shortness of molecular length, entanglements, etc. The fact that a single molecule could be a part of more than one crystalline domain was of great usefulness in explaining some of the mechanical properties of polymers, and for quite some time the concept of the fringed micelle was the accepted one. Later, however, certain new experimental evidence began to cast doubt upon the validity of this model. It has been shown that single crystals of certain polymers could be prepared by crystallizing them from solution. l6 One of the worrisome details associated with the fringed micelle model was that as the crystal grew in size there naturally accumulated a large amount of strain at the ends of the crystal where the molecule emerged again into the amorphous matrix. Since this strain would be cumulative, the crystals by necessity would be limited in “girth.” But now there was evidence that, at least from solution, single crystals of appreciable size could be prepared. Finally, in 1957 Keller“ showed that polyethylene could be crystallized from a dilute xylene solution in the form of singlecrystal platelets of the order of 130 A thick, with electron diffraction patterns indicating conclusively that the polymer chains were perpendicular to the flat surfaces of the single crystal. It became apparent,17-19 since the length of the average polyethylene molecule was many times that of R. Jaccodine, Nature (London) 176, 305 (1955). A. Keller, Philos. M a g . [S] 2, 1171 (1957). la E. W. Fischer, Z . Naturforsch., Teil A 12, 753 (1957). lo P. H. Till, Jr., J . Polym. Sci. 24, 301 (1957). l6 l’

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the 130 8, found for the crystals, that a single molecule must fold back into the crystal many times, the number of folds being limited only to the molecule’s length. At the same time other investigations were showing through the use of electron microscopy and X-ray techniques that, even in bulk crystallized polymers, the same type of structures were present, i.e., radiating lamellar crystals wherein the molecules were perpendicular to the flat planes of the lamellae, which were found to be of the order of 50-150 A thick. It is interesting to note that as early as 1938, one investigator seriously considered a folded crystal as being a good model from the available experimental evidence.20 In the 1950s, when this new information regarding polymer solution-grown single crystals became available, several models involving chain folding were proposed. The most simplistic model was that of a crystal composed of polymer molecules with exclusive adjacent reentry of folds (Fig. 2C), resulting in a regularly folded surface that, with the tilted structure of the four-sectored single crystals, resembled a terra cotta roof. The only defects were the ends of the molecules, which either formed as short-length flexible rods or cilia above the folded surface or else folded back into the crystal. Of course, the crystal had to accommodate certain defects such as small degrees of atacticity in the molecule body or ends, the problem of matching ends of molecules, and multiple nucleation of the same molecule. At the other extreme is the model that was proposed by Florya21 In this model it was postulated that folding exists, but that the molecule reenters the crystal somewhat randomly. The fold surface would be very rough with loops of various lengths due to this random reentry, entanglements, etc. This model was termed the “switchboard” model in analogy to the old telephone switchboards and their entangled or over-lapping plug-in leads. Despite the details of the,controversy, the question still arose as to whether any model that explained the relatively perfect sectorized solution-grown single crystal could be applied to crystallization in the bulk. Over the intervening years from the time of Keller’s first monumental work, many capable investigators have used a variety of techniques in an attempt to conclusively demonstrate the true nature of the polymer crystals in bulk. In particular, infrared spectroscopy, small-angle light scattering, nuclear magnetic resonance, and neutron scattering have been employed. The results of these studies are again not conclusive in providing a unique model consistent with all the experimental observations. *O

21

K. H.Storks, J . A m . Chem. SOC. 60, 1753 (1938). P. J. Flory, J . A m . Chem. SOC. 58, 2857 (1962).

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GENERAL BACKGROUND O N SEMICRYSTALLINE POLYMERS

345

The general structure of spherulites and axialites (open-structure spherulites) is now thought to be due to long, ribbonlike lamellar, chain-folded crystals, which exhibit small-angle branching as they grow (Fig. 1). Electron-microscopic investigations have clearly shown that the fibrils or lamellar crystals rotate or twist about their long-growth axes (Fig. 2A). A number of investigators notable among whom were Keller,22 Point,23Keith and Padden,24*25 and Pricez8have observed and shown how this crystal rotation about the radii of the growing spherulite is responsible for the characteristic extinction patterns observed in the spherulites. In the classical work of Keith and Padden,27-2sthey were able to provide a phenomenological explanation of spherulitic growth in polymers, the basic problem being to explain how the growth front degenerates into a group of fibrils. In polymers it is recognized that the growth kinetics result not from diffusion of material away from the growing interface, but rather from a nucleation process occurring at the tip of the lamellar crystal. Keith and Padden conclude that the morphological character of the growing spherulite is controlled by the rejected material concentrated at the growth front. In their terms the impurity produces cellulation wherein there is produced, at the growth front, fibrils having a width S = D / G , where 6 is of the order of 100 nm, D is a diffusion constant, and G is the growth rate. The theory proposed by Keith and Padden explains the branching and ribbonlike character of the lamellar crystals and, in addition, provides a mechanism for the fractionation of material as a result of the crystallization process. Materials unused in the initial crystallization process are deposited between growing fibrils. This rejected portion may crystallize much more slowly, may crystallize at a lower temperature, or may not crystallize at all. The explanation still allows the radial growth rate to be a constant as more and more of the liquid nutriment is used until such time as neighboring spherulitic structures impinge. Keith, Padden, and Vadimsky30have performed a series of crucial experiments showing the existence of tie molecules and/or crystallized columns of molecules (probably crystallized in an “extended-chain” structure) holding together the ribbonlike lamellae. They performed a series of A. Keller, J . Polym. Sci. 39, 151 (1959). J. J. Point, Bull. Cl. Sci., Acud. R . Belg. [ 5 ] 41, 974 (1955). 24 H. D. Keith and F. J . Padden, Jr., J . Polym. Sci. 39, 101 (1959). 25 H. D. Keith and F. J . Padden, Jr., J . Polym. Sci. 39, 123 (1959). 26 F. P. Price, J . Polym. Sci. 39, 139 (1959). 27 H. D. Keith and F. J . Padden, Jr., J . Appl. Phys. 34, 2409 (1963). H. D. Keith and F. J. Padden, Jr., J . Appl. Phys. 35, 1270 (1964). 29 H. D. Keith and F. D. Padden, Jr., J . A p p l . Phys. 35, 1286 (1964). 30 H. D. Keith, F. J. Padden, Jr., and R . G . Vadimsky, J . Appl. Phys. 42, 4585 (1971). 22

23

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10. NUCLEATION

A N D CRYSTALLIZATION

experiments wherein polyethylene was diluted with dotriacontane. The n-alkane was subsequently leached out of the crystalline polyethylene and a series of electron micrographs clearly showed the existence of large numbers of these intercrystalline links between the lamellae. The theory does not explain the reason for the twist in lamellae as they grow. However, the general details of lamellar crystals connected through single molecules entering into the crystallization of two or more chain-folded lamellae, is an accepted fact. The bulk of the experimental and theoretical evidence supports the idea of adjacent reentry with relatively uniform folds. The questions of how much adjacent reentry occurs and the tightness of the fold loops, etc., is still not answered conclusively. 10.2.2.2. Theoretical Basis for Crystallization Studies. The major thrust of this part will be devoted to an understanding of nucleation and crystallization as it applies to the bulk polymer systems. The view we choose to take here is that in such systems the primary nucleation process is heterogeneous, i.e., the formation of a stable nucleus as aided by foreign particles or surfaces. In general, such a nucleus is envisioned as being of a folded nature, similar to that of the crystal lamella that will be produced. This ribbonlike lamella will have a fold distance, i.e., the distance from the top folded surface to the bottom folded surface, which is dependent upon the temperature of crystallization (Fig. 2D). The fold surface will be rough, implying that some of the folds will be “floppy,” that there will be extending ends or “cilia” protruding above the fold surface, that there will be some molecules or portions of molecules adsorbed on the fold surface, and that there be tie molecules between one fold surface of one lamella and the fold surfaces of adjacent ones. However, the fold surfaces will show a marked preference for adjacent reentry and a general regularity of fold. The fact that crystals have an amorphous content has required the concepts of adsorbed molecules, cilia, tie molecules, etc., between adjacent lamellae to be introduced. In addition, even in the most crystalline of the semicrystalline polymers, there will be a substantial percentage of the total molecular population that does not crystallize in this initial crystallization process or in fact does not crystallize at all. Recent developments in etching and staining of samples for the electron microscope reveal a wide variety of crystallized entities in the bulk samples.31 In addition to the major lamellae discussed above, there also exists crystallized material whose top to bottom measurements are much smaller than the fold distances associated with the larger lamellae. Such structures primarily reside between the larger lamellae and may or may not be folded. They certainly represent material that has crystallized at a later time. Finally, because of lack of required stereo regularity, or entan31

D. C. Bassett and A. M. Hodge, Pruc. R . Soc. London. Ser. A 359, 121 (1978).

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GENERAL BACKGROUND O N SEMICRYSTALLINE POLYMERS

347

glement, molecular size, etc., a significant portion of the polymer does not crystallize. The primary crystallization process, nucleation controlled, consists of the formation of these lamellae, radiating outward and branching in such a fashion that the overall crystal density of the growing structure remains nearly constant. In some cases the ribbonlike lamellae become twisted as they grow, giving rise to the spectacular ringed spherulites where variations of this rotating nature accounts for the production of ruffles and rings in the spherulitic pattern. Again, both the outward radial growth rate and the lamella thickness is a function of the growth temperature. The magnitude of the radial growth rate is itself strongly dependent upon the molecular weight of the growing species. The descriptions of the techniques used in the measurement of such radial growth by either observing a few spherulites separately or by observing the bulk growth changes resulting from many such spherulites will occupy the major portion of this part. In respect to the material covered herein, we first discuss bulk crystallization and then crystallization from solution. As far as the development of crystallization theory, and specifically as it applies to chain folding, the study of the growth of single-polymer crystals from solution has immense historical importance. The exact nature of such crystals is still involved in a bit of controversy, but for our part, we envision crystallization from solution as, under carefully controlled conditions, producing chainfolded, single crystals having a high degree of regularity. The folded surfaces will have some degree of roughness caused, in general, by the same mechanisms that caused the roughness and amorphous nature of the surfaces between the lamellae produced in bulk crystallization. Considerably less detail will be given to two other important crystallization processes, namely, crystallization induced by hydrodynamic flow (Pennings’ shish kebob type of crystals), and crystals produced by the application of high pressures. The latter produces crystals having extremely large fold distances, which may be viewed as being much closer to extended-chain crystals than to the folded ones. The former may be viewed as being composed of a central filament composed of nearly extended-chain crystallized material upon which at somewhat regular intervals there appear buttons composed of folded crystals with the fold surfaces perpendicular to the central filament. As is the case with all other viable experimental fields there is a great deal of controversy associated with the exact explanations of such growth. 10.2.2.2.1. THE AVRAMITHEORY.Avrami published two paper^^^,^^ 32 33

M . Avrarni, J . Chem. Phys. 7 , 1103 (1939). M. Avrarni, J . Chern. Phys. 8, 212 (1940).

348

10. NUCLEATION A N D CRYSTALLIZATION

relating to the kinetics of phase change in general. His fundamental relationship as applied to the crystallization process was

x = 1 - exp(-Ktn),

(10.2.1)

where x is the degree of crystallinity, K the crystallization rate constant, t the elaspsed time, and n an integer characteristic of the type of nucleation. In Avrami’s development and the extension of this work by Morgan,34n may take on values of 1, 2, 3, and 4,depending on the nature of the nucleation and growth process. For spherical growth n is 3 or 4; for platelike growth n is 2 or 3; and for fibrillar growth n is I or 2. Later it will be shown that for some polymers spherulitic growth may exhibit an n of 3 of 4, depending upon whether or not nuclei appear sporadically (homogeneous nucleation) in time or are in existence from the beginning of the crystallization process (heterogeneous nucleation). With other polymeric systems it will be shown that n can exhibit values from less than l to more than 4, these experimentally determined values not necessarily being integers. The complexity of the problem arises from the fact that there may be two or more processes taking place simultaneously, that n may be a function of the degree of crystallization, and that the degree of crystallization itself is not well defined. However, the use of the Avrami relationship can produce a much needed insight to the understanding of the crystallization process. The rate constant K in the Avrami equation is equal to 4rvoG3/3, and n = 3 when the spherulites formed were “born” at the same time. We can use the relationship to determine G, the radial growth rate, providing we know the number of heterogeneous nucleation centers per unit volume, vo. 10.2.2.2.2. KINETICGROWTHTHEORY.Providing the spherulites are formed through the process of heterogeneous nucleation, the primary nucleus is developed from a solid surface such as a dirt particle. It is then postulated that additional molecules are laid down and become a part of the chain-folded lamellar structure by first having a portion of the molecule form a stable folded nucleus on the existing crystal substrate. Following the formation of such a nucleus the rest of the long-chain molecule is “reeled in,” crystallizing with a fixed, temperature-controlled fold distance. This type of nucleation-controlled growth seems very consistent with our knowledge of the fibrillar growth of the spherulites. Applying the theoretical idea of Turnbull and Fisher,35the concept of formation of L. B. Morgan, Philos. Trans. R . Soc. London, Ser. A 247, 13 (1954). s5

D.Turnbull and J. C . Fisher, J .

Chem. Phys. 17, 71 (1949).

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349

critical-sized nuclei can be expanded to give the following growth relations hip:

G = Goexp

[

-

g] [

exp - K ,

1 (10.2.2) (m)],

where AF* is the activation energy for transport of segments to the site of crystallization, T the temperature of crystallization, k the Boltzmann constant, and AT = Tmo - T , where T , is the melting point of a defect-free, large extended-chain crystal of the polymer. With the use of a reasonable approximation for A F * , it is possible to compute the growth rate constant K , . Using the flux equation approach Hoffman36 calculated this rate constant to be 4houueTmo/kAhf in one limit (see later). In this relationship Ahf is the heat of fusion, 6 0 the monomolecular layer thickness, and (+ and u, the lateral surface and fold surface free energies respectively. Optical microscopy can be used to determine radial growth rates, and Eq. (10.2.2) allows observations made from measurements on single spherulites to be compared with the calculated growth rates from such experimental techniques as dilatometry , which provide a measure of the bulk crystallization rates, Eq. (10.2.1.). Relying on these two relationships, we can calculate the free-energy product from the computed rate constant. 10.2.2.2.3. HOMOGENEOUS NUCLEATION THEORY.The measurement of the growth rates allow one to determine values of (+re. If one then determines the value of v2(+,from homogeneous nucleation experiments, the values of both (+ and U, can be determined. The generalized equation for the homogeneous nucleation rate I is defined as

I (nuclei ~ e c - k m - ~ ) = I. exp[- ( A F * / k T ) ] exp[-K,(l/T A T 2 ) ] , (10.2.3) , is where, according to Fisher and T ~ r n b u l lI,, ~=~ ( N / v m ) ( k T / h ) which approximately equal to for materials such as polyethylene; N is Avogadro’s number; 7, the molar volume of a sequence of chain segments whose length is approximately that of the crystal nucleus; k T / h the usual frequency factor, where k is Boltzmann’s constant, T the temperature, and h Planck’s constant; AF* the activation energy for transport of crystallizing molecules; and K i is the homogeneous nucleation rate constant. K i is defined by Lauritzen and Hoffman3’ as 30.2u2(+, ( T m 0 ) 2 / K ( A h f ) 2ATP f , where Tmo, k , A h f , and AT have the same 38 37

J. D. Hoffman, SPE Trans. 4, No. 4, 315 (1964). J. I . Launtzen, Jr. and J. D. Hoffman, J . Res. Nor/. Bur. Stand.. Sect. A 64,73 (1960).

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NUCLEATION A N D CRYSTALLIZATION

meanings as in Eq. (10.2.2.). The factor 30.2 is obtained when it is assumed that the homogeneous nucleus builds upon the orthorhombic unit cell. Because of the very large undercoolings encountered in homogeneous nucleation experiments, the thermodynamic driving force must be corrected by a factorf, which is equal to 2T/(Tm - T ) as demonstrated by Hoffman and Hoffman et al. 39 have shown that it is reasonable to assume that cr and cre vary with temperature and suggested a dependence of the form

( 10.2.5)

10.2.2.2.4. THEORYRELATING TO Tmo. As an adjunct in determining the surface free energies from the growth and nucleation constants, Ahf and Tmoare necessary input information. In the case of simpler materials, the experimental determination of the heat of fusion and the melting points from the known crystal form is straightforward, using timehonored physical methods. However, in the case of semicrystalline polymers the determination of either quantity is not easy. From the description of “semicrystalline,” it is obvious that accurate heats of fusion cannot be obtained by a simple determination of the melting heat from any given sample. This problem is discussed in Part 9 of this volume. It is sufficient here to state that the problem is not a trivial one. The determination of the quantity Tmois also discussed in Part 9. However, we briefly discuss the problem here. We have previously defined Tmoas being the melting temperature of a large perfect crystal in an extended-chain configuration. Obviously, it must also be the same crystalline form as the bulk crystal. Extended-chain crystals are not easily obtained experimentally. The usual form of crystals in bulk polymers is that of chain-folded lamellae, but the experimental melting point from such crystals is severely depressed by the smallness of the fold distance (i.e., the thinness of the crystal). W ~ n d e r l i c hhas ~ ~shown that for linear polyethylene, crystallization under conditions of high pressure produces crystals that in the extreme may closely resemble extended-chain crystals. Experimental melting points from such materials are indeed much higher than those obtainable from crystals of the same material grown under atmospheric pressures. In certain cases an extrapolated J. D. Hoffman and J. J. Weeks, J . Chem. Phys. 37, 1723 (1962). J. D. Hoffman, J. I. Lauritzen, Jr., E. Passaglia, G . S. Ross, L. J. Frolen, and J. J. Weeks, Kolloid-2. & Z . Polym. 231, 564 (1969). B. Wunderlich and T. Arakawa, J . Polym. Sci., Part A 2, 3697 (1964). 38

38

10.2.

GENERAL BACKGROUND ON SEMICRYSTALLINE POLYMERS

35 1

melting point has been computed from data obtained from experimental melting points of simple molecules, which may be considered as precursors to the high-molecular-weight polymers, e.g., the use of such properties of the linear normal alkanes to predict the melting points of polyethylene by Flory and Vrij4I and B r o a d h u r ~ t . It ~ ~has been shown that the fold-to-fold distance of crystallizable polymers is a function of the temperature of crystallization. It has also been shown that when such crystals remain at the crystallization temperature, this fold-to-fold distance 1 increases as a function of time, the increase having the general form of l ( t ) = 12

+ B log f.

(10.2.6)

It has further been shown that when crystals grown at a lower temperature are raised to and remain at a high temperature, again there is an increase in the fold-to-fold distance. By Eq. (10.2.6.) it is seen that the thickness (fold-to-fold distance) at any time, i(t),increases from the initial thickness 12 logarithmically with time. In general, this relation can be simplified 1 = yl2.

(10.2.7)

This equation then presumes that at some final time, the initial thickness times the constant y will give the final fold distance. With certain other simplifying assumptions it can be shown that ( 10.2.8)

where T , is the experimentally observed melting point obtained by crystallizing at T , and allowing the crystals to remain at T , for sufficient time as to change 12 by a factor of y. Then a plot of T , vs. T , can be extended linearly until intersection with the line T , = T , to obtain Tm0,the melting point of the large extended crystal. There will be different lines corresponding to different y values, but all will intersect at the same temperature. It is therefore presumed that no further thickening occurs during the melting process. t P. J. Flory and A. Vnj, J . Am. Chem. SOC. 85, 3548 (1963). M. G . Broadhurst, J . Res. Natl. Bur. Stand., S e c t . A 70, 481 (1966). J. D . Hoffman and J. J. Weeks, J . Res. Nut/. Bur. Stand., S e c t . A 66, 13 (1962). 1p J. D. Hoffman, G . T. Davis, and J. I. Lauritzen, Jr., Treatise Solid Stnte Chem. 3, 497 (1976). 41

42

t Experimental measurements of T,,, vs. T , for a given polymer are much easier and less time consuming than direct measurements of I by electron microscopy or small-angle X-Ray diffraction. However, the experimenter should be aware that the value of y may be dependent on both time and temperature, resulting in errors in the determination of Tm0for some systems. For a further discussion of problems in the use of either Eq. (10.2.8) or (10.2.9)the reader is referred to Hoffman et a/.,44p. 528.

352

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NUCLEATION A N D CRYSTALLIZATION

An interesting melting point relationship that involves fewer assumptions and that has a straightforward theoretical basis is the following:43

T , = Tm0[l - 2ce/(Aht)l].

(10.2.9)

Since by plotting T , vs. 1/1 one obtains an intercept of Tm0and a slope of - 2ae Tmo/Ahf,this single relationship not only allows calculation of the required melting point, but in addition provides a value for ve if Ahf is known. Again, one of the chief experimental problems is assuring that 1 does not change during measurement of length or melting point. One group of investigator^^^ used moderate gamma-ray dosage to induce cross-linking. They measured both dissolution temperature and the melting point for polyethylene single crystals. The effect of the radiation has received some comment, but the technique as used by these and other observers seems to be a reasonable compromise solution to the problem of thickening while making the required measurements. The usual methods used to measure 1 are small-angle X-ray measurements and direct length measurements using electron microscopy. The whole problem of determining the melting point of the infinite polymer crystal has been developed in Part 9 of this volume. In this part we restrict our remarks to some experimental work that has been hone in our laboratory using differential thermal analysis.

10.3. Experimental Methods for Measuring Crystallization Ratest 10.3.1. General Considerations

There are several hundred papers in the literature that report the study of crystallization in a wide variety of polymers using a large number of different methods. In general, there is a lack of experimental detail with regard to the characterization of the actual samples and the actual techniques used (both with regard to equipment and handling techniques); in many cases the temperature control of the system is poor. It is hoped that this section will provide the reader with some insight regarding the care needed in performing experiments using a variety of methods. The experimental observations of the development of crystallinity in polymer samples fall into two groups. The first type of measurement is that of direct observation of the development of spherulites or single crystals under isothermal conditions. The methods used in this type of 45

H. E. Bair, W. Huseby, and R. Salovey, J . Appl. Phys. 39, 4969 (1968).

t See also different chapters in Vols. 6A and B of this series,

10.3.

MEASURING CRYSTALLIZATION RATES

353

experiment are optical microscopy, transmission electron microscopy, and some small-angle scattering techniques. These experiments require a sample that is prepared in the thin-film configuration. The second type of observation consists of following the development of the total crystallinity of a sample under isothermal conditions. This type of experiment requires the measurement of a physical property which is very sensitive to changes in crystallinity such as specific volume or density, infrared absorption bands, X-ray intensities, dielectric properties, and thermal properties. These methods include dilatometry, density balance and density gradient methods, differential thermal analysis, and various infrared, X-ray scattering, and dielectric techniques. These techniques usually require a sample that is in what we shall call the bulk configuration. Before we become involved with the details of sample preparation, there are several general considerations to be discussed. In addition to having a distribution of molecular weights that influences the primary nucleation and subsequent crystallization, a typical polymer has many other nonpolymeric materials that have been deliberately added or inadvertently found their way into the polymer. Examples of deliberately added materials include plasticizers, antioxidants and other stabilizers, other polymeric materials, and solid fillers of various types. Material left from the polymerization or subsequent working of the sample as well as dust or other solid debris are examples of the types of impurities that are added to the sample inadvertently. Before one decides how best to prepare the sample for study, the scientific purpose of the experiments must be determined. If, in fact, the purpose is to study the “as is” whole polymer then, although thorough characterization may be required, the study should be performed on the material as received. If, however, information of a more fundamental nature is desired, then a detailed study of the effect of molecular weight and molecular-weight distribution on crystallization is needed without the sometimes enormous but usually unknown effect of the foreign materials. In the latter case, fractionation, characterization, and -a general “cleaning up” procedures are necessary prerequisites to the actual crystallization study. In terms of time, cost, and scientific ingenuity, this step is often the most demanding one. Although some polymers can be synthesized in such a way that the resulting polymer is nearly monodisperse (e.g., the anionic polymerization of polystyrene), usually the polymerization procedure results in an uncontrolled mixture of molecular weights ranging from the terminated monomer to molecular weights ranging into the millions. The most widely used methods of fractionation at the present time are column elu-

354

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NUCLEATION A N D CRYSTALLIZATION

tion techniques and gel permeation chromatography (GPC). The determination of the molecular-weight distribution is most easily done by GPC if such a calibrated device is available. In addition to GPC the classical methods of viscometry, osmometry, light scattering, or ultracentrifugation may be used to determine molecular weights as well as molecularweight distribution (see Part 2 in this volume, Part A, for detailed discussion of these techniques). 10.3.2. Methods Using a Thin-Film-Type Specimen? 10.3.2.1. Sample Preparation. A thin film of polymer (of the order of 20-50 pm thick) sandwiched between two thin transparent covers is perhaps the most popular sample configuration when direct observation of crystal growth is required. This technique was used extensively by the authors in a study on the growth kinetics of polyethylene. Although the details described here refer directly to that work, the descriptions are quite general in nature and apply directly to other polymeric materials. In our study on polyethylene, we wished to prepare samples that contained a small number of nucleation centers so that individual spherulites could be observed for longer periods of time prior to impingement with neighboring spherulites. Further, we wished to use the same field for all of the observations on a single sample. In some materials, the sample degrades at the temperature of the experiment and this effect can be detected by irreproducibility of the results on subsequent runs. When this is the case a new sample must be prepared prior to each run. In our work, the samples were cleaned according to the flow chart shown in Fig. 3. The filtering techniques used were similar to those used for preparing a sample for light-scattering studies in solution. Regenerated cellulose filters of controlled pore size were used throughout the experiments. The filtration was done in a stagnant air box and the solution was filtered very slowly (about one drop per second). Faster filtration rates resulted in less clean solutions. Once the filtering was complete the sample was collected on a clean filter and dried at 80°C under vacuum to assure that all solvent had been removed. Part of this sample was used to characterize the material and the rest of the sample was used to prepare the thin-film specimen. Between 100 and 300 mg of the sample was prepared for viewing by lightly pressing the molten sample between two 1-in. square microscope cover glasses. This was done at 150°C in a vacuum oven. The use of spacers between the cover slips allowed the preparation of samples of unit See also Volume 6A, Chapter 2.5, as well as Volume

1 1 in this series, Part 12.

F I L T E R XYLENE 0.2 p F I L T E R

DISSOLVE PE FRACTION I N XYLENE ( . l % by w e i g h t )

MEASURE MOLECULAR WEIGHT D I S T R I B U T I O N BY GPC

T VACUUM

FIG.3. "Preparation" of polyethylene samples for growth studies.

356

10. NUCLEATION A N D CRYSTALLIZATION

form thickness. The thin film should nearly cover the entire surface of the cover glasses. The cleaning pretreatment of the cover glasses is very important since surface nucleation of the sample is not desirable. In our studies the cleaning method used was as follows: (1) cleaning with hot chromic acid solution, (2) rinsing with distilled water, and finally (3) further cleaning in distilled water using ultrasonic scrubbing techniques. After complete drying in a vacuum, cover slips treated in this way were found to cause little or no surface nucleation in the polyethylene samples. In the case of polyethylene, it was found that although some thermal or oxidative degradation occurred at the edges of the polymer film, the usual temperatures employed during the melting and crystallization sequences caused no noticeable changes in the crystallization kinetics of the central portion of the polymer film. Measurement of the molecular-weight distribution in the central portion of the film, using GPC showed no change in the distribution even after heating the sample to 150-155°C for as long as 200 hours. The choice of the thickness of the sample to be employed is dictated by the experimental method used. In the case of optical microscopy, the optical amplification most generally used is of the order of 3 0 0 ~ . In this case the smallest crystal aggregate that shows any detail is approximately 5 pm in its largest dimension. The field of observation is approximately 500 p n in diameter. Even when there are only a few active nuclei in the field, by the time the spherulites have grown to approximately 20 pm one or more of their surfaces must impinge on the glass cover slips. Thus, for most of the growth process one observes growth that is restricted to two dimensions. Although one would like to eliminate this problem by using thicker samples, the numerous nucleation centers that are scattered through the specimen in proportion to the sample thickness cause rapid overgrowth, which makes accurate measurements impossible. 10.3.2.2. Optical Microscopy. Two methods are available for measuring the growth rates of polymer crystals using the optical microscope. The first is the growth in situ on the stage of the microscope using a suitable hot or cold stage. This method has one major disadvantage; namely, a microscope and the auxiliary equipment can be tied up for weeks or even months on a single experiment. However, it offers the tremendous advantage that individual spherulites can be observed from their first development up to the time when impingement occurs. The second method consists of preparing a large number of samples, which can be crystallized away from the microscope, for instance, in a temperaturecontrolled bath. In this technique the samples are heated to a temperature above the melting point for a period of time, and then placed in another bath, which is set at the desired crystallization temperature. The

10.3.

MEASURING CRYSTALLIZATION

RATES

357

samples are removed from the bath at specified time intervals and quenched rapidly to a lower temperature. Self-decoration occurs and the size of each spherulite can be measured with a microscope as a function of residence time at the given crystallization temperature. This method has the advantage that equipment is available for other uses while the experiment is in progress, but has the disadvantage that sample-to-sample variations can occur and each spherulite can be examined only once. For a complete study on a given material a combination of the two methods is recommended. 10.3.2.2.1. TEMPERATURE-CONTROLLED STAGES.Regardless of the particular technique employed, the most obvious prerequisite is that there be a temperature environment with the capability of constant temperature control for long periods of time. For a polymer like linear polyethylene, the experimentally accessible temperature range may only be from 3 to 6", but over this range the growth rate will change by a factor of 1000. To make accurate growth measurements, it is necessary that the temperature bath or hot-cold stage should vary by no more than +O.O5"Cfor periods of time up to several weeks in duration. There are several constant temperature baths available commercially that will easily satisfy this specification. However, there are to our knowledge no microscope hot-cold stages commercially available that satisfy this requirement. In our own work we made stages that controlled the sample environment to within & O.O2"C, and where the sample temperature was accurately measured to +O.Ol"C. There are many proportionating temperature controllers commercially available that have the ability to sense and control temperature to 2 0.005"C, using either thermocouple junctions or thermistor sensors. The major problem in constructing the hot-stage environment is that of spatial dimensions, particularly the limitation of only a 2 cm clearance between the stage platform and the objective. Four our studies of the crystallization of linear polyethylene, we chose to use hot stages that were electrically heated, and two types were constructed. For those experiments in which only constancy of temperature was a prerequisite, we constructed stages having a high thermal mass. For those experiments demanding quick changes in temperature as well as eventual constancy, we constructed stages having such small thermal inertia that the thin fdm of sample itself was the limiting thermal mass. In both cases a square box having external dimensions 12.5 x 12.5 x 2.5 cm was constructed to house the sample and heating environment. The box was constructed of balsa wood planks 0.80 mm thick with sheets of aluminum foil laminated between them. Three balsa wood layers and two aluminum foil layers were used to construct the top, bottom, and sides of the box. The

358

10. NUCLEATION A N D CRYSTALLIZATION

layers of balsa were cross-grained from layer to layer and the whole sandwich was held together with silicone cement. Even though the temperature of the sample inside was above 150"C, the outer temperature of the top or bottom of the box was less than 40°C. A slow flow of room temperature air was passed between the condenser and the bottom of the box as well as between the top of the box and the objective. This air flow was sufficient to prevent any appreciable rise in temperature of the optics. While externally similar, the two types of cells were very different. For the low-thermal-inertia cell, the thin film of sample was positioned between two parallel grids each 10 x 10 cm. These heating grids were composed of a large number of taut parallel 0.025 mm diameter Advance metal heating wires, each wire being separated from the next by approximately 0.05 mm. Each wire was attached at its ends to a brass strip and the entire grid network held taut and parallel by springs. The sample itself was, of course, in the focal plane of the microscope being held and positioned by a micromanipulator attached to the metal stage of the microscope. The two heating grids above and below the sample were sufficiently far out of the focal plane as to remain invisible when the sample was observed through the microscope optics. A small bead thermistor on the sample was used as the temperature controller sensor. By far the largest thermal mass was in the sample itself, and the system was capable of exceedingly rapid temperature changes while still capable of excellent constancy of temperature when desired. In the other type of cell, relatively thick copper plates were used as a support for electrically insulated heating wire. The wire was in grooves cut into the copper plates, and the temperature sensing and control thermistors were embedded in the top surface of the bottom plate, next to the sample. Rapid temperature changes were impossible with this cell, but the constancy of temperature was very good. In both cases copperconstantan thermocouple junctions were used to measure the sample temperature. The temperature of the sample was continuously recorded using thermocouples that had been previously calibrated. A suitable reference such as a water-ice bath is needed. In our case, a water triplepoint cell was used. In both types of cells the two heating grids were electrically connected as a parallel resistive network, with a variable trim resistor between the two. The control sensor was always near the bottom grid, and the trim resistor was adjusted so as to have both grids at precisely the same temperature when operating at constant temperature. A differential thermocouple pair ofjunctions was used to make certain that the top grid had the same temperature as did the controlled bottom one. When using the low-thermal-mass cell, the sample temperature could

10.3.

MEASURING CRYSTALLIZATION RATES

359

be lowered in a controlled fashion at rates in excess of 10”per minute. Of course, the temperature could be raised at a much faster rate. With the cell having high thermal mass, the sample temperature could only be dropped at a rate somewhat less than 1” per minute. The sample temperature could be raised more rapidly but still at a much smaller rate than that obtained with the previously described cell. 10.3.2.2.2. THE“BATH” TECHNIQUE. There are several advantages to using the constant temperature bath over using an in situ microscope hot-cold stage: 1. The method does not require continuous use of a polarizing microscope or camera equipment. The measurements of the spherulite or axialite dimensions can be made at one’s own convenience. Even if a camera is employed it too is only in use for short periods of time. 2. The method requires many samples. Therefore, possible sample to sample variations are considered in the resulting growth statistics. 3. Because of the large thermal mass of the bath, short-term electronic variations or even complete power outages of a very short duration can be tolerated. With the hot stages any controller difficulties are almost immediately reflected by a change in temperature of the sample. 4. The shorter periods of time required for thermal equilibrium make it possible to determine the crystallization rates at larger undercoolings. 5 . It is easier to employ the self-seeding technique. However, there are also certain disadvantages in using the constant temperature baths:

1. Much larger quantities of sample are required. 2. Three baths are needed: one to melt the sample, one to crystallize the sample, and one to quench the sample. 3. It must be assumed that all spherulites commence to grow at the same time. 4. There is greater uncertainty with regard to the actual periphery of the growing crystal, i.e., measurement errors are usually greater. This technique is relatively simple and straightforward. The thin-film sample specimens are melted and then placed directly in the constanttemperature bath. A continuously running timer is used so that the removal of each sample can be chronologically logged. At appropriate time intervals a sample is removed from the bath and quickly quenched in to order to decorate the growth periphery by rapid crystalization producing a different morphology or by quenching the remaining liquid to a glass. The time that the sample was in the crystallizing bath is noted, and at the experimenter’s leisure, the dimensions of the spherulites can be deter-

3 60

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NUCLEATION A N D CRYSTALLIZATION

mined. Wherever possible, radial measurements should be made, several for each spherulite or axialite. By this technique, the entire area of the sample can be surveyed and a large number of measurements made. A plot of the average radial growth vs. time of crystallization should be linear. The intercept on the time axis is the induction time and will be zero if crystallization commenced immediately after temperature equilibrium occurred. The linearity of the growth vs. time plots is usually good providing that no measurements are made of spherulites that are close to impinging upon each other. 10.3.2.2.3. THE in Situ TECHNIQUE. This technique is similar to that employed with the constant-temperature baths, except that only one sample and one thermally controlled hot stage are used. The sample is melted by raising the temperature of the cell. After melting, the cell temperature is lowered to the desired crystallization temperature. In a relatively short period of time the cell is in thermal equilibrium and crystallization commences. If the type of cell having a large thermal mass is used, it is better to preset the cell temperature to the desired crystallizing temperature and to melt the sample using another thermal environment such as a hot bar. The sample is then inserted into the controlled cell and attached to the micromanipulator. With cells of low thermal mass the change in temperature from that required for melting to that for crystallizing can be performed quite rapidly. The need for having the sample come to the desired crystallizing temperature rapidly is that, with large undercoolings, no appreciable crystallization occurs prior to thermal equilibrium being attained. Unlike the method employing the constant temperature bath, when using the microscope, and a temperature-controlled stage, only a limited number of spherulites are observed, but each is observed during the entire time of crystallizing, i.e., the field is not changed. The measurement of the dimensions of such spherulites can be made visually employing a bifilar eyepiece, but it is usual to record the growth sequences on film. Again, a camera back using some type of Polaroid@film can be employed, but this is also a manual operation. It is therefore customary to use some kind of automated camera. Many researchers have found that a 16 mm camera-microscope system is best. In this system an intervalometer is used to advance the film and operate the shutter sequencing. The use of photographic techniques does impose some limitations on the experiment. One wishes to take many pictures and to have each exposure time as short as possible. This implies the use of a high-intensity light source, usually having associated with it a large amount of infrared illumination. To avoid heating of the sample by such radiation, the sample should only be illuminated during exposure of the film. This means that either some type of flash illumination should be employed or that an auxiliary shutter

10.3.

MEASURING CRYSTALLIZATION RATES

36 1

that only allows illumination of the sample during the film exposure be used. A xenon flash controlled by the intervalometer is ideal for this purpose. Heat filters should still be used. A high-speed film having small grain structure is advisable. As in any good cinephotomicrographic setup, the camera-microscope equipment should be protected from the adverse effects of vibration. In general the observations are done using a polarizing microscope. With crossed Nicol prisms (film-type elements deteriorate under intense illumination) the crystal appears brightly (due to birefringence) against the otherwise black (liquid) background. For the purposes of accurate spherulitic measurement it is more important to have the boundary of the crystal properly defined than to have internal spherulitic structural details visible on the film. For this purpose a high-resolution, high-contrast film has been found to be most satisfactory. The resulting photographs produced from such film show the spherulites’ boundary in sharp relief against a black background. In each isothermal growth experiment at least ten pictures taken at fixed intervals during the experiment should be used. Any deviation from linearity in the resulting growth vs. time plot should be viewed with suspicion. In the case of linear polyethylene, it was noted that the thin films, providing one did not use the growth near the periphery of the film, showed little or no effect of thermal degradation even when heated at crystallizing temperatures for several hundred hours. However, the experimenter must always be on the lookout for such effects. With either the in situ measurements or with the constant-temperature bath experiments, it is wise to repeat the crystallization studies at one temperature frequently in order to detect degradation in the sample. By this technique any change in growth characteristics of the sample can be easily noted. If degradation occurs it is often necessary to use a new set of sample preparations for each growth experiment. 10.3.2.2.4. ANALYSIS OF D A T A . No attempt has been made in this part to rigorously drive the equations relating to the kinetics of the crystallization of chain-folded polymers nor to examine critically the assumptions used in deriving the working relationships. For these details the reader is referred to the excellent review on the subject by Hoffman et ~ 1 . ~ The working relationship used in the analysis of growth rates obtained by any of the methods described in this section is Eq. (10.2.2) with the F*/kT factor replaced by U * / R ( T - T,) and a temperature correctionf to the driving force, as follows: G = Go exp[- U * / R ( T - T,)] exp[-K,(I/T

AT’],

(10.3.1)

where T, is taken to be the glass transition temperature minus 30°K. For

~

3 62

10.

NUCLEATION A N D CRYSTALLIZATION

some polymers the entire growth rate curve can be measured [e.g., isotactic p o l y ~ t y r e n enylon , ~ ~ 6,4Tpoly(tetramethy1-p-silpheylene siloxane) (TMPS),48 poly(chlorotrifluoroethylene),49 and poly(o~ypropylene)].~~ For other materials such as p ~ l y e t h y l e n e ,p~ly(oxyrnethylene),~~ ~~ and selenium,53growth rates near the maximum become so rapid that no measurements can be made and the low-temperature side of the curve has been inaccessible to date. Before any data for a given polymer can be analyzed, reasonable estimates for Tgand Tm0must be made. As mentioned previously, the experimentally observed melting points for polymers tend to be well below the value for a defect-free, extended-chain crystal both because of defects and because of the thinness of the crystal. The best methods presently available for determining Tm0are the construction of a T , vs. T , plot and extrapolating to the T , = T , line or using x-ray or electron-microscopic techniques to obtain values of I that are used in constructing a plot of T , vs. I//. Both methods are difficult and time consuming. If these types of data are not available one can “guess” at a reasonable value for Tm0by assuming that it is 6 4 ° C above the highest experimentally observed melting point for the polymer crystallized under normal conditions. A method developed in this laboratory for constructing T , vs. T, plots using differential thermal analysis is described in some detail in Section 10.3.4.3.5. Gopalan and Mandelkern” have also studied in detail the effect of molecular weight and the level of crystallinity on T , vs. T , for fractions of linear polyethylene using dilatometry . Estimates for values of T , and U* can be made by a method similar to that used by Suzuki and K o v a c if~ the ~ ~ entire growth curve is accessible. In this case, the curve for G(T) rises rapidly as the temperature is lowered, passes through a maximum, and then rapidly drops as the temperature is lowered even further (Fig. 4). The rise is controlled by the factor exp[ - ( K g / T ATf)and the fall at temperatures below the G ( T )maximum is largely controllkd by the factor exp[-(U*/R(T - Tm)].It is for this case that the values for U* and T can be determined with some confidence. T. Suzuki and A. J. Kovacs, Polym. J . 1, 82 (1970). J. H. Magill, Polymer 3, 655 (1962). 48 J. H. Magill, J . Polym. Sci.,Part A 27, 1187 (1969); ibid. 25, 89 (1967);J . Appl. Phys. 35, 3249 (1964). 4 9 J. D. Hoffman and J . J. Weeks, J . Chem. Phys. 37, 1723 (1962). J . H. Magill, Makromol. Chem. 86, 283 (1965). 51 J. D. Hoffman, G. S. Ross, L. J. Frolen, and J . I. Lauritzen, Jr., J . Res. Natl. Bur. Stand. 79, 671 (1975). 52 E. Baer and D. R. Carter, J . Appl. Phys. 35, 1895 (1964). R. G. Crystal, J . Polym. Sci.,Part A-2 8, 2153 (1970). M. Gopalan and L. Mandelkern, J . Phys. Chem. 71, 3833 (1967). 48 47

10.3.

36 3

MEASURING CRYSTALLIZATION RATES -3

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Briefly, the technique used, once the value of Tm0is determined, is to obtain the best fit of a linear plot of In G + U * / R ( T - T,) vs. 1/T AT’by varying values for U* and T,, where G is the measured linear growth rate. Hoffman et al.“ found that for a fairly large number of polymers, values of U* in the vicinity of 1000-1600 cal/mole and T, values about 30°K below Tg give excellent fits. Even in those cases where the entire growth curve is not accessible, a reasonable analysis can be carried out. At temperatures near Tmo, the slope K , is relatively insensitive to the selection of values for U* and T and values of 1500 cal/mole and Tg - 30”K, respectively, appear to be reasonable choices for most of the materials studied to date. Due to the

3 64

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NUCLEATION A N D CRYSTALLIZATION

very long extrapolations involved under these circumstances, one should not put too much reliance on the values for Go obtained. In the case of dilute solutions, Eq. (10.2.2) can be used to analyze the growth rate data in the same way as for crystallization from the melt. One simply replaces the factor U * / R ( T - T,) by AH*/RT, where AH* is the activation energy of diffusion in solution (= 1-3 kcal/mole), and replaces Tmoby the dissolution temperature T D o . TDois obtained by plotting the dissolution temperature vs. 1/1 and extrapolating to an infinite I value, i.e., 1 / 1 = 0.55 When dilute solution data are plotted in this way, the slopes of the lines obtained are nearly the same as those obtained for bulk samples, indicating that the fold surfaces that are obtained in both types of crystallization are thermodynamically similar. Once a value for G has been determined experimentally by any method such as optical microscopy, light scattering, or dilatometry (either in solution or bulk), the manner for obtaining u and ceis similar. One plots In G + U * / R ( T - T,) vs. 1/T ATf (In G + AH*/RT in the case of solutions) and determines the slope K , and the intercept Goby a least-squares fitting procedure. L a ~ r i t z e nand ~ ~'Hoffman and c o - w ~ r k e r shave ~ ~ shown that the value for K , (for bulk crystallization) depends upon whether growth occurs by a single nucleation act followed by rapid completion of the growth strip or by multiple nucleation acts followed by slow spreading. This behavior has been termed regime I and regime I1 growth, respectively. For regime I, K , = 4b0crueTmo/Ahfk,and for regime 11, K , = 2 b o a a eTmo/Ahfk. Here as before, bo is the monomolecular layer thickness, Ahf the heat of fusion, k the Boltzmann factor, and cuethe product of the lateral and end surface free energies. L a u r i t ~ e nhas ~ ~ devised a test to determine which type of nucleation occurs in a given polymer. The reader is referred to this paper for the details of the test. Polyoxymethylene and polypropylene conform to regime I, Poly(chlorotrifluoroethy1ene) has been shown to conform to regime I, at least ir. the nucleation-controlled growth region (i.e., at temperatures near the melting point) whereas isotactic polystyrene follows regime I1 type behavior. In polyethylene both types of behavior can be observed in the molecular-weight region from about 20,000 to 100,000. For a molecular weight of around 30,600 the transition occurs sharply with an attendant change of morphology (see Fig. 5 ) . Above 127°C axialites are formed and regime I1 type kinetics are observed. Maxwell and Mandelkern5' have observed these same effects in polyethylene fractions. ss T. W.

Huseby and H. E. Bair, J . Appl. Phys. 39, 4969 (1968). J. I. Lauritzen, Jr., J . Appl. Phys. 44, 4353 (1973). 57 J. Maxwell and L. Mandelkern, Macromolecules 10, 1141 (1977).

10.3.

MEASURING CRYSTALLIZATION RATES

365

FIG.5. Transition in growth behavior of a sample of polyethylene, M, = 30,600 as a function of the crystallization temperature (see also Fig. l).44

However, in their fractions the shift of regimes was more gradual as was the change in morphologies. Up to the present time no other polymer has been found that shows both behaviors in a single sample. However, the experimenter should be aware of the possibility and not just assume that measurements at two extreme temperatures will necessarily define the growth curve. Once the regime is established, it is a trivial matter to determine the product cuefrom K , . 10.3.2.3. The Intensity of Depolarized Light as a Measure of Crystallinity. When the average size of the spherulites is too small to be measured directly, or when the crystallization rate is too large to allow camera-microscope techniques to be used, it is still possible to modify the apparatus so as to obtain a great deal of useful information. A photomultiplier tube is substituted for the microscope eyepiece and/or cinephotographic equipment and the intensity of the depolarized light is measured or recorded as a function of time or of temperature. The technique is not only useful in determining isothermal growth rates, but also can be used to determine experimental melting points or solidsolid transition temperatures.

366

10. NUCLEATION A N D CRYSTALLIZATION

The experimental details are similar to those described previously using the cinephotographic equipment. The only major difference lies in the need to first calibrate the photomultiplier. For this, a series of calibrated neutral filters is used, measuring the transmitted intensity with a fixed source and only the polarizer in place. Once linearity is established, or departures from linearity known, the growth experiment is conducted in exactly the same fashion as if a camera were being used. The resulting sigmoidal plot of intensity vs. time provides in itself much useful information regarding the crystallization process. For a quantitative interpretation of the data, the intensity function is normalized. The Avrami equation (10.2.1.) is used, where (1 - x) is defined as ( I , - Zt)/(Zc - lo). I, is the intensity at any time t , I , the initial intensity of the depolarized light when the sample is liquid, and I, the intensity of the light scattered from the crystallized sample. The normalization cornpensates for both changes in illumination intensity and changes in sample thickness as different samples are employed. The above definition of 1 - x implies that the intensity of the depolarized light is directly proportional to the quantity of crystalline material present. Such an assumption is reasonably well satisfied if the final spherulitic size is less than the thickness of the sample, a condition that requires a profuse spherulitic growth. The Avrami integer n is the slope of the plot of In[ - In( 1 - x)] vs. In t. The rate constant K is the slope of the plot of [-In( 1 - x)] vs. tn. The rate constant can also be calculated from the half-time of crystallization tl,z,using the relationship K = In 2/(t1,#. This technique has been used in investigating the growth kinetics of many polymers including nylon 6,5* nylon 6,6,5g poly(3-methyl butenepoly(Cmethy1 pentene- 1),61 isotactic p o l y ( p r ~ p y l e n e ) , ~and ~-~~ poly(ethy1ene t e r e ~ h t h a l a t e ) . ~ ~ 10.3.2.4. The Measurement of Crystal Growth by Means of Light Scattering. The technique of measuring the growth of spherulites by correlation with intensity curves from light scattering is a very powerful one. The polarization direction is quite important; different information is obtainable depending on whether the scattered light is observed using the J. H. Magill, Polymer 3, 43 and 655 (1962). J. H. Magill, Polymer 2, 221 (1961). 6o 1. Kirshenbaum, R. B. Isaacson, and W. C. Feist, J . Polym. Sci., Part B 2, 897 (1964). 1. Kirshenbaum, W. C. Feist, and R. B. Isaacson, J . Appl. Polym. Sci. 9, 3023 (1965). J . H. Magill, Polymer 3, 35 (1962). J. H. Magill, Nature (London) 191, 1092 (1961). C. W. Hock and J. F. Arbogast, Anal. G e m . 33,462 (1961). 65 K. G . Mayhan, W. J. James, and W. Bosch, J . Appl. Polym. Sci. 9, 3605 (1965). 58 58

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H, arrangement (incident light is vertically polarized, whereas the analyzer is horizontal) or by using the V, configuration (both polarizer and analyzer vertical). The apparatus is basically very simple. A light source is polarized. The polarized beam travels through a thin-film sample in which scattering occurs. This scattered light travels through the analyzer and onto a screen perpendicular to the optic axis. The spherulitic radius is defined by the following relationship:

R

=

U h [ h sin(em/2)],

(10.3.2)

where is the average spherulitic radius, A the wavelength of the light source, emthe scattering angle in the polymer sample, and U a form factor having a value close to 4. When the H, mode of analyzer-polarizer is used, the observed scattering pattern is that of a cloverleaf with four lobes. 8, is the scattering angle corresponding to the point of maximum light intensity in the lobes. The light source should be of high intensity; mercury lamps and laser sources are perhaps the most common. Either high-speed photography or a mobile photocell scanner may be used. The scanning photocell moves back and forth between two opposite leaves of the cloverleaf. With the scanning device it is also possible to have a cell permanently fixed at the optic axis, which continually records the intensity of the depolarized light. Thus, a plot of intensity vs. time can be used to determine the overall crystal growth as was described previously in Section 10.3.2.3, while at the same time determining the radial growth of the average spherulite. The use of wide apertures allows the measurement to be averaged over many spherulites; conversely, very small apertures or slits may allow observations to be made on sections of single spherulites. When the light-scattering technique is used as well as measurements using the optical microscope, excellent agreement exists between the two methods, particularly in the range of spherulitic size from 1-50 pm. The method is particularly valuable when the profuse growth of spherulites prevents the attainment of the size necessary for the observations using the optical microscope. In practice, the apparatus usually has the thin-film sample mounted in a thermostated environment that includes an additional compartment for temperature preconditioning. This separate compartment may be a hot oven used for melting the sample or may be double-compartmented, having one compartment for melting and the other for quick quenching in order to produce a glass. The apparatus is very useful in observing the entire growth curve, which in some instances is experimentally accessible for over 100". In

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those instances where a glass can be formed, the growth can be studied when the sample is warmed from the glassy state as well as when it is cooled from the liquid state. At any given growth temperature the resulting growth rates may be different depending on whether the growth temperature was approached from a molten or a glassy sample. There are many references to the use of the light-scattering technique to the problems of crystal growth in polymers. For further information on the development of theories and applications of the method, the reader is referred to Stein et ul.sa-72 10.3.2.5. Infrared Absorption Technique. Infrared studies can be carried out on any good spectrophotometer using prisms of sodium chloride, potassium bromide, calcium fluoride, etc. The usual technique is to place a film of polymer along with a thermocouple between sodium chloride or other suitable plates and to enclose the specimen in a heating cell mounted on the spectrophotometer. (The sample can also be ground up and a potassium bromide disk containing approximately 1% polymer can be prepared. The disk can be placed in a temperature-controlled environment and the spectrum scanned at different temperatures. Care must be taken in the grinding process since it is known that this process can lead to changes in the polymersample.) Using an instrument that has both preand postsample chopping of the beam to be certain that radiation from the sample at elevated temperatures does not lead to errors in the absorbance measurements is also recommended. Initially, the spectrum is run as a function of temperature, the sample being cooled from above its melting point to room temperature. Generally, bands can be assigned to crystal and amorphous regions. Once these bands are assigned the intensity of each band or bands can be measured as a function of temperature and time. The intensity D is defined as D = log(Zo/Z), where Zo is the height of the background trace and Z the height of the band peak. The thickness of the sample, of course, must be constant. If one defines 0,as the measured intensity of the crystalline sample, D, as the intensity for the melt or amorphous region, and D t as the intensity of the band at time t, the Avrami equation becomes 1 - y, = (D,- D t ) / ( D ,- 0,). Frequently, a ratio is taken between the intensities of the crystalline and amorphous bands. This procedure elimiA. Plaza and R. S. Stein, J . Polym. Sci. 40, 267 (1959). R. S. Stein and M. B . Rhodes, J . Appl. Phys. 31, N o . 1 1 , 1983 (1960). Ba R. S. Stein and P. R. Wilson, J . Appl. Phys. 33, 1914 (1972). V. G . Baranov, A. V. Kenarov, and T. I. Volkov, J . Po/ym. Sci., Purr C 30,271 (1970). F. van Antwerpen and D. W. van Krevelen, J . Polym. Sci., Purr A-2 10,2409 (1972). 71 F. van Antwerpen and D. W. van Krevelen, J . Polym. Sci., Purt A-2 10, 2423 (1972). 72 R. S. Stein and A. Misra, J . Polym. Sci. 11, 109 (1973). O7

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MEASURING CRYSTALLIZATION RATES

369

nates the problem of variations in the thickness of the samples. The crystallization rate constant can be calculated from the half-time of crystallization fIl2 using the usual relationship K , = In 2 / ( 1 ~ / ~ The ) ~ . K , values are then analyzed in the usual way. Several studies comparing the results of an Avrami-type analysis on data obtained from density m e a s u r e m e n t ~dilat~metry,’~ ,~~ and x-ray diffraction showed essential agreement between the results of these measurements and those obtained by careful infrared techniques. 10.3.2.6. The Use of Dielectric Measurements for Determining Crystallization Rates.? In 1963 it was that the measurement of the relative permittivity (dielectric constant) as a function of the fraction melted for a pure material was a very effective method for measuring purity in such systems. For most polymers there exists a linear relationship between the relative permittivity and the fraction of polymer crystallized. In general, the relative permittivity has its maximum value for the amorphous phase and its minimum value for the crystalline phase. It is therefore possible to measure the development of crystallinity in a sample at different temperatures either by measuring the change of the relative permittivity or the capacity of a capacitor filled with sample as a function of both time and temperature. The resulting data are analyzed using the usual Avrami relationship, Eq. (10.2.1.). In this case 1 - x = (C, - C , ) / ( C , - Ca),where Ct is the relative permittivity at time t , C , the relative permittivity for the pure crystalline phase, and C , the relative permittivity for the melted sample. The value of 1 - x is related to the growth rate constant K in the usual way. An application of this technique to a study of the crystallization of Neoprene is described in a paper by Simek and M i i l l e ~ - . ~ ~ 10.3.3. Methods Using a Bulk-Type Specimen 10.3.3.1. Sample Preparation. Generally, “as-received” samples are not suitable for dilatometric or other related techniques that require a bulk-type sample. The method required to clean the sample is highly dependent on the material to be studied but generally requires dissolution in a suitable solvent, filtration, precipitation, and drying under vacuum T. Okada and L. Mandelkern, J . Polym. Sci., Purr A-2 5, 239 (1967). M. Hatano and S. Kambara, Polymer 2, 1 (1961). 75 C. Baker, W. F. Maddams, and J. E. Preedy, J . Polym. Sci., Polym. Phys. Ed. 15, 1041 (1977). 76 G . S. Ross and L. J . Frolen, J . Res. Nntl. Bur. Stand., Sect. A 67, 607 (1963). 77 I. Simek and F. H. Muller, Kolloid-Z. & 2. Polym. 234, 1092 (1968). 73 74

t See also Volume 6B, Chapter 7.1.

3 70

10. NUCLEATION A N D CRYSTALLIZATION

conditions to remove the solvent. Frequently, at the high temperatures needed to dissolve the samples, oxidative degradation occurs and one must always check with GPC,viscosity, or other similar measurements to be certain that the sample has not been adversely affected during the process. Addition of a suitable antioxidant or operation under vacuum or in an inert atmosphere is required if degradation is likely to occur. As mentioned previously, the crystallization usually proceeds via a heterogeneous nucleation step. The number of heterogeneities can be controlled to some extent. Heating the sample in the apparatus to a temperature well above the melting point destroys some of these nuclei. The cleaning and filtering techniques described in Section 10.3.2.1 are also effective in reducing the number of active nuclei. Reducing the number of active sites produces fewer crystals in the sample and consequently allows one to follow the crystallization process for a longer period of time prior to the impingement of adjacent crystals. However, particularly at crystallization temperatures near the melting point, the process becomes very slow and one often wishes to have many crystallization sites in order to shorten the period needed for observation. This can be accomplished using the self-seeding technique described in Section 10.3.5.1. Once the pretreatment is completed the sample must be prepared in a suitable shape and size for use. In the case of dilatometry the sizes of the sample can be varied from as small as 50 mg to as large as 100 g depending on the design of the dilatometer, i.e., the size of the sample bulb and the length and diameter of the capillary tube, the thermal properties of the confining liquid, and the thermal properties of the polymer itself. Once the size of the sample has been set, the sample is generally molded at a suitable temperature into the desired shape, preferably under vacuum to assure that no voids are contained in the sample. Sheets of rolled material can be cut into strips, polymer can be extruded in the form of strings, or samples can be molded into various shapes and sizes using commercially available heated presses. 10.3.3.2. Dilatometric Techniques. Volume dilatometers have been found to be very useful in studying the development of crystallinity in a large number of polymer systems. The method is relatively simple and the volume changes are a very sensitive measure of changes in crystallinity. It has the further advantage that the temperatures can be controlled for the long periods of time required when polymers are crystallized at temperatures near their melting points. The apparatus and procedure most generally used in performing dilatometric experiments are similar to the ones developed by BekkedahP in his early studies on the crystallization of natural rubber. The published N. Bekkedahl, J . Res. Natl. Bur. Stand. 42,

145 (1949).

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MEASURING CRYSTALLIZATION RATES

37 I

work on the use of dilatometry generally tends to emphasize the results obtained by the method rather than a detailed description of the method. Bekkedahl's paper is highly recommended to anyone contemplating the use of dilatometry in that it describes construction of the dilatometer, calibration techniques, the method for adding the sample, as well as the choice of and the method for adding the confining liquid. In addition, the experimental technique is presented including a detailed procedure for performing the necessary calculations (well illustrated using the data he obtained for butyl rubber). A detailed discussion of errors involved as well as the overall precision of the method is also included. Once the dilatometer is loaded with the polymer specimen and a confining liquid such as mercury, the dilatometer is heated (usually in an oil bath) to a temperature TI, which is 10-20" above the melting point for the period of time necessary to be certain that all of the sample has melted. The dilatometer is then placed in a second thermostated bath, which is set at the desired crystallization temperature T , . Once thermal equilibrium is established, the height of the mercury in the capillary is read with a good cathetometer as a function of time. Initially as the polymer crystallizes, the mercury level drops fairly rapidly (state I, primary crystallization) but, as the polymer approaches its maximum level of crystallinity, the mercury level changes very slowly and for a long period of time (stage 11, secondary crystallization). The specific volume of the polymer can be calculated from the changes in height of the mercury. Once the values f o r v are determined, x,the degree of crystallinity, can be calculated for each measured point using the relationship 1-x = Herev, is the specific volume of the supercooled amorphous phase,vc the specific volume of the crystal (usually determined as a function of temperature from X-ray data), a n d v , the specific volume of the polymer at any time t (usually obtained from density measurements). In the bulk crystallization of polymer systems the Avrami relationship holds reasonably well only for the initial stage of the crystallization (stage I) prior to the time when the crystals impinge, as long as no new nuclei are formed as a function of time. As mentioned previously, the Avrami n depends on the type of nucleation that occurs as well as the shape of the crystalline entity formed. For this reason, it is recommended that a thin-film specimen of the polymer be examined with an optical microscope in the. desired range of crystallization temperatures to find out whether homogeneous or heterogeneous nucleation occurs, as well as to gain information concerning the crystal forms that may be present at different temperatures. For spherulitic growth, where the growth rate G is constant with time

v

(v, vc)/(v,v,).

372

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NUCLEATION A N D CRYSTALLIZATION

and the Avrami n is 3 when the nucleation is heterogeneous, the relationship x = xw[l - exp(-Kt3)] describes each sigmoidal-shaped isotherm and the isotherms are superimposable. Here xw is the value of x at the onset of stage I1 crystallization; K is the bulk crystallization rate constant and is defined as 4rv,,G3/3, where yo is the number of nucleation centers per cm3 at the beginning of the experiment. G can be determined from a set of experiments at several temperatures if vo is known or if vo does not If one then plots change as a function of temperature. log 4rvoG3/3 + U*/2.303R(T - T,) vs. 1/T AT, the slope is equal to -4bocr~eT,0/2.303 Ahfk and the value for C T C T ~can be calculated (see Section 10.3.2.2.4). The dilatometric technique has been applied to determine the bulk crystallization rates for a very large number of polymers. Wunderlich has collected information on many of these systems along with the appropriate references in Table VI.8, p. 216, in his recent book.4 Mandelkern’O has examined in detail the application of the Avrami equation to partially crystalline polymers. Kovacs and MansonaOhave used a differential-type system to study crystallization from solution. MagilP has studied a polymer system that contains two or more types of spherulites at the same time, each with its own nucleation and growth kinetics. Doll and Lando,a2Armeniades and B a e ~ and ,~~ Maeda and Kanetsuna,fflto mention a few, give examples of the use of high-pressure dilatometry in studies of the effect of pressure on bulk crystallization and melting behavior. 10.3.3.3. The Density Balance Technique. In this method a fairly large sample (approximately 4 g) is melted while surrounded by an inert contact liquid, and then transferred to a second bath, which is maintained at the crystallization temperature. The weight of the sample in liquid is measured continuously as crystallization proceeds by means of a suitable balance. The density of the sample is calculated at any time from the equation

+

d = dLWO/(WO - W ) ,

(10.3.3)

where d Lis the density of the contact liquid, Wothe weight of the polymer sample in V C I C U O , and W its weight in the oil. L. Mandelkem, “Crystallization of Polymers,” Chapter 8. McGraw-Hill, New York, 1964. A. J. Kovacs and J. A. Manson, Kolloid-2. & Z . Polym. 214, 1 (1966). J. H. Magill, J . Polym. Sci.. Part A 4, 243 (1966). 82 W. W. Doll and J . B. Lando, J . Macromol. Sci.. Phys. 2, 219 (1968). C. D. Armeniades and E. Baer, J . Macromol. Sci., Phys. 1, 309 (1967). Y. Maeda and H. Kanetsuna, J . Polym. Sci., Polym. Phys. Ed. 13, 637 (1975).

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373

The experimental technique most commonly used is as described by Keller, Lester, and Morgane5 in their study on poly(ethy1ene terephthalate). The contact liquid used in their experiments was silicone oil. A container that consisted of two concentric. cylinders mounted on a common base was constructed from fine-mesh stainless steel gauze. This fine gauze was not only a suitable container for the molten polymer, but ensured good contact between the surface of the polymer under study and the contact liquid, making it possible to obtain an accurate record of the change in weight of the polymer as crystallization proceeds. Since polymers frequently react with moisture when they are fused it is usually important to remove all traces of moisture. This is done by heating the polymer, preferably in a powdered form, to an elevated temperature (below the melting point) under vacuum for several hours. After drying, the polymer is placed in the stainless steel container and again evacuated to remove all entrained air. The gauze container is then placed in a closely fitting, easily detachable glass tube filled with silicone oil or other suitable contact liquid while still under vacuum. Two silicone oil baths are placed under the balance, one for melting the sample and the other for crystallizing it. A convenient method for maintaining the oil at a constant temperature is by using vapor baths. Any temperture can be maintained by boiling a suitable liquid in an outer jacket surrounding the oil. The pressure on the boiling liquid is adjusted and maintained with suitable manostates and the temperature can be easily controlled to within +O.l"C. The sample, dried and immersed in silicone oil as described above, is immersed in the melting bath for approximately 15 minutes, and then quickly transferred to the second bath, which is maintained at the preselected crystallizing temperature; the glass tube is detached and allowed to sink to the bottom of the boil bath, and the gauze container and the polymer sample are suspended from the arm of the balance. Thermal equilibrium is generally attained within 10 minutes. After equilibrium is reached, the apparent weight of the sample is measured at suitable time intervals (2 to 3 minutes) until it becomes constant. Weighings should be to the nearest milligram. A semiautomatic balancing device, incorporating a photocell that has been described by Allen and Wrighte6can be used to greatly reduce the tedium of these measurements. When the experiments are complete, density-time curves for each experimental crystallization temperature can be constructed by plotting the fraction phase change 1 - x against time. Here xI is the fraction of 85 A . Keller, G . R . Lester, and L. B. Morgan, Philos. Trans., R . Soc. London, Ser. A 247, l(1954). 86 P. W. Allen and R. A . Wright, J . Sci. Insirurn. 29, 235 (1952).

3 74

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NUCLEATION A N D CRYSTALLIZATION

remaining original amorphous phase and is calculated from the density pt at time t , and the initial and final density for the crystallization, pi and pf, from the expression 1 - x = ( pf - p t ) / (pf - p i ) . The initial density pi is taken to be the density of the molten polymer prior to the onset of crystallization and the final density pf is the density of the sample when crystallization is effectively completed. Once the density-time curves are obtained, the crystallization constant K in the expression 1 - x = exp( - K t n )can be determined. The detailed analysis of the experiments are the same as those discussed in Section 10.3.3.2. 10.3.3.4. The Density Gradient Technique. In the late 1930s, Linderstrgm-Lang and c ~ - w o r k e r s ~developed ~ - ~ ~ the density gradient method to a high degree of perfection. The usual form of the apparatus consists of two fairly large glass reservoirs connected by a vertical glass tube whose volume is relatively small. The bottom reservoir and lower half of the tube are filled with a high-density solvent or solution and the upper half of the tube is filled with a less dense solvent, which is miscible with the first material. The density at either end of the vertical tube is controlled by the density of the original liquids in the reservoirs and an essentially linear, vertical density gradient develops in 24-48 hours, which is stable for weeks and months, through the process of mutual diffusion of the liquids. By means of partial mixing or layering solutions of different densities, a stable gradient can be developed in a period of approximately 6 hours.OO The steepness of the gradient can be controlled by the choice of liquids having the proper densities and under ideal temperature conditions and in the absence of vibrations, densities can be determined accurately to six decimal places.88 The gradient tube must be carefully calibrated before use with materials of known specific gravity (i.e., aqueous zinc chloride solutions, glass floats). When the organic liquids are used in the gradient tube one must be certain that the liquids are neither adsorbed nor react with either the amorphous or the crystalline component of the polymer under study, for otherwise large errors would result. The samples used in this method are generally small pieces of quenched amorphous polymer that are crystallized at elevated temperatures and removed periodically from the crystallizing bath, washed and dried at room temperature, dropped into the density gradient column, and the density K. Linderstrgm-Lang,Nature (London) 139, 713 (1937). K. Linderstrgm-Langand H. Lanz, Jr., Mikrochim. Acra 3, 210 (1938). K.Linderstrgm-Lang,0. Jacobson, and G . Johansen, C . R. Trav. Lab. Carlsberg, Ser. Chim. 23, 17 (1938). R. F. Boyer, R. S . Spencer, and R. M. Wiley, J . Polym. Sci. 1, 249 (1946). sB

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3 75

determined. The theory and the analysis of the data obtained by this method are discussed in detail in Section 10.3.3.2. 10.3.4. Differential Thermal Analysis (DTA) as Applied to the Determination of Tm0

The difficulties associated with experimentally determined the melting point Tm0of a semicrystalline polymer have been discussed in Section 10.2.2.2.4. The procedure involves crystallizing a sample at temperature T, , and then determining its melting point T , . A plot of these T , ,T , pairs is supposed to result in a linear plot that intersects the line T, = T , at the temperature Tm0,which is the melting point of the perfect extended-chain crystal.43 Such a series of experiments can be accomplished using differential thermal analyzers. The experiments can be performed in several different ways, some of which are described below. 10.3.4.1. Crystallization Procedures. 10.3.4.1.1. COMPLETECRYSTALLIZATION. A few milligrams of powdered sample are placed in a DTA cell, melted, and then crystallized in situ at the desired T,. The thermogram is recorded. In our experiments with linear polyethylene the sample was held at the crystallizing temperature for 12 hours, the time necessary for the sample to reach its maximum crystallinity at the highest crystallizing temperature employed. The assumption was that most of the annealing or thickening would be accomplished during this 12 hour period. 10.3.4.1.2. PARTIALCRYSTALLIZATION. In this series of experiments, samples were melted and brought to the crystallization temperature. When the thermograph trace showed that from 0.5 to 1% crystallization had taken place, the melting sequence was started. In these instances the sample was held at the crystallization temperature for only the time necessary to achieve the partial crystallization. 10.3.4.2. Melting Procedures. 10.3.4.2.1. FROMCOMPLETE CRYSTALLIZATION WITH CONTINUOUS HEATING.The samples were first lowered to a temperature approximately 50" below T , . During this time appreciable crystallization takes place. The sample is then heated at a constant rate until the thermogram indicates that all melting has taken place. Figure 6 depicts such a melting sequence. 10.3.4.2.2. FROMCOMPLETECRYSTALLIZATION WITH DISCONTINUOUS OR STEPWISE HEATING.The sample may be lowered in temperature as described above, or the heating sequence can commence from the crystallization temperature T , . In either case the sample is heated very rapidly from an initial temperature TI to a temperature Tz , this temperature not being sufficiently high to melt the sample. After the thermogram

376

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FIG.6. Thermogram of the melting curve of a fully crystallized sample of polyethylene. The sample was heated at a constant rate.

trace has returned to the baseline, the sample is again rapidly raised to a temperature T 3 . Again, after the trace returns to the baseline, the process is continued until finally the melting is complete. The process is continued for one or two more steps until the experimenter is certain the thermogram shows only the heat capacity of the liquid (see Fig. 7). 10.3.4.2.3. MELTINGFOLLOWING PARTIAL CRYSTALLIZATION. In this technique one tries to find the temperature at which the sample can be set to just produce complete melting. This requires a series of experiments at each T, . The melting point is verified by incrementally raising the temperature afterwards and only producing the thermogram corresponding to the liquid heat capacity (see Fig. 8). Note: The technique of melting all of the crystallized sample by immediately raising the sample to its melting point may also be applied to a sample that has been completely crystallized. Figure 9 shows such a sequence using a pure indium sample. 10.3.4.3.Analysis Of Data. 10.3.4.3.1. WHEN THE SAMPLE IS COMPLETELY CRYSTALLIZED A N D THEN MELTED WITH CONTINUOUS HEATING.The heating rates used ranged from 1 to 10"per minute. Once temperatures were properly corrected for the effect of rate, there was little change in the determined melting point that could be associated with the rate of heating. As shown in Fig. 6 approximately 20% of the sample crystallized during the cooling from T , to T , - 50°C (the shaded area of the thermogram). Polymers behave as if they are impure, showing a

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377

FIG.7. Stepwise melting of a completely crystallized sample of polyethylene, first heating from T to T I producing area 1. After equilibration the temperature is raised to T 2 ,producing area 2, etc. The equilibration curves (times) have been shortened in this schematic representation. The shaded areas represent recrystallization.

I

time

-+

time

-+

FIG.8. Stepwise melting (heating) curves for partially crystallized (x 5 0.01) polyethylene. Both figures are obtained for the same sample, crystallized at the same temperature.

378

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change in melting temperature as more and more of the material is melted. The two extreme melting points that can be uniquely defined are T , , the temperature obtained by extrapolating the melting ramp back to the baseline, and Tb,the temperature corresponding to the melting temperature of the last crystals. In addition to these two extreme temperatures, it is also possible to determine the melting point of any fraction melted, this fractionfbeing defined as the ratio of the area of the thermogram representing that portion melted to the area representing the complete melt. In Fig. 6 the ratio of the area B, which includes the shaded area, against the total area A of the thermogram would be one such fraction frozen, f. In analogy with simple materials that contain a moderate amount of impurity, a plot of l/f; versus Tiis linear over llfvalues between 1.5 and 5 . If the polymer system behaved as the simple systems, T , would be that temperature corresponding to the l/f = 0 intercept. Such polymeric systems most certainly are not behaving as do the nearly pure small-molecule crystalline systems, but it is interesting that (a) the llfplot is quite linear, and (b) as will be shown later, the T, vs. T, plots are also linear. 10.3.4.3.2. DISCONTINUOUS (STEPWISE) HEATING. In this type of experiment, the value of the temperature at which the last crystals are melted is taken as T,. Figure 7 clearly shows that there is a great deal of recrystallization occurring (shaded areas) as the sample is partially melted. While it is obvious that the shaded areas (exothermic portions) represent some type of recrystallization process, the shapes of endothermic portions also suggest that there is a rather complex process going on during the partial melting sequence. Even when the melting sequence begins at the temperature of crystallization T , , the accumulative area of the thermograms, excluding the shaded areas, is still larger than that obtained during the crystallization process. However, as with melting that occurs by a continuous heating, a l/fvs. Tplot can be made. Here again, such a plot is quite linear, and a similar T , is obtained at the l/f = 0 intercept. Such a plot is reconstructed after first subtracting that portion of area due to the simple heating of the sample, i.e., the heat capacity. PROCEEDING TO THE T,. 10.3.4.3.3. MELTING BY IMMEDIATELY When samples are immediately raised to their melting point, a second peak or shoulder appears on the forward ramp of the thermogram. As shown in Fig. 9, this secondary peak is due to the heat capacity of the system. When this portion, area a is subtracted from the thermogram, the resulting area b represents the heat of fusion for the sample. The heat of fusion determined and that determined by the continuous heating method differ from each other by less than 0.5% in the case of indium. When the process is applied to polymers that have reached their maximum level of crystallinity, the resulting areas after subtracting that por-

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379

time -+ FIG.9. Melting of a pure indium sample. Starting at a temperature of T , - 1O"C, the sample was completely melted by rapidly raising the temperature to T, . The thermogram of area a alone was produced by starting at a temperature of T , - 10.2"C, and rapidly raising the temperature to T , - 0.2"C.

tion due to the heat capacity of the crystal is still larger than the area under the crystallization thermogram even when melting proceeds directly. Again this implies that the melting process is not a simple one. In the case of polymers even the shape of the melting curve changes as a function of T , . 1 0 . 3 . 4 . 3 . 4 . PARTIALLY CRYSTALLIZED SAMPLE. A series of experiments are needed to find the exact temperature at which the sample is completely melted. As stated previously, this is verified by the size and shape of the subsequently determined liquid heat capacity peaks (see Fig. 8). The melting curves are always double-peaked, again due to the heat capacity. Subtraction of this portion from the area under the thermogram gives an area identical to that found during the partial crystallization, indicating that little or no change occurs during the melting process. Figure 8A,B depict partial and complete melting on the first stepwise rise in temperature from samples crystallized to a x of 0 . 0 1 . The figures show the results of only 0 . 1 " C change in the initial temperature rise. The sensitivity of the method is clearly shown. 10.3.4.4. The T , vs. T, Plot. Figure 10 shows the results of plotting T,,, vs. T, for the various methods described. For this particular polyethylene fraction, M , = 30,600 and the polydispersity M , / M , = 1 . 1 9 . Anywhere from three to eight crystallization temperatures were used. There was an appreciable amount of scatter of the points for any particular method. Some of this work is still in progress and Fig. 10 only serves to

10.

3 80

NUCLEATION A N D CRYSTALLIZATION

FIG. 10. (1) Peak Tms vs. T , (continuous heating); (2) baseline Tms vs. T , (continuous heating); (3) l/f(stepwise heating); (4) I/f(continuous heating); (5) rapid, single-step heating from completely crystallized sample; and (6) single-step heating from partial crystallization (x < 0.01).

show the type of results that can be obtained. The temperature intercepts of the various plots and their T , - T , relationships as shown in Fig. 10 are given by: (a) From complete crystallization (1) Using a continuous heating technique (i) T , from baseline intercept:

T , = 47.457 (ii)

T,

+ O.6571Tx,

Tm0 = 138.40"C.

T , from peak melting temperatures: =

85.119

+ O.3935Tx,

Tm0 = 140.34"C.

(iii) T , from llfplot:

T , = 82.151

+ 0.4O9Tx,

Tm0= 139.00"C.

(2) T , from stepwise heating (l/f): Tm0 = 140.13"C. T , = 86.812 + 0.38O5Tx,

10.3.

MEASURING CRYSTALLIZATION RATES

38 1

(3) T, from complete crystallization, one-step melting: T, = 70.1

+ 0.501Tx,

(b) From partial crystallization T, = 52.54

Tm0= 140.48"C.

(x 5 0.01), one-step melting:

+ 0.6214Tx,

Tm0= 138.77"C.

It should be noted that using the theoretical predictions of Flory and Vrij4I with Tm0= 146.5"C,the T, value for a polyethylene sample of this molecular weight would be 144.6"C. Gopalan and MandelkernS4 in a study of the effect of crystallization temperature and molecular weight on the melting temperature of polyethylene, examined the behavior of several different fractions using dilatometry. In a sample (M,+20,000) they found extrapolated melting points of 141.3 - 0.3" and 137.7 & 0.2" for crystallinities of 0.05 and 0.8, respectively. These authors also reported a value for Tm0of 146.0° & 0.5 for a high-molecular-weight sample having a low degree of crystallinity. WeeksQoahad previously reported a value of 1453°C for a high-molecular-weight sample of polyethylene using dilatometric techniques. The particular differential thermal analyzer that was used in the experiments described herein was the Mettler TA2000.t The minimal temperature difference that can be set is 0.1"C. Consequently, none of the T,s obtained by the incremental heating can be reported with better precision than 0.1". In general, new samples were used for each melting experiment. No visible evidence of sample deterioration was observed as a result of the heating or crystallization process. Even with replicate experiments there was an unexpectedly large variation in the T,s obtained, as much as 0.5"Cbetween duplicate experiments. This is not an uncertainty that can be attributed to the apparatus and again shows the complexity of the crystallization process that occurs in even so-called simple linear polymers. The DTA itself is a very useful tool in the study of crystallization ofpolymers, but there still remains agreat deal that is not understood. 10.3.5. Crystallization from Solution 10.3.5.1. The Self-seeding Technique in Solution and in the Melt. When it is necessary to measure growth rates, either in solution or in Boa

J . J. Weeks, J . R E S .Natl. Bur. Stand., Sect. A 67, 441 (1963).

t Certain commercial equipment, instruments, or materials are identified in this part in order to adequately specify the experimental procedure. In no case does such identification imply recommendation or endorsement by the National Bureau of Standards, nor does it imply that the material or equipment identified is necessarily the best available for the purpose.

382

10. NUCLEATION A N D CRYSTALLIZATION

bulk, close to the dissolution temperature or the melting point of a sample, the experiments often must extend for weeks or months by the usual techniques. In 1966, Blundell, Keller, and Kovacsgl published a note describing the self-seeding technique in the growth of polymer crystals from solution. Briefly, they found while doing a series of dilatometer runs on dilute solutions of polyethylene and a two-block copolymer sample of poly(ethy1ene oxide-b-styrene) that the overall rate of crystallization was strongly dependent on T,, the temperature to which the suspension was heated. The rate of crystallization decreased with increasing T, up to a limiting dissolution temperature T beyond which it remained constant. Their conclusion was that there must be nucleation centers that survive the heating to T, , which speed up the subsequent crystallization, and the total number of nuclei decreases with increasing T , up to T, where they all disappear or at least reach a limiting value. Further work by Blundell and KellerQ2 showed that the nuclei consist of highmolecular-weight material that is stabilized by refolding during the period of time required to heat the sample to T,. In a later paper, Blundell and Kellerg3describe in detail the experimental technique developed for the preparation of polyethylene crystals from dilute solution that were uniform, single-layer crystals of controlled crystal habit. The development of the use of the self-seeding technique in bulk polymers was largely due to the work of Kovacs and c o - w o r k e r ~ . ~In* ~ ~ some very beautifully done studies on poly(ethy1ene oxide), it was found that growth starts simultaneously from all the persisting nuclei and gives rise to crystalline units (single crystals, hedrites, or spherulites) of identical shape and size. The concentration of the nuclei can be controlled between large limits by an appropriate thermal treatment of the sample. The experimental technique used for measuring the growth rates of poly(ethy1ene oxide) from the melt was as follows. A 5-10 mg sample contained between 'two microscope cover glasses was melted at about 80°C and subsequently crystallized at room temperature. The sample was then heated in a reproducible way to a temperature 1-2" below the melting point, where the major part of the sample melted, leaving seed nuclei that represent a volume fraction of the order of lo-* or less of the sample. The sample was then rapidly cooled to the desired crystallization temperature and allowed to crystallize for varying periods of time. The growth was stopped by quenching the sample in a dry ice-acetone D. J. Blundell, A . Keller, and A. J. Kovacs, J . Polym. Sci., Part B 4, 481 (1966).

* D. J. Blundell and A. Keller, J . Macromol. Sci., Phys. 2, 301 (1968). (L9

O5

D. J. Blundell and A. Keller, J . Macromol. Sci., Phys. 2, 337 (1968). G . Vidotto, D. Levy, and A. J. Kovacs, Kolloid-Z. & Z . Polym. 230, 1 (1969). A. J. Kovacs and A. Gonthier, Kolloid-Z. & Z . Polym. 250, 530 (1972).

10.3.

MEASURING CRYSTALLIZATION RATES

383

mixture. This resulted in a self-decorated crystal that was very easy to measure. Since all crystals are initiated at the same time by the selfseeding technique, this treatment results in a sample containing many crystals of the same size. These crystals were then measured to obtain the overall growth rate as a function of time and temperature. The data obtained are analyzed by the methods described in Section 10.3.2.2.4. It should be pointed out that when a sample is carefully cleaned by the methods described in Section 10.3.2.2.1, a very large fraction of the heterogeneous nuclei are removed. Since, as can be seen in Fig. 12, different nuclei are activated at different temperatures, this cleaning also results in a small number of nuclei that tend to become active at the same temperature. Under the conditions described for polyethylene fractions in Section 10.3.2.2.1, all crystals started to grow at essentially the same time (no induction time) and there were very few nuclei. Thus the crystals could be observed for long periods of time, i.e., until impingement with neighboring spherulites occurred. In this case the self-seeding technique offers no particular advantage. 10.3.5.2. Kinetics of Crystallization from Solution. 10.3.5.2.1. THE GENERAL CASE. At the present time, the number of studies on the growth rate of polymers from solution is very limited, primarily being restricted to polyethylene. The experimental methods used generally involve either optical microscopy, transmission electron microscopy, or solution dilatometry. Usually the polymer is dissolved in a suitable solvent, the number of growth centers set by “self-seeding,” the solution cooled to the crystallization temperature, and finally crystals are removed from the solution at set time intervals and their dimensions measured. In dilute solutions, nearly perfect single crystals are formed at temperatures near the dissolution temperature Td . As the concentration increases or the crystallization temperature decreases, the crystals formed tend to become more and more dendritic and finally approach the type of crystallization that occurs in bulk, i.e., spherulites. As discussed in Section 10.3.2.2.4,Eq. (10.2.2.) can be used to analyze the data obtained from solution. The appropriate growth rate equation is

G

=

Go exp[- A H * / R T ] exp[- K , l / T ( T d o- T , ) f l ,

(10.3.4)

where AH* is the activation energy of diffusion in solution, Tdothe dissolution temperature (from Td vs. 1 / 1 plots), and T , the crystallization temperature. In dilute solutions the growth rate is linear and constant until the solution is depleted of polymer. In solution dilatometry only primary crystallization occurs, i.e., there is no indication of secondary crystallization or

3 84

10. NUCLEATION A N D CRYSTALLIZATION

crystal perfecting. The Avrami type of equation also holds for dilute solution studies and can be applied to dilatometry results. Blundell and Kellerg6observed that in dilute solutions and in the temperature range where single crystals are formed, the growth is nucleation controlled. Concentration dependences have been studied by Seto and M ~ r i , Keller ~’ and P e d e m ~ n t eand , ~ ~Cooper and M a n l e ~ .They ~ ~ found that G is proportional to C“. The value for (Y is 0.2-0.5 for relatively low crystallization temperatures and high molecular weights and approaches 2 at temperatures near Td for low molecular weights. Furthermore, as the crystallization temperature is increased the upswing occurs at lower concentrations. Dilatometric studies at higher concentrations have been performed by Mandelkern. loo~lol He found that the Avrami isotherms were superimposable initially by a shift along the log t axis. However, the growth rate was found to decrease as more polymer crystallized. Yeh and Lambert102analyzed the systems isotactic polysytrene -atactic polystyrene and polycaprolactone -poly(vinyl chloride). They found in the polystyrene case that the linear growth rate decreased almost linearly with increasing atactic component; however, the presence of the atactic component did not affect the temperature at which the maximum growth rate occurred. 10.3.5.2.2. THEEFFECTOF STIRRING. When apolymer solution is rapidly stirred, crystallization occurs at lower undercoolings for a given concentration than in the unstirred systems, as predicted by Keller.lo3 The morphologies of the crystals produced are very interesting. A central fiber, which has been shown to be extended-chain material, forms first, followed by the formation of the familiar lamellar chain-folded crystals perpendicular to the central fiber and spaced at fairly regular intervals (200-1000 A). This type of morphology has been called shish kabob by Pennings et al. lo4-lo6 D. J. Blundell and A . Keller, J . Polym. Sci., Part B 6, 433 (1968). T. Set0 and N. Mori, R e p . Prog. Polym. Phys. Part Jpn. 12, 157 (1969). ga A. Keller and E. Pedemonte, J . Crysf. Growth 18, 111 (1973). M. Cooper and R. St. J. Manley, J . Polym. Sci., Polym. Lett. Ed. 11, 363 (1973). L. Mandelkem, J . Appl. Phys. 26, 340 (1955). lol L. Mandelkem, Polymer 5, 637 (1964). lo’ G . S. Y. Yeh and S. L. Lambert, J . Polym. Sci., Part A-2 10, 1183 (1972). lO3 A. Keller, J . Polym. Sci. 17, 291 (1955). Iw A. J. Pennings, in “Crystal Growth” (H. S. Peiser, ed.), p. 389. Pergamon, Oxford, 1967. loS A. J. Pennings and A. M. Kiel, Kolloid-Z. & Z . Polym. 205, 160 (1965). m A. J. Pennings and M. F. J. Pijpers, Macromolecules 3, 261 (1970). ga

10.4.

NUCLEATION

385

The shish kabob-type growth is also observed when a bulk sample is crystallized under stress such as that induced by stretching. This process is called row nucleation and the number of crystals that develop perpendicular to the applied stress increases as a function of stress.lo7 It is not within our scope here to go into all the details concerning crystallization under stress. For further discussion of experimental techniques and theories regarding this interesting and important type of crystal growth, the reader is referred to the recent book by Magill'O* and the references contained therein. 10.3.5.2.3. CRYSTALLIZATION UNDER PRESSURE. W u n d e r l i ~ hhas ~~~ shown that when a polymer solution is crystallized under different pressures, the usual chain-folded type of crystal is produced and there are no changes in crystal thickness at constant undercoolings. The primary effect of hydrostatic pressure is a slight change (increase) in dissolution temperature and a decrease in the volume of melting with increasing pressure. The effect of pressure on crystallization of a polymer from the melt is more dramatic. In the case of polyethylene, Maeda and Kanetsunallo showed that at pressures above 4000 kg/cm2, two types of extendedchain crystals were formed with different thermal stabilities. These two forms were described as ordinary and highly extended chain crystals. It is thought that extended-chain crystals are formed by a stepwise unfolding of normal chain-folded crystals. Another effect of pressure is, of course, the formation of different crystal forms. In general, for most polymers the effect of pressure is to increase the melting point and to decrease the volume of melting. For further discussion and references on pressure crystallization the reader is referred to Wunderlich's book.4

10.4. Nucleation 10.4.1. Homogeneous Nucleation

For a large number of years there has been a lively amount of interest in the precise manner in which polymers crystallize. Fundamental to the understanding of such a process is the necessity of accurately determining the values of u and ce,the lateral and end surface free energies. One method at our disposal, namely, the direct measurement of I and T , from crystals obtained at different crystallization temperatures [Eq. (10.2.1 .)I, E. H . Andrews, Proc. R . Soc. London, Ser. A 270, 232 (1962). J. H. Magill, Treatise Muter. Sci. Techno/. 10, 261 (1977). *Og B . Wunderlich, J . Polym. Sci., Part A 13, 1245 (1963). 'lo Y. Maeda and H. Kanetsuna,J. Polym. Sci., Polym. Phys. Ed. 13, 637 (1975). lop

Io8

386

10.

NUCLEATION A N D CRYSTALLIZATION

provides for a direct determination of ue. Another technique, the determination of the growth rate at different crystallization temperatures [Eq. (10.2.2.)], allows the calculation of the product uge. Finally, we have the determination of the homogeneous nucleation rate at various temperatures [Eq. (10.2.3.)], which provides a measure of the product cr2ue. Most of the work on homogeneous nucleation in polymer systems has been done on samples of linear polyethylene. The first published work on polyethylene was that of Cormia, Price, and Turnbull.lll Since then other groups112-115have reported on the homogeneous nucleation in both whole polymers and carefully prepared fractions. In all cases the investigations were carried out using small liquid droplets of the sample. The small droplets of polymer are dispersed in a suitable nonsolvent, and a small portion of this dispersion is placed in a cell on the hot stage of a polarizing microscope. When the sample is viewed through the crossed polarizer and analyzer of the microscope, only those droplets which are crystalline are seen. Against the black background the droplets appear as stars when they nucleate and immediately crystallize. The sample is first warmed to melt the droplets and then either (1) continuously cooled at a constant rate until all of the droplets are crystalline or (2) immediately cooled to some temperature within the temperature range of homogeneous nucleation. In either case a series of photographs is taken at specified time intervals. Comparison of the number of “stars” appearing in consecutive photographs thus allows the nucleation events per time and/or per degree of temperature to be calculated assuming that only a single nucleation event was responsible for each crystallized particle. Vonnegut116was the first to suggest that the existing particulate matter responsible for heterogeneous nucleation could be isolated by subdividing the bulk sample into many small pieces. If the particles were small enough, a significant portion of them would be free of foreign matter. The methods used in achieving this subdivision differ depending upon the investigators. One method is to simply grind up the sample to the desired particle size. This method was used by Turnbull and Cormia in their classic work on the homogeneous nucleation of certain n-alkanes117and R. L. Cormia, F. P. Price, and D. J. Turnbull, Chem. Phys. 37, 1333 (1962). F. Gornick, G . S. Ross, and L. J. Frolen, J . Polym. Sci., Part C 18, 79 (1967). J. D. Hoffman, J. I. Lauritzen, Jr., E. Passaglia, G. S. Ross, L. J. Frolen, and J. J. Weeks, Kolloid-Z. & Z. Polym. 231, 565 (1960). 114 J. A . Koutsky, A. G . Walton, and E. Baer,J. Appl. Phys. 38, 1831 (1967). G . S . Ross and L. J. Frolen, J . Reu. Nut/. Bur. Stand., Sect. A 79, No. 6 , 701 (1975). ll@B. J. Vonnegut, J . Colloid Sci. 3, 563 (1948). D. Turnbull and R. L. Cormia, J . Chem. Phys. 34, 820 (1961). Il2

10.4. NUCLEATION

387

subsequently employed by Uhlmann et a / . and later by Oliver and Calvertllg in homogeneous nucleation work involving a large number of normal alkanes. The other technique used was to dissolve the sample in a suitable solvent, then to subdivide the solution into small droplets, and finally to remove the solvent. Koutsky et in using this technique, achieved the initial subdivision by atomizing the hot polyethylene solution onto a suitable liquid substrate. The other investigator^^^^-^^^ used the technique first reported by Cormia et a / .l l 1 The polyethylene was dissolved in hot nitrobenzene and then slowly cooled. Upon reaching the consolute temperature, a new phase, rich in polyethylene, was formed. Upon subsequent cooling these small liquid droplets crystallized, expelling the remainder of the nitrobenzene. These micron-sized droplets were then separated from the nitrobenzene and dispersed in a suitable nonsolvent. In the cases involving the normal alkanes, the subdivision and dispersal of the particles in the nonsolvent was achieved in one step. The alkanes and the nonsolvent were mechanically blended until an average particle size of a few microns was produced. Regardless of the particular method utilized in obtaining the small droplets, it is obvious that the ultimate success of the experiment is dependent upon having as few a number of heterogeneities as possible in the final droplet population. This can only be accomplished by rigorous cleaning techniques. In the case of Ross and Frolen115the two solvents used for their nucleation work on polyethylene were xylene and nitrobenzene. In order to remove as much particulate matter as possible, filtration, distillation, crystallization, and ultracentrifugation were used. Each solvent was filtered twice through two regenerated cellulose filters, each having a porosity of 0.2 pm. Next, each solvent was distilled under reduced pressure. Only the center cut was selected. Following the distillation each solvent was cooled a few degrees below its melting point and crystallization was induced by rapid shaking. The resultant slurry of small crystals was filtered off and discarded. This type of crystal removal was performed twice. The technique is sometimes referred to as “snowing out,” the rationale being that any dirt particles present will act as heterogeneous nuclei. Then by filtering and removing the resultant crystals, the solution remaining would be dirt free. Finally, each solvent was ultracentrifuged, and only the central portion was kept. It was later found that this centrifugation had little or no beneficial effect. Indeed early experiments were performed in a “dust-free’’ white bench area. This was likewise shown D. R. Uhlmann, G . Kritchevsky, R. Straff, and G. Scherer, J . Chern. Phys. 62, 4896

llE

( 1975).

M. J. Oliver and P. D. Calvert, J . Crysr. Growth 30, 343 (1975).

388

10. NUCLEATION A N D CRYSTALLIZATION

to be of little benefit. At the end of the above processes the solvents were shown to be free of particles normally "visible" using light-scattering techniques. While purity was not the main reason for the above procedures, analysis by gas chromatography indicated a chemical purity of at least 99.5 mole% for each solvent. The liquid dispersing agent used was either a silicone oil1l40r111-113*115 isooctylphenoxy-poly(ethyleneoxy1)ethanol. This chemical is a product of the General Aniline and Cilm Company under the trade name of Igepal CA-630.t The silicone oil was preheated before use to 290°C to flash off high-vapor-pressure materials, The Igepal was cleaned by vacuum distillation and filtration. A solution of polyethylene and the clean xylene was prepared and filtered through the 0.2 pm cellulose filters. In all of the filtering operations the solutions or solvents were allowed to flow through the filter network without applying any pressure. In previous experience with preparing polymer solutions prior to analysis of molecular weight using lightscattering techniques, it was found that this very slow filtering was necessary if the samples were to be free of particulate. Each solution was filtered twice. A portion of the l% by weight, polymer-xylene solution was then poured into approximately 200 ml of nitrobenzene heated to 200°C. The polymer was immediately dissolved, with the xylene rapidly boiled away. Sufficient xylene solution was used to introduce approximately 30 mg of polyethylene into the nitrobenzene. When this solution was cooled a new liquid phase, rich in polyethylene was formed. The small liquid droplets (Fig. 11A) so formed crystallized as the temperature of the nitrobenzene was further lowered. Upon cooling to room temperature the frozen droplets were separated from the nitrobenzene by filtering. Finally, a small portion of the droplets was redispersed in approximately 10 ml of clean Igepal. It has been previously demonstrated by Cormia et al."' that polyethylene was not soluble in Igepal even after standing for days at elevated temperatures. When droplets were prepared in this fashion115from the nitrobenzene solution, it was found that the droplets tended to cluster (Fig. 11B). These clusters were not broken up by ordinary mechanical agitation when dispersed in the Igepal. Ultrasonic treatment was found to be necessary in order to assume complete break up of these clusters (Fig. 11C). Koutsky and c o - w ~ r k e rused s ~ ~ microscope ~ slides having shallow concavities to hold their dispersions. In their case no cover was used and the droplets floated at orjust below the surface of the silicone. The other investigators used cells formed by fusing a 0.2 mm thick glass washer with t See footnote on bottom of p. 381.

10.4.NUCLEATION

389

FIG. 1 1 . Photographs of typical polyethylene droplet populations. (A) Electron micrograph of droplets as they are precipitated from nitrobenzene showing size distribution. (B) Droplet population dispersed in Igepal using only mechanical (shaking) agitation. ( C ) Droplet population in Igepal using ultrasonic dispersion. Note better uniformity and smaller droplet size.

an inner diameter of 0.5 cm to an 18 mm square microscope cover glass. The sample in turn was contained within the washer and a disk made from a cover glass. Gornick and co-workers112 had previously shown that cleaning of glass cells by treatment with chromic acid, followed by washing with distilled water and clean, filtered ethanol, was sufficient. The cells, including their covers, were ultrasonically scrubbed in distilled water. The resulting glass surfaces exhibited no nucleating effects upon the liquid-polymer droplets in contact. When dispersions of liquid droplets of polyethylene were cooled, it was observed that there are temperature regions where heterogeneous nucleation occurred. There were other temperatures at which no nucleation occurred. It was noted that successive preparations of droplets using the same nitrobenzene showed that a much larger percentage of the droplets nucleated at lower temperatures. Thus, it appeared that even in the nitrobenzene a snowing out process was necessary in order to effectively scavenge those particles effective in heterogeneous nucleation. The result of such consecutive droplet preparations using the same nitrobenzene are shown in Fig. 12. All of the data reported in Ross and Frolen115resulted from nucleation experiments performed on droplet populations representing the fourth precipitation from the nitrobenzene solutions. Further repetitions of the process did not seem to result in an increase of the droplet population nucleating homogeneously. When a sample is melted and then slowly cooled, there are temperature regions of relative quiescence and then other narrow temperature regions of furious activity. This is evidenced by crystallized droplets appearing

3 90

10.

NUCLEATION AND CRYSTALLIZATION

0

TEMPERATURE ("C) FIG. 12. Nucleation of polyethylene fraction (M, = 23,000) showing the effect of repeated droplet preparation using the same solvent. 1 - n/no is the fraction of droplets nucleated as a function of temperature (cooling rate is O.I"C/minute). Population A is the droplets prepared from the fourth precipitation, which is assumed to be homogeneously nucleated.

as stars in the otherwise dark background of a sample being observed between the crossed analyzer and polarizer. The most primative nucleation experiment is to slowly cool the specimen, noting the lowest-temperature region at which a large amount of nucleation-crystallization is observed in the droplet population. This is assumed to be the homogeneous nucleation region. Taking the average temperature in this region, assigning a probable value for I , the nucleation frequency, and either a most probable or else a theoretically calculated value for Z,, the preexponential factor, one can then calculate by using Eq. (10.2.3.) the nucleation rate constant Ki. Thus, with a single measurement it is possible to determine a value of the free energy product uzu,. This approach was used, not with droplets, but with a thin-film specimen of polychlorotrifluoroethylene to determine the nucleation rate constant by Hoffman and Weeks.lz0 They were able to determine that J. D. Hoffman and J. J. Weeks, J . Chem. Phys. 37, No. 8, 1723 (1962).

10.4.

N UCLE AT10 N

39 1

154°C was the average temperature of homogeneous nucleation for such a bulk sample. When a molten film was rapidly cooled down, in the vicinity of 154°C it was observed that 35-40% of the crystallization occurred in approximately 5 sec at this temperature, but that no spherulites were observed by using an ordinary optical microscope. It is unusual to have polymer systems that allow this type of observation to be made, since crystal growth is normally so rapid that by the time the sample has attained the homogeneous nucleation temperature range, the sample appears completely crystalline. However, the same experiment now applied to small droplets is quite simple to do. Indeed, in their work on normal alkanes, this was the technique used by Uhlmann and coworkers,ll* although the type of treatment is considerably less satisfactory than the other two methods to be discussed. In general such experiments furnish valuable information as preliminary experiments prior to performing the more detailed ones described below. The most important assumption concerning homogeneous nucleation experiments is that indeed such nucleation can be seen and measured. In the droplet experiment it is automatically assumed that the lowest temperature region at which nucleation-crystallization is observed corresponds to homogeneous nucleation. As with studies on crystallization it is also assumed that there are no competing reactions. Consequently, an experiment similar to that described above clearly identifies the temperature region of interest. Knowing this, we can quickly bring our molten droplet population down to within a few degrees of this temperature region and then slowly cool from this point, eventually reaching the temperature at which all droplets have crystallized. If the rate of cooling is constant and if, during the course of the nucleation experiment, we have taken pictures at known time intervals, then we have the experimental data necessary to use the second method. Usually a 16 or 35 mm cinephotographic assembly is used, although a Polaroid? camera can be used. The latter is somewhat less satisfactory in that fewer pictures are taken. The film must be of a high speed, high resolution, and preferably medium contrast type, i.e., all of the crystallized droplets must appear on the film and yet the exposure time should be small in respect to the time of the experiment. Depending upon its orientation, a crystallized droplet may appear as a very bright “star” or a barely visible one. Both extremes must be clearly discernible in any photograph. Ideally the speed of the film should be such that during the required exposure, no nucleation events occur. t See footnote on bottom of p. 381.

392

10.

NUCLEATION A N D CRYSTALLIZATION

Referring to Fig. 12, we see that by the time that the temperature region of homogeneous nucleation is reached, the beginning of region A, 350% of the total droplet population has already frozen. This population is considered as a blank, to be subtracted from the total number of droplets visible at any time during the homogeneous nucleation experiment. The portion of the experiment representing the homogeneous nucleation is subdivided into at least 10 and preferably 20 or more equal time periods and pictures corresponding to these times are printed from the many frames taken of the experiment. The total number of visible droplets on each picture is determined. The difference between this number and the blank is the number of droplets that have nucleated homogeneously for that portion of the experiment. Since the cooling rate is constant, and providing we have measured T , we have data sets of T i , ti, and n , these being the temperature, the time, and the number of droplets that have homogeneously nucleated by time ti. Similar sets of data are obtained by repeating the experiment, using a different constant cooling rate. Plots of the fraction of droplets frozen vs. temperature for a series of experiments produce a series of S-shaped curves whose major difference corresponds to a shift of temperature. From such plots for identical fraction frozen lines, we can obtain AT,,,, which is the difference in temperature between two experiments. Cormia et al. (CPT) derived the following relations hip: ln(r1/r2) = 2W* ATD/kT AT,,, ,

(10.4.1)

and consequently were able to determine W * , the work necessary to form the critical nucleus, from a plot of ln(rl/r2) vs. ATD. rl and r2 are two different experimental cooling rates and ATD is the temperature difference at the midpoint of the curves produced when n/nois plotted vs. T , the undercooling corresponding to each cooling rate. In their terminology, I = K, exp[- ( W * / k T ) ] , where K, is a constant dependent upon the molecular and transport properties of the system. W* is defined as ~ T ( ~ ~ ~ , T , , , ~ / A H , ~From ( A TW ) ~* . they were able to calculate the surface free-energy product v2a,. Equation (10.2.3.) shows a I / T ( A Y ) 2 dependence, and therefore a W * from this equation would be somewhat different, but still a derivation similar to that of CPT would allow calculation of v2vefrom the plot of ln(rl/r2) vs. A T D . In such a set of experiments the difference between the smallest and largest cooling rate should be of the order of 100. By restricting the experiments to isofractions frozen of from 0.3 to 0.7, a very linear plot of ln(rl/r2) vs. AT, is obtained, and the value of v2wedetermined from this and similar treatments such as that to be described below is very reasonable. The third method111-l15 of determining homogeneous nucleation fre-

10.4.

NUCLEATION

393

quencies or, specifically, m2ue from homogeneous nucleation experiments involves measurement of such frequencies from experiments run at a constant temperature. Again, there is a series of pictures taken of the droplet population at different times. As before, one has a blank representing the nucleation-crystallization occurring prior to the homogeneous nucleation. In this type of experiment it is not necessary to run the experiment the length of time necessary to nucleate all the droplets. Instead, one can determine the total droplet population by simply dropping the temperature to a much lower one and after a brief period of time take the resulting picture; the difference in droplet count between this picture and the blank represents the total number no of droplets that could have participated in the homogeneous nucleation experiment. In describing this technique we are specifically describing that published in Ross and Frolen,l15 although the general technique is that discussed in Cormia, Ross, and a s s o c i a t e ~ . " ' - ~ ~The ~ results of preparing the sample by ultrasonic dispersion as compared to mechanical stirring is vividly shown in Fig. 11C. The resulting uniformity of droplet size is to some degree responsible for the precision of the results described herein. In as much as the entire droplet population remains fixed in space, i.e., the droplets do not change position from experiment to experiment or during a single experiment, it is possible to divide up the two-dimensional droplet space and to compare different subpopulations. For instance, comparison of those droplets in the upper half of the field with the behavior of the droplets in the lower half of the field is useful in determining that there is a uniformity of temperature. The behavior of these subpopulations was identical, showing that within the sensitivity of the experiment, there was no discernible temperature difference. Another test of droplet uniformity or of a single rate constant is to divide up the experiment into two or three equal times. Each should give the same slope and the same precision if indeed all the droplets conform to the same statistics. The first step in the analysis of data uses the relationship

n

=

no exp(- Zvt),

(10.4.2)

where no is the total number of droplets, n the number of droplets that remain uncrystallized after any time t , v the average droplet volume. A typical plot of In n / n o vs. t is shown in Fig. 13. The data are extremely good, having correlation coefficients of 0.998 or greater. The typical experiment contained from 500 to 1000 droplets which nucleated homogeneously. By either dividing the field into three sectors or by taking the first, second, and final one-third of the droplets as they nucleated, statistics were produced that showed little deterioration from the treatment

394

10.

-4

0

NUCLEATION A N D CRYSTALLIZATION

50

100

Time (minutes) FIG.13. Typical isothermal homogeneous nucleation experiments showing the temperature dependence of the nucleation rate for a polyethylene fraction ( M , = 30,600). n/no is the fraction of droplets that remain unnucleated at any time t .

of the population as a whole. While the nucleation experiments cover only a temperature range of a few degrees, this remarkable conformity to Eq. (10.2.3.) allows considerable testing of the ensuing nucleation theories. One of the major reasons for doing the work described in Ross and Frolen115was to see if there was any molecular-weight dependence of the homogeneous nucleation rate of linear polyethylene. The actual temperature range of homogeneous nucleation was of course dependent upon the molecular weight of the sample, being directly related to AT or Tm for each sample. However, in the range of 24,000-250,000 molecular weight on the molecthere was only a very slight dependence of the product a2ae ular weight of the sample. In fact, over this range of molecular weight the data would support the contention that there was no molecular-weight dependence. To make this analysis Eq. (10.2.3.) was modified as follows: Z = l o exp(-[U*/k(T - T,)]} exp{-[30.2 (~1~~.,(k)Tm~/k(Ahf)~]} (10.4.3) X [(l + x AT)'(l + y AT)/T(AT)'.P].

10.4.

NUCLEATION

3 95

As shown by Hoffman and L a ~ r i t z e nthe , ~ ~term AF*/kT can be substituted by U * / R ( T - T , ) for a large number of polymers, including polyethylene. U* has the same value as described in Section 10.3.2.2.4. In addition, the fact that homogeneous nucleation occurs some 65" below T,,, required that some consideration be given to the dependence of u and ue on temperature. A linear d e p e n d e n ~ was e ~ ~assumed for both u and ue, the coefficients beingx and y, respectively. The quantity? is the correction term to the free energy and is equal to 2T/(T,,, + T ) . Again, this term becomes of increasing importance with the high undercoolings. When the data were fitted with the unreasonable assumption that x = y = 0 the fit was quite good, giving an average value of the surface it free energy product v2we= 19,000 erg3 cm-6. From previous had been shown that the best value for y was 0.014. In further analysis this value of y was assumed as was the theoretical value of Zo, i.e., 1X The value of x was chosen by allowing x to vary until the best correlation coefficient of fit was obtained. Again for molecular weights above 25,000 this value was found to be nearly constant, having an average value of - 0.0073. The value of the product u2uedid not appreciably change. From these results, it is our opinion that the theoretical value of Zo should be included as input data. From this work, and that where the values of u and cewere determined by other methods, a very uniform picture emerges. The consistency of such surface-free energies when determined from vastly differing techniques suggests that the underlying theories herein used are indeed sound. Unfortunately, data on a large number of polymers are not readily available. Indeed, the work of Koutsky et af.lI4 on polyethylene oxide, polyoxymethylene, nylon 6, poly(3,3-bis-chloromethyloxacyclobutane),isotactic polypropylene, and isotactic polystyrene appears to represent the sum total of homogeneous nucleation work of a definitive nature on polymers other than linear polyethylene. Even Koutsky et af. feel that their results unequivocably represent homogeneous nucleation behavior for only polyethylene and isotactic polypropylene. Experimentation involving investigation of the homogeneous nucleation behavior of other polymers, including the effect of molecular weight, is needed. There are two methods that were used with the study of normal alkanes, but that have not been employed in the study of homogeneous nucleation in polymers; namely, dilatometry 117 and differential scanning calorimetry (DSC). In particular, Oliver and C a l ~ e rwere t ~ ~able ~ to obtain the same type of data by taking the liquified-droplet dispersions and cooling at known rates in a differential scanning calorimeter. By proper treatment of the resulting thermograms they were able to obtain relative nucleation rates and hence u3that agreed with results obtained by earlier methods in-

3 96

10. NUCLEATION A N D CRYSTALLIZATION

volving droplet counting. This method could perhaps be fruitfully applied to polymeric systems. Another method attempted by the authors of this part was to cool the liquid-droplet population while listening to the noise produced by nucleation-crystallization of the individual droplets. The result of the very rapid crystallization following nucleation of an individual droplet produces sufficient noise to be detected by ultrasensitive noise-detecting systems. Preliminary work suggested that the approach was quite reasonable, although the work was not actively pursued because of the difficulty of producing a sensor (acoustic transducer) that was not adversely effected by the elevated temperatures encountered. 10.4.2. Heterogeneous Nucleation

As mentioned in Section 10.4. l . , most polymers nucleate heterogeneously at temperatures near the melting point, and different types of heterogeneities become active at different temperatures. At the present time the nature of the foreign surfaces are not understood and the mechanism of the process has not been explained, although some interesting work on the subject has been d ~ n e . ' ~ l - 'Several ~~ experimental methods have been applied to the examination of the nucleation efficiencies of foreign materials added to the polymer samples under investigation (inorganic salts, catalysts such as titanium, and other polymers). Three such methods have been the depolarization of polarized light,122-124OPtical r n i c r ~ ~ c ~and p y differential ,~~~ thermal analysis. 126;127 The effect of substrates on solution c r y ~ t a l l i z a t i o n ' ~and ~ - ~on ~ ~crystallization from the melt131-133has also been investigated. The procedure described by Chatterjee et al. lZ5is straightforward. A polarizing microscope is focused on the interface between the polymer and the freshly cut substrate. In this way the pure polymer can be observed as well as the interface. The sample and substrate are placed on a F. L. Binsbergen, Ph.D. Thesis, Groningen, The Netherlands (1969). F. L. Binsbergen, Polymer 11, 253 (1970). lZ3F. L. Binsbergen, J . Polym. Sci., Polym. Phys. Ed. 11, 117 (1973). lZ4M. Inoue, J . Polym. Sci., Part A 1, 2013 (1963). lZaA. M. Chattejee, F. P. Price, and S. Newman, J . Polym. Sci., Polym. Phys. Ed. 13, 2369, 2385, and 2391 (1975). lZeH. N. Beck and H. D. Ledbetter, J . Appl. Polym. Sci. 9, 213 (1965). lZ7H. N . Beck, J . Appl. Polym. Sci. 11, 673 (1967). lZ8E. W. Fischer, Kolloid-Z & Z . Polym. 159, 108 (1958). J . A. Koutsky, A. G. Walton, and E. Baer, J . Polym. Sci., Part A-2 4, 611 (1966). 130 J. A. Koutsky, A. G. Walton, and E. Baer, J . Polym. Sci., Part B 5, 177 (1967). 131 3. A. Koutsky, A. G. Walton, and E. Baer, J . Polym. Sci., Part B 5, 185 (1967). H. Schonhom, Macromolecules 1, 145 (1968). 133 D. R. Fitchmun and S. Newman, J . Polym. Sci., Part A-2 8, 1545 (1970). lZ1

lZ2

10.4.

NUCLEATION

397

hot stage, heated to a temperature well above the melting point of the sample, and then cooled to the desired crystallization temperature. Data on the growth rates of spherulites nucleated by the substrate and nucleated in the bulk sample are compared by direct observation, (generally photographically). Melting temperatures for the spherulites nucleated in both ways can be determined by observing the temperature at which birefringence disappears. Nucleation densities at the interface and in the bulk can also be determined. The nucleation densities are calculated using the area of the substrate interface and the volume of the bulk polymer. The larger the ratio ns/nb, where n, is the nucleation density at the substrate and n b the nucleation density in the bulk sample, the greater the nucleating power of the substrate. Using a model of folded-chain nuclei, the dependence of the rate of heterogeneous nucleation I, on the undercooling AT is described by the equation134 log I,

=

log I% - (U*/2.3kT) - [l6uue A u T , ' ~ / ~ . ~ T ( AAhr'], T)~~

(10.4.4)

where I, is an essentially temperature-independent constant, acreare the usual surface free energies for the polymer, and A a = u + uc + urn. Here u is the lateral surface free energy for the polymer, uc the substrate-crystal interfacial energy, and r, the substrate-melt interfacial energy. If the product mue is known from other measurements on the polymer (growth rates as a function of temperature), the value for A u can be obtained from the slope of the curve when log I, + U*/2.3kT is plotted vs. 1/T AT2. A u is then a measure of the nucleating power of the substrate. Other methods for preparing samples and measuring heterogeneous nucleation rates are described in the references cited in this section. The study of heterogeneous nucleation is very important and many more systematic studies are needed to explain the mechanism of the process and to predict the effect of additives on the polymer systems. Acknowledgment We wish to thank Dr. John D. Hoffman for the benefit of many helpful discussions, comments, and suggestions.

134

F. P. Price, in "Nucleation" (A. C. Zettlemoyer, ed.), Chapter 8. Dekker, New York,

1969.

This Page Intentionally Left Blank

AUTHOR INDEX FOR PART B Numbers in parentheses are footnote reference numbers and indicate that an author’s work is referred to although the name is not cited in the text.

A

Adachi, K., 214 Adam, G., 324 Agboatwala, M. C., 276, 277(160) Akahori, T., 269 Albers, J. H.’M., 308 Alcock, T.C., 342 Aleksandrov, A. A., 213 Alexander, L. E., 5 , 20, 67, 74, 118, 129, 133(2), 137, 138(13), 139 Alexander, W. J., 270, 273 Ah, M. S., 328 Allegra, G. E., 93, 94, 95, 96 Allen, P. W., 373 Allison, S. K., 27, 30 Allou, A. L., 325 Allou, Jr., A. L., 295, 300, 305, 306 Anderegg, J. W., 180, 181(91) Anderson, F. R., 217, 238, 244 Anderson, H. C., 334 Andrews, E. H., 385 Aqua, E. N., 144 Arai, N., 253, 254(80) Arakawa, T., 253, 254, 308, 350 Arbogast, J. F., 366 Anmoto, H., 82 Armeniades, C. D., 372 Arnott, S., 97, 113 Atkinson, C. M. L., 325 Averback, B. V., 147 Avrami, M., 347, 348

B Bacon, R. C.. 256, 262(91) Baer, E., 238, 245(27), 247, 249, 250, 251, 399

252, 253, 254(27), 304, 308, 327(72), 362, 372, 386, 387(114), 388(114), 392(114), 393( 114), 395(114), 396 Baev, A. S., 213 Bagchi, S. N., 102(65), 103, 105, 106, 109(65,68), 112(65),144, 147(26), 148(26), 164(26), 171(26) Bailey, G . W., 240 Bair, H. E., 254,300,305(57), 314,315,331, 352, 364 Baker, C., 369 Baker, P. N., 208 Ballard, D. G. H., 269 Balta Calleja, F. J., 299 Balwit, J. S., 279 Baranov, V. G., 368 Barrales-Rienda. J. M., 337 Barrett, K. E. J., 332 Barton, D. H. R., 268, 269 Barton, J. M., 329, 334 Bascom, W. D., 332 Bassett, D. C., 216, 217(21), 229, 233(21), 238, 245, 248, 297, 299, 312, 315, 339, 346 Bassett, G. A., 217,’222, 237 Battista, 0. A., 276 Baudisch, J., 269 Baumeister, W., 213 Beacheli, H. C., 256, 262(93) Beck, H. N., 396 Beech, D. R., 308 Bekkedahl, N., 370, 371 Bell, A. T., 241 Bell, D., 175 Bell, J. P., 255, 274, 276, 277, 306, 307, 308

400

AUTHOR INDEX FOR PART B

Benedetti, E. 93, 94(59, 61), 95(59), 96(61) Berghmans, H., 301, 303(59) Bergmann, K., 237 Berkelhamer, L. H., 291 Bernard, M. Y.,186 Bertein, F., 186 Bezruk, L. I., 244 Bhat, N. V.,244 Bibikov, V. V., 279 Billmeyer, F. W., 151, 325 Billmeyer, Jr., F. W., 18 Binsbergen, F. L., 396 Blackadder, D. A., 229, 325 Blaine, R. L., 325 Blais, P., 241, 242(51) Blanchard, L. P., 336 Blazek, A , , 287 Blundell, D. J., 230, 256, 260, 262(97, 100, 105), 263(105), 307, 382, 384 Blundell, D. S., 169 Bobalek, E. G., 256, 262(92) Bodily, D. M., 279, 280, 281, 328 Bodnar, M. J., 241, 242(50) Boersma, S. L., 291 Bohlin, L., 2% Bolz, L. H., 242, 243 Bonart, R., 144, 147(27), 148(27), 172 Boon, J., 306 Booth, A , , 336 Booth, C., 307, 308 Borchardt, H. J., 291, 292(9) Borkowski, C. J., 46, 61, 155 Bosch, W., 366 Boyer, R. F., 252, 331, 374 Bragg, W. H., 1' Bragg, W. L., 1 Bramer, R., 164 Bravais, A., 3 Brennan, W. P., 318, 319,320, 321,322, 324 Brenner, N., 289, 290(5), 321 Broadhurst, M. G., 325, 351 Broers, A. N., 228 Brumberger, H., 182, 330 Buben, N. Y.,279 Buchanan, D. R.,148, 149, 150 Budzol, M., 279 Buerger, M. J., 11, 71, 72, 77, 92, 93 Bullough, R. K., 92 Bunn, C. W., 15, 74, 79, 80, 81, 82, 88, 89, 90(39), 93, 95, 325, 342 Burmester, A., 163, 167(59), 299

C Calvert, P. D., 387, 395 Canterino, P. J., 245 Carlson, G. L., 241, 242(49) Carrlson, D. J., 241, 242(51) Carroll, B., 332 Carter, D. R., 362 Castaing, R., 186 Ceccorulli, G., 336 Cernee, F., 332 Chabre, M., 155 Challa, G., 121, 255, 303, 304(66), 306 Chan, K. S., 330 Chandler, L. A., 306 Chang, M., 260(121), 261(121), 264, 269(121), 270, 271, 272 Chapiro, A., 278 Charlesby, A., 278, 279, 280 Chatani, Y.,93, 94(58), 95, 96, 97, 98 Chatterjee, A. M., 3% Chaudhuri, A. K., 256 Chiang, R., 237, 298, 300 Chiu, J., 334, 335 Christiansen, A. W., 304, 327(72) Cirlin, E. H., 334 Clampitt, B. H., 305 Clark, D. T., 245 Clark, E. S., 20, 83, 85, 87, 88, 90, 113, 114 Cobbold, A. J., 238, 255, 262(31) Cochran, W., 83, 84, 92, 113 Cogswell, F. N., 307 Cohen, J. B., 142 Coleman, M. M., 174 Collins, E. A., 306 Compton, A. H., 27, 30 Connor, T. M., 256, 262(100) Cooper, C. W., 248 Cooper, M., 384 Cooper, S. L., 307, 335 Cooper, W., 304, 305 Corey, R. B., 90, 91 Cormia, R. L., 386, 387, 388, 392, 393 Cormier, C. M., 325 Corradini, P., 16, 17(14), 37, 78, 81, 88, 89 Cox, W. P., 330 Cox, W. W., 304, 305, 327(71) Crescenzi, V., 332 Crick, F. H. C., 83, 84(41), 113(41) Crist, B., 167, 171, 172(69) Critchley, J. P., 329

40 1

AUTHOR INDEX FOR PART B

Cruickshank, D. W. J., 92 Crystal, R. G., 362 Cullis, A. G., 222, 224, 227(29) Cullity, B. D., 40, 160 Curtiss, E. M., 334

D Daniels, F., 291, 292(9) Danusso, F., 327 Darwin, C. G., 1 Davies, B., 291 Davies, D. R., 83, 85 Davis, G. T., 342(44), 351, 361(44), 363(44), 365(44) Davis, H. J., 208 Davis, T. G., 308 Dawkins, J. V., 269 De Angelis, R. J., 137, 142 Debye, P. P., 140 Dechant, J., 269 Delben, F., 332 Denison, R., 303, 325(65) Devine, A. J., 241, 242(50) Devoy, C. J., 307 DeWolff, P. M., 140 D'Illario, L., 260(126), 261(126), 280( 126), 281(126), 315 Dioro, A. F., 300 Dlugosz, J., 170, 235 Dodgsen, D. V., 307, 308 Dole, M., 278, 279, 280, 325, 326 Dolemetsch, H., 270 Doll, W. W., 372 Dolmetsch, H., 270 Donati, D., 324 Doyle, C. D., 334 Dreyfuss, P., 300 Dumbleton, J. H., 307, 308 DuMond, J. M. W., 183 Dunn, P., 335 Dunning, W. J., 186 Duong, D. T., 274, 276 Dupont, Y., 155 Duswalt, A. A., 304, 305, 327(71) Dweltz, N. E., 146, 151(34)

E Eastbrook, J. N., 138 Eaves, D. E., 304 Ebdon, J. R., 332

Eberlin, E. C., 329, 33Q(180) Edwards, B. C., 307 Eisenberg, A., 330 Elias, H. G., 270 Engelter, A., 324 Ennis, C., 335 Enns, J. B., 331 Era, V. A., 326, 327 Ergun, S., 142, 177 Erhardt, P. F., 293, 301(18) Ewald, P. P., 1, 34(4)

F Faget, J., 223 Fagot, M., 223 Fallgatter, M. B., 279 Faro, H. P., 269 Farrow, G., 274, 275 Fatou, J. G., 303, 325(65), 337 Faucher, J. A., 330 Fava, R. A., 334, 339 Feast, W. J., 245 Fedorova, I. S., 175 Feist, W. C., 366 Fen, C., 223 Fielding-Russell, G. S., 336 Fischer, E. W., 168,237,274, 299, 325,343, 396 Fisher, J. C., 348, 349 Fitchmun, D. R., 396 Flory, P. J., 18,237,238,298, 300,330, 344, 351, 381 Flynn, J. H., 322, 323 Folkes, M. J., 235 Foltz, C. R., 328, 335 Fournet, G., 172, 173(71), 181(71), 183 Fox, T. G., 330 Frank, F. C., 229, 312 Franklin, R. E., 83, 85 Fraser, G. V., 170 Fredericks, R. J., 306 Frey-Wyssling, A., 270 Fritschi, J., 1 Frolen, L. J., 350, 362, 369, 386, 387, 388(112,113,115),389,392(112, 113,115), 393, 394, 395(39), 396(115) Frosch, C. J., 93, 94(59) Fukami, A., 214 Fukuda, T., 274 Fuller, C. S., 93, 94(59) Furuta, M., 228

402

AUTHOR INDEX FOR PART B

G Gabriel, A., 155 Gangulee, A., 137 Ganis, P., 37, 78, 81, 88, 89 Gamer, E. V., 88, 89 Gaulin, C. A., 336 Gauzit, M., 186 Geacintov, C., 327 Cede, I., 244 GeiL P. H., 163, 167(59), 186, 238,308,339, 341 Geiss, R. H., 227 Georgiadis, T., 230 Cianotti, G., 327 Gieniewski, C., 230 Gilbert, M., 336 Gill, P. S., 334 Gillham, J. K., 331 Godard, P., 332 Goddar, H., 237 Goggin, P. L., 170 Goldberg, R. N., 291, 292(16) Goldey, R. N., 256, 262(94) Goldfarb, L., 335 Gonthier, A., 382 Goodhew, P. J., 244 Gopalan, M., 325, 362, 381 Gornick, F., 386, 387(112), 388(112), 389, 392(112), 393(112) Gottfried, B. S., 291, 292(10), 332(10) Could, A. R.,279 Gray, A. P., 316, 322, 323(124), 324 Griffith, J. H., 330 Grivet, P., 186 Groves, T., 227 Grubb, D., 170 Guarini, G. G. T., 324 Guinier, A., 102(65), 103, 106, 109(69), 130, 140, 172, 173(71), 177, 181(71), 183 Gulik-Krezywicki, T., 155 Guttman, C. M., 322, 323 H Haberfeld, J. L., 326 Habrle, J. A., 238, 255(29), 269(29) Hachiboshi, M., 274 Hahn, M., 213 Hall, C. E., 186 Hall, J. R., 241, 242(50)

Hall, W. G., 186, 187(9a) Hamada, F., 325 Hammer, C., 120 Hann, V. A., 258 Hannon, M. J., 274 Hansen, R. H., 241, 242(48), 245 Hara, K., 325 Harget, P. J., 157 Hari, Y., 280 Harker, D., 92 Harland, W. G., 298, 307,308(36) Harrison, I. R.,151, 152, 174, 175,230,238, 245, 247, 249, 250, 251, 252, 253, 254(27), 301, 309, 310, 311, 312, 316 Hartman, P., 186 Hasuda, H., 241, 242(42) Hatano, M., 369 Hausen, E. A., 1 Hay, J. N., 296, 325(27), 336 Hayashi, J., 269 Hayashi, S., 297, 300, 314(30), 325 Haynes, S . K., 274, 276 Heidenreich, R. D., 186 Heine, S., 182 Hellman, M. Y., 256, 262(99), 280, 281(201) Hellmuth, E., 301, 313 Henderson, J. N., 256, 262(92) Hendricks, R. W., 61, 159, 178, 180, 181(91) Hendus, H., 237, 325 Henry, N. F. M., 2, 11(10), 18(10), 23(10), 24(10), 29(10), 30(10), 43(10), 47(10), 48(10), 60(10), 61(10), 67(10), 68(10), 70(10), 78(10) Hermans, P. H., 119, 121 Hess, P. H., 334 Hettinger, W. P., 325 Heuval, H. M., 321, 322 Hillier, I. H., 307 Hinrichsen, G., 237, 325 Hirai, N., 299, 300(45) Hirsch, P. B., 186 Hoashi, K., 306 Hoback, R. F., 256, 262(94) Hobbs, S. Y., 325 Hock, C. W., 256,257(96), 262(96), 263, 366 Hodge, A. M., 216, 217(21), 233(21), 346 Hoffman, J. D., 296,300,330,342,349,350, 351, 352(43), 361, 362, 363, 365(44), 375(43), 386, 387(113), 388(113), 390, 392(113), 393( 113), 395

403

AUTHOR INDEX FOR PART B

Hohne, G., 148 Holden, H. W., 301, 303(62) Holdsworth, P. J., 256, 262(108, 109), 263(108), 267, 306, 307 Holik, A. S . , 215 Hollahan, J. R., 241, 242(49) Holland, L., 208 Holland, R. F., 250, 251(73) Holland, V. F., 238, 244 Holmes, D. R., 82, 90(39), 93, 95 Hone, K., 332, 334 Hosemann, R., 100, 101(64), 102(65), 103, 105, 106, 109(65, 68), 111, 112(65), 123, 144, 147(26, 27), 148, 164(26), 171, 172 Howie, A., 186 Hrdlovic, P., 256 HSU,N. N.-C., 335 Hubbell, D. S., 335 Hudis, M., 241, 242 Huges, A. J., 277 Huges, R. E., 20 Hughes, R. H., 305 Huira, H., 334 Hungerford, G. P., 248 Hunt, B. J., 332 Huseby, T. W.,254, 300, 305(57), 314, 315(57), 352, 364 Hybart, F. J., 303, 336

I Igarashi, S., 336 Ikeda, M., 306 Ilicheva, Z. F., 279 Illers, K. H., 237, 256, 25,7(90), 262(90), 325 Ingram, P., 339 Inoue, M., 3% Isaacson, R. B., 366 Ishibashi, T., 280 Iwasaki, M., 245

J Jaccodine, R., 343 Jackson, J. B., 237, 300 374 Jacobson, 0.. Ja!Te, M., 306 James, R. W.. 30, 38, 65, 66(24), 67(24), 102(24) James, W. J., 366 Jauhianen, T., 327

Jellinek, H. H. G., 256 Jenkins, H., 278, 280(170) Jenkins, L. T., 287 Joffre, S. P., 245 Johansen, G., 374 Johnsen, U., 337 Johnson, D. R., 280 Johnson, G. B., 334 Johnston, J. H., 336 Jones, F. W.,137 Jones, L. D., 325 Justin, J., 289, 290(5), 303, 325(65)

K Kaelbe, D. H., 334 Kagan, D. F., 240 Kaji, K., 148, 151 Kajikawa, N., 2W, 312(40) Kakudo, M., 84, 85, 109, 112, 129 Kambara, S., 252, 369 Kambe, H., 332, 334, 336 Kambour, R. P., 215 Kamide, K., 301, 302(60) Kanetsuna, H., 372, 385 Kanig, G., 232 Kaplan, M. H., 328 Karasz, F. E., 325, 335 Kardos, J. L., 304, 308, 327(72) Kargin, V. A., 240, 279 Kasai, N., 84, 85, 109, 112, 129 Kashiwabara, H., 280 Kashmin, M. I., 328 Kasper, J. S., 92 Kato, K., 216 Kaufman, F., 242 Kavesh, S., 127, 163, 167(58) Kawai, T., 280, 297, 299, 300(37), 308, 315 Kawamura, Y.,248, 250(70a) Kay, D., 186 Ke, B., 295, 330, 331, 335 Keating, D. T., 139 Keavney, J. J., 329, 330(180) Kefeli, A. A., 256, 262(86, 131), 267 Keith, H. D., 186, 231, 232, 339, 345 Keller, A., 151, 152, 153, 163, 170, 174, 222, 229,230,235, 237,238, 245(26), 247, 248, 249. 254(26), 256, 257(107), 260, 261(117, 118, 119, 120, 123, 124, 125, 126, 127, 128, 129). 262,263(101,102, 103, 104, 105, 107, 108, 117, 118, 119, 120). 264, 265, 266,

404

AUTHOR INDEX FOR PART B

267,268,278,279,280,281,282,283,284, 285,297,299,300,312,315,339,343,345, 373, 382, 384 Keller, L. A., 238 Kenarov, A. V., 368 Keniry, J. S., 325 Kent, P., 182 Kenyon, A. S., 326, 336 Kessis, J. J., 291, 292 Khadr, M. M., 298, 307, 308(36) Khoury, F., 339, 341 Kibler, C. J., 308 Kiel, A. M., 384 Kim, H. G., 307 Kirdon, A., 137, 142 Kirshenbaum, I., 366 Kiselev, A. G., 213 Kleinman, L., 307 Klemperer, O., 186 Klug, A., 83, 85, 118 Klug, H. P., 5, 129, 133(2) Klute, C. H., 332 Kobayashi, M., 20 Kobayashi, S., 274 Kobayashi, Y.,151 Koch, J., 120 Koenig, J. L., 245, 248, 251, 274, 276, 277(160), 308 Koleske, J. V., 330 Kooistra, T., 303, 304(66) Kopp, M. K., 46, 61, 155 Korenaga, T., 297, 314(30) Koritskii, A. T., 279 Kortleve, G., 167, 177 Kothari, N. R., 146, 151(34) Kotoyori, T., 336 Koutsky, J. A., 386, 387, 388, 392(114), 393(114), 395, 396 Kovacs, A. J., 192, 229, 230, 237, 260, 362, 372, 382 Kramer, E. J., 174 Kratky, O., 157, 159(51), 179 Kritchevsky, G., 387, 391(118) Krsova, A., 277 Kubat, J., 296 Kubota, K., 240, 241(38) Kulshreshtha, A. K., 146, 151 Kusumoto, N., 280 Kusy, R. P., 280

Lagow, R. J., 245 Lajiness, W.G., 334 Lake, J. A., 182 Lamaze, C. E., 241, 242 Lambert, S . L., 384 Lando, J. B., 372 Langford, J. I., 139, 143, 146(24) Lanz, Jr., H., 374 Lanzetta, N., 306 Lapanje, S., 332, 333(210) Larson, N. R., 325 Lauer, J. L., 20 Laurenson, L., 208 Laurent, G., 252 Lauritzen, Jr., J. I., 342(44), 349, 350, 351, 361(44), 362, 363(44), 364, 365(44), 386, 387(113), 388( 1131, 392(113), 393( 113), 3 95 Lawton, E. J., 279 Lebedev, Y.V., 244 Le Chatelier, H., 287 Ledbetter, H. D., 396 Ledbury, K. J., 279 Lemstra, P. J., 255, 303, 304, 306 Lenk, C. T., 334 Lester, G . R., 373 Lety, A., 303 Levy, D., 382 Li, L. S., 240 Liberti, F. N., 306 Lin, J. S., 182 Lind, K. C. J. B., 321, 322 Linderstr@m-Lang,K., 374 Lipatov, Y. S., 244 Lipson, H., 92 Liquon, A. M., 93, 94, 95 Lonsdale, K., 2, 11(10), 18(10), 23(10), 24(10), 29(10), 30(10), 43(10), 47(10), 48(W, 60(10), 61(10), 67(10), 68(10), 70(10), 78(10) Lorenz, R., 237 Lotz, B., 192, 229, 237 Lovering, E. G., 327 Luthy, M., 1 Lundberg, R. D., 330 Luttinger, M., 248 Lyons, B. J., 278, 280

AUTHOR INDEX FOR PART B

M McCullough, R. L., 144, 147(27), 148 Machizuki, T., 306 McKenna, L., 326 Mackie, J. S., 240 McKinney, P. V.,328 MacKnight, W. J., 326, 335 Maddams, W. F., 369 Maeda, M., 280 Maeda, Y.,372, 385 Magill, J. H., 362, 366, 372, 385 Maglio, G., 306 Maher, D. M., 222, 224, 227(29) Makhlis, F. A., 278 MaKuchi, K., 245 Manley, T. C., 258, 260(121) Malavasic, T., 332, 333 Malhotra, S. C., 336 Manche, E. P., 332 Mandelkern, L., 237, 238, 280, 295, 300, 303,305,306, 307,309,315,325, 326,330, 362, 364, 369, 372, 381, 384 Manescalchi, F., 336 Mankin, G. I., 325 Manley, R. St. J. 230,261(121), 264,269(121), 270, 271(121), 272(121), 273, 274, 301, 303(63), 384 Manson, J. A., 372 Mantell, R. C., 241, 242(47) Marchessault, R. H., 305 Marchetta, C., 306 Marchetti, A,, 307 Mardon, P. G., 180, 181(91) Margrave, J. L., 245 Mark, H., 1 Martin, G. M., 237 Marton, C., 186, 187(9a) Marton, L., 186, 187 Martuscelli, E., 153, 256, 257(107), 260(119, 120, 126) 261(119, 120, 126), 262, 263(107, 119, 120). 264(120), 280(126), 281(126), 307, 315 Marx, C. L., 307 Matheson, L. A., 252 Matthews, J. L., 121 Matreyek, W., 238, 245(26), 247(26), 248(26), 249(26), 254(26), 256, 262(99), 280, 281(201)

405

Matsumoto, K., 280 Maurer, J. J., 329 Maxwell, J., 364 Mayhan, K. G., 366 Mazur, J., 182 Menta, R. E., 255 Meinel, G., 256, 262(98), 325, 326 Meltzer, T. H., 256, 262(94) Mercier, J. P., 332 Messersmith, D. C., 335 Miles, R. B., 327 Millard, M. M., 241 Miller, B., 318,319(126), 320(126), 321(126), 322( 126), 324( 126) Miller, D. C., 279 Miller, G. W.,306, 329 Miller, R. C., 148, 149, 150 Miller, R. L., 148, 238 Millett, M. A., 270 Miron, R. R., 334 Misra, A., 368 Mita, I., 332, 334 Mitchell, R. L., 270, 273 Mitsuhashi, S., 163 Mitsuhata, T., 299, 300(45) Mittal, P. K., 244 Miyagi, A., 274, 275, 301, 302 Mocherla, K. K., 274, 276(157) Moore, W. E., 270 Morgan, L. B., 348, 349(34), 373 Mori, N., 384 Mori, S., 256, 262(106), 263(106) Morosoff, N., 169, 172(69) Morrow, D. R., 307 Moze, A., 332, 333(210) Miiller, F. H., 369 Muggli, R., 270 Muhlethaler, K., 270 Muller, F. H., 324 Murphy, C. B., 334 Musgrave, W. K . R., 245 Muus, L. T., 20, 83, 85, 87, 113, 114 N

Nachtrab, G., 337 Nagai, H., 299, 312(40) Nagai, L., 20 Nagata, H., 280

406

A U T H O R INDEX FOR P A R T B

Nagatoshi, F., 253, 254(80), 299, 300 Nakagawa, T., 248, 249 Nakahara, S . , 222 Nakajima, A., 297, 300, 314, 325 Nara, S., 280 Natta, G., 16, 17(14), 81 Nawotki, K., 237 Nealy, D. L., 308 Nebuloni, M., 295 Neck, H. N., 396 Newman, S., 330, 396 Nghi, D., 269 Nicholson, R. B., 186 Nickerson, R. F., 238, 255(29), 269(29) Niegisch, W. D., 327 Nielsen, J. R., 250, 251(73) Nishimura, H., 300 Noel, C., 303 Nolands, J. S., 335 0

O’Connor, A., 163, 297 Odajima, A., 280 O’Dell, J. A., 170 Okada, M., 245 Okada, T., 237, 369 Okazaki, S., 245 Okui, N., 308 Olf, H. G., 256, 262(98), 326 Oliver, M. J., 387, 395 Olley, R. M., 216, 217(21), 233 O’Neill, M. J., 289, 290(5), 291,292(15), 321 Oplinger, G., 246 Ormand, W.L., 241, 242(47) Ormerod, M. G., 280 Osugi, J., 325 Ott, E., 1 Overbergh, N., 300, 303(59) Overton, J. R., 274, 276 Oyama, T., 248, 250

P Paciorek, K. L., 334 Padden, Jr., F. J., 231, 232, 345 Pae, K. D., 308 Painter, P. C., 174 Pal, K. D., 304, 327(73) Palm, J. A., 334

Palmer, R. P., 216, 238, 255, 262(31) Pals, D. T. F., 308 Palumbo, P., 306 Pande, M. R., 244 Pape, N. R., 93, 94(59) Parrod, J., 252 Pashley, D. W., 186 Pask, J. A., 291 Passaglia, E., 237, 339, 341, 350, 386, 387(113), 388(113), 392(113), 393(113), 39339) Patel, G. N., 230, 260(117, 118, 119, 122, 123, 126, 127, 128, 129, 130),261(117, 118, 119, 122, 123,261(126, 127,128, 129, 130), 262,263(117, 118, 119), 264,265,266,267, 268,279,280(126, 127, 128, 129, 130), 281, 282, 283, 284, 285, 286, 315 Patel, R. D., 230 Patterson, A. L., 92 Patterson, G. D., 331 Pauling, L., 90, 91 Pavlath, A. E., 241 Pavlinec, J., 256 Pedemonte, E., 384 Pedone, C., 93, 94(59, 61), 95(59), 96(61) Peiper, P., 332 Peiser, H. S., 121 Pella, E., 295 Pelzbauer, Z., 301, 303(63) Pennings, A. J., 384 Percival, D. F., 334 Perret, R., 175 Peterlin A., 168, 237, 256,262(98), 299, 300, 325, 326, 339 Peters, R. H., 298, 307, 308(36) Pfeiffer, H. C., 225, 227(30) Phillipp, B., 269 Pierce, E. M., 306 Pierret, S., 276 Pijpers, M. F. J., 384 Pillai, P. S . , 336 Pinkerton, D. M., 280 Pinner, S. H., 280 Platt, J. D., 303 Plaza, A., 368 Point, J. J., 345 Popova, L. A., 240 Porod, G., 173 Porter, R. S . , 256, 262(106), 263(106) Posner, A. S., 300

407

AUTHOR INDEX FOR PART B Pound, T. G., 260(121), 261(121), 264, 269(121), 270(121), 271(121), 272(121) Powell, R. S., 279 Predecki, P., 237 Preedy, J. E., 369 Price, F. P., 345, 386, 387(111), 388(111), 392(11l), 393( 11l), 3%, 397 Priest, D. J., 153,238,255, 260,261,262(32) Pnngle, 0. A., 175 Prosen, E. J., 291, 292(16) Prudhomme, R. E., 305

113, 115), 389, 392(112, 113, 115), 393, 394, 395(39), 396(115) Rossman, K., 241, 242(45) Roth, J. P., 252 Rudin, A., 240, 335 Rudnaya, G. V., 279 Ruhle, M.,224 Rugg, F. M.,256, 262(91) Ruland, W., 120,122, 124, 125, 126, 164, 175 Runt, J., 151, 152(45), 174, 175 Ruscher, C., 269

Q

S

Quinn, F. A., 325

R Rachinger, W. A., 137 Rafi Ahmad, S., 280 Rinby, B. G., 330 Rathje, J., 175 Rau, R. C., 130 Ravens, D. A. S., 255, 274, 275(154) Razumovskii, S. D., 256, 262(86, 131), 267 Read, B. E., 330 Reding, F. P., 240 Reed, R. L., 291, 292(10), 332(10) Rees, D. V., 339 Refner, J. A., 326 Reich, L., 291, 336 Reinhold, C., 168 Rempp, P., 252 Reneker, D. H., 242, 243, 300 Rhodes, M.B., 368 Rich, A., 83, 85 Richards, R. B., 121 Richardson, G. C., 307 Richardson, J. J., 325 Richardson, M. J., 228 Riew, C. K., 216 Rijke, A. M.,280, 315 Ritchie, I., 245 Roberts, D. E., 300 Roberts, R. C., 306, 307, 328, 329 Roberts, T. L., 229, 325 Rochas, P., 276 Roe, R. J., 230 Roppert, J., 182 Ross, G. S., 350,362,369,386,387,388(112,

Sabir, M., 336 Sacher, E., 334 Sadler, D. M.,152, 256, 262(103), 263(103) Sakaoka, K., 216,242 Salovey, R., 254, 256, 262(99), 280, 281(201), 300, 305(57), 314, 315, 352 Salyer. I. O., 326 Samuels, S. L., 307 Santer, Jr., E. R., 330 Sass, S. L., 174 Sauer, J. A., 308 Savolainen, A., 336 Sawada, M.,334 Sawada, S., 256, 262(95) Sax, N. I., 258 Saxon, R., 335 Scalisi, F. P., 235 Schechter, E., 155 Scherer, G., 387, 391(118) Schimada, S., 280 Schinohara, H., 245 Schmidt, G. F.,237, 299 Schmidt, P. W.,175, 178, 182, 183 Schmitt, J. M.,335 Schonfeld, A., 148 Schonhorn, H., 241, 242(48), 245, 396 Schotland, R. S., 327 Schouten, A. J., 255, 303, 306 Schreiber, H. R., 335 Schultz, J. M., 123, 163, 164, 165(61), 167(58), 175 Schwenker, Jr., R. F.,329 Sears, W.C., 279 Septier. A., 186 Serafiny, T. T., 256, 262(92) Serebryakov, E. P., 268(135), 269

408

AUTHOR INDEX FOR PART B

Seto, T., 384 Seymour, R. W., 307 Shalby, S. W., 306 Sharma, R. K., 300 Sharpe, R. D., 308 Sharples, A., 238, 255(30), 269(30) Sheldon, R. P., 328 Shelton, J. R., 256, 262(92) Shen, M. C., 330 Sherliker, F. R., 328, 329 Shin, C. N., 252 Shimada, S.,280 Shinoharra, I., 252 Shiokawa, K., 248, 250(70a) Siegel, B. M., 228 Simek, I., 369 Slade, P. E.,287, 307 Slayter, H. S., 214 Slichter, W.P., 238 Slovokhotova, N. A., 279 Smets, G., 301, 303(59) Smith, D. J., 82, 90(39) Smith, J. J., 256, 262(9) Smith, R. K., 305 Smith, R. W., 216 Snyder, R. G., 250, 251(74) Sohma, J., 280 Soule, J. L., 291 Speil, S., 291 Spencer, R. S., 374 Spinicci, R., 304 Spit, D. J., 244 Spruiell, J. E.,82, 89, 91 Stadnicki, S . J., 331 Stanislow, L., 175 Statton, W. O., 51, 121, 151, 153(42), 163(42), 237 Staudinger, H., 1 Stehling, F. C., 330 Stein, R. S., 368 Stejny, J., 260(122), 261(122), 264 Stenger, V. A., 246 Sterzel, H. J., 274 Stille, J. K., 213 Stivala, S. S., 291 Stoelting, J., 335 Stokes, A. R., 142, 145 Storks, K. H., 344 StratT, R., 387, 391(118) Stratta, J. J., 330

Strella, S., 293, 301(18), 329 Stroble, G. R., 175 Stutzman, G. L., 312, 316 Sueman, J. F., 270 Sumida, T., 297, 314(30), 325 Suzuki, T., 362 Sweet, G. E.,306

T Tadokoro, H., 80,81, 93,94(58), 95,96, 97, 98 Taggart, W. P., 326 Takayanagi, M., 280, 299, 300 Tamamoto, T., 280 Takokoro, H., 80 Tamura, Y., 299, 300(45) Tanaka, T., 93,94,95, 96, 97, 98 Tarbet, G . W., 256, 262(93) Tarin, P. M., 233 Temussi, P. A., 81 Thielke, H. G., 151 Thomas, E. L., 152, 174, 233 Thomas, H. R., 332 Thompson, K. R. L., 280 Thomson, J. J., 27 Till, P. H., 237 Till, Jr., P. H., 343 Tonelli, A. E., 328, 331 Tsuchida, E., 252 Tsuji, K., 380 Tsujimura, S . , 245 Tsvankin, D. Y., 169, 171(66) Tukiya, M., 280 Turnbull, D. J., 348, 349, 386, 387(111), 388(111), 392(111), 393(111) Turner, D. T., 278, 279(168), 280 Turner-Jones, A., 306 Tutiya, M., 280

U Udagawa, Y.,153, 174, 268 Uematsu, I., 330 Uematsu, Y., 330 Uhlmann, D. R., 387, 391

V Vadimsky, R. G., 230, 231, 232, 345 van Antwerpen, F., 368

AUTHOR INDEX FOR PART B

Vand, V., 83, 84(41), 113(41) Van Der Zee, P., 308 Vankrevelen, D. W., 306 van Krevelen, D. W., 368 Van Vlack, L. H., 5 Vaughan, G., 304 Venalainen, H., 336 Ver Strate, G.. 326 Vidotto, G., 382 Viehmann, W., 332 Vizovisik, I., 332, 333(210) Vogel, W., 148 Vold, M.J., 291 Volkov, T. I., 368 von Bastian, C. R., 182 Vonk, C. G., 167, 177, 182, 183 von Laue, M.,1 Vonnegut, B. J., 386 Vo Van, K., 336 Vrij, A., 351, 381

W Wagner, C. N. J., 144, 146 Wakelyn, N. T., 326 Waldman, M.H., 335 Waller, I., 140 Walter, E. R., 238 Walton, A. G., 386, 387(114), 388(114), 392(114), 393(114), 395(114), 396 Wang, J . 4 , 174 Ward, I. M.,255,256,262(97, 101, 102, 103, 104,105, 108). 263(101, 102, 103, 104, 105, 108), 264(104), 267, 268, 274, 275(154) Waring, J. R. S., 308 Warren, B. E., 30, 38, 65, 66(23), 67, 102(23), 133, 139, 147 Watanabe, K., 245 Watanabe, S., 269 Watson, E. S . , 289, 290 Weber, L., 291, 292(10), 332(10) Weeks, J. J., 296, 330, 350, 351, 352(43), 362, 375(43), 381, 386, 387(113), 388(113), 390, 392(113), 393(113), 395(39) Weeks, N. E., 256, 262(106), 263(106) Wegner, G., 274 Weidinger, A., 119, 121 Weininger, J. L., 241, 242(46) Weir, F. E., 278

409

Wells, 0. C., 227 Wen, W. Y.,280 Wendlandt, W. W., 293 Wenner, W. M.,306 Westerdahl, C. A., 241, 242(50) Wethington, J. A., 325 Whaley, T. P., 258, 259, 260 Whelan, M.J., 186 White, J. L., 82, 89, 91 White, J. R., 229 Whitney, J. F., 120 Whitwell, J. C., 318, 319(126), 320(126), 321( 126), 322( 126), 324( 126) Wiesener, E., 329 Wilchinsky, Z. W.,326 Wiles, D. M.,241, 242(51) Wiley, R. M.,374 Wilke, W., 144, 148 Wilkes, G. L., 307 Willard, P. E., 334 Williams, T., 256, 262(101, 102, 103, 104, 105, 108), 263(101, 102, 103, 104, 105, 108), 264(104), 267, 268, 279 Willmouth, F. M.,256, 262(102), 263(102), 307 Wilson, A. J. C., 102(65), 103, 133, 134(8), 135, 138(8), 139, 143, 145, 146 Wilson, 111, C. W.,330 Wilson, F. C., 88,90 Wilson, J. E., 278 Wilson, P. R., 368 Windle, A. H., 152 Windle, J. J., 241 Winslow, F. H., 238, 245(26), 247(26), 248(26), 249(26), 254(26), 256, 262(99), 280, 281(201) Witenhafer, D. E., 245, 248, 251 Wonacott, A. J., 97 Wittmann, J. C., 192 Wooden, D. L., 327 Woods, D. W., 330 Woodward, A. E., 307 Woolfson, M. M.,92, 93 Woolsey, N. F., 268(135), 269 Wrasidlo, W., 295 Wright, R. A . , 373 Wunderlich, B., 255, 274, 275, 288, 295, 297, 300, 301, 302, 306, 308, 313, 317, 318(125), 325, 328, 336, 339, 341, 350, 372, 385

410

AUTHOR INDEX FOR PART B

Y Yamada, S., 248, 249 Yamaguchi, K., 301, 302(60) Yamashita, Y., 299, 300(45) Yasuda, H., 241, 242 Ydagawa, Y., 256, 262(104), 263(104), 264( 104) Young, P. R., 326

2 Zachariasen, W. H., 30, 102(25) Zachmann, H. G., 295, 313(21), 337 Zaikov, G. E., 256, 262(86, 131), 267 Zuccarello, R. K., 329 Zworykin, V. K., 186

SUBJECT INDEX FOR PART B A

B

Aberrations, in the electron microscope, Background fog, from multiple electron image, 199-205 scattering, 190 Activation energy, for crystallization, 349 Barrel distortion, 204 from melt, 363 Baseline correction, in DSC melting curve, from solution, 364 318-324 Acylation of polystyrene, 255 Block copolymer, 340 Adjacent reentry model of folding, 229-230, electron microscopy studies, 234-235 238, 342, 344 Body-centered cubic lattice, Aminolysis of polyesters, 275-276 definition, 5 Amorphous material, structure factor for, 37 in cellulose, selective hydrolysis, 269-273 Bragg’s law of X-ray diffraction, 32, 161 in melt-crystallized polymers, removal by Branches in polymer chains, location of solvent-etching, 239-241 using degraded samples, 266-267 in poly(ethy1ene terephthalate), selective Bravais lattice, 3, 5 hydrolysis, 274-275 Bright-field image, in electron microscopy selective halogenation of, 245-251 description, 190- 192 selective oxidation of, 261-267 mode of operation, 211-212 Amorphous peak, in X-ray diffraction pattern, 119 C Anisotropic distortion, of electron microCalorimetry, see Differential scanning scope image, 204-205 Annealing, 299-300 calorimetry of polymer single crystals, 230, 231 Carbon film, for electron microscope speciprevention of crystal thickening, 254 men support, 213 Aperture, in electron microscope, Catalytic hydrolysis of polyesters, 274 diffraction at, 2OOT201 Cauchy function, to describe X-ray line selected area, 212 shape, 143, 145 Arrhenius equation, for polymerization rate Cellulation during spherulitic growth, 345 constants, 332 Cellulose Astigmatism, in electron microscope image, chain conformation in crystalline regions, 205 270, 272-273 Atactic chain configuration, 19 selective degradation by hydrolysis, 269Atomic scattering factor, 273 definition, 29 Cellulose triacetate, single crystals, selecof various atoms, 30 tive degradation, 273 Chain-folding model, 342, 343-344 Avrami equation, 336, 348 applications of, 366, 368, 369, 371-372 Chromatic aberration, in electron microsAvrami integer, for various nucleation COPY, 190, 201-203 modes, 348 Chromatic aberration coefficient. 202 Cilia, 342, 344, 346 Avrami theory of crystallization, 347-348 41 1

412

SUBJECT INDEX FOR PART B

Circle of confusion, at a point image, 200 determination by calorimetry (DSC), 3 16Cloverleaf pattern, in light scattered from 326 thin polymer films, 367 determination by X-ray diffraction, 114Collimation of X rays, 41, 156-159 127 slit-smearing effects at small angles, 164nature of, 99 167 Crystallinity index, 117-118 correction for, 177-183 determination by X-ray diffraction, 118Compatibility in polymer blends, 335 127 Compton scattering, correction in X-ray dif- Crystallite shape, effect on X-ray line fraction, 65 broadening, 133- 135 Condenser lenses, in an electron micro- Crystallite size, by wide-angle X-ray diffracscope, 207 tion, 129-153 Configuration Crystallization of crystallizable polymers, 340 Avrami theory, 347- 348 definition, 18 in bulk-type specimens, 369-375 Conformation density balance technique, 372-374 definition, 19 density gradient technique, 374 helical nomenclature, 20 dilatometric techniques, 370-372 Constant-temperature bath technique for sample preparation, 369-370 crystallization in thin films, 359 configurational requirements for, 340 Contamination, from organic molecules, in DSC studies, 336 an electron microscope, 208, 221 growth rate Contrast effect of molecular weight, 364 amorphous-crystalline, in an electron miequation for, 361 croscope, 189 homogeneous nucleation theory, 349 defocusing, 219 kinetic growth theory, 348-349 topographical, imaging, 222-224 primary, 371 Copper K a X radiation, 42, 47, 136-138, rate, experimental methods, 352-385 161 secondary, 303, 371 Correlation function, 175- 177 self-seeding technique, 381-383 Critical nucleus, work of formation, 392 from solution, 381-384 Crosslinking by radiation, 278, 352 kinetics, 383 location of, in polyethylene single crysin thin film specimens, 354-369 tals, 282-286 analysis of data, 361-365 Crystal distortions (defects) depolarization technique, 365-366 of the first kind, 100 dielectric measurements, 369 effect on X-ray diffraction pattern, 102infrared absorption technique, 368 105 light scattering technique, 366-368 of the second kind, 100-102, 128 optical microscopic studies, 356-365 effect on X-ray diffraction pattern, 105under pressure, 385 111 various polymers, supercooling depencrystallite size effects, 143- 149 dence, 363 typical corrections for, 149- 153 Crystallography, 2-25 Crystal structure Cubic crystal system determination by X-ray diffraction, 68-98 definition, 4 of polypropylene, effect on melting beinterplanar spacing formulae, 26 havior, 304 reciprocal lattice relationships, 24 Crystal systems, 2-5 Cubic lattice, definition, 5 Crystallinity Curing reaction, DSC studies, 334 determination, comparison of methods, Curved crystal X-ray monochromator, 49 326 Cylindrical film X-ray camera, 56

413

SUBJECT INDEX FOR PART B

D Dark-field image, in electron microscopy description, 194- 195 mode of operation, 211-212 Debye ring, 51, 53 Debye-Schemer powder camera, for X rays, 52, 59 Debye- Waller factor, 112, 140 Deconvolution procedures, in X-ray diffraction line analysis, 141 Defocusing contrast, in electron microscope image, 219 Degradation of polymers, thermal analysis studies, 335 Degree of polymerization, 340 Dehydrohalogenation, of poly(viny1idene chloride), 252-253 Dendritic crystals, melting behavior, 313 Density balance, for crystallization studies, 372-374 Density gradient column, 374 Detectors for X rays, 42-46, 154- 156 Dielectric measurement, of crystallization in thin polymer films, 369 Diffraction, see X-ray diffraction, Electron diffraction Diffraction contrast, in electron microscopy, 190 Differential scanning calorimetry (DSC) baseline correction, 318-324 crystallinity determinations, 3 16-326 glass transition determinations, 327-33 1 isothermal crystallization studies, 336337 melting studies, 294-316 polymerization kinetic studies, 33 1-334 principles, 289-291 Differential thermal analysis (DTA) apparatus, 288 equilibrium melting point determination, 375-381 instrumental factors affecting curve, 293 principles, 287-289 sample factors affecting curve, 293-294 sample holder (block), 293 theory, 291-292 Diffusion-controlled crystal growth, 345 Dilatometer, for crystallization studies, 370-372 Direction indices, crystallographic, 6

Disorder in crystalline polymers, 98- 114 analysis in terms of helical transforms, 113-114 calculation of scattered X-ray intensities, 1 1 1-112 one-dimensional models used in smallangle X-ray scattering, 167- 171 Distortions, geometric, of electron microscope image, 203-205 Droplet formation, 388-389 Droplet technique, for homogeneous nucleation, 385-396 DSC, see Differential scanning calorimetry DTA, see Differential thermal analysis

E Electromagnetic lens, 196- 197 Electron particle-wave concept, 186 scattering, by an atom, 187 multiple, 190 Electron density distribution, Fourier series representation, 38-39 Electron diffraction mode of operation, in electron microscope, 211-212 patterns, 192-194 small-angle, 222 Electron microscopy beam damage, 190 bright field image, 190-192, 211-212 dark field image, 194, 211-212 diffraction pattern mode, 21 1-212 electron optics, 195-205 focusing, 217-220 image degrading factors (aberrations), 199-205 instrumentation, 206-212 low-loss, 227-228 magnification calibration, 221 observation of single polymer molecules, 228 resolution, 220-221 scanning, 224-227 scanning-transmission, 227 specimen preparation, 212-217 topographical contrast imaging, 222-224 Electron source (gun), 206-207 End-group analysis by infrared, 251 Epoxy resins, curing studies by DSC, 334

414

SUBJECT INDEX FOR PART B

Equator of X-ray fiber pattern, 55 Equilibrium melting point, 296, 350, 362, 375-381 Etching of electron microscope specimens, 216, 240, 242-245 with plasma, 241 -245 with solvent, 239-241 Ewald sphere, 34-35 Extended-chain crystals, 347, 350, 385 melting behavior, 308

F Face-centered cubic lattice, 5 Fiber, X-ray patterns, 56-60 indexing, 71 -76 Fibrillar crystals, in spherulites, 342 Field-emission source, 207 Filter, nickel foil, for X-ray monochromatization, 47-48 Focusing action of a lens, 196 in electron microscopy, 217-220 electrons for X-ray production, 42 an X-ray beam, 157, 158 Fold period deduced from GPC of degraded samples, 263-266 definition, 162 dependence on crystallization temperature, 351 from diffuse X-ray scattering, 172-175 distribution in cellulose triacetate, 273 effect of annealing on, 300, 351 Fold surface chlorination and bromination of, 248 nature of, 128, 229, 238, 342, 344, 346 selective oxidation with nitric acid and ozone, 261 thickness of, 150 Fold surface free energy, 349 effect on melting point, 352 Fractionation, 353 Freeze-drying, of single crystal suspensions, 259 Fresnel fringes, in electron microscope image, 218-219 Friedel-Craft acylation, 255 Fringed micelle model, 342, 343

G

Gauche conformation, in polyethylene, infrared bands, 251 Gaussian distribution, of sizes in semicrystalline polymers, 169 Gaussian function, to describe X-ray line shape, 145 Geiger (Geiger-Muller) counter, 44,45, 155 Gel permeation chromatography (GPC) of hydrolyzed cellulose, 272 of oxidatively degraded crystalline polymers; 263 effect of crosslinking radiation, 282286 Glass-transition temperature determination by calorimetry (DSC), 32733 1 definition, 328 effect of heating rate and sample size, 329 effect on crystallization rate, 361-363 effect of molecular weight, 330-331 of various polymers, 327 Glide plane, in crystal symmetry, 9, 10 Glow-discharge etching, 242 Gold nuclei decoration, of electron microscope specimens, 217, 234 Goniometer, X-ray, 61, 63 GPC, see Gel permeation chromatography Grids, for specimen support, in electron microscopy, 213

H Halogenation, of crystalline polymers, 24525 1 effect on small-angle X-ray scattering, 249 Heat of fusion, of polyethylene single crystals effect of bromination, 250 effect of irradiation, 315 Heat of polymerization, by DSC, 332 Heat of reaction, from DTA curve, 292 Helical conformations, analysis by X-ray diffraction, 83-87 Helical point net system, 20 Heterogeneous nucleation, see Nucleation Hexagonal crystal system definition, 4 interplanar spacing formulae, 26 reciprocal lattice relationships, 24

SUBJECT INDEX FOR PART B

Hexagonal lattice definition, 5 reciprocal, 23 Holey film, specimen support in electron microscopy, 214 Homogeneous nucleation, see Nucleation Hot stage for optical microscope construction of, 357-359 crystallization of thin polymer films in, 360 Hydrogen-bonded sheets, in polyamide crystals, 88-90 Hydrolysis, selective degradation by, of cellulose, 269-274 of polyesters, 274-277

I Impurities in polymers, 353 clean-up operation, 354, 355 Infrared absorption, as measure of crystallization in thin polymer films, 368 Infrared analysis of irradiated polyethylene, 286 of polyethylene, effect of halogenation, 250-251 of unsaturation and branch location in polymers, 267 Intercrystalline links, 223, 231 -233, 234, 346 see also Tie molecules Inversion center, in crystal symmetry, 8, 9 Isotactic chain configuration, 19, 340 Isothermal crystallization Avrami equation for, 336 effect of molecular weight, 337

K Kinetics of polymerization, DSC studies, 332 Kratky small-angle X-ray collimation system, 157, 178-179

L Lamellar crystals core thickness from X-ray line shape, 150 in melt-crystallized polymers, 170, 244, 345 thickness from diffuse X-ray scattering, 172- 175 two-phase model, 128

415

Lamellar thickening, 230, 300 Lamellar thickness effect of annealing on, 299 effect on melting point, 351-352 effect of molecular weight on, 297 see also Fold period Lattice, crystallographic, various types, 5 Lattice plane indices, 7 Laue equations, 32, 34 for oriented fiber, 56 Laue group, in crystal symmetry, 18 Layer lines, 53-59 Lens theory, 196 Light scattering, from thin polymer films, 366-368 Liquid-liquid t_ransitions, in amorphous polymers, 331 Lithium-drifted silicon detector, for X rays, 45 Lorentz polarization factor, 66 correction to X-ray diffraction line profile, 139-140

M Maltese cross extinction pattern in spherulites, 341 Melt-crystallized polymers, morphology, 163, 230-233, 238, 341-346 Melting behavior of polymers, 294-316 dependence on crystallization conditions, 297 effect of heating rate, 300-302, 313-314 effect of irradiation, 305 effect of molecular weight, 298 effect of sample mass, 308-312 Melting point definition on DSC curve, 295 effect of crystal morphology, 298 equilibrium, 296, 350 determination, 362, 375-381 of lamellar crystals, 296, 352 multiple peaks, 303-308 Melting transition, DSC/DTA trace, 289 Meridian, of X-ray fiber pattern, 55 Methyl methacrylate, bulk polymerization, DSC study of, 332-333 Microtoming (sectioning) polymer specimens for electron microscopy, 215 Miller indices, 7 Mirror plane, in crystal symmetry, 9, 10

416

SUBJECT INDEX FOR PART B

Monochromatization of X rays, 46-50, 136 Monoclinic crystal system definition, 4 interplanar spacing formulae, 26 reciprocal lattice relationships, 24 Monoclinic lattice, definition, 5 Monomer, heat of polymerization by DSC, 332 Morphology of block copolymers, 234-235 of melt-crystallized polymers, 163, 230233, 238, 341-346 by solvent-etching, 239-241 interior, by plasma-etching, 241-245 one-dimensional model in small-angle Xray scattering, 167-171 of polyethylene, effect on melting point, 298 of polymer single crystals, 128, 174, 229230, 237, 342, 343-344 control by self-seeding, 382 deductions from halogenation, 248-25 1 deductions from oxidation, 261 -267 Mosaic block model, of paracrystallinity in single crystals, 151 Multiple melting peaks, 303-308 Multiplicity factor, in X-ray peak intensity, 67

N Nitric acid degradation of polymers, 256257 Nucleation effect on melting point, 303 effect of mode on crystallization, 364 heterogeneous in Avrami theory, 348 measurement of, 396-397 homogeneous in Avrami theory, 348 measurement by droplet technique, 385-396 rate constant, 349, 390 theory of crystallization, 349 Nylon 6 drawn, true melting point by methoxymethylation, 253-254 a form, repeat distance and unit cell, 8182

y form

crystallite thickness from X-ray line broadening, 151 repeat distance and unit cell, 82 selective degradation by hydrolysis, 276 solvent-etching, 240 spherulitic growth rate, 363 typical crystallinities, 126 Nylon 6,6 etching with argon and nitrogen plasma, 244 a form, unit cell and crystal structure, 8890 selective degradation by hydrazinolysis, 277 selective degradation by hydrolysis, 276 0

Objective lens, in an electron microscope, 208-211 Oriented polymers, electron microscopic studies, 233-235 Orthorhombic crystal system definition, 4 interplanar spacing formulae, 26 reciprocal lattice relationships, 24 Orthorhombic lattice definition, 5 reciprocal, 23 Orthorhombic unit cell, direction indices in. 6 Oxidation of polyethylene, selective, with ozone, 260-261 Oxidation of polymers, studied by thermal analysis, 335 Ozone degradation with, 257-261 generation of, 258-259 properties of, 257-258

P Packing of chains, in polymer crystals, 8790 Paracrystalline lattice factor, 106 for one-dimensional lattice, 110 Paracrystallinity, 128 effect on X-ray diffraction pattern, 1051 1 1 , 144-149

SUBJECT INDEX FOR PART B

one-dimensional lattice distribution function, 106-108 n-Paraffins crystallization of polyethylene from dotriacontane, 346 long-chain, preparation from degraded polyethylene crystals, 268-269 Partial crystallization, in DTA technique, 375, 376, 379 Phase contrast, in electron microscope image, 219 Photographic recording, of X rays, 46-53, 154 Photographic techniques in optical microscopy, 360 Photomultiplyer, in crystallization studies, 366 Pin-cushion distortion, 203, 204 Plasma-etching, 241 -245 Pleated sheet structure, of polypeptide chains, 90-91 Point group, crystallographic, 17 Polarizing microscope observation of spherulites in, 341 in thin film crystallization, 361 Polyacrylonitrile, glass-transition temperature, 327 Polyamide, see Nylon Polybutadiene, staining for electron microscopy, 216 Polychloroprene, glass-transition temperature, 327 Polychlorotrifluoroethylene crystallization behavior, 364 spherulitic growth rate, 363 Poly(dimethy1 siloxane), glass-transition temperature, 327 Polyethylene amorphous/crystalline contrast enhancement, in electron microscopy, 232 chain conformation, 20 crystal growth rate, 365 crystallite size in highly-drawn samples, 149 crystallization from dotriacontane, 346 DTA melting curve, 376 equilibrium melting point, 379-381 etching with oxygen plasma, 242-245 etching for electron microscopy, 216, 240

417

heat of fusion, 325 homogeneous nucleation, droplet experiments, 386-396 calculation of surface free energies, 395 intercrystalline links in, 231 -233, 234 irradiation with X rays, 281 infrared bands, 251 melt-crystallized, small-angle X-ray pattern, 168 morphology, effect on melting point, 298 photochlorination of, 248, 249 purification and thin film preparation, 355 repeat distance, 79 single crystals annealed in suspension, 231 branching location in, 267 bright-field electron micrograph, 190191, 192 bromination of, 250 core thickness by X-ray diffraction, 152-153, 161 dark-field electron micrograph, 195 DSC melting curve, 295, 306, 309 oil suspended, 311 effect of irradiation on melting, 305 electron diffraction pattern, 194 fold structure, 229-230 from GPC of degraded samples, 263266 heat of fusion, 250, 315 irradiation with y rays, 282-286 mats of, small-angle X-ray scattering pattern, 162 mosaic blocks in sedimented mats, 152 in suspended samples, 174 ozone degradation, .260-261 preparation of n-paraffins from, 268-269 self-seeding, 260 twinned, 229 solvent-etching, 240 space group and crystal structure, 15, 16 spherulites in, 341, 342 stepwise melting of, 377 unit cell dimensions, 75 X-ray fiber pattern, 73-74 indexing of, 75 Poly(ethy1ene oxide) glass-transition temperature, 327 self-seeding crystallization, 382

418

SUBJECT INDEX FOR PART B

Poly(ethy1ene terephthalate) Poly(artho-methyl styrene) crystallinity by X-ray diffraction, 121 crystal structure, 88-89 crystallization of, density balance density, 88 method, 373 repeat distance, 88 degradation by aminolysis, 276 unit cell dimensions, 78 DTA melting point, effect of heating rate, Pol yoxy methy lene 302 crystallization behavior, 364 etching with argon and nitrogen plasma, glass-transition temperature, 327 244 helical chain conformation, 20-21 selective degradation by hydrolysis, 274unoriented, X-ray pattern, 54, 59 X-ray fiber pattern, 53, 55, 57 275 Polyisobutylene X-ray peaks, crystalline and amorphous, density, 93 119 helical conformation and crystal struc- Polypropylene ture, 93 atactic, glass-transition temperature, ef1,4-Polyisoprene, cis, melting behavior, 307 fect of molecular weight, 331 Polymer mixtures, identification by thermal isotactic analysis, 335-336 crystallization behavior, 364 Polymer reactions, 335 DSC melting curve, 305 Polymer single crystals effect of crystal form on melting, 304 bromination of, 247 nitric acid degraded, molecular weight, chlorination of, 247-248 263 crosslinking by electron beam, in electron photochlorination of, 248, 249 microscope, 190 repeat distance and chain conformagaseous halogenation of, 248 tion, 80 halogenation in suspension, 246-248 solvent-etching of, 240 melting behavior typical crystallinities, 125 irradiated, 315 unit cell and crystal structure, 16- 17 in oil suspension, 308-312 syndiotactic melting point, 296 repeat distance and chain conformamorphology, 128, 174, 229-230, 237, 342, tion, 81 343-344 single crystals, melting behavior, 307 precipitated mats Poly(propy1ene oxide) crystallinity profile, 161 isotactic, melting behavior, 304 small-angle X-ray scattering in, 162 spherulitic growth rate, 363 self-seeding, 382 Polystyrene “true” melting point, 313-316 glass-transition temperature, 327 Polymer surfaces isotactic chemical modification, 245-255 crosslinked film, for electron microdegradation with chemicals, 255-277 scope specimen support, 213 modification by plasma-etching, 241 -245 crystallization behavior, 364 Polymerization, of organic molecules in an DSC melting curve, 304 electron microscope, 208, 221 single crystals, 255 Polymerization kinetics, calorimetric (DSC) spherulitic growth rate, 363 studies, 331-334 photochlorination of, 248, 249 Poly(methy1 methacrylate), X-ray diffrac- Polytetrafluoroethylene tion pattern (amorphous), 54 chain conformation, 20 Poly(4-methyl pentene-1). chlorination of, phase transformations of the chain helix, 248-249 1 I4

419

SUBJECT INDEX FOR PART B X-ray fiber pattern and helical conformation, 86-87 Poly(tetramethy1ene-p-silphenylene siloxane), spherulitic growth rate, 363 Polythioethylene, glass-transition temperature, 327 Poly(viny1 alcohol), glass-transition temperature, 327 Poly(viny1 chloride), glass-transition temperature, 327 Poly(viny1 fluoride), glass-transition temperature, 327 Poly(viny1idene chloride) dehydrohalogenation of, 252-253 glass-transition temperature, 327 infrared bands, 253 single crystal, fold surface, 253 Poly(vinylidene fluoride), glass-transition temperature, 327 Position-sensitive detector, for X rays, 46, 155-156 Powder diffractometer, X-ray, 61-63 Powder pattern, X-ray, 51, 53 Precession camera, X-ray, 60, 69 Primary nucleus, 348 Projection system, in an electron microscope, 21 1-212 Proportional counter, for X rays, 44,45-46,

I55

Q Quadrants, of X-ray fiber pattern, 55

R Radiation-induced chemical changes in polymers, 278-280 location of, 28 1-286 Radius of confusion, of a point image, from spherical and diffraction errors, 201 Reciprocal lattice, 22-25, 193 in cylindrical coordinates, for fiber pattern indexing, 71-73 relationship to real lattice vectors, 24 Reinhold distribution, of sizes in semicrystalline polymers, 168- 169 Reorganizational processes, during melting, 313

Repeat distance, of a polymer chain, 19 determination by X-ray diffraction, 79 Repeat unit, 19 chemical, 18 of a crystal, 3 Replication, of specimens for electron microscopy, 217 Resolution of electron microscope image, 185, 22022 1 of scanning-electron microscope image, 225 Rhombohedral crystal system definition, 4 interplanar spacing formulae, 26 reciprocal lattice relationships, 24 Rhombohedral lattice, definition, 5 Ribbonlike lamellae in melt-crystallized polymers, 345, 346 twisting in, 347 Rotating anode, in X-ray tube, 41, 158 Rotation axis, in crystal symmetry, 8, 9 Ruland’s method, for X-ray crystallinity, 122- 127

S

SAXS, see Small-angle X-ray scattering Scanning-electron microscopy (SEMI, 224227 Scanning-transmission electron microscopy, 227 space charge effects in, 205 Scattering, of electrons, 187-190 Scattering theory, classical, 27 Scherrer equation, 130 Scherrer shape constant, 130, 133- 135 Scintillation counter, for X rays, 43-45, 155 Screw axis, in crystal symmetry, 9, 10 conventional notation, 21 Secondary crystallization, 303 Secondary electron imaging, in scanningelectron microscopy, 225 Selected-area aperture, 212 Self-seeding technique, of crystallization, 260, 381-383 SEM (scanning-electron microscopy), 224227

420

SUBJECT INDEX FOR PART B

Shadowing, of specimens for electron microscope, 191, 214-215 “Shish kebob” crystals, 347, 384 Single crystal, polymer, see Polymer single crystal Single-crystal orienter, X-ray goniometer, 63 Single-crystal X-ray monochromator, 48 Small-angle electron diffraction, 222 Small-angle X-ray scattering collimation, 156-159 focusing methods, 157, 158 detectors, 154-156 diffuse, 172- 175 instrumentation, 153-161 intensity changes, during polymer halogenation, 249 line-broadening, due to crystal disorders, 171-172 in melt-crystallized polymers, 163- 164 modeling lamellar crystal morphology, 167-171 in polymer single crystal mats, 161-163 sample geometry, 160 slit-smearing, 164- 167 correction for, 177- 183 Solvent-etching, 239-241 Space-charge effects, in electron imaging, 205 Space group, crystallographic, 11 Space lattice, 3, 5 Specific volume, in crystallinity determination, 120 Sphere of reflection, 34 Spherical aberration, in electron microscope, 199-200 coefficient, 200 Spherulites, 34 1 - 346 in polyethylene, revealed by plasma-etching, 243 ringed, 347 size from light scattering, 367 Spherulitic growth, nature of, 345 Staining specimens, for electron microscopy, 216 STEM (scanning-transmission electron microscopy), 227 Stereopair, 222 Strain in crystals, effect on X-ray line shape, 144-149

Structure factor for body-centered lattice, 37 definition, 36 experimental evaluation, 65-68 of a helix, 83-84 in terms of electron-density distribution, 39 Superheating, in crystalline polymers, 295 Surface degradation, techniques, 255-277 Surface free energy of lamellar crystals, 296 in lamellar growth theory, 349 temperature dependence, 350 Surface modification of polymers, 245-255 Switchboard model, 344 Symmetry, crystallographic, 3, 8- 18 Syndiotactic chain configuration, 19, 340 Systematic extinctions (absences), in X-ray diffraction, 37, 69 criteria for various symmetries, 77-78

T Tacticity, 18, 340 Tetragonal crystal system definition, 4 interplanar spacing formulae, 26 reciprocal lattice relationships, 24 Tetragonal lattice, definition, 5 Thermal analysis, instrumentation, 287-291 see also Differential thermal analysis, Differential scanning calorimetry Thermal history, of poly(ethy1ene terephthalate), effect on melting point, 302 Thermal motion effect on X-ray diffraction line shape, 140- 14 1 effect on X-ray scattering, 38 lattice distortions from, 101 Thermogravimetry , 336 Thin film specimens, for crystallization, preparation of, 354-356 Through-focus series, of electron micrographs, 209, 210 Tie molecules, 11 1 , 345 see also Intercrystalline links Topographical contrast imaging, in electron microscopy, 222-224 Toxicity of halogens, 246

SUBJECT INDEX FOR PART B

Trans conformation, in polyethylene, infrared bands, 251 Transmission-electron microscopy, see Electron microscopy Triclinic crystal system definition, 4 interplanar spacing formulae, 26 reciprocal lattice relationships, 24 Triclinic lattice, definition, 5 Tsvankin distribution, of sizes in semicrystalline polymers, 169 Two-phase model, of crystallinity in polymers, 99, 116, 175, 373-374 size distributions, 167- 171

U Ultrasonic dispersion, of polymer droplets for nucleation studies, 389, 393 Unit cell, 3, 5 Unsaturation in polymers in irradiated polyethylene, 286 location of, using degraded samples, 266. 267

W Weissenberg camera, 60, 69 Wobbler focusing, in electron microscopy, 218

X

X-ray cameras, 50-60 X-ray detection systems, 42-46

42 1

X-ray diffraction background correction, 64-65 crystal structure determination, 68-98 crystallinity determination, 118- 127 crystallite size determination, 129- 153 fiber patterns, 56-60 instrumentation, 39-64 integrated intensity, 65-68 powder (polycrystalline) patterns, 51, 53, 54,59 small-angle, see Small-angle X-ray scattering theory, 25-39 trial structure technique, 91-93 X-ray diffractometers, counter, 60-64 X-ray line broadening effect of crystal distortions, 143- 149 effect of crystallite shape, 133-135 effect of crystallite size, 129-135 instrumental broadening types, 135-143 typical corrections for, 149- 153 separation from crystal size effects, 141-143 at small angles, 171-172 X-ray monochromatization, 46-50 X-ray scattering by an array of atoms, 30-34 by an atom, 28-30 by complex crystal structures, 35-38 by electrons, 25-28 effect of thermal motion, 38 parasitic, removal of in diffraction work, 159-160 small-angle, see Small-angle X-ray scattering X-ray source (tube), 39-42 for small-angle scattering, 157- 159

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