Polymers. Pt C: Physical Properties (Experimental Methods in the Physical Sciences)

Methods of Experimental Physics VOLUME 16 POLYMERS PART

c : Physical Properties

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

Volume 16

Polymers PART C: Physical Properties

Edited by

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

I980 ACADEMIC PRESS

@

A Subsidiary of Horcourt Brace jovonovich, Publishers

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COPYRIGHT @ 1980, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. N O PART OF THIS PUBLICATION MAY B E REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RFTRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

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Library of Congress Cataloging in Publication Data Main entry under title: Physical properties. (Methods of experimental physics : v. 16C) Includes bibliograpliical references and index. 1 . Polymers and polymerization. I . Marton. Ladislaus Laszlo, Date 11. I'ava, Ronald A. i l l . Series. TA455.PS8P47 620.1'92 79-25995 ISBN 0-1 2-475958-0

PRINTED IN THE UNITED STATES OF AMERICA

80 81 82 83

9 8 7 6 5 4 3 2 1

CONTENTS CONTRIBUTORS . .

. . . . . . . . . . . . . F O R E W O R D ... . . . . . . . . . . . . . . P R E F A C E . .. . . . . . . . . . . . . . . . OF VOLUME16, PARTSA AND B . CONTENTS

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xiii xv xvii xix

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xxiii

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xxv

TO VOLUME 16, PARTSA CONTRIBUTORS

VOLUMES I N SERIES

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AND

B

11. Viscoelastic and Steady-State Rheological Response by DONALDJ. PLAZEK

11.0. Introduction

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1 1 . 1 . Linear Viscoelastic Behavior . . . 1 1 . 1 . 1 . Definitions and Background 11.1.2. Instrumentation . . . . .

1 1.2. Steady-State Response

1

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3 3 21

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44 44 45

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46

11.2.1. Practical Solids. . . 11.2.2. Viscoelastic Liquids

1 1.3. Nonlinear Viscoelastic Behavior V

vi

CONTENTS

11.3.1. 11.3.2.

Nonlinear Steady-State Behavior of Viscoelastic Liquids . . . . . . . . . . . . 47 Nonlinear Transient and Dynamic Properties . 49

11.4. Pressure Effects on Viscoelastic Behavior .

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51

1 1.5. Sample Handling . . . . . . . . . 11.5.1. Molding . . . . . . . . . 11S . 2 . Solution Mixing . . . . . 11S . 3 . Molecular Weight Blending 11.5.4. Film Casting . . . . . . .

52 52

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Introduction . . . . . . . . . Immersion Apparatus . . . . . Other Experimental Techniques Molecular Interpretation . . . . Conclusions . . . . . . . . .

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56 57 57

12. Further Mechanical Techniques

12. 1. Ultrasonic Measurements by BRUCEHARTMANN 12.1.1. 12.1 .2 . 12.1.3. 12.1.4. 12.1.5.

59 60 75 79 89

12.2. Static High-Pressure Measurements on Polymers by R . W . WARFIELD 12.2.1. Introduction . . . . . . . . . . . . . . . . 91 12.2.2. Types of Equipment . . . . . . . . . . . . 92 12.2.3. Response of Polymers to Static High Pressure 104

12.3. Stress-Strain Yield Testing of Solid Polymers

by JOHNL . RUTHERFORD AND NORMAN BROWN

12.3.1. Introduction 12.3.2. Definitions .

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117 117

vii

CONTENTS

12.3.3. Methods for Measuring Strain 12.3.4. Test Method. . . . . . . . 12.3.5. Significance of Results . . .

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119 121 129

13. Production and Measurement of Orientation

by IAN L . HAY

13.1. Introduction

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138

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146

13.2. The Production of Orientation

13.3. Description of Orientation

137

13.4. Measurement of Orientation . . . . . 13.4.1. Wide-Angle X-Ray Diffraction 13.4.2. Birefringence. . . . . . . . 13.4.3. Sonic Modulus . . . . . . . 13.4.4. Infrared Dichroism . . . . . 13.4.5. Small-Angle X-Ray Scattering

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150 150 161 167 173 175

14. ESR Study of Polymer Fracture

by TOSHIHIKO NAGAMURA 14.1. Introduction

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14.2. Basic Theory and Experimental Techniques . 14.2.1. Principle of ESR Method . . . . . 14.2.2. Radical Concentration . . . . . . . 14.2.3. System for Observing Mechanically

Generated Radicals .

185

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186 186 187

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188

viii

CONTENTS

14.3. Radical Formation by Mechanical Fracture of Polymers . . . . . . . . . . . . . . . . . . . . 14.3.1, Radical Species . . . . . . . . . . . . . 14.3.2. Reaction and Location of Radicals . . . . . 14.3.3. Radical Concentration and Fracture Surface .

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195 i95 197 202

14.4. Radical Formation during Tensile Deformation and Fracture of Oriented Crystalline Polymers . . . . . . . 203 14.4.1. Radical Species . . . . . . . . . . . . . . 203 14.4.2. Reactivity and Location of Radicals . . . . . 205 14.4.3. Radical Concentration. . . . . . . . . . . . 206 14.4.4. Constant-Rate and Stepwise Stretching . . . . 207 14.4.5. Effects of Temperature and Heat-Treatment . 210 14.4.6. Effects of Strain Rate and Cyclic Loading . . . 212 14.5. Fracture in Elastomers . . . . . . . . . . . . . . . 213 14.5.1, Ozone-Stress Cracking . . . . . . . . . . . 214 14.5.2. Low-Temperature Deformation of Preoriented Rubbers and Granular Filled Rubbers . . . . . 216 14.6. Molecular Mechanism of Deformation and Fracture of Polymers . . . . . . . . . . . . . . . . . . . . 217 14.6.1. Some Models of Polymer Fracture and Polymer Morphology . . . . . . . . . . . . 217 14.6.2. Molecular Models of Deformation and Fracture Mainly Based on ESR Results . . . . . . . . 219 14.7. Limitations of ESR Method and Comparison with Associated Studies . . . . . . . . . . . . . . . . . 225 14.7.1. Problems in ESR Investigations . . . . . . . 225 14.7.2. Other Methods for Studying Micromechanism of Polymer Deformation and Fracture . . . . . 226

15. Methods by Studying Crazing by NORMANBROWN

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233

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237

15.1. Introduction 15.2. Structure

ix

CONTENTS

15.2.1. Optical Methods . . 15.2.2. Electron Microscopy 15.2.3. The Stress Field . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15.3. Initiation and Growth . . . . . . . 15.3.1. Stress Criteria for Initiation 15.3.2. Growth of Crazes . . . . .

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15.4. Environmental Effects in Liquids and Gases .

237 242 245 249 249 252

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255

15.5. Relationship of Crazing to Macroscopic Mechanical Behavior . . . . . . . . . . . . . . . . . . . . . . 15.5.1. The Stress-Strain Curve . . . . . . . . . . 15.5.2. Creep . . . . . . . . . . . . . . . . . . . 15.5.3. The Size Effect . . . . . . . . . . . . . . . 15.5.4. Shear Flow and Crazing . . . . . . . . . . . 15.5.5. Fracture . . . . . . . . . . . . . . . . . . 15.5.6. High-Impact-Strength Polymers . . . . . . .

262 263 265 268 269 270 271

16. Polymeric Alloys

by J . ROOVERS

16.1. Introduction

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16.2. Thermodynamics . . . . . . . . . . . . . . . 16.2.1. Polymer Mixtures . . . . . . . . . . 16.2.2. Block Copolymers . . . . . . . . . . 16.2.3. Polymer-Polymer Interphase . . . . . 16.2.4. Segmental Polymer-Polymer Interaction Parameter . . . . . . . . . . . . . . 16.3. Direct Observation . . . . . 16.3.1. Visual Observation . 16.3.2. Optical Microscopy . 16.3.3. Electron Microscopy

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275

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276 276 278 282

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283

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287 287 288 291

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C0N TEN TS

X

16.4. Scattering Techniques . . . . . . . . . . . . . . . . 16.4.1. Small-Angle Light Scattering (SALS) . . . . . 16.4.2. Small-Angle X-Ray Scattering (SAXS) . . . . 16.4.3. Small-Angle Neutron Scattering (SANS) . . .

16.5. Glass Transition Temperature Measurements 16.5.1. Tg of Mixtures of Polymers . . . . 16.5.2. TBof Block Copolymers . . . . . . 16.6. Conclusion.

299 299 300 305

. . . . . 306 . . . . . 307 . . . . . 311

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314

17. Permeation. Diffusion. and Sorption of Gases and Vapors by R . M . FELDERA N D G. S. HUVARD

17.1. Introduction

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17.2. Historical Perspective . . . . . 17.2.1. Theory . . . . . . . 17.2.2. Experimental Methods

315

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17.3, Phenomenology . . . . . . . . . . . . . . . . 17.3.1. Correlation and Estimation of Transport and Solubility Coefficients . . . . . . . . 17.3.2. Effects of Polymer Composition and Morphology on Transport Rates . . . . . 17.3.3. Transport of Water Vapor . . . . . . . . 17.3.4. Concentration-Dependent Fickian Diffusion in Rubbery Polymers . . . . . . . . . . 17.3.5. Dual-Mode Sorption and Diffusion in Glassy Polymers . . . . . . . . . . . . 17.3.6. Anomalous Transport of Vapors in Glassy Polymers . . . . . . . . . . . . . . . 17.3.7. Two-Stage Sorption of Swelling Penetrants in Glassy Polymers . . . . . . . . . . .

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316 316 319

324

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328 331

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332

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333

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334

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337

xi

CONTENTS

17.4. Categories of Experimental Methods

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338

17.5. Pressure Measurement and Temperature Control . . . . 340 17.6. Sorption Methods . . . . . . 17.6.1. Experiments and Data 17.6.2. Calculations . . . . . 17.6.3. Experimental Methods

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17.7. Integral Permeation (Closed Receiving Volume) Methods . . . . . . . . . . . . . . . . . . . . . . 17.7.1. Experiments and Data . . . . . . . . . 17.7.2. Calculations . . . . . . . . . . . . . . 17.7.3. Experimental Methods . . . . . . . . .

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. .

342 342 343 349

356 356 357 362

17.8. Differential Permeation and Weighing Cup (Open Receiving Volume) Methods . . . . . . . . . . . . . 17.8.1. Experiments and Data . . . . . . . . . . . 17.8.2. Calculations . . . . . . . . . . . . . . . . 17.8.3. Experimental Methods . . . . . . . . . . .

367 367 369 371

17.9. Sources and Minimization of Errors . . . . . . . 17.9.1. Operating Procedures . . . . . . . . . . 17.9.2. Data Analysis . . . . . . . . . . . . . 17.9.3. System Dynamics . . . . . . . . . . . .

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372 372 373 375

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379 381 395

18. Electrical Methods

18.1. Dielectric Constant and Loss by RICHARD H. BOYD 18.1.1. Introduction . . . . . . . . 18.1.2. Phenomenology of Dielectrics 18.1.3. Experimental Procedures . .

xii

CONTENTS

18.2. Static Electricity by D . KEITHDAVIES 18.2.1. 18.2.2. 18.2.3. 18.2.4. 18.2.5.

Introduction . . . . . . . . Methods of Measuring Charge Contact Electrification . . . Radiation Charging . . . . . Charge Migration . . . . . .

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424 428 435 439

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443 444 451 462 489

18.3. Electric Breakdown by B . R . VARLOW 18.3.1. 18.3.2. 18.3.3. 18.3.4. 18.3.5.

AUTHORINDEX

Introduction . . . . . . . Mechanisms of Breakdown Specimen Preparation . . . Experimental Methods . . High-Field Conduction . .

. . . . .

. . . . . . . . . . . . . . . . . . . . . . .

SUBJECTINDEX.

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499 519

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

RICHARD H. BOYD,Department of Materials Science and Engineering, University of Utah, Salt Lake City, Utah 84112 (379) NORMAN BROWN,Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19174 (1 17, 233)

D. KEITHDAVIES,Electrical Research Association Limited, Leatherhead, Surrey, KT22 7SA, England (421)

R. M. FELDER,Department of Chemical Engineering, North Carolina State University, Raleigh, North Carolina 27650 (315) BRUCEHARTMANN, Naval Surface Weapons Center, White Oak, Silver Spring, Maryland 20910 (59)

IAN L. HAY,Celanese Research Company, Summit Laboratory, Summit, New Jersey 07901 (137) G. S . HUVARD,Department of Chemical Engineering, North Curolina State University, Raleigh, North Carolina 27607 (315) TOSHIHIKO NAGAMURA,* Department of Mechanical and Industrial Engineering, Department of Materials Science and Engineering, College of Engineering, University of Utah, Salt Lake City, Utah 84112 (185) DONALDJ. PLAZEK,Department of Metallurgical and Materials Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261 (1) J . ROOVERS,Division of Chemistry, National Research Council of Canada, Ottawa, Ontario, K I A OR9 Canada (275)

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

xiv

CONTRIBUTORS

JOHN L. RUTHERFORD, Kerrrfott Division, The Singer Compciny, Little Fcills, N e ~ Jersey t 07424 (1 17)

B. R. VARLOW, Electrical Engineering Ltrboratory, University of ManChester, Manchester M13 9PL. Englrind (444) R.

W . WARFIELD, Nui~mlSurfice

Spring, Mnryland 20910 (91)

Weapons Center, White Ook, Silver

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

xv

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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 specifically 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

xvii

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CONTENTS OF VOLUME 16, PARTS A AND B

PART A: Molecular Structure and Dynamics

1. introduction by R. A. FAVA 1.1. 1.2. 1.3. 1.4. 1.5. 1.6.

Historic Development Definitions Formation and Conformation The Solid State Orientation Impurities

2. Polymer Molecular Weights by DOROTHY J. POLLOCK A N D ROBERT T. 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. WHITEand T. WOLFRAM 3.3. Rayleigh-Brillouin Scattering in Polymers by G. D. PATTERSON 3.4. Inelastic Neutron Scattering Spectroscopy by C. V. BERNEYand SIDNEY YIP xix

xx

CONTENTS OF V O L U M E

16,

PARTS A A N D B

4. High-Resolution Nuclear Magnetic Resonance Spectroscopy by J. R. LYERLA 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. DuPRE 5.3. Paramagnetic Probe Techniques by PHILIP L. KUMLER 5.4. Small-Angle Neutron Scattering by J. S. KING AUTHORINDEX-SUBJECT INDEXFOR PARTSA, B,

AND

C

PART B: Crystal Structure and Morphology

6. X-Ray Diffraction

6.1. Unit Cell and Crystallinity by JOSEPHE. SPRUIELL and S. CLARK EDWARD 6.2. Crystallite Size and Lamellar Thickness by JING-IWANG and IANR. HARRISON 7. Electron Microscopy by RICHARD G. VADIMSKY

7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.7.

Introduction Fundamentals Electron Optics The Instrument Operational Considerations Other Microscopy Techniques Applications

8. Chemical Methods in Polymer Physics by G. N. PATEL

8.1. Disorder in Polymer Crystals and Chemical Methods 8.2. Solvent-Etching

CONTENTS OF VOLUME

16,

PARTS A AND B

xxi

8.3. Plasma-Etching 8.4. The Surface Modification Techniques 8.6. Irradiation and Selective Degradation 9. Thermal Analysis of Polymers by JAMES RUNT and IAN R . HARRISON

9.1. 9.2. 9.3. 9.4. 9.5. 9.6. 9.7. 9.8.

Introduction Iristrumentation and Method Theory Basic Factors Affecting the DTA/DSC Curve Melting Behavior of Polymers Quantitative Methods Other Applications Summary

10. Nucleation and Crystallization by GAYLONS. Ross and LOISJ . FROLEN

10.1. 10.2. 10.3. 10.4.

Introduction General Background on Semicrystalline Polymers Experimental Methods for Measuring Crystallization Rates Nucleation

AUTHORINDEX-SUBJECT INDEX FOR PARTB

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CONTRIBUTORS TO VOLUME 16, PARTS A AND B Part A

C. V. BERNEY,Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 L. LAWRENCE CHAPOY,Instituttet for Kemiindustri, The Technical University of Denmurk, 2800 Lyngby, Denmark DONALD B. DuPRB, Department of Chemistry, University of Louisville, Louisville, Kentucky 40208 R. A. FAVA,ARCO Polymers, Inc., 440 College Park Drive, Monroeville, Pennsylvania I5146 J . S. KING,Nuclear Engineering Department, University of Michigan, Ann Arbor, Michigan 48109 ROBERTF . KRATZ,ARCO Polymers, Inc., Research and Development Department, P.O. Box 208, Monaca, Pennsylvania 15061 P. L. KUMLER, Department of Chemistry, State University of New York, College of Fredoniu, Fredoniu, New York 14063 J. R. LYERLA,IBM Reseurch Laboratories, Sun Jose, California 95193 G. D . PATTERSON, Bell Telephone Laboratories, Murray Hill,New Jersey 07974 DOROTHY J . POLLOCK, ARCO Polymers, Inc., Research and Development Department, P.O. Box 208, Monaca, Pennsylvania 15061 R. G. SNYDER, Western Regional Research Center, Science and Education Administration, U.S. Department of Agriculture, Berkeley, California 94710 J. R. STEVENS, Department of Physics, University of Guelph, Guelph, Ontario, N I G 2WI, Canada H. W. WHITE,Depurtment of Physics, University of Missouri, University Park, Columbia, Missouri 65211 T . WOLFRAM,Department of Physics, University of Missouri-Columbia, Columbia. Missouri 65211 xxiii

xxiv

CONTRIBUTORS TO VOLUME

16,

PARTS A AND B

SIDNEY YIP,Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Part B

EDWARDS. CLARK,Polymer Engineering, University of Tennessee, Knoxville, Tennessee 37916 LOISJ. FROLEN, National Measurement Laboratory, National Bureau of Standards, Washington, D. C. 20234 IAN R . HARRISON, College of Earth and Mineral Sciences, The Pennsylvania State University, University Park, Pennsylvania 16802 G. N . PATEL,Corporate Research Center, Allied Chemical Corporation, Morristown, New Jersey 07960 GAYLON S . R o s s , National Measurement Laboratory, National Bureau of Standards, Washington, D. C. 20234 JAMES RUNT,Polymer Science Section, Material Sciences Department, The Pennsylvania State University, University Park, Pennsylvania 16801 JOSEPH E. SPRUIELL, Polymer Engineering, Universiry of Tennessee, Knoxville, Tennessee 37916 RICHARDG. VADIMSKY, Bell Telephone Laboratories, Murray Hill, New Jersey 07974 JING-I WANG,College of Earth and Mineral Sciences, The Pennsylvania State University, University Park, Pennsylvania 16802

METHODS OF EXPERIMENTAL PHYSICS Editors-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 two parts) Edited by Benjamin Bederson and Wade L. Fite

Interactions (in

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 Volume 12. Astrophysics-Part A: Optical and Infrared Edited by N. Carleton Part B: Radio Telescopes, Part C: Radio Observations Editedby M. L. Meeks Volume 13. Spectroscopy (in two parts) Edited by Dudley Williams xxv

xxvi

METHODS OF EXPERIMENTAL PHYSICS

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 Experimental Physics VOLUME 16 POLYMERS PART

c: Physical Properties

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11. VISCOELASTIC AND STEADY-STATE RHEOLOGICAL RESPONSE

By Donald J. Plazek 11.O. Introduction This part, on mechanical methods, deals with the determination of the character of polymeric rheological response including creep, stress relaxation, dynamic-mechanical properties at audio and ultralow frequencies, and steady-state shearing response. The experimental techniques used to make these determinations have not changed dramatically in kind in the recent past, and several reviews give summaries of the instrumentation that has contributed significantly in the area of mechanical polymer characterization. We have no desire to simply repeat material that is already available, but endeavor to supplement the description of the primary methods with principles and details that can be helpful in maximizing the possibility of obtaining useful and accurate results. The sources of greatest errors will be emphasized when known to us. Our necessarily limited view of rheology, the study of the deformation of matter, will cover principally linear viscoelastic response with brief reference to the nonlinear domain of response. Steady-state and equilibrium response, both linear and nonlinear, is treated as the long-timelimiting response of viscoelastic liquids and solids. Viscoelastic Behavior Definition

Viscoelastic behavior is a time-dependent mechanical response, which represents characterizing material behavior as opposed to system response. It changes with temperature but, by definition, is not the temperature dependence of any property, although, as will be seen below, temperature variation of some properties, such as a modulus, can reflect the presence of viscoelastic behavior. The time dependence referred to here is the noninstantaneous response of a homogeneous material body to variations of applied tractions or deformations. The delayed response referred to is that which cannot be attributed to the inertia of the body. In addition, it is assumed that the I METHODS OF EXPERIMENTAL PHYSICS, VOL.

16c

Copyright @ 1980 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12475958-0

2

11.

VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

thermodynamical state of the material is not changing with time, that is to say, aging effects are time effects that are not included in the definition of viscoelastic behavior. If a viscoelastic body is instantaneously deformed and held fixed thereafter, the forces generated by the body after the deformation is achieved will not result in any work because of the lack of any concurrent associated displacement, and therefore the involved energy will be dissipated into heat as the forces decay. Likewise, when such a previously deformed body is released any delayed change in shape will take place without any attendant stress; again, work is not done and all of the mechanical energy involved is dissipated into thermal energy. Finally, if one considers the deformation of a Newtonian liquid, where the shearing stress is proportional to the rate of strain, it is clear from its definition that when applied shearing tractions are removed deformation immediately ceases and the shearing stress is zero. No work at all is recovered frdm the sample. All of the energy utilized in bringing about any preceding shear strain is dissipated. The lack of complete simultaneity of,stress and strain in any deformation process necessarily implies that one is not dealing with a mechanically conservative process. Time dependence implies mechanical energy losses into thermal energy at least to some extent. In addition to fulfilling the requirements for characterizing the intkinsic response of an isotropic elastic material? (i.e., accounting for the effect of geometrical shape and the nature of the deformation process used) to characterize a viscoelastic material it is necessary to specify the effective stress or strain history of the viscoelastic body being investigated. To explain what is meant by the “effective” stress or strain history it is helpful to note that viscoelastic materials have been said to have a memory, albeit a fading one. The state of stress or strain of a viscoelastic body is a function of its history; the strength of the dependence is greatest for events in the most recent past and diminishes as they become more remote in time. Therefore, if such a body is unperturbed for a sufficiently long time it can be considered to be devoid of previous impressions and ready to receive a controlled pattern of stimulus. The forgetting process can be accelerated by heating the material, if necessary. Criteria for the judgement of the degree of relaxation are given below. The simplest of patterns of mechanical stimulus are used to characterize viscoelastic materials. A constant state of stress “instantaneously” produced in a previously relaxed specimen, with the resulting increasing t ‘No attempt is made here to deal with the complications associated with the presence of anisotropy, which in the case of polymers is most often encountered as the result of orientation processes on amorphous polymers below their glass temperature or on polycrystalline polymers.

1 1 . 1 . LINEAR VISCOELASTIC BEHAVIOR

3

strain being monitored as a function of the time following the creation of the stressed state, is the description of the creep experiment. Alternatively one can use the stress relaxation experiment in which a rapidly imposed fixed strain is created in a previously relaxed specimen and the diminishing stress resulting is followed as a function of the duration of the applied strain. The third of the most commonly used viscoelastic measurements is that which yields dynamic moduli or compliances. A steady-state sinusoidal strain is imposed on a specimen and the ratio of the maximum magnitudes of the stress and strain and the phase angle between them is determined as a function of frequency. It is assumed that further details and references will be sought out by the reader concerning his particular interest or problem in the principal instrumentation references known to us.l-*'

11.1 Linear Viscoelastic Behavior 11.1.1. Definitions and Background

11.1.1.1 Creep and Creep Recovery. 1 1.1.1.1.1. VISCOELASTIC LIQUIDS.If a shearing stress alztis rapidly created and maintained in A. J. Staverman and F. Schwarzl, in "Die Physik der Hochpolymeren" (H. A. Stuart, ed.), Vol. IV, Chapter 1. Springer-Verlag. Berlin and New York, 1956. J. D. Ferry, Rheology 2, 433 (1958). J. D. Ferry, "Viscaelastic Properties of Polymers," 1st ed. Wiley, New York, 1%1; 2nd ed., 1970. B. A. Toms, Rheology 2, 475 (1958). A. Jobling and J. E. Roberts, Rheology 2, 503 (1958). K. Walters, "Rheometry." Chapman & Hall, London, 1975. " S. Middleman, "The Flow of High Polymers." Wiley (Interscience), New York, 1968. I. M. Ward, "Mechanical Properties of Solid Polymers." Wiley (Interscience), New York, 1971. * J. R. Van Wazer, J. W. Lyons, K. Y. Kim, and R. E. Colwell, "Viscosity and Flow Measurement." Wiley (Interscience), New York, 1963. D. A. Thomas and S. Turner, Test. Polym. 4, 73 (1%9). S. Oka, Rheology 3, 18 (I%O). I* L. E. Nielsen. "Mechanical Properties of Polymers and Composites," Vol. 1. Dekker, New York, 1974. Is W. Philippoff, Phys. Acoust. 2B, 1 (1965). R. S. Marvin and J. E. McKinney, Phys. Acoust. 2B, 165 (1965). I5 B. E. Read and G. D. Dean, "Determination of Dynamic Properties of Polymers and Composites." Wiley, New York, 1979. J. Koppelmann, Kolloid-Z. 144, 12 (1955); Rhcol. Acra 1, 20 (1958). I' F. H. Gaskins and W. Philippoff, J . Appl. Polym. Sci. 2, 143 (1959). t The convention is followed where the first subscript indicates the direction of the surface normal and the second indicates the direction of the applied traction.

4

11.

VISCOELASTIC A N D STEADY-STATE RHEOLOCICAL RESPONSE

an isotropic viscoelastic material a shear strain ylz(t) results, which grows in time, indefinitely if the material is liquid or toward a long-time asymptotic limit if the material is a solid. In the case of the viscoelastic liquid, a constant velocity of deformation or rate of strain is ultimately reached. It is found as a sufficient condition that, if the strain is small, the strain at any given time t is proportional to the stress. Therefore, the ratio of the strain divided by the stress is a unique function of time, which is characteristic of the material for at least all strains less than that measured. This function J ( t ) is called the shear creep compliance and has been found for linear and branched amorphous polymers that are viscoelastic liquids to be the sum of three contributions18 ~ l Z ( r ) / ~ l Z

J ( t ) = Jg

+ Jd$(r) + t/r),

(1 1 . 1 . 1 )

where J g the glassy compliance arises from an apparently instantaneous contribution to the deformation; r) is the limiting low-rate-of-shearviscosity; $(t) is the normalized retarded recoverable creep compliance function, which is zero when the time r is zero (i.e., the time at which the stress is created in the sample) and is equal to one when t = 00; and J d is the normalization constant, the retarded compliance. The units of the compliance are mZ/Nor cm2/dyne. The first two terms on the right-hand side of Eq. ( 1 1 . 1 . 1 ) are proportional to the recoverable portion of the deformation accumulated during creep extending to steady-state conditions. Steady-state conditions are obtained when $(t) becomes equal to one. The only way that one can prove that a condition of steady state has been achieved is by measuring the total recoverable deformation following each of a series of creep experiments of ever greater duration. When it is established that the total recoverable deformation no longer increases no matter how prolonged the creep experiment is, a time that is effectively infinite can be denoted. The measurement of the time-dependent recoverable deformation following the “instantaneous” removal of the stress after steady state has been achieved is called creep recovery or more simply recovery. The recovered strain divided by the stress that was applied during the creep portion of the experiment is the recoverable creep compliance .I,( It? is obvious ). from the preceding statements that the remaining term in Eq. (1 1 . 1 . 1 ) is proportional to the permanent deformation accumulated during creep. When $0)= 1, J(f) = J g

+ Jd

-t f / r ) ,

(1 1 . 1 . 2 )

and (11.1.3) H.Leaderman, Rheology 2, 1 (1958).

11.1. .LINEAR VISCOELASTIC BEHAVIOR

5

That is to say, the creep velocity measured in steady state is simply the reciprocal of the viscosity. The sum of the two recoverable compliance terms J , + Jd is called J e , the steady-state recoverable shear compliance. It is often assumed that a constant creep velocity is proof that steady-state conditions have been reached. This assumption is usually borne out by experience but several cases have been e n c ~ u n t e r e d ' ~ -where ~l direct recovery tests clearly showed that, in spite of a velocity that was constant within fine limits, steady state was far from being attained. Measurement of the terminal creep velocity is by far the most dependable method of determining the shear viscosity, but the viscosity can also be determined by examining the difference between the recoverable and creep deformation, where the creep run might have fallen short of achieving steady state but was still five to ten times longer than the recovery involved (see the Boltzmann superposition below). NinomiyaZ2has shown that a plot of [ J ( t ) / t ] [ dlog J ( t ) / d log r] vs. l / r yields a relatively linear extrapolation to the intercept. This intercept is the limiting value of dJ(r)/dr at infinite time, which is the reciprocal of the viscosity, namely, Eq. (1 1.1.3). Finally, determinations of the viscosity may be made by measuring the amount of permanent deformation accumulated during a relatively short creep run by waiting for complete recovery. If the entire process were to be carried out isothermally no time would be saved, but if the temperature were raised during the recovery a saving of time might be possible. When the time to reach steady state is excessive and it is desired to determine the viscosity from the terminal velocity or to get on with the measurement of a unique recovery curve (recovery curves following nonsteady-state creep are not unique and do not yield the recoverable compliance curve) a technique suggested by Leaderman er a/.23is invaluable. They proposed that once steady state is achieved, so long as the state of stress remains unchanged, the temperature of the sample can be changed and the steady-state condition is maintained. If this is true one can choose a higher creep temperature, at which one attains steady state in a convenient length of time, and subsequently cool to the temperature of interest for the determination of the viscosity and the recoverable compliance. A creep run made on a linearly viscoelastic liquid starts at its highest velocity and monotonically decreases to its terminal velocity, reK . E. Van Holde and J . W. .Williams, J . Polym. Sri. 11, 243 (1953). D. J. Plazek, Trans. Sac. Rheol. 9, 119 (1965). E. Riande, H. Markovitz, D. J . Plazek, and N. Raghupathi, J . Polym. Sci.. Polym. Symp. 50, 405 (1975). 22 K . Ninomiya, J . Phys. Chem. 67, 1152 (1963). H. Leaderman, R . G . Smith, and R . W. Joncs, J . Polym. Sci. 14, 47 (1954). 2o

6

11. VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

flecting only viscous flow. Recoverable deformation processes enhance the velocity at all times preceding the attainment of steady state. Any premature determination of the viscosity from the creep velocity yields an answer that is erroneously low. It can be estimated that it would take over 100 years to achieve steady-state creep in a polymer with a molecular weight of about 1 x lo5at its glass temperature T R . It would therefore be impossible to determine the viscosity of such a polymer at Tgrbut with Leaderman's technique a determination should be possible in a day. We have concluded from a similar line of reasoning that any viscosities greater than 1014Pa sec ( l O I 5 poise) that have been reported in the literature would be incorrect on the basis that the deformation involved was not all viscous unless Leaderman's technique had been employed. To emphasize how difficult it can be to decide whether steady state has been achieved by observing creep behavior instead of creep recovery we refer to an extended creep run carried out on a poly dimethylsiloxane gum, where from the second day to beyond two weeks the velocity remained constant within experimental certainty. The creep run was extended to 36 days by which time it appeared clear, that from.the second hour of creep to the end, the deformation had been linear with the cube root of time, a form called Andrade creep, and was completely recoverable.24 Instead of coming close to steady-state creep no viscous deformation had been observed at all, but the continuing deformation was direct evidence that the normalized compliance function had not reached unity. The viscosity also may be obtained, as will be seen below, from stress relaxation measurements as well as from dynamic-mechanical measurements and a host of steady-state deformations. Rotational, capillary, and falling-ball viscometers are several common examples involving the latter. 11.1.1.1.2. VISCOELASTIC SOLIDS.If, in extensive creep and creep recovery experiments on a viscoelastic body, all of the deformation is found to be ultimately recoverable (i.e., no viscous flow is observed), the material is called a viscoelastic solid. Equation (11.1.1) and the associated remarks hold except for the fact that the viscosity coefficient r) is operationally infinite and so r/q = 0. Instead of reaching a limiting creep velocity, all creep runs made on viscoelastic solids, at least in principle, decelerate from the initial highest velocity to a zero velocity and an equilibrium deformation is reached. As far as most polymeric systems are concerned, if at high temperatures (say 100°C above T g )no viscous deformation is discernible it can be concluded that an effective molecular network is present in the material. Such molecular networks, which preD. J. Plazek, W. Dannhauser, and J. D. Ferry,J. Colloid Sci. 16, 101 (1961).

11. I

7

LINEAR VISCOELASTIC BEHAVIOR

-

0

TIME t FIG.I . Linear creep response of a viscoelastic solid.

clude the permanent displacement of polymeric chains past one another, are usually present by virtue of crystallites or covalently bonded units (cross linkages), which connect neighboring polymer chains together. A more unusual case can occur with block copolymers, where the blocks are thermodynamically incompatible and microphase separation occurs. At temperatures where one phase is glassy, for many purposes the copolymer behaves as a viscoelastic solid.25 Like many definitions, that for the viscoelastic solid has practical limitations. A vulcanized rubber, where the molecular network is held together by covalently bonded linkages, is certainly categorized as a viscoelastic solid; but it can exhibit a permanent deformation called “set” or “cold flow.” This deformation occurs because all of the covalent bonds do not maintain their integrity at all times and when a covalent bond opens, with the material in a deformed state, the two loose chain ends are free to diffuse into a relaxed state before possible reaction of the free radicals involved occurs. This phenomenon has been treated by Tobolsky under the name of chemorheology.26 11.1.1.1.3. BOLTZMANNSUPERPOSITION. If a viscoelastic material is linear not only will the creep strain observed at a given time of creep be proportional to the stress but the corresponding strain arising from any increment of stress will add to strains resulting from stresses previously created in the body. The proportionality is illustrated in Fig. 1, where 0 is the time at which the state of stress is created. If a stress w1is created in a viscoelastic body at a time el and is changed at a later time e2, during the period between + 0, the body deforms simply as in a creep run where the stress is v1. Then yl(t) = J(t -

el)ul,

(1 1.1.4)

m G. Holden, E. T. Bishop, and W. R. Legge, J . Polym. Sci., Part C 26,37 (1969). *O

A. V. Tobolsky, “Properties and Structure of Polymers.” Wiley, New York, 1960.

8

11. VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

where f is universal time and J ( t - 6,) is defined to be zero for all negative time, i.e., when t < 8,. After the stress is changed at 6, the deformation resulting can be thought of as a superimposed creep run that is proportional to the stress increment and is added to the on-going deformation at all subsequent times. The stress increment is c2: ytot(t)= y,(r)

For the case that

U, =

+ y 2 ( t ) = J ( t - el)U1 + J ( t - e2)c2.

(11.1.5)

-el,

ytot(t)/cl=

-

el) - J ( t

-

(1 1.1.6)

6,).

Substituting from Eq. (1 1.1. l), (11.1.7)

When

el is set equal to 0 and the subscript 2 is dropped, Y ( t ) / c l = Jd[+(f)

-

- 6)1 + 6/71.

(11.1.8)

If steady state had been reached before the equal and opposite stress was introduced, $(r) would be unity and y(f)/Ul= J g

+ J d + -e - [ J g + J d $ ( f 71

-

6)].

(1 1.1.9)

The first three constants multiplied by U,add up to the total creep deformation accumulated during the presence of v1alone, i.e., for 6 sec. The creation of an equal and opposite stress at O2 is equivalent to removing the traction that gives rise to elin the body being studied. Therefore, the preceding stress history is the important one used in performing the creep and creep recovery experiment. The bracketed compliances in Eq. (1 1.1.9) are obtained directly from the “instantaneous” and delayed recoverable deformations measured following the removal of stress in a specimen creeping in steady state. The decomposition of the total creep deformation into permanent and recoverable components, using Eq. (1 1.1. I), by determining 71 from a constant terminal velocity and then subtracting r/q at different times to ascertain the recoverable compliance J , = J g + JdNf), has often been suggested in monographs on viscoelasticity. We agree that in principle the procedure is correct but in practice we have found it to be a dubious procedure, which can yield highly erroneous recoverable compliances in the terminal zone of response as often as not. Small changes in the geometry of a specimen and in the temperature during a prolonged creep run can lead to J ( t ) - 2 / 7 1 differences that are grossly in error long before 2/71 >> J , + J d $ ( f ) . Even without the mentioned perturbances, $ ( t ) can

11.1

9

LINEAR VISCOELASTIC BEHAVIOR

TIME

FIG.2. Decomposition of a time-varying stress into incremental steps.

be far from unity when the experimental resolution in creep is of the same order as or larger than the differences noted. A given continuously varying stress history as shown in Fig. 2 can be considered as a succession of increments, and the resulting strains can be added in the general sense of mathematical linearity, i.e., extending Eq. (1 I. 1.5) to the general case, (1 1.1.10)

In the limit where the stress increments become infinitesimal, we have

which is one form of Boltzmann superposition. This is a complete one-dimensional statement of viscoelastic linearity aside from finite jumps in the stress, which can be accounted for with additional finite terms as expressed in Eq. (1 1.1.10). Given a known stress history then the accompanying strain can be calculated at any time t. 11.1.1.2. Stress Relaxation. The corresponding expression of Boltzmann superposition for the stress at any given time as a function of a known strain history involves the stress-relaxation modulus function. This function is defined in shear as the decreasing time-dependent stress crlz(t), which follows the sudden imposition of a fixed shear strain yyz divided by rt: uiz(t)/Y?z = G(th

(1 1.1.12)

The rate of decrease in stress is a monotonically decreasing function of time. The initial time derivative is the greatest and the ultimate rate is zero for both viscoelastic solids and liquids. The equilibrium longtime-limiting stress level u r Z (is~finite ) for a viscoelastic solid and is zero for a viscoelastic liquid. Since r!z is a constant, the same statements are true for the stress-relaxation modulus. Because the stress-relaxation

10 1 1 .

VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

modulus is not the reciprocal of the creep compliance, it is customary to adhere to this nomenclature to avoid confusion in the literature. The use of the expression “creep modulus” for this reason should be avoided. Following a development parallel to that shown above for J ( r ) , one obtains an alternative form of the Boltzmann superposition principle,

( 1 1.1.13) 11.1.1.3. Dynamic Mechanical Properties. In principle, only one parameter determined over the entire time or frequency scale is necessary to completely characterize the shear response of a viscoelastic material. However, since different standard deformation histories are used in studying viscoelastic response, different response curves are measured, which tend to be complementary. Creep and stress relaxation measurements are convenient means of studying the long-time behavior, i.e., from to lo7 sec. Creep results are conveniently applied to problems where the stress history of a material is controlled and stress relaxation behavior readily helps the determination of response to controlled strain histories. There is some overlap, but dynamic (sinusoidal) methods can extend characterization curves to equivalently short times by measurements at very high frequencies, up to loB Hz. Dynamic measurements usually yield two quantities: one reflecting the maximum energy stored during a cycle and represented by G’(w), which is that part of the stress in phase with the strain divided by the strain; the other reflecting the energy dissipated per cycle and represented by G”(w), which is that part of the stress 90” out of phase with the strain divided by the strain. G’ and G” are, respectively, the real and imaginary parts of the complex shear modulus G* = G’ + iG” (Nm-2). The complex shear compliance is J* = J ’ - jJ” = l/G*, where J ’ ( w ) , the storage compliance, is the component of the strain in phase with the stress divided by the stress and J ” ( w ) , the loss compliance, is that part of the strain 90” out of phase with the stress divided by the stress. The complex dynamic shear viscosity r)* = 77’ - ir)” = G*/w has not infrequently been reported. The angular frequency w = 27w, where u is the ordinary frequency in hertz. The absolute modulus JGI= (Gf2+ G”z)1/2and the absolute compliance IJI = ( J f 2 + J”z)112are rarely referred to but the loss tangent tan 6 = G ” / G ‘ = ,”/, isI probably ’ the most frequently reported parameter. The loss angle 6, which is the phase difference between the stress and the strain, ranges from 0 to ~ / and 2 therefore tan 6 takes on only positive values between 0 and of. The stress always leads the strain. The reason for some of the above redundancies is that some instruments more readily yield compliances and some rigidities. The loss tangent, as will be seen

11.1

11

LINEAR VISCOELASTIC BEHAVIOR

M A G IN A RY C

I

-REAL

FIG.3. Rotating vector relationship between the stress, strain, and rate of strain.

below, comes directly from the damping envelope of a free-oscillation instrument. Nomenclature recommended by the Society of Rheology for linear functions derived from different modes of deformation such as shear, elongation, and hydrostatic compression have been reported by Leadermanz7in 1957. Sieglaffz8has reported an extended list of recommendations covering steady-state flow and linear viscoelastic behavior. The algebraic relations between all of the above-mentioned dynamic quantities are readily seen in Fig. 3, where the stress u,strain y , and rate of strain i. = dy/dt are represented by vectors rotating together at constant angular velocity o in a complex plane. The components projected along the real axis correspond to the actual values of all the sinusoidally varying quantities at any time. Since G' = OD/OB, G"

=

AD/OB,

v"

J'

=

OE/OA.

= AF/OC,

J" = EB/OA,

[GI = OA/OB,

v' = OF/OC,

IJI = OB/OA

tan 6 = A D / O D = EB/OE all of the relationships follow. Note that (GI = l/IJI. Also

IG12 =

(3'2

+ G"2

= OAZ/OBZ= OD2/OB2

+ AP/OBZ,

since by the Pythagorean theorem OA2 = ODz + ADZ,and G'

*'

=

OD OB

OA OB

- = -cos 6

=

H. Leaderman, Tnrns. Soc. R h d . 1, 213 (1957). C. L. Sieglaff, Trans. Sac. Rhea/. 20, 311 (1976).

[GI cos 6.

12 11.

VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

But J‘ =

6 - G’ 6 = cos -

OE = -cos OB -

OA

ICl -

OA

Iclz’

Similarly, J” = G”/lGIZ,

G’

=

J’/IJIz

G“ = J”/1512.

If y = OB sin or, then dy/dr = y = w(OB)cos w t ; but since the amplitude of y is OC, w = OC/OB. Now r)‘ = OF/OC = A D / O C . It follows therefore that

Utilizing the Boltzmann superposition relations, G’ and G” can be related to the stress-relaxation modulus G ( t )and J’ and J” can be related to the creep compliance J ( r ) . Thorough coverage of the various interrelations is given by Ferry.3 11.1.1.4. Response to an Imposed Constant Rate of Strain. Measurements of stress made while the rate of strain is held constant are not often carried out in the range of linear viscoelastic response simply because it is substantially more difficult to remain in the linear range of response while the strain is being increased at a constant rate. The omnipresent universal tensile testing machine operates at a constant crosshead speed, which approximates a constant strain rate in a tensile specimen at small strains. To obtain measurable stresses at short times, high strain rates must be used and, to obtain stresses within the linear limit at long times, low rates must be applied. However, all the stresses measured divided by the strain rates employed fall on a common curve as a function of time of straining creating yet another viscoelastic function. This stress growth function or stressing viscosity measured in the shearing mode, r ) ( t ) = & ) / y o , we emphasize, should not be confused with a steady-state viscosity r ) anymore than the dynamic viscosity r)’ should. As the name indicates, r) is a steady-state parameter and in fact is the long-time-limiting value of q ( t ) for a viscoelastic liquid, i.e., lim t + 03 q( t) = q0. To distinguish the tensile or elongational viscosities from viscosities obtained in shear, an overbar will be used: q(r),?j’(o), Qll .3 Smith has shown the general forms of the response expected from viscoelastic solids, elastomers,2g and viscoelastic liquids such as a highmolecular-weight linear polymer, polyis~butylene.~~ He has also pointed out that the slope of q(r) is the stress relaxation function E(r), i.e., 2s

30

T.L. Smith, J . Polym. Sci. 32, 99 (1958). T. L. Smith, J . Polym. Sci. 20, 89 (1956).

1I . 1

LINEAR VISCOELASTIC BEHAVIOR

13

log t / a T (mc) FIG.4. Schematic creep and creep recovery behaviors for high-molecular-weight linear amorphous polymers with narrow and broad molecular-weight distributions. Typical directly measurable curves are shown with the corresponding reduced compliance curves, the temperature of reduction being chosen as TB.

E(r) = d?j(t)/dr. Since remaining in the linear range of response with a constant shear rate is difficult, this deformation is applied more often in the nonlinear regime. However, at large elongational deformations an increasing cross-head speed is necessary to maintain a constant rate of Hencky strain (d In l/dt, where I is the instantaneous specimen length). 11.1.1.5 Dispersions and their Origins. The shapes of the various viscoelastic functions are conveniently illustrated by Ferry3 for eight representative polymeric materials. One example of a polymeric glass, poly(methy1 methacrylate)(PMMA), is shown and so is one example of a highly crystalline polymer, high-density polyethylene (HDPE). The remaining examples depict the primary softening dispersion along with either the flow-dominated terminal zone of response for the viscoelastic liquids or the approach to equilibrium of the cross-linked viscoelastic solids. McCrum, Read, and Williams31present many examples of secondary dispersions that are observed in glassy and crystalline polymers along with the proposed assignments for the responsible loss mechanisms. In this necessarily limited presentation, only creep and recoverable creep compliance curves will be used to illustrate the features most commonly encountered in the response of a high-molecular-weight linear amorphous polymer near or above its glass-transition temperature T,. Figure 4 semiquantitavely represents the behavior of such a polymer with a weight-average molecular weight in the neighborhood of 2 x lo5. Remember, as mentioned above, that the principal effect of N. G . McCrum, B. E. Read, and G . Williams, "Anelastic and Dielectric Effects in Polymeric Solids." Wiley, New York, 1967.

14 11.

VlSCOELASTlC A N D STEADY-STATE RHEOLOGICAL RESPONSE

increasing temperature is to shift the time scale of response to shorter time. The reduced curve (heavy solid line), which is constructed from the more restricted curves, obtained at higher temperature than the temperature of reduction (or reference temperature) To over the usual time scale range of 1 sec to 1 day, is depicted as it would appear with To = T,. The procedure of obtaining the composite curve is called timetemperature reduction. It is referred to again in Part 12 of this volume and in extensive detail by Ferry.3 This time-temperature principle is one of our most valuable analytical tools and can be as illuminating in its violation as in its c o m p l i a n ~ e . ~ ~ ~ ~ The principal temperature dependence of the viscoelastic response is measured by the shift factors uT, which are the ratios of the velocities of response at different temperatures. In Figure 4 the logarithms of these factors, log uT, can be seen to be equal to the horizontal time scale displacements necessary to superpose each curve onto an arbitrarily chosen reference curve measured at the temperature To (hereafter called the reference temperature). For linear viscoelastic materials the same shift factor curve necessarily holds for all the viscoelastic function^.^^^^ One important feature with experimental ramifications is the growing severity of the temperature sensitivity with decreasing temperature. Near Tga temperature change of 2-3°C will change the speed of material response by an order of magnitude. Therefore instruments operated near T g must be thermostatted to within 0.01"C. The small short-time dispersion shown between log fluT = - 5 and 0 is normally not seen at or above Tgbecause of its experimental inaccessibility. We have shown only one such secondary process when in fact several are normally observed. However, they are usually observed dynamically at approximately constant frequency as a function of temperature. They are all ascribed to relatively local molecular motions, i.e., the vibration of side groups or crankshaftlike motions of very short segments of the polymer chain b a ~ k b o n e . ~ ~ . ~ ~ The principal dispersion seen between log ?/aT = 0 and 6 is called the primary softening dispersion or the transition from glasslike to rubberlike behavior. The use of the word transition here implies a change in the magnitude of the property and not a change in the material, i.e., no thermodynamic transition of any kind or order. Also note that the level designated J g [as in Eq. (1 1.1.1)] can be seen to be the long-time limit of the preceding secondary dispersions. The primary dispersion appears to a D. J. Plazek, J . Phys. Chem. 69, 3480 (1965); J . Polym. Sci., Purr A-2 6, 621 (1968).

D. J. Plazek and V. M. O'Rourke, J . Polym. Sci., Part A-2 9, 209 (1971). H. Markovitz, J . Polym. Sci., Polym. Symp. 50, 405 (1975). ss T. F. Schatzki, J . Polym. Sci. 51, 4% (1962). 36 R. F. Boyer. Rubber Rev. 34, 1303 (1963). sI

1 I . 1 LINEAR VISCOELASTIC BEHAVIOR

15

originate from two different groups of viscoelastic mechanisms. Near the glassy level of response Andrade creep is usually found, i.e., the creep strain is a linear function of the cube root of the time.37-39 The nature and origin of this form of deformation remains a mystery. However, the larger portion of the deformation in this dispersion certainly arises from the retarded normal mode motions of the polymer chain b a ~ k b o n e .The ~ rubbery plateau seen in Fig. 4 at times between log t / a T = 6 and 11 is believed to be the result of the presence of a transient network temporarily held together by the entanglements between the long, chainlike molec u l e ~ .The ~ ~ permanent deformation contributed by slippage of molecules past one another, leading to the growing separation of their tenters of gravity, is measured by t / r ] and is represented by the straight line with a unit slope. The recoverable compliance curves shown represent typical behavior for a narrow distribution polymer (dashed line) and one with an ordinarily broad distribution: M , / M , = 2-3 (dotted line). The long-time-limiting value, the steady-state recoverable compliance J , for narrow-distribution polymers is independent of molecular weight for samples with M 2 1 x lo5 and in fact often has a value of 1.5 x m2/N.41 However, J , is a strong function of the molecular-weight distribution; hence the substantial difference between the rubbery plateau compliance and the steady-state value indicated for the broad distribution polymer. The normally observed jump is between 10 and 100-fold in value. To understand the apparently paradoxical fact of Je’s insensitivity to molecular weight but extreme sensitivity to the molecular-weight distribution one must first be able to explain the molecular-weight independence. The modest increase of J,(t) by a factor of two to three in the final dispersion between the rubbery plateau value JN and the steady-state response of narrow-distribution polymers, coupled with molecular weight-independence, indicates that the entanglement network determines both JN and J,. But since we are referring at the moment only to linear behavior we know that the entanglement concentration is unaffected by the creep deformation. Therefore at least some entanglements must become less effective. The entanglement concentration is maintained by the dynamic equilibrium between the diffusion processes of entangling and disentangling. Permanent deformation can only occur if neighboring molecules disentangle so that they can, as a whole, move ST E. N . da C. A. Andrade, Proc. R. Soc. London, Ser. A 84, I , (1910);Philos. Mag. [8] 7 , 2003 (1962). s8 D. R. Reid, Br. Plasr. 31, 2 (1959). D. J . Plazek, V. Tan, and V. M. O’Rourke, Rheol. Acfa 13, 367 (1974). W. F. Busse, J . Phys. Chem. 36, 2862 (1932). ‘I W. W. Graessley, Adv. Polym. Sci. 16, 1 (1974).

16 11. VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

apart in the shear field. The onset of the final dispersion in the approach to steady state is usually seen in the same time scale region wherein the viscous deformation becomes a measurable fraction of the total deformation, suggesting that the increase of deformation above the rubbery level is also determined by the disentangling process. The combination of the above observations leads us to the conclusion that, whereas the entanglement population is constant, polymer chains involving newly formed entanglements, arising from diffusion processes, to a good approximation do not bear any of the load until they have existed long enough to sense the shear field. That is, they have to experience the softening dispersion themselves. The final dispersion is spread out on the time scale to such an extent because of the distribution of lifetimes of various entanglements. Those involving central segments of the polymer chains have to diffuse further to reach the chain ends and so are the longest lived. Under steady-state conditions then, since the recoverable compliance has risen by a factor of two to three times that of the rubbery plateau, only i to f of the entanglements can be considered effective. The much larger J , exhibited by polymer samples with broad molecular distributions can be understood by noting that shorter molecular chains will be involved with smaller terminal retardation times than their longer brothers. Hence as the distribution of terminal times is being passed, the remaining chains (elastic elements) that are short of steady-state orientation carry a disproportionately large fraction of the stress in the sample. Alternatively, one can view the process as one where the longer molecules see a strain rate that is substantially higher than it would be had they been surrounded by molecules of equal molecular weight. Hence they become correspondingly more oriented and a larger recoverable deformation is observed. 11.1.1.6. Spectra and Interrelations. It is possible in principle to deal only with the measured viscoelastic functions and the relations between them: but most often the dependence of the viscoelastic response of a material upon various experimental and molecular parameters is considered in terms of the distribution function of retardation times (the retardation spectrum) L(ln T ) or the distribution function of relaxation times (the relaxation spectrum) H(ln T ) . They can be defined by (1 1.1.14)

G(O) =

1-r

H d In

T,

(1 1.1.15)

where G(0) is the instantaneous stress-relaxation modulus and T is either the retardation time or the relaxation time, depending on whether L or H

11.1 LINEAR VISCOELASTIC BEHAVIOR

17

is involved. Mechanical analogs to the above integrals are given elsew h e ~ e . ' ~Compliances ,~~ functions involve L(ln r)and rigidity functions involve H(ln 7). With the spectra, the common viscoelastic functions are calculated using the following expressions:

J'(w) = J ,

+

rm J-m

I L

1

+

Lwr

+a

0272

d In r, 1

dlnr+-, 07

( 1 1.1.17)

( 1 1.1.18) (1 1.1.19)

G'(w) =

j+-+ -m

1

+

llW2? din C,, 029

( 1 1.1.20)

(11.1.21)

G,, the equilibrium modulus, is zero for a viscoelastic liquid. To utilize these expressions the spectra must be known over wide regions of the time scale. For any particular dispersion, if values of the spectrum are known about the associated maximum down to levels two or more orders of magnitude less than the value of that maximum, accurate values of the viscoelastic functions can be obtained using reasonable extrapolations to the extremes of the time scale. A fairly complete compilation of known interrelations, exact and approximate, are given by Ferry.3 Many attempts have been made to invert the integral relations given above to obtain the spectra from experimentally obtained viscoelastic functions. Inevitably the curves analyzed have been composite curves obtained by the temperature reduction procedure. Curves obtained at a single temperature normally cover too small a portion of the needed time scale range to make the analysis practicable. Usually one of two schemes is used in the inversion. One approach is to use approximation expressions, which have been analytically determined. They are called first-, second-, and third-level approximations and require first, second, and third derivatives of the measured curves, respectively. Since few data are precise enough to yield third derivatives, second approximations are most often used. It should be appreci-

'*

T. Alfrey, Jr., "Mechanical Behaviour of High Polymers." Wiley (Interscience), New

York, 1948.

I8 1 1 .

VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

ated that in time regions of rapid property changes the resulting second The alternaapproximation spectra are usually in error by 60tive approach has been to attempt iterative solutions on a ~ o m p u t e r . ~ ~ - ~ ~ Success here has been demonstrated on test analytical functions, but not infrequent instabilities in the solutions occur when real data are analyzed. The ultimate achievable test of a spectrum is how well it reproduces the viscoelastic data from which it was derived. We have used this test in a simple scheme of altering an approximate spectrum with human estimates arrived at from consideration of deviations between original creep recovery data and curves calculated via Eq. ( 1 1.1. 16).21 The necessary spline function fits to the data and the computer programs were written and checked by Stephen J. Orbon. A 7.0 wt% solution of an anionically polymerized polystyrene (M,,, = 4.40 x lo5) in a mixed solvent (65% m-tri cresyl phosphate and

Log (1)

FIG.5. Recoverable compliance curves for NBS Nonlinear Test Fluid XI determined at temperatures between -45 and - 10°C. presented logarithmically as functions of time. 43

D. L. Phillips, J. Assoc. Compur. Much. 9, 84 (1962).

* R . 1. Tanner, J . Appl. Polyn. Sci.

12, 1619 (1968). J . F. Clauser and W. G. Knauss, Truns. Soc. Rheol. 12, 143 (1968). I s P. Linden, Proc. IEEE 58, 1389 (1970). “J. E. Soussou, F. Moavenzadeh. and M. H. Gradowczyk. Truns. Soc. Rheol. 14, 573 45

(1970). 48

M. Shen and R. T. Jamieson, J. Polym. Sci., Purr C 35, 23 (1971).

11. I LINEAR VISCOELASTIC BEHAVIOR

19

35% Aroclor) has been widely circulated by E. A. Kearsley of the U.S. National Bureau of Standards. Its linear and nonlinear rheological properties and rheooptical properties have been determined in many laboratories with the goal of providing a reference material for the comparison of various techniques being used or developed to rheologically characterize materials, especially in the nonlinear region of response. It is most often referred to as NBS Nonlinear Test Fluid # I . Measurements of the limiting low rate of shear viscosity and the recoverable shear creep compliance in the linear range of response have been made by Pawan K. Agarwal in our laboratories. The results and analysis are presented here to illustrate an extensive softening dispersion. The recoverable compliance curves J,(t) obtained at temperatures between -45 and - 10°C are presented logarithmically in Fig. 5 . Since J , (1 sec, T = -44.6"C) is m2/N), close to the usual glassy level of 1.0 x 10-lo cm2/dyne (1.0 x the glass temperature Tg must be within a few degrees of - 45°C. These curves are shown successfully reduced with a simple time scale shift to an arbitrarily chosen reference temperature of -25°C in Fig. 6. The complete dispersion is clearly seen to cover a millionfold change in J,(r) over nearly 12 decades of time scale. The value of log J , is - 3.79, which reflects a dependence, relative to the pure polymer, that is close to the ex-

-4

-5

-6

0

-7

0

-0

-9

-10 -6

-4

-2

2

0

LOP

4

6

~/QT

FIG.6. Logarithmic plot of the recoverable compliance curves from Fig. 5 reduced to - 25°C.

20 1 1 .

VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

-6

-4

-2

0

2

4

6

Log T/at

FIG.7. Logarithmic presentations of the retardation spectrum L at -25" obtained from the log J , ( t ) curve in Fig. 6.

pected square dependence on the concentration C . In spite of the relatively high molecular weight of the polystyrene, no rubbery plateau is observed because of the dimunition of the entanglement concentration by the solvent. The second or long-time peak in the retardation spectrum L, which identifies the presence of an entanglement dispersion, can be seen to be absent in Fig. 7. This spectrum of retardation times was obtained with the above-mentioned iteration technique. With this L, the dynamic compliances and rigidities were readily calculated and are presented in Fig. 8. It is interesting to note that the log G' curve suggests the presence of an entanglement plateau, while the compliance curves log J' and log J,(r) along with log L categorically deny its existence. The temperature dependence of the viscosity between - 32 and 24°C has been found to be log 7 = -5.77 + 1087/(T - T,), where T , = - 103.0"C. The temperature variation of the shift factors aT required to reduce the curves in Fig. 5 is slightly different: log aT = - 13.84 + 1024/(T - T,) where T , = -99.0"C. This difference in temperature dependencies is far more pronounced in bulk polystyrene. When such a difference is found neither stress relaxation nor total creep compliance curves are properly reduc-

11.1 LINEAR VISCOELASTIC BEHAVIOR

21

Log waT FIG.8. Components of the dynamic rigidity G* and dynamic compliance J * , calculated with the retardation spectrum shown in Fig. 7. The dashed curve, J" - I / q , represents the loss compliance without the simple contribution from viscous flow.

ible. The measured J,(r) and 7 values for the NBS Nonlinear Test Fluid #1 are given in Tables 1-111. 11.1.2. Instrumentation

It is rare indeed when a rheological result can be expected to be in error by less than 1%. The most convincing evidence of this claim is our state of knowledge concerning the viscosity of water at 20°C. Three independent determinations have been made at the U.S. National Bureau of Standards with the greatest manageable care in three different in~trurnents:'~ two involved measuring the pressure drop caused by laminar flow in well-defined tubes50*51 and the third was a torsion pendulum with a precisely fabricated hollow sphere, which was filled with liquid and suspended from a torsion wire.52 Complete error analysis yielded the expectation in each determination that the results should be in error less than 0.1% but the spread in these determinations was 0.5%. R. S. Marvin, J . Res. Nutl. Bur. Stand Sect. A 75, 535 (1971). J . F. Swindells, J . R. Coe. Jr., and T. B. Godfrey,J. R e s . Nut/. Bur. Stand. 48, I (1952). R . W. Penn and E. A . Kearsley, J . Res. Nut/. Bur. Stand., Sect. A 75, 553 (1971). a H. S. White and E. A . Kearsley, J . R e s . N d . Bur. Stand.. Sect. A 75, 541 (1971).

22 1 1.

VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

TABLEI.

Recoverable Creep Compliances of NBS Nonlinear Test Fluid # I J A 1 ) (crne/dyne) log J A r )

log

1

-0.35 -0.25 -0.15 -0.05 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05

1.15 1.25 1.35 1.45 1.55 I .65 I .75 1.85 I .95 2.05 2.15 2.25 2.35 2.45 2.55 2.65 2.75 2.85 2.95 3.05 3.15 3.25 3.35 3.45 3.55 3.65 3.75

- 10.9"c

-18.4"C

-4.863 -4.803 -4.747 -4,694 -4.640 -4.587 -4.536 -4.486 -4.437 -4.390 -4.341 -4.298 -4.254 -4.214 -4.173 -4. I34 -4.104 -4.073 -4.039 -4.007 -3.978 -3.951 -3.926 -3.903 -3.882 -3.865 -3.850 -3.837 -3.826 -3.818 -3.810 -3.805 -3.800 -3.7% -3.792 -3.790 -3.717 -3.786 -3.785 -3.785

-5.602 -5.522 -5.447 -5.380 -5.305 -5.240 -5.172 - 5 . I08 -5.044 -4.981 -4.927 -4.865 -4.806 -4.748 -4.694 -4.638 -4.586 -4.535 -4.483 -4.435 -4.386 -4.334 -4.295 -4.253 -4.21 I -4.172 -4. I33 -4.097 -4.062 -4.035 -4.000 -3.970 -3.943 -3.917 -3.894

-26.1"C

-3 1 . 8 T

-6.099 -6.003 -5.917 -S.830 -5.74s -5.668 -5.592 -5.516 -5.441 -5.370 -5.301 -5.229 - 5 . I48 -5.098 -5.032 -4.973 -4.909 -4.849 -4.792 -4.735 -4.680 -4.626 -4.574 -4.522 -4.472 -4.424 -4.376 -4.330 -4.285 -4.242 -4.201 -4.161 -4.124

-7.514 -7.416 -7.330 -7.236 -7. I47 -7.054 -6.%5 -6.874 -6.783 -6.694 -6.608 -6.516 -6.426 -6.337 -6.250 -6. I63 -6.078 -5.9% -5.913 -5.838 -5.780 -5.700 -5.625 -5.548 -5.476 -5.403 -5.332 -5.264 -5. I97 -5.130 -5.066 -5.01 1 -4.943 -4.888 -4.830 -4.775 -4.721 -4.667

1 1.1 TABLEI .

LINEAR VISCOELASTIC BEHAVIOR

(continrwd)

log J J t )

log I

-38.4"C

log I

0.226 0.326 0.426 0.526 0.626 0.726 0.826 0.926 I .026 1.126 1.226 1.30 I .40

-8.718 -8.634 -8.561 -8.477 -8.399 -8.319 -8.236 -8.154 -8.067 -7.980 -7.893 -7.830 -7.741 -7.650 -7.582 -7.490 -7.408 -7.304 -7.214 -7.180 -7.088 -7.000 -6.909 -6.818 -6.727 -6.639 -6.552 -6.464 -6.376 -6.293 -6.208 -6.126 -6.045 -5.%5 -5.888

0.0% 0.1% 0.2% 0.3% 0.4% 0.5% 0.6% 0.7% 0.8%

1S O

I .60 1.70 1.80 1.90 2.0 2.05 2. I5 2.25 2.35 2.45 2.55 2.65 2.75 2.85 2.95 3.05 3.15 3.25 3.35 3.45 3.55

23

0.996 1 .o%

I . I96 I .27 1.37 1.47 1.57 1.67 I .77 1.87 1.97 2.07 2. I7 2.27 2.37 2.47 2.57 2.67 2.77 2.87 2.97 3.07 3.17 3.25 3.35 3.45 3.55 3.65 3.75 3.85 3.95 4.05 4.15 4.25 4.35 4.45

log J A 1 ) -44.6"C

-9712 -9.680 -9.662 -9.633 -9.599 -9.566 -9.529 -9.491 -9.445 -9.41 1 -9.350 -9.303 -9.267 -9.217 -9.148 -9.090 -9.206 -8.%8 -8.899 -8.830 -8.762 -8.688 -8.612 -8.533 -8.452 -8.369 -8.277 -8.186 -8.129 -8.015 -7.921 -7.824 -7.751 -7.666 -7.578 -7.492 -7.399 -7.313 -7.223 -7.142 -7.052 -6.942 -6.860 -6.771 -6.681

24 11.

VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

TABLE11. Shift Factors from Recoverable Compliance Reduction log

T W)

-2.25 -1.15 0.00 +0.33 1.47 3.11 5.00

- 10.9 -18.4 -25.0 -26.1 -31.8 -38.4 -44.6

With this kind of uncertainty involved in our viscosity standard, the necessarily more compromised routine measurements on the more difficultly handled polymers and their solutions should not ordinarily be expected to be less uncertain than several percent. Systematic errors usually predominate, so that statistical analysis of results is usually fruitless. In almost all cases if a better answer is needed more careful determinations with internal checks are in order. When material characterization is the aim of a rheological determination it is wise to heed the warning often made by Hershel Markovitz that easy experiments are not usually simple experiments. In most instances the results that can readily be explained or analyzed are obtained only with appreciable difficulty. A single pointed example is found in one of the most frequently utilized instruments, the extrusion viscometer. Polymers are simply pushed through capillary dies and the .results obtained are still often cause for debate. A few commercially available instruments are commonly utilized in the determination of viscoelastic and steady flow response, but most data in TABLE111. Temperature Dependence of Viscosity (sec-')

1% 1)

T ("C)

fmax

2.52 2.89" 3.42 3.88O 4.80 5.97 7.04 8.43 9.46

30.5 23.4 14.8 8.9 -1.1 - 10.9 -18.4 -26. I -3 I .8

4.35 x 101

Falling-ball determination.

5.60 x 10" 2.32 9.41 2.43 6.81 1.46

x 10x 10x lo* X

lod

x lo4

11.1 LINEAR VISCOELASTIC BEHAVIOR

25

the literature have been obtained from individually constructed specialpurpose instruments. We wish to distinguish between techniques or methods that have only been established and the relatively few that have been effectively utilized and are responsible for most of the existing data. It is hoped that most of the latter will be mentioned below. 11.1.2.1. Creep and Creep Recovery. Elongational creep is often the mode chosen for glassy or crystalline polymers. Since large deformations are not usually encountered with these materials in their linear range of response an effectively constant stress is maintained in the specimen if a constant tensile force is applied. A hanging weight has inevitably provided the tensile force. The convenience, cost, and constancy of this source of force are its principal merits. The elongational creep compliance D(r)is obtained from measurements on rods or strips of the material of interest. When obtained directly from tensile loading, small deformations have to be detected. The linear range may not extend beyond 2% strain for a glassy or crystalline polymer. The maximum deformation measured in a 10 cm long specimen would therefore be 2 mm. Direct optical measurements of the displacement made at a distance of 20 cm from the specimen would allow only a resolution of about 10 pm, or one part in 200. If the deformation changed by a modest factor of 10 during creep, 5% uncertainties would be encountered early in the determination and derivatives of the data would be virtually useless. Resolutions of 1 or even 0.1 p m are required for reasonably precise measurements. The temperature fluctuations must be kept in corresponding bounds so that thermal expansions and contractions do not hide the decelerating creep at long times. If the above specimen has a typical linear expansion cm/cm deg, the length will change 20 pm per decoefficient of 2 x gree. Temperature control within 0.1 or even 0.01"C may be necessary for this reason alone. If weights were not used to supply the tensile force, fluctuations in the sample load would cause similar concern. When dealing with samples that do not deform appreciably under their own weight elongational moduli may be determined from the bending of bars or rods, since bending is a combination of elongation and compression. Relatively large flexural displacements are obtained for a given maximum level of strain and therefore detection is less demanding. In instruments where the atmosphere is controlled or measurements are carried out in vucuo feed throughs or remote weight handling are required. Probably the greatest drawback in the utilization of weights in creep or stress relaxation instruments is the difficulty of obtaining the starting time of the experiment within 1 sec. Two or more decades of possible measurement time are thus lost. An even greater loss of information is suf-

26 1 1 .

VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

FULCRUM

FIG.9. An inverted tensile creep apparatus displaying some widely used features.

fered if a traveling microscope is used to monitor the deformation in creep. Such readings usually take from 10 to 20 sec. Figure 9 is a schematic sketch of a tensile creep apparatus in which the sample is inverted, i.e., fixed at the bottom with weights hung over a balance fulcrum. A pulley with a gas bearing can be used alternatively to reverse the direction of the force. This arrangement allows one to counterbalance extraneous weights and allows for more convenient thermostatting including the use of a liquid bath.=*” The high-sensitivity displacement detector most commonly used is a linear variable differential transformer, LVDT, which is available from numerous sources. Distance resolution of 1 x lo4 cm is achievable. When measurements are made where the specimen being studied remains rigid, high resolutions can also be obtained using an optical e x t e n ~ i o m e t e r .An ~ ~ instrument as depicted could be built within the confines of a standard bell jar vacuum H. Odani, N . Nemoto, S. Kitamura, M. Kurata, and M. Tamura, f o l y m . J . 1, 356 (1970).

H. Miinstedt, Rhcol. A r m 14, 1077 (1975). = C . M. R. Dunn, W. H. Mills, and S.Turner, Er. flasr. 37, 386 (1964). sI

11.1

LINEAR VISCOELASTIC BEHAVIOR

27

system. Tensile creep of rubbery materials as well as of highmolecular-weight linear amorphous polymers have been studied where the limiting low strain rate viscosity of the polymer is sufficiently high (10' Pa sec or greater).M,s-58 When deformations are encountered where changes in the cross-sectional area are no longer negligible, some scheme is necessary to make a corresponding decrease in load so that a constant stress can be maintained. Cam-shaped pulleys have been used where the lever arm decreases with times*5e-61or specially shaped weights have been made to sink into a buoyant liquid to reduce the tensile load.62.s3 The direct determination of any recoverable tensile compliance that includes the terminal response zone may not yet have been successfully accomplished because of the following two experimental difficulties. First, characteristic recoverable deformations are obtained only following steady-state creep. The recoverable portions of the total deformation are always less than 10% after steady state is established,t thereby making very stringent demands on the detection system. Second, since characteristic recovery is only observed when the specimen is in a stress-free state, friction and sagging of the sample from its own weight are difficult effects to overcome. The shear creep compliance is readily determined on glassy, crystalline, or rubbery materials in torsion. Liquids are most often measured in shear while contained between coaxial cylinders (Couette geometry) (Fig. lOa), parallel plates (Fig. lob), or cone and plate (Fig. 1Oc). End and edge effects must be minimized. We have had appreciable success working with short liquid cylinders in the parallel plate configuration where the sample is held within the gap by means of surface tension.32 If the plate separation is greater than 1 mm it can be measured optically with a traveling microscope with fair accuracy. At plate separations of 2 mm slight sagging of the sample-air interface can be observed microscopically. If the edges of the sample plates are moderately sharp, liquids in the gap will remain there indefinitely, i.e., will not flow out, so long as no liquid is forced over the edge by bringing I F. 57

Bueche,J. Appl. Phys. 26, 738 (1955). R. F. Landel and T. L. Smith. J . Am. Rocket Soc. 31, 599 (1961). H. Odani. N. Nemoto, and M . Kurata, Bull. Inst. Chem. Res., Kyoto Univ. 50, 117

(1972).

H. Leaderman, Trans. Soc. Rheol. 6, 361 (1962). E. N . da C. Andrade and B. Chalmers, Proc. R . Soc. London, Ser. A 138,348 (1932). 61 W. L. Holt, E. 0. Knox, and R. L. Roth, J . Rcs. Natl. Bur. Stand. 41, 95 (1948). E. N . da C. Andrade, Proc. R . Soc. London, Ser. A 84, 1 (1910); 90, 339 (1914). as C. A. Dallquist, J . 0 . Hendricks. and N. W. Taylor, Ind. Eng. Chem. 43, 1404 (1951). (lo

t In some instances much less.15J6

28 1 1 .

VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

a.

b.

C.

FIG.10. A selection of some sample geometries that are commonly used in rheological instruments. (a) Rotational concentric cylinders (Couette), (b) rotational parallel plates, ( c ) cone and plate.

the platens too close together. This reluctance of liquids to go around corners is enhanced by machining knife edges on the platens as shown in Fig. lob. Since the geometrical factor involved in calculating the compliance of a right circular cylinder from the applied torque M and the angle of twist 8 is proportional to the fourth power of the radius r , i.e., ( 1 1.1.22) more accurate geometrical factors, usually called sample coefficients, are obtained using the density of the material p along with the mass of material in the gap m and the platen separation h. A partial correction for a slightly bulging, concave, or sagging surface is thus obtained. A limited number of geometrical factors are given in Table IV, and more can be found in many of the primary reference^.^^^^^*-'^ In instruments used for the determination of the shear properties of viscoelastic liquids with low viscosities, there is inevitably a rotating or oscillating element. Bearings are necessary for a rotating element and any frictional drag or other residual instrumental torques will interfere with the stress-free condition required for recovery. These instrumental torques determine the extent of available recoverable compliance information. Precision ball bearings, at best, will give rise to fluctuating torques in the neighborhood of 100 dyne cm. Gas bearings usually have a degenerate turbine torque that is 1 dyne cm or larger. A magnetic levitation bearing can be made to operate with residual torques between lo-' and dyne cm.

29

11.1 LINEAR VlSCOELASTlC BEHAVlOR TABLEIV. Selected Geometrical Factorsa Yln/oll = bO/M or b X / f

Experimental configuration Sliding parallel plate Rotating parallel plate (Rod) Cone and plate Rotating concentric cylinder (Couette) Axial translation of coaxial cylinders (Pochettino) a

Sample coefficient b

Maximum stress

Maximum strain (linear range)

A h

f A

X h

TR‘ 2h

2M aR3

OR h

2 a ~ 3 3a

3M 2rr~s

e -

4ah I/Riz - IlR.1

M

2ah In(R.IRi)

f

X

2aRih

R1In(R,IRJ

2ahRt

a

2eR: R,2 - R?

e is the angle of twist or rotation; M the torque: X linear displacement; f applied force;

A area; a cone angle; h height or length; R, inner radius; R , outer radius.

If large amounts of the material being studied are available, the effects of the instrumental residual torques can be minimized to a large extent by using sample geometries and dimensions that result in a stiff sample. With limited detector sensitivity, care must be taken to see that the strains necessary to yield measureable deformations are not so great that nonlinearity results. 11.1.2.1.1. ANGLEDETECTORS.Virtually without exception, light levers have been used to monitor the deformation in rotary creep instruments. The detection of the displacement of thz light lever beam has been accomplished by various methods. It has been followed visually on a ground-glass scale and on a rotating-drum camera.B4 Most kinds of photodetectors have been tried. Focused and shaped light beams that vary the amount of light impinging on a photocell linearly with the angle of rotation have been s u ~ c e s s f u l .The ~ ~ most sensitive of all angle detectors is that designed by R. V. Jones,6Bwhich employs the image of one Ronchi grating passing over an interceptor grating. Most angle detectors resolve between and 10” deg, while the Jones detector has been made to resolve deg. Probably the fastest responding device, offering convenience and flexibility, is a Schottky diode in a linear configuration that yields a voltage output that is a linear measure of the position but is senW. Lethersich, J . Sci. Instrum. 27, 303 (1950). D. J. Plazek, N. M. Vrancken, and J. W. Berge, Trans. SOC. Rheol. 2, 39 (1958). BB R . V. Jones, J . Sci. Instrum. 38, 37 (I%]). 85

30 11. VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

sibly independent of the beam size.s7 Since this device has a response time approaching 1 nsec, torsional creep information at short times is limited only by inertial effects and in some instances the speed of torque application. The most convenient and dependable device for monitoring the angular deformation is a light-spot-following recorder. Only one such recorder is apparently available at the present time.sa Faster potentiometric strip chart recorders are readily converted into light spot followers by mounting pairs of photocells on their pen carriages and utilizing them electrically so that a balance is achieved when the light lever beam illuminates both cells equally. Inexpensive low-power helium-neon lasers prove to be the most reasonable choice for the light lever source. PRODUCTION. Precise torques over an unlimited 11.1.2.1.2. TORQUE range of rotational angle can be produced with a drag cup m o t ~ r . ~ ~ ~ ~ Weights and pulleys can be used to produce constant torques for up to several rotations. Only limited rotation is encountered when measuring polymer glasses or elastomers. Therefore, in addition to the above devices, torque motors (i.e., coils in permanent magnetic fields) and torsion wiress5can be considered as constant torque producers. If torsion wires are used, effective torque removal can be accomplished for the measurement of creep recovery by using a torsionally very compliant fiber in series above the torsion wire and a release mechanism.'O Practical torsion wires with fixed torsional stiffness can be made by attaching sturdy cylinders to the ends of the spring wire, which can then be readily clamped into the instrument. To be stable, the effectively noncompliant end fixtures must be secure. Rarely can hard solder be used to attach the end cylinder because the high temperatures involved (800-900°C)usually cause ductility in most spring materials by recrystallization or solution of precipitates. Tungsten wire is one exception and can be safely soldered with silver solders. Beryllium-copper, phosphor-bronze, and music wires cannot. Soft solder should not be used since it will creep. Effective clamping of the ends is difficult and usually complicated. We have found that slipping the wire through a neatly fitting hole in the end cylinder, deforming the end slightly on an anvil, and subsequently pulling it back into the cylinder using a lathe until the deformed section just starts to come through, secures the wire better than any form of clamping we have tried. Torsion wires are usually calibrated by timing the period of oscillation of carefully machined internal disks, which are attached to their end.s5 The torsional constant k (Nm/rad) = I d , where I (kg mZ)is the United Detector Technology, 1732 21st St., Santa Monica, California 90404. Graphispot Recorder, Sephram, Paris, France. 89 G . C. Berry and C.-P. Wong, J . Polym. Sci., Polym. Phys. Ed. 13, 1761 (1975). 70 K. Osaki, Y. Einaga, M. Kurata, and M. Tamura, Macromolecules 4, 82 (1971). O7

11.1 LINEAR VISCOELASTIC BEHAVIOR

31

moment of inertia, the angular frequency w = 27r/P, and P is the period (sec). A schematic ghost sketch in Fig. 11 shows most of the essential features of a recent design of a torsional creep apparatus, which achieves frictionless operation with magnetic levitation of the rotor, and in which torques are induced with a drag cup motor and rotation is monitored by means of a light lever.32 Although a liquid cylinder sample is indicated, a Couette geometry can and has been used. Long strips or rods can be

I

SUPPORT SOLENOID

COUPLINGROD MIRRORS SAMPLE

-

-

FIG. 1 1 . A frictionless, magnetic-bearing, torsional-creep apparatus.

32 1 1.

VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

clamped or glued in place for measurement. The two mechanical in siru adjustments indicated are essential for efficient and convenient operation. 11.1.2.2. Stress Relaxation. The elongational stress relaxation modulus E ( t ) is most often determined directly from elongational measurements when the material being studied is as elastomer, since accurate elongations of 10% and greater are easily applied to rubbery specimens. Resulting stresses generally yield moduli still in the linear region of response. A schematic drawing of a stress relaxometer that incorporates

DISPLACEMENT

BALL BUSHING

-BALL BUSHING

FIG.12.

A tensile stress relaxometer for elastomers.

11.1

LINEAR VISCOELASTIC BEHAVIOR

33

several of the features that have proved s u c ~ e s s f uis1 shown ~ ~ ~ ~in~ Fig. 12. The fast initial deformation can be accomplished with a solenoid and the stress can be monitored by a linear variable differential transformer with the core supported by cantilever springs that are more than 200 times stiffer than the initial stiffness of the specimen. Since the range of linear response of glassy materials extends only to several percent deformation, measurement of E(r) by elongation becomes exceedingly difficult. The changing stress has to be detected while the change in length is being held constant to within one part per thousand. Tobolsky championed and exploited elongational stress relaxation for many years, but finally turned to torsion to characterize the glassy region of response. However, E ( t ) should still be obtainable from glassy materials using bending experiments. To determine the shear relaxation modulus G(r)of a rigid polymer, torsion in a rod is the method of common choice. Figure 13 shows a possible arrangement that makes reasonable demands in the application of the required fixed strain and the stress measurements. A cylindrical specimen, 4 mm in diameter and 10 cm long, when twisted through an angle of 0.1 rad has a maximum strain ymaxat its surface of 2%: ymax= R 8 / h , where R is the radius and h the height. With a 10 cm lever arm the LVDT core can move through 0.01 cm before the applied strain changes by 1%. Most often, when the stress relaxation of polymer flm has been studied, balance relaxometers have been used. The stretched length of the sample is usually maintained by varying the weight opposite the sample by means of an automatic adjustment of a hanging chain (as used in chain balance^).'^-^^

Shear stress relaxation of non-cross-linked amorphous polymers, which are viscoelastic liquids, have been measured in the terminal zone of response, not only following the usual sudden imposition of strain but also following the cessation of steady-state hearing.^^,^^ Viscoelastic mechanisms with longer relaxation times are weighted more heavily in this stress relaxation measurement. Ball bearings used in Menefee's instrument,77which offers a convenient stress-measuring technique, must limit the range of accurately measured stresses in a manner similar to the restriction of creep recovery by friction. J . R. McLoughlin, Rev. Sci. Insfrum. 23, 459 (1952). R. F. Landel and P. J. Stedry, J . Appl. Phys. 31, 1885 (1960). 7s R. D. Andrews, N . Hofman-Bang, and A. V. Tobolsky, J . Polym. Sci. 3, 669 (1948). " N . Ninomiya and H. Fujita, J . Colloid Sci. 12, 204 (1957). 75 K. Fujino, K. Senshu, and H. Kawai, J . Colloid Sci. 16, 262 (l%l). 70 F. W. Schremp, J. D. Ferry, and W. W. Evans, J . Appl. Phys. 22, 711 (1951). 77 E. Menefee, J . Appl. Polym. Sci. 8, 849 (1964). 71

7p

34 1 1.

VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

.STRAINING SOLENOID

FIG.13. A torsional stress relaxometer for elastomers, glasses, and crystalline polymers.

The stress decay following steady-state flow is given by

With the initial stress equal to 79 and with this time function, J , fan be obtained:

11.1 LINEAR VlSCOELASTlC BEHAVIOR

35

When ordinary stress relaxation is measured in the terminal region following the rapid application of the strain to a visco-elastic liquid, r] and J , can be determined from the following equations: +m

77

=

J,

=

tG(r) d In 1,

1 r]2

+m

tzC(t)d In t.

(1 1.1.24)

(1 I. 1.25)

11.1.2.3. Dynamic-Mechanical Properties. All of the geometries that have been mentioned in the preceding discussions concerning creep and stress relaxation can be utilized in the determination of dynamicmechanical properties. Twisting and bending are the preferred modes of deformation for relatively long thin rods and bars of glassy or polycrystalline polymer^.^.^.^* However, many such systems in film form have been studied in elongation with the Rheovibron, an instrument designed and first utilized by T a k a ~ a n g i . It ~ ~currently offers a frequency range from 0.01 to 110 Hz. Significant sources of error and corrections for its operation have been reported by M a ~ s a . ? ~ Most viscoelastic measurements made in the past have been obtained on instruments designed by the investigator but a small number of instruments seem to have established themselves as permanent fixtures on the scene. One group of multiple purpose instruments includes the Weissenberg Rheogoniometer, the Mechanical Spectrometer, and the Kepes Balance-Rheometer. These and other related instruments are well referenced by Walters6 The most recent design of a multipurpose rheometer is that of Drislane et U ~ . ~ OThe most demanding requirement in the measurement of dynamic-mechanical properties with forced oscillation techniques is the determination of the phase angle between the stress and the strain. Limitations in system phase angle determination are encountered near 0, 90, and 180". Among the best dynamic measurements made with the Weissenberg Rheogoniometer are those of Rest and O'Reilly,81"2 who employed a digital phasemeter with it.83 Most recently the most accurate phase angle measurements are being made with the aid of 78 M. Takayanagi, Japanese Patent 591,286; U.S. Patent 3, 132,509; Mem. Fac. Eng., Kyushu Univ. 23, 41 (1963). 70 D. J. Massa, J . Appl. Phys. 44, 2595 (1973). C . J . Drislane, J . P. De Nicola, W. M. Wareham, and R. I. Tanner, Rhea/. Acra 13,4 (1974). J. M. O'Reilly and W. M. Prest. Jr., Pap. SOC. Rheol. Meet., 1966; Pap. A m . Phys. SOC. Meet.. 1968. W. M. Prest, Jr., J . Polym. Sci., Part A-2 8, 1897 (1970). M . H. Birnboim, U.S. Patent 3,286,176 (1%6).

36 1 1 .

VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

b

a.

'

FY

t FIXED OBSERVER LOOKING FROM ABOVE

s d.

..

3 ROTATING OBSERVER LOOKING FROM ABOVE

FIG. 14. Eccentric rotating-disk geometry; see text for details.

cross-correlation equipment or computer averaging,84 which can include cross-correlation averaging.85 Phase angle measurements are avoided when dynamic-mechanical properties are determined in the eccentric rotating-plate mode,sas where forces generated in the direction of orthogonal axes, which are mutually perpendicular to the axes of rotation, are measured (see Fig. 14). Equivalent limitations are encountered when sample phase angles approach 0 or 90" in the nature of transducer cross-talk and orthogonality deviations. Usually only one rotating shaft is driven directly, which allows for a slower angular velocity of the other shaft (called slippage) if the sample is a liquid, even in the absence of bearing friction. Payvar and TanneF' have found this slippage to be slight and independent of chemical character for Newtonian liquids. They found the slippage measurable only when I . M . Krieger and T.-F. Niu, Rheol. Actr~12, 567 (1973). M . H . Birnboim, J . S. Burke, and R. L. Anderson, Proc. Int. Congr. Rheol.. 5th. 1968 Vol. I , p. 409 (1969). B . Maxwell and R. P. Chartoff, Trans. Soc. Rheol. 9, 41 (1965). P. Payvar and R. I . Tanner, Truns. Soc. Rheol. 17, 449 (1973). IM

11.1

LINEAR VISCOELASTIC BEHAVIOR

37

the offset distance of the axes d was greater than the separation of the plates. Slippage can therefore be considered negligible. If, in addition, the frequency of rotation is low enough so that inertial forces on samples have no measurable effect, then in the linear range of response the two orthogonal forces are each proportional to one of the components of the complex dynamic rigidity, G* = G' + ic"?

(1 1.1.26)

where a and h are the plate radius and separation respectively, A is the area, and y = d / h . The deformation of the material under the above restrictions can simply be visualized as shown in Fig. 14. Relative to a fixed point and direction on the bottom surface of the sample a point on the top surface (ignoring the synchronous rotation) executes a circular path. Imagine yourself standing on the point on the lower surface looking radially outward like the symbol b in Fig. 14b. You will then see the corresponding point on the top surface circle about you in a clockwise direction as the two plates rotate in the opposite direction (see Fig. 14b-d). This is equivalent to the material being sinusoidally sheared by two simultaneous simple shears of equal amplitude orthogonally oriented and 90"out of phase in time. It is most important to note that the sample is not experiencing any shearing that is continuous in any direction. Any rate of shear calculated from the rate of rotation is as superfluous as a rate of shear calculated from a derivative of sinusoidal simple shearing. In simple shearing or torsional determinations of the dynamic moduli or compliances, in the linear range of response, the values obtained are independent of the amplitude of the deformation and therefore at fixed frequency they are independent of the rate of shearing, which is sinusoidal in time. As appealing as simple shearing between parallel plates can appear, caution with this geometry must be exercised because of significant edge effects when the sample thickness approaches the magnitude of the sample diameter or width.88 The effect of bulging on the effective geometrical coefficient has also been considered.8g When extensive viscoelastic characterization is not desired and only a fingerprint or a cataloging of viscoelastic loss peaks is required many inW. T. Reid, J . Appl. Mech. 17, 349 (1950). E. R . Fitzgerald and J . D. Ferry, J . Colloid Sci. 8, 1 (1953).

38 11.

VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

vestigations resort to isochronal or almost isochronal measurements of dynamic-mechanical moduli as a function of temperature. An extensive literature, which can be effectively sampled in McCrum, Read, and Williams' book,31 has developed on the single or natural frequency response of glassy and crystalline polymers. To obtain results in the form of the *~' reed (clampedtensile modulus E*(w), simple e x t e n s i ~ n ~a~ vibrating ~ , ~ bar) - ~has been used. To obfree bar)02or a resonant r ~ d ~(free-free tain values of the shear modulus G*(w),the free oscillation torsional pendulum has been exclusively Any free oscillation or resonance instrument changes its frequency with the change in the storage modulus of the specimen, which varies with the temperature. One therefore detects the change in storage modulus by determining the change in frequency. In such measurements then, one has neither constant temperature nor frequency. If the instrument is driven at a constant frequency w,, a detailed quantitative analysis is still not possible because the shape of the log G"(w,) or tan 6(0,) versus temperature curves change with frequency.? If the relaxation times of all of the measurable viscoelastic mechanisms have the same temperature dependence the material is called "rheologically simple"' and the shape of viscoelastic response curves remains nearly the same as a function of logarithmic time orfrequency. This single temperature dependence is a measure of how these curves shift along the time or frequency scale axis toward shorter times or higher frequencies with increasing temperature. Two other temperature dependences are measures of the magnitude of the viscoelastic response curves, such as J'(o).and J"(o). On log J , vs. log w plots the magnitude changes are seen as relatively small vertical shifts of the curves. In terms of Eq. (1 1.1.1) one temperatture dependence would be that of J d , which is largely entropically determined, and the other is that of J,. Different signs and magnitudes of these temperature dependences result in changes in shape of the viscoelastic curves with temperature. However, even if the material is rheologically simple and the changes in 8o

M. Takayanagi, Proc. Int. Congr. Rheol., 4th, 1%3 Part I , p. 161 (1965). M. Nakatani, K. Iijima, A. Suganuma, and H. Kawai, J . Macromol. Sci.. Phys. 2, 55

(1%8). 82

D. W. Robinson, J . Sci. Instrum. 32, 2 (1955). J. Heijboer, P. Dekking, and A. J. Staversman, Proc. Int. Congr. Rheol., 2nd. 1953 p.

123 (1954). B1 D. E. Kline, J . Polym. Sci. 22,449 (1956). L. E. Nielsen, "Mechanical Properties of Polymers," Chapter 7. Van NostrandReinhold, Princeton, New Jersey, 1%2.

t McCrum, Read, and Williamss'show in pp. 121-127 of their book what approximations are possible.

11.1 LINEAR VISCOELASTIC BEHAVIOR

39

magnitude can be considered negligible, the fixed-frequency plots become more gradual with increasing frequency because viscoelastic mechanisms are encountered at relatively high temperatures where the temperature dependence of the time scale shift factor uT is less intense, i.e., the log uT vs. temperature curve diminishes in slope with increasing temperat~re.~ In general, mechanisms responsible for different loss tangent peaks have significantly different temperature dependences. The "apparent activation energies" obtained from Arrhenius plots are systematically greater for loss peaks found at higher temperature (see reciprocal temperature plots in McCrum er ~ 1 . ~ ' ) . In spite of the shortcomings of isochronal measurements, when large numbers of materials have to be characterized in a short period of time such measurements appear to be necessary. When possible, it is highly desirable to check on any change in the character of the sample with temperature by making measurements through a complete temperature cycle, i.e., while cooling as well as heating. Any nonuniqueness in the curves obtained is usually indicated with this procedure. It is possible to obtain changes in the level of G ' ( o )with temperature that appear to indicate the presence of a relaxation process when actually the level of crystallinity in the sample is changing. This is an example where temperature cycling would help in analyzing the material response. The simplest form of torsion pendulum is shown in Fig. 15, where a rod-shaped sample is simply clamped in place with a known inertial disk and the damped, free oscillations are followed by the recording of the mo-

FIG. 15. A simple free-oscillation torsion pendulum.

40 1 1.

VISCOELASTIC AND STEADY-STATE RHEOLOGICAL RESPONSE

SUPPORT WIRE

INERTIAL ARM

FIG. 16. An inverted free-oscillation torsion pendulum.

tion of an optical lever beam. Here the specimen supports the inertial disk and is in a state of tension, which may or may not significantly influence the results. The damping can be seriously affected before the modulus changes appreciably.@ In any case the simple torsion pendulum falls apart when the sample melts and/or the viscosity of the material drops below lo7 Pa sec. These problems can be eliminated by using an inverted torsion pendulum with squat samples (Fig. 16).65 A wire, whose torsional stiffness can usually be kept negligible compared to the sample stiffness, supports the inertial member and the tension in the sample can be monitored from belowaB7If the sample is short enough ( € 2 mm) its surface tension will keep it between the sample platens regardless of how low its viscosity. Viscoelastic liquids above their Tg and initially fluid curing systems can be readily measured within the restrictions of the free oscillaH. Nakayasu, H. Markovitz, and D. J. Plazek, Trans. Soc. R h d . 5, 261 (1961). A. J. Kovacs, R. A. Stratton, and J. D. Ferry, J . Phys. Chem. 67, 152 (1%3).

1I. 1

LINEAR VISCOELASTIC BEHAVIOR

41

tion technique. The principal practical limitation is that results cannot be obtained when the loss tangent of the pendulum becomes greater than 0.8. However, when the sample loss tangent becomes too high the system loss tangent can be diminished by using a stiffer wire and the sample behavior can be followed through any ordinary loss peak. Since the torsion wire is mechanically in parallel with the sample the appropriate equation of motion for free oscillation iss8 d2e

I - + - - bG’’ de dt2 o dt

+ (bc’ + ko)e = 0.

(1 1. I .27)

The usual solution taken for this equation is that of a linear second-order equation with constant coefficients. To the extent that G’ and G” change with frequency this solution is an approximation, but allowing for a drifting base line the approximation is a good one. Equation (11.1.27) yields G’ = (Iw2/b)[l + (A2/4#)] G” = (Io2/b)(A/7r),

tan 6 = A/.n[l + (A2/47?) - (ko/lw2)1,

-

ko/b,

( 1 1.1.28)

(11.1.29) (1 1.1.30)

where I is the moment of inertia, b the sample coefficient, 8 the angle through which the inertial member oscillates, ko the torsional constant of the support wire, and A the logarithmic decrement (the natural logarithm of the ratio of two succeeding peak a m p l i t ~ d e s ) .The ~ ~ sample coefficient or geometrical factor b is given in Table IV for a rod. It is pcd3/16h for a rectangular strip ( p , numerical factors5; c width; d thickness). Note that the damping of free oscillations is the most accurate and sensitive measure we have of very small loss tangents. The popular torsional braid instrumentee is a simple torsion pendulum where the material of interest, be it polymer or low-molecular-weight organic glass, is coated on a glass braid that supports the inertial disk. The “sample” being measured is therefore a composite material. Only qualitative information concerning the material being investigated is obtained. The decreases in apparent stiffness and loss tangent intensity as a function of temperature are attenuated. At relatively high temperatures, if the material is a viscoelastic liquid, G’ can decrease sufficiently so that the torsional stiffness of the glass braid takes over and a maximum in the system loss tangent is observed where there is none exhibited by the material of i n t e r e ~ t . ~ ~ ~ ~ . ’ ~ ~ OB

loo

T. E. Morrison. L. J . Zapas, and T. W. De Witt, Rev. Sci. Instrum. 26, 357 (1955). A. F. Lewis and J . K . Gillham. J . Appl. Polym. Sci. 6, 422 (1%2); 7, 685 (1963). C. A. Glandt, H. K . Toh, J. K. Gillham, and R. F. Boyer,J. Appl. Polym. Sci. 20,1245

( 1976).

42 1 1.

VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

Quantitative results can be obtained from a different kind of composite system: a low-loss metal strip with a uniform coating of the viscoelastic liquid of i n t e r e ~ t . ~ ~ JMaterials ~ ~ - ' ~ with high loss and with moduli too low for self-support can be measured in free or forced vibration at and around resonance. Forced oscillation nonresonance instruments designed to operate over a wide frequency range are l e g i ~ nbut , ~ few have been used in more than an isolated reported study or two. The most prolific have been the apparatus of Fitzgerald and Ferry,*s Horio and O n ~ g i ,and ~ ~Birnboim ~ . ~ ~and ~ Ferry. 107~108 The frequency range of nonresonance instruments is limited on the low end by either operator impatience or detector cutoff. Since steady-state oscillation is required, a measurement at o = 0.001 rad/sec requires at least half a working day. The amount of information obtained for the time and effort expended at such low frequencies is marginal at best, and so few such measurements are made.24,96,106 Creep and stress relaxation measurements are more rewarding in this region of the time scale. The usual low-frequency detection limitation is one of phase angle discrimination. Since phase angles can rarely be measured with a resolution of less than 0.1", when angles of 1 or 89" are encountered accurate measurements cease to be possible. Some instruments approach Oo109 at low frequencies while others approach 90°08~110 when the material being studied is a viscoelastic liquid. The high-frequency limit of possible measurement can also be due to phase angle resolution but more often wavelength limitations or inertia domination occur first. Most forced-oscillation nonresonance instruments yield dependable results at frequencies only below that of the system resonance. In most measurements the effect of the sample's own inertia is usually ignored and it is assumed that the wavelength of the signal propagated through the sample is much greater than the size of the sample in the direction of propagation. An alternative expression of this requirement is that the applied deformation suffers negligible phase shift as it is transmitted through the sample. Derivations including the effect H. Oberst and K . Frankenfeld, Acusticu 4, 181 (1952). W. P. Van Oort. Microtecnic, 'I, 246 (1952). Io3 F. Schwarzl, Acusticu 8, 164 (1958). T. Nicholas, Shock Vibrution Bull. 38, 13 (1%8). Io5 M. Horio, S. Onogi, and S. Ogihara, J . Jpn. Soc. Test. Muter. 10, 350 (I%]). IW M. Horio, T. Fujii, and S. Onogi, J . Phys. Chem. 68, 778 (1964). lo' M. H. Birnboirn and J . D. Ferry, J . Appl. Phys. 32, 2305 (1961). N . W. Tschoegl and J . D. Ferry, Kolloid-Z. 189, 37 (1963). H. Markovitz, P. M. lavorsky. R. C. Harper, L. J. Zapas, and T. W. De Witt, Rein. Sci. Instrum. 23, 430 (1952). 110 D. 0. Miles, J . Appl. Phys. 33, 1422 (1%2). Iol

IM

11.1

43

LINEAR VISCOELASTIC BEHAVIOR

of sample inertia have been given by Markovitz for two kinds of rbeometers: coaxial cylinders with torsional and axial vibration.ll’ To understand the ways inertial effects of added masses influence the response of many instruments it is important to note whether sample excitation and response are observed at the same end or at opposite ends, i.e., through the sample. In the forced-oscillation torsion pendulum of Morrison et the torque and the angular motion are determined at the top of the rod-shaped sample. A sinusoidal torque is produced by a coil in a permanent magnetic field and the motion is followed with a light lever. Here the contribution of the torsional constant of the support wire, ko, and the moment of inertia Z of the sensing coil and top clamp enter into the equation for the storage modulus G’ additively: G‘ =

[F

cos

1

a1 + lo2 - ko ,

( 1 I . 1.31) ( 1 1.1.32)

where b is the sample coefficient (see Table IV), M oand 8, are the maximum values of the torque and angle, respectively, and 6, is the system phase angle. The dynamic rheometer of Markovitz et al. log is an example of the contrasting type of instrument. In its operation a sinusoidal angular oscillation of a rod-shaped sample is imparted through the bottom clamp and the oscillation of the top of the sample is followed by a detection coil that is attached to the top sample clamp and is in a permanent magnetic field. The moduli are calculated from m(m - cos 6,)

I G’ = - [ZW2 - k,] b 1 G” = - [ l o 2

b

-

k,]

m sin 6,

1

+ mz - 2m cos 6,

1,

(11.1.34)

where m is the ratio of the maximum amplitude of oscillation of the top to the bottom of the sample. It can be readily seen that in the torsion pendulum at high frequencies the additive inertial term Io2grows rapidly, ko becomes insignificant, 8, approaches 18V, and therefore O0 must decrease rapidly. G’ is then determined by a relatively small difference between two large numbers, so that the uncertainty in I alone gives rise to large errors. At the same time G” values become meaningless because of phase angle uncertainty. This is not the pattern of response in the dynamic rheometer. Both moduli become uncertain when Zw2 = ko,sbut H.Markovitz, J . Appl. Phys. 23, 1070 (1952).

44 11.

VlSCOELASTlC A N D STEADY-STATE RHEOLOGICAL RESPONSE

at high frequencies, where Iw2 >> ko, since lozis a factor and not an additive term its rapid growth is compensated for principally by a diminution of m , i.e., the amplitude of response becomes small. A sensitive detector can help extend the frequency range upwards. Ultimately, but not so soon as with the torsion pendulum, S, nears 180°, its sine becomes too small to measure and G" is unobtainable.

11.2.

Steady-State Response

The onset of a steady-state rheological condition is usually judged by the indication of a limiting stress or rate of shear. The time-independent stress or rate of strain should be that of the material and not just at a point in laboratory space. These criteria are usually adequate, but a dominant contribution of viscous deformation to the total deformation can mask the fact that not all of the viscoelastic mechanisms may have yet contributed their bit. If only the viscosity is desired or if the characteristic amount of recoverable deformation and/or residual stress are inconsequential for practical purposes, time steady state as described in Chapter 11.1 may not be required. However, the condition that viscous contributions are overwhelmingly dominant must be provable. We fear that recoverable deformation has frequently been included with viscous deformation for the calculation of the viscosity with the consequence that the calculated value is erroneously low. In our experience, as related above, we have calculated viscosities from deformation that was completely recoverable. There can be no greater error than this. The time of steady shearing or of applied constant tractions for a steady-state condition can be judged by the time that it takes the retardation function $(r) to reach the value of unity or equivalently by the terminal region of the retardation spectrum. 11.2.1. Practical Solids

In addition to true viscoelastic solids, elastomers and thermoset resins (where a molecular network guarantees the existence of an equilibrium response if degradation processes are negligible), polycrystalline polymers and polymeric glasses are also effectively solid in that their deformation processes responding to stimuli applied months or years earlier are insignificant. Our reduced viscoelastic functions indicate that, for a high-molecular-weight polymer held at its glass temperature, steady-state deformation would be achieved in about a million years. In so far as polycrystalline polymers are concerned no one has, nor probably ever will, establish a long-time-limiting deformation or stress. One expects an equilibrium response on the basis that the crystallites act as cross links but no equilibrium deformation has ever been observed.

17.2.

STEADY-STATE RESPONSE

45

Even the equilibrium response of elastomers is surprisingly difficult to reach. Lightly cross-linked natural rubber does not reach its equilibrium deformation or stress before degradation or crystallization allows additional The higher the cross-linking level the closer one can approach equilibrium. This equilibrium behavior will not be commented on further here. Its treatment is covered in the realm of rubber elasticity and has been recently reviewed by T r e l ~ a r . ~ ~ ~ 11.2.2. Viscoelastic Liquids 11.2.2.1. Measurement of the Shear Viscosity. The limiting lowrate-of-shear viscosity q is most often determined in one of the three following modes. First, if the viscosity is below 1 0 0 Pa sec (lo3 Poise) a simple glass-bulb capillary viscometer of the Ostwald or Ubbelohde type is used. Second, viscosities from 1 to lo4 Pa sec can be conveniently measured by means of the falling-ball technique. This range can be covered by a low-pressure gas-driven capillary viscometer.l16 Third, a wide variety of rotational viscometers and rheometers have been used that cover various ranges, but rarely reaching lo0 Pa sec. The ordinary glass capillary viscometer is usually used to obtain molecular-characterization information on polymers in dilute solution or to characterize oils. Shear rate effects on the intrinsic viscosity of polymers are not usually encountered until molecular weights approaching a million or higher are being studied. Then rotational viscometers of the Zimm-Crothersll' or Eisenberg-Frei1lB type can be used. A number of possible sources of errors encountered in capillary viscom. ~ reported ~ ~ eters are discussed by Van Wazer et a1.O Cannon et ~ 1 1have in some detail on the kinetic-energy correction required for short flow times. The falling-ball method is convenient and yields precise values for the viscosity of moderately viscous materials. The familiar Stokes equation q s = (2/9)(ps - p ) g R Z / u

(1 1 . 2 . 1 )

yields viscosity from the terminal velocity u of fall of a sphere of density p s through an infinite medium of density p. The sphere's radius is R and g A . N . Gent, J . Appl. Polym. Sci. 6, 442 (1962). R . Chasset and P. Thirion, Proc. I n t . Conf. PhyAics of Non-Cryst. Solids. 1964 p. 345 (1965). D. J Plazek, I . P o l y m . Sci.. Port A-2 4, 745 (1966). L. R . G. Tieloar, "The Physics of Rubber Elasticity," 3rd ed. Oxford Univ. Press (Clarendon), London and New York. 1975. l l s T . G Fox, S . Gratch. and S. Loshaek, Rheology 1, 431 (1%5). 11' B . H. Zimm and D. M. Crothers, Proc. Null. Acad. Sci. U . S . A . 48, 905 (1962). H . Eisenberg and E. H. Frei. J . Pulym. Sci. 14,417 (1954). llS M. R. Cannon, R. E. Manning, and J . D. Bell, Anal. Chem. 32, 355 (1960). 11*

113

46 1 1 .

VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

is the local gravitational constant. Ball velocities are measured with traveling microscopes or calibrated lines on the glass tube. If the liquid is opaque, capacitive, or inductive, electrical switching and timing circuits can be used. Under many practical conditions, end, wall, and inertial effects require corrections to Stokes' law. The most recent exhaustive investigations into the necessary corrections have been reported by Sutterby.I2O Previous developments are discussed by him and Van Wazer et a1.O The wall correction is needed most frequently, and the Faxen expression for this, which is effective up to values of d / D = 0.3, ist r) =

qS[l - 2.104(d/D) + 2.09(d/D)3- 0.95(d/D)'],

(11.2.2)

where d is the diameter of t h e sphere falling down the axis of a tube with diameter D . Sutterby presents a table of correction factors for wall and inertia effects and another for wall and end effects. Most of the wide variety of available viscometers and rheometers are designed to characterize non-Newtonian fluids. Newtonian fluids exhibit viscosities that are insensitive to the shear rate of measurement and, of course, can also be characterized with such viscometers; but, in general, they have to be considered overdesigned and too costly for the study of Newtonian fluids. However, most fluids with high viscosities, including polymers, exhibit complex rheological behavior and require characterization over the wide range of shear rates that can be produced in the more complex rheological instruments. The easiest and most common method for obtaining a range of reproducible shear rates has been to drive a changeable gear train with a synchronous motor. The result is that most rotational instruments have operated at a constant strain rate. Weak structures in materials, which give rise to yield points, and other small strain effects are often obscured by the necessity of operating at a constant strain rate. During the last decade, however, dc motors driven by servomechanism amplifiers have been supplanting the synchronous motor-gear systems. The new systems make the application of a number of stress histories convenient.

11.3. Nonlinear Viscoelastic Behavior The vast developing area of nonlinear viscoelastic behavior cannot be covered adequately by the space allotted here nor by this author, whose J. L. Sutterby, Trans. SOC.Rheol. 17, 559 and 575 (1973).

t The correction factor in Eq. ( 1 I .2.2)is erronously presented by Van Wazer e f ul. as its reciprocal.

11.3.

N O N L I N E A R VISCOELASTIC B E H A V I O R

47

interest and activities have been in polymer molecular-structure correlations and not in continuum mechanics. Walterss has recently reviewed much of the current status with analyses of many of the currently used techniques and instruments employed in studying nonlinear rheology of viscoelastic liquids. Included in his references are most of the pertinent monographs to which one can go for background information. In some cases the instrumentation and geometries discussed in preceding chapters can be and have been utilized beyond the linear range of response. 11.3.1. Nonlinear Steady-State Behavior of Viscoelastic Liquids

Non-Newtonian shear flow and the presence of normal stresses have been at the center of interest in nonlinear liquid behavior for more than three decades. A shear viscosity that decreases with increasing shear rate is called “shear thinning” (in the older literature, pseudoplastic) and one that increases is “shear thickening” (dilatant). Suspensions can be shear thickening but polymers and their solutions are inevitably shear thinning and exhibit normal ~ t r e s s e s . ~Nearly ~ ~ - ~ all ~ ~of the data measured at the highest shear rates have been obtained with extrusion capillary visc~meters.~ There have been notable exceptions where rotational viscometers with extremely small annular gaps have been used. The narrow gaps are essential to carry away the appreciable heat generated at the high shearing rates.124-12s In an extrusion viscometer, pressure is applied to a polymer melt in a reservoir barrel by a compressed gas or a piston driven at a constant force or rate. The polymer is forced into and through a capillary die. Viscosities are calculated from the shearing stress and the measured throughput per unit time. This simple-looking procedure is subject to several requirements or corrections. Viscous heating as well as piston friction should be negligible and the pressure should not be so great that the viscosity is increased by densification. Since channeling of the polymer occurs in the barrel just before the capillary, a capillary length enhancement correction is necessary. A Bagley plot127of pressure P vs. the ratio of capillary length to radius ( L / R ) at constant Q / R 3 , where Q is the Iz1

A . S. Lodge, ”Elastic Liquids.” Academic Press, New York, 1964.

B. D. Coleman, H. Markovitz, and W. Noll, “Viscometer Flows of Non-Newtonian Fluids.” Springer-Verlag Berlin and New York, 1966. Iz3 W. L. Wilkinson, “Non-Newtonian Fluids.” Pergamon, New York, 1960. Iz4 E. W. Merrill, J . Insrrum. SOC. Am. 3, 124 (1956); J . Colloid Sci. 9, 7 (1954). I Z 5 E. M. Barber, J . R. Muenger, and F. J. Villforth, Jr., Anal. Chcm. 27, 425 (1955). lZ6 R. S. Porter and J. F. Johnson, J . Appl. Phys. 32, 2326 (1961). lZ7 E. B. Bagley, J . Appl. Phys. 28, 624 (1957).

48 1 1 .

VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

throughput (cm3/sec), yields a negative intercept at P = 0, which is interpreted as the necessary entrance correction L/Ren,for the calculation of the maximum stress at the wall uw,i.e.,

Capillaries with L / R ratios ranging from less than 1 to over 200 have been used. Throughputs are usually measured by cutting off and weighing sections of extrudate that have emerged during a known period of time. The viscosity is usually non-Newtonian, q(+), i.e., the ratio u/+is not a constant but a function of the rate of shear ?, and it cannot be calculated immediately from capillary flow data because the shear rate at the wall pW cannot be calculated directly without the velocity profile. The profile is more blunted for a shear-thinning fluid than the well-known parabolic profile for a Newtonian fluid. The required qW can be obtained from the Weissenberg equation1** (1 1.3.2)

where -jlN = 4Q/7rR3 sec-I, the shear rate at the wall for a Newtonian liquid. A computer analysis has been suggested for accomplishing the preceding two corrections with a minimum of data.lZn Since uw and pw can be experimentally determined, their ratio uw/pw= q(pw)is known in spite of the fact that the fluid in the capillary is not sheared uniformly. If the viscosity is very high and capillaries with high L / R ratios are used, the hydrostatic pressure developed in the capillary can increase the viscosity measurably.130 Corrections for this abberation have been proposed.131 Since steady state is implicitly assumed when viscosities are calculated from capillary data it could be expected that the residence time of the polymer in the capillary, i.e., the time of shearing, would be at least as long as the time required to reach viscoelastic steady state. This requirement is rarely checked, simply because this time is usually not known for most materials. The determination of a shear viscosity function of a bulk high polymer is usually limited at high shear rates by the onset of extrudate distortion which is called melt fract~re.~ B. Rabinowitsch, Z . Phys. Chcm.. A h . A 145, 1 (1929). S. Negami and R. V. Wargin, J . Appl. Po/ym. Sci. 12, 123 (1968). 130 R. C. Penwell and R. S.Porter, J . Appl. Polym. Sci. 13, 2427 (1969). lS1 R. C. Penwell, R. S. Porter, and S. Middleman, J . Po/yrn. Sci., Pnrf A-2 9.73 I (I97 I ).

11.3.

NONLINEAR VISCOELASTIC BEHAVIOR

49

11.3.2. Nonlinear Transient and Dynamic Properties

Stress relaxation measurements following steady-state flow usually involve nonlinear results, partly because the useful ranges of the instruments utilized to date have been largely, if not completely, in the nonlinear domain.76,77*132 The limited results available from this method have shown that viscoelastic mechanisms with the longest relaxation times are preferentially eliminated76and that the recoverable strain per unit stress decreases with increasing shear stress for polymers with broad molecular-weight distributions and tends to be stress-independent for narrow-distribution polymers, even when the shear rate is high enough for the viscosity to have dropped nearly an order of magnitude from its limiting low-shear value. It is widely known that the onset of nonlinear behavior is partly manifested by the appearance of normal stresses.6*121*122 Two normal stress functions and the shear viscosity function suffice to characterize the steady-state response of “general simple fluids”.lZ2 The treatment of normal stresses and a recent review covering the techniques used in attempts to measure them is given by Walters.6 He includes discussion of the “hole errors” that appear in conjunction with the use of manometer tubes and remote sensing diaphragms to measure the normal forces. 133-135 The cone and plate has been most often the choice for the determination of high non-Newtonian viscosities in the low to medium shear-rate range, i.e., y = to lo2 sec-l. The shear rate is virtually uniform throughout the sample and sample installation is rapid and convenient, whereas the Couette geometry (rotating concentric cylinders) is not convenient to fill when the viscosity is greater than lob Pa sec. Higher viscosities are usually determined from the torsional flow of a rod or from samples between parallel circular plates at the end of a creep experiment. Because of the converging flows and drawing operations that are frequently encountered in the polymer industry, a substantial interest in ex13* R. A. Stratton and A. F. Butcher, J. Polym. Sci., Part A-2 9, 1703 (1973); J . Polym. Sci.. Polym. Phys. Ed. 11, 1747 (1973). J. M. Broadbent, A. Kaye, A. S . Lodge, and D. G . Vale, Nature (London) 217, 55 (1%8). 134 A. Kaye, A. S . Lodge, and D. G . Vale, Rheol. Acra 7, 368 (1968). R. I. Tanner and A. C. Pipkin, Trans. SOC. Rheol. 13, 471 (1969). H. Markovitz, L. J. Elyash, F. J. Padden, Jr., and T. W. De Witt, J . Colloid Sci. 10, 165 (1955). 13‘ R. McKennel. Anal. Chem. 28, 1710 (1956). 13’ J . C. Slattery, J. Colloid Sci. 16, 431 (1961). E. 0. Forster and H. H. Horowitz. Am. SOC. Test. Muter., Spec. Tech. Pub/. 299,85 (1961).

50 1 1 .

VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

tensional flows has developed in the past decade. A number of devices have been designed and developed to carry out the difficult determination of the stressing viscosity function over a range of strain r a t e ~ . l ~ ~In- l ~ ~ shearing experiments carried out at a constant rate of strain in rotation, an unlimited time is available over a wide range of shearing rates so that steady-state response is achievable. In elongational measurements the sample is continually thinning down so that steady state cannot be reached at high strain rates when the viscoelastic memory of the materials extends for days or even hours. Biaxial extension is also of interest academically and industrially because of the film-blowing process. It is being studied via the deformation of a blown b~bb1e.l~' Beyond a shear-thinning viscosity and the presence of normal stresses, a limited number of general features identified with the nonlinear shear response of polymers and their solutions have been accumulating. The steady-state recoverable strain per unit stress J , ( y ) has been found to be a decreasing function of rate of hear.^^,^^.^^^.^^^ This appears to be a consequence of the loss of the viscoelastic mechanism with the longest relaxation or retardation times. Such a loss of mechanisms is compatible with a lower non-Newtonian viscosity and the fact that linear dynamic measurements made orthogonally to a steady rotational shearing flow show a long time truncation of the relaxation s p e ~ t r u r n . ~Other ~ ~ - evidence ~~~ indicates that nonlinear polymer response commences above a critical train.^^.^^^*^^ Thus linearity occurs at all stress levels (below fracture) at short times and small strains. If the molecular weight of an amorphous polymer sample is high enough the stress-overshoot phenomena may be seen. These are step-function steady-shearing maxima in the normal and shearing stresses generated in the samples, and found before steady state J . Meissner, Trans. Soc. Rheol. 16, 405 (1972). G. V. Vinogradov, V. D. Fikhman, and B. V. Radushkevich, Rheol. Acra 11, 286 ( 1972). F. N. Cogswell, Plast. & Polym. 36, 109, (1968). J. Dealy, R. Farber, J. Rhi-Sausi, and L. Utracki, Trans. Soc. Rheol. 20, 455 (1976). 144 N. E. Hudson, J. Ferguson, and P. Mackie, Trans. SOC.Rheol. 18, 541 (1974). 145 R. L. Ballman, Rheol. Acra 4, 138 (1965). 1 4 ~E. A. Everage, Jr. and R. L. Ballman. J . Appl. Polym. Sci. 20, 1137 (1976). 14' D. D. Joye, G . W. Poehlein, and C. D. Denson, Trans. Soc. Rheol. 17, 287 (1973). W. Philippoff, F. H. Gaskins, and T. G. Brodnyan, J . Appl. Phys. 28, 1118 (1957). R. 1. Tanner and J. M. Simmons, Chem. Eng. Sci. 22, 1803 (1967). 150 J . Simmons, J . Sci. Insfrum. 43, 887 (1966). IS1 R. I . Tanner and G. Williams, Rheol. Acfa 10, 528 (1971). C. P. Wong and G. C. Berry, Polym. Prepr.. Am. Chern. Soc.. Div. Pol.vm. Chem. 15, No. 2, 126 (1974). IPJ. Meissner, J . Appl. Polym. Sci. 16, 2877 (1972). 140

141

11.4. PRESSURE EFFECTS ON VISCOELASTIC BEHAVIOR

51

is a c h i e ~ e d . ' ~ ~ * 'There ~ ~ ' " is also a corresponding maximum in the recoverable strain as a function of the time of These maxima are interpreted by some as indicating a "structure breakdown," a structure that is self-healing, i.e., upon sitting, the sample regains its original properties. lS5 Advocates of the influence of entanglements between polymer molecules in the determination of mechanical response believe that the concentration of entanglements is reduced by the hearing.^^.'^^ This breakdown in structure, or reduction in entanglement concentration, occurs at the time of shearing where the stress maximum is seen and is largely responsible for the decrease in viscosity with increased rate of shearing. Because the material may still not have reached viscoelastic steady state even after the entanglement concentration has adjusted to the applied shear rate or stress, it is possible under the proper conditions to observe a minimum following any of the above-mentioned maxima. When measuring these transient phenomena it is important to establish that instrumental compliance does not cause or contribute to the measured effects.lS3 One can most easily be assured that no such compliances contribute when a Couette geometry is used.'" However, the possibility of secondary flows has been called upon to cast doubt on the interpretation of the stress-overshoot effect.lJ6 Very few dynamic (sinusoidal) measurements have been made in the nonlinear range of response where the response is not sinusoidal like the input, but is distorted. A computerized operation and analysis of the harmonic content of the response appears to be the only reasonable approach to such measurements. To obtain a reasonably wide range of frequencies in spite of the large amplitudes of oscillation required to reach the nonlinear range of response, low-inertia printed-circuit motors with fast servoamplifiers are needed. The oscillatory rheometer of Krieger and N ~ u , *which ~ employs the Couette geometry, uses the above-mentioned features.

11.4. Pressure Effects on V iscoe Iast ic Behavio r Most studies of the viscoelastic behavior of polymers use temperature as a principal variable: however, interest in the effects of pressure is I M Von K.-H. Hellwege, W. Knappe, F. Paul, and V. Semjonow, Rheol. Acru 6, 165 (1967). A. Ya. Malkin, B. V. Yarlykov, and G . V. Vinogradov, Rheol. Acra 9, 329 (1970). 168 F. N . Cogswell, Plusr. & Polym. 41, 39 (1973).

52 11.

VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

growing. In terms of free-volume theories, an increase of pressure is qualitatively equivalent to a decrease of temperature, which causes decreases in transport property rates. Hence pressure variations of the viscoelastic response should help test the free-volume and competing theories. The effects of pressure are obviously also of interest in understanding the industrial processes of extrusion and injection molding of polymers. Pressure studies of viscoelastic behavior are still rare, and cost of equipment and experimental difficulties are probably the principal reasons why very few extensive polymer characterizing studies have been carried out. Marvin and McKinney14have reviewed the status of volume relaxati9ns in amorphous polymers more than a decade ago. Pressure effects were highlighted. Since then a milestone investigation by Goldbach and Rehage15' has shown that linear volume creep measurements are possible. Torsion pendulum and torsional stress relaxational measurements have been made by Zosel at pressures up to 1000 atm.IJB Several viscometers have been constructed that are capable of operating at up to several thousand atmospheres. 158*158*160 The most comprehensive studies of the stress relaxation response of elastomers under pressure appear to be those of Tschoegl.161

115.Sample Handling 11.5.1. Molding

Preformed pellets, wafers, or rods of polymer are required as the specimens for investigation in many instruments. Most often, molds involving pistons and cavities are heated in laboratory hot presses; pressures from 5 x lo3 psi (3.5 x 10' Pa) to 2 X 104 psi (1.4 x 108 Pa) are reported as being used in the molding of polymer specimens. In the past we followed many of the standard molding practices, often with the resultant loss of valuable materials via extrusion at high rates of shear through fine slitlike gaps. However, for more than a decade we have been molding specimens without the aid of presses. A simple vacuum mold shown in its latest form in Fig. 17 has been used with great success. It was first devised by V. M. O'Rourke, and during our studies together we have found that the use of precision-bore borosilicate tubing as the mold barrel guarantees G . Goldbach and G. Rehage, Rheol. Acra 6, 30 (1967). A. Zosel, Kolloid-Z. 199, 8 (1964). 16* R. J. McLachlan,J . Phys. E. 9, 391 (1976). IM V. Semjonow, Rheol. Acfa 2, 138 (1962). lE1 R. W. Fillers and N . W. Tschoegl, Trans. SOC.Rheol. 21, 51 (1977).

16'

11.5. SAMPLE HANDLING

53

a reproducible fit to the machined stainless steel piston plungers following the replacement of broken sections. Polymer in fiber, powder, or granular form is placed in the precision-bore tubing with the bottom piston secured and sealed with the compressed Teflon sleeve. The tubing is then attached to the vacuum-tee with the top piston fixed loosely to the ramrod in the elevated position. The mold is then evacuated and heated. The

54 1 1 .

VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

heating is most simply accomplished in a small vertically oriented tube furnace. A ceramic power resistor with its bottom plugged with rolled asbestos tape has served successfully to provide the required heating. A variable autotransformer (e.g., Variac, Powerstat) provides the power to reach the desired temperature for molding. The temperature is conveniently monitored with a thermocouple, embedded in the bottom cap or held to the glass tube at sample level with Teflon tape. For crystalline polymers, temperatures 40°C above the melting point should provide molded specimens that are independent of previous thermal histories (see Wunderlich162on self-nucleation). Glassy polymers should usually be heated between 50 and 100°C above their Tg, depending on their molecular weight. In both of the above cases, with the exception of extremely high-molecular-weight materials and/or broad distributions of molecular weight, the polymers should be in their terminal zone of response at times greater than an hour, i.e., + ( t ) should be close to unity and viscous deformation should predominate. Under these conditions hand pressure applied to the ramrod will be all that is necessary to finally shape the cylindrical specimen. The success of the application of the pressure can be assessed visually immediately after, and in most cases it will be found that no more than a few milligrams of material are lost in the molding process. Before the application of pressure it will usually be observed that under vacuum conditions, when a temperature above 100°C is reached where the viscosity of the polymer falls to the neighborhood of lo4 Pa sec, the polymer will burst into a foam from the evolution of absorbed vapors. Eventually the foaming will cease and the quiescent material will slowly flow toward the bottom of the mold. For high-molecular-weight specimens (M > lo5)the foaming can take place for several hours. We believe that most often it is steam that is being evolved since simply exposing the specimen to the atmosphere at room temperature (and subsequently heating under vacuum conditions) will yield a repeat of the foaming. This foaming phenomenon cannot be considered to be trivial since in many instances when measurements in vucuo are attempted it can occur in the measuring instrument. Thus the sample's shape can be lost and in fact the sample can blow itself out of position. Measurements are then impossible unless the specimen shape can be restored in s i t ~ . ~ Under ~3~ conditions where it is known that, once disturbed, the sample cannot be reshaped without being removed from the measuring device, the foaming must be avoided. Although an inert-gas atmosphere such as nitrogen or argon does not appear to be as effective as a vacuum in reducing thermal degradation, B. Wunderlich, "Macromolecular Physics," Vol. 2, p. 52. Academic Press, New York, 1976.

11.5.

SAMPLE HANDLING

55

one can be used to inhibit the foaming phenomenon. However, the rate of creep of the polymer, most notably near its Tg,will be enhanced by the plasticizing effect of the dissolved vapors. We have unpublished results on polyvinylacetates similar to and, in one case, the same material as that reported on by Ninomiya and FerryIg3where our creep velocities determined on degassed samples in vacuo were sixfold slower near 30°C than the published results, which were determined in air on samples that were equilibrated with the ambient moisture. The effect of absorbed water is even more severe in polymethylmethacrylate where degassing in vucuo has been seen to slow creep by one to three orders of m a g n i t ~ d e . ~In~ , ~ ~ general, it is expected that highly polar polymers would absorb water to a greater extent and thereby experience a greater degree of plasticization than polymers of low polarity, but even polystyrene is significantly affected. If, of course, mechanical characterization is made for the purpose of engineering design, the influence of expected absorbed water on the response had best be included. The variability of performance with changing humidity should be expressly indicated. Yoshitomi et al. Ig5 have reported reduced stress relaxation curves for dry nylon 6 and for specimens that have been equilibrated at 75% relative humidity to about 5% moisture content. The wet sample appeared to relax at a rate of 15 orders of magnitude faster. A substantial decrease in Tg was obviously involved. Additional benefits often accrue from the use of vacuum during sample installation since any gas present when a glassy or crystalline sample softens to make initial contact with an instrument platen can lead to the mundane but disturbing trapping of bubbles in the samples near or at the interface, bubbles that often cannot be removed and lead to erroneous results because of uncertainties in geometrical factors. Under conditions where foaming cannot be tolerated (as in highmolecular-weight specimens where the time for the material to flow together again at the accessible temperatures is prohibitive) it is still possible to make measurements in uacuo by degasing the specimen at temperatures above Tgwhere the viscosity is still high, e.g., greater than lo8 Pa. At very high molecular weights it is not clear if such a viscosity criterion is sufficient but nonetheless an effective temperature not too far above Tg can be found. Following the lowering of the ramrod in the molding procedure it is important to allow the sample to relax in its new shape, for otherwise geometrical distortion can occur in the measuring instrument upon heating K . Ninomiya and J. D. Ferry, J . Phys. Chem. 67, 2292 (1963). 1wJ . R. McLoughlin and A. V. Toboisky, J . Colloid Sci. 7, 555 (1952). 165

T. Yoshitomi, K. Nagamatzu, and K. Kosiyama, J . Polym. Sci. 27, 335 (1958).

56 1 1. VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

after installation. Some experience with the material being molded is required before a sufficient relaxation time can be estimated. In many cases of nonvacuum molding, even when vented molds are used, an appreciable amount of gas is entrapped by the polymer melt. The molded specimen can only become free of bubbles by the process of dissolution of the gas under pressure. The need of pressure to dissolve gas may be the principal reason that high pressures are often called for to achieve successful molding. 11 5 2 . Solution Mixing

The preparation of dilute solutions poses no particular problems except when the polymers being dissolved have extremely high molecular weights (A4 > lo6). In these cases gentle and patient treatment must be afforded the materials lest the long polymeric chains literally be torn apart by shearing forces. This danger has been known for some time where enormous biological molecules have been involved. More recently synthetic polymers with molecular weights greater than 10 million have been obtained and it has been reported that complete and uniform dissolution can take three weeks even at low polymer concentrations.188 At high polymer concentration levels one is dealing with solutions that can have enormous viscosities (7 > lo8 Pa sec). Mechanical agitation under these conditions cannot be carried out without shear degradation in most polymers. The rolling mill used in the rubber industry is the most common device used for blending in various components that are at times particulate as well as soluble. It is certain that nonlinear viscoelastic response as well as some degradation is involved in such blending. The plasticization of many commercial goods would pose appreciable problems, to say the least, were it not for the fact that high viscosities can often be circumvented if the homogenization process of blends of particles or slurries takes place before dissolution is allowed to occur. The production of a single-phase polymer solution is usually brought about by a subsequent change in temperature. In the preparation of concentrated high-polymer solutions for research purposes two other less practical procedures have been utilized. If the chosen solvent is a plasticizer, i.e., it is a liquid with an extremely low vapor pressure, a ternary dilute solution can be prepared where the polymer and plasticizer are dissolved in a low-boiling liquid. Subsequent vaporization of the highly volatile liquid, it is hoped, will yield the desired homogeneous concentrated polymer solution. Unfortunately, such is not generally the case. It has been our experience to obtain clearly segI8O

D. McIntyre, L. J . Fetters, and E. Slagowski,Science 176, 1041 (1972).

11.5. SAMPLE HANDLING

57

regated samples, even when the vaporization has been carried out in a dropwise fashion under partial vacuum in a rotary evaporator. Successful separation of the volatile component without segregation has been accomplished by means of lyophilization (freeze drying) but only when the temperature of the sample flask was kept at all times at a temperature below the Tgof the final solution.167 The process can take several days if the sublimation temperature is -78°C (dry ice cooling). The final condensation trap then is most likely one filled with liquid nitrogen. If the polymer and plasticizer are not both soluble in benzene or water, the most commonly used frozen components, the problem of searching out a suitable sublimation candidate remains. All else failing, one has to resort to diffusion processes to obtain a homogeneous concentrated polymer solution. This can be exceedingly time consuming but can be expedited after the initial absorption of the plasticizer with slow compression into a sheet that is subsequently folded. This blending step should be repeated as often as is convenient. One general effect that we have observed of inhomogeneity on the viscoelastic response of polymer solutions is a spreading out of the primary softening dispersion on the time scale. 11.5.3. Molecular Weight Blending

The study of the effects of the molecular-weight distribution on the viscoelastic behavior of polymers will be with us for some time to come and, as long as it is, alteration of the distribution by blending different species will be necessary. The most obvious and safe way to blend small samples is dissolve the components together in solution. The mixed polymer should be recovered by means of freeze-drying if possible, since molecular-weight segregation is possible if recovery from solution is by phase separation and drying. 11.5.4. Film Casting

Specimens of glassy or crystalline polymers in the form of fdms are required or are convenient for some commonly used instruments or techniques. Dynamic measurements on the widely used Vibron apparatus or tensile creep or stress relaxation measurements are examples. The usual method of forming these films on a laboratory scale is by casting from solution. The drying of solutions on glass or mercury surfaces is most commonly reported. If the film is other than a single component, possible segregation is a concern as with the preparation of concentrated solu-

'" N. Raghupathi, Ph.D. Thesis, University of Pittsburgh, Pittsburgh, Pennsylvania (I975).

58 11.

VISCOELASTIC A N D STEADY-STATE RHEOLOGICAL RESPONSE

tions. The lack of anisotropy has to be established for any films after their preparation. Acknowledgments The author wishes to gratefully acknowledge the support of both the Science Research Council of Great Britain and the U.S.National Science Foundation. who made the writing of this chapter possible. NSF support has been provided by the Chemical Processes Program of the Engineering Division and by the Polymer Program of the Materials Division.

12. FURTHER MECHANICAL TECHNIQUES 12.1. Ultrasonic Measurementst

By Bruce Hartmann 12.1.l.Introduction

The ultrasonic properties of polymers are influenced strongly by their molecular structure, and therefore the measurement of such properties is useful for the further understanding of polymers. This chapter is aimed at the polymer physicist who wishes to relate ultrasonic properties to molecular structure. The presentation is intended to be instructive rather than exhaustive. For example, no effort is made to make the references historically complete. Older references are given only if they have particular significance and not just to establish priority, and current references will be representative and not comprehensive. Since the goal of this chapter is to gain structural information about polymers, the amplitude of the ultrasonic waves used is restricted to values small enough that no permanent changes are brought about in the polymer. The chapter will be further restricted to solid isotropic polymers. This is not a major restriction since most bulk polymers are isotropic either because they are amorphous or because they are polycrystalline, with random orientation. In an unbounded isotropic solid, two types of ultrasonic waves can be propagated. In the first type, the polymer particles vibrate along the direction of propagation. This is called a longitudinal wave. In the second type, the particle motion is perpendicular to the direction of propagation. This is called a shear wave. Longitudinal waves are also sometimes called dilational, compressional, or irrotational waves. Shear waves are also sometimes called distortional, isovoluminous, or transverse waves. The terms longitudinal and shear are the most commonly used and are adopted here. These two types of waves propagate independently of one another and are the only two types possible. Associated with each of the two modes of propagation there is a speed t See also Volume 1 I (Solid State Physics) of this Series, Part 7. 59 METHODS OF EXPERIMENTAL PHYSICS, VOL. 16c

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

ISBN 0-12-4759584

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and an absorption. Thus, four parameters are required to specify the ultrasonic properties of a solid isotropic polymer: longitudinal speed, shear speed, longitudinal absorption, and shear absorption. (Note that speed is the scalar magnitude of the velocity vector.) These four parameters generally vary with the measurement frequency as well as the polymer temperature and pressure. A complete description of polymer properties then requires measurements to be made of all these variables over suitable ranges. Nominally, the term ultrasonic implies frequencies higher than can be heard by the human ear: above about 20 kHz. It is sometimes convenient to call even lower frequencies ultrasonic, as set by the lower limit of a given ultrasonic apparatus. Much lower frequency measurements (< lo2 Hz) are made with very different equipment. These low frequency measurements are covered in Part l l (this volume). The upper limit of ultrasonic frequencies is nominally taken as 1000 MHz. Frequencies above this value are called hypersonic and are also measured with very different equipment, covered in Chapter 3.3 (this volume, Part A) on Brillouin scattering. Ultrasonic measurements in polymers cover temperatures ranging from liquid helium temperature, 4.2 K, to the degradation point of the polymer, generally less than 500 K. Thus, the temperatures measured are not particularly high compared to what is used with many other solids. Ultrasonic measurements as a function of pressure are seldom made in polymers and, as a rule, pressures do not exceed 1000 bar ( I bar = 0.987 atm). This chapter is organized in the following manner: a particular experimental apparatus is described in detail as an illustrative example, then other experimental techniques are discussed briefly, then typical results of experimental measurements on various polymers are presented and compared with other materials, and finally the molecular interpretation of these results is discussed. The intention is to progress from giving enough information for setting up an experimental apparatus, to using that apparatus to make ultrasonic measurements, to interpreting the results obtained. A short summary is given at the end of each section to reemphasize the main ideas discussed. 12.1.2. Immersion Apparatus As a starting point for making ultrasonic measurements in polymers, a particular apparatus' is described in detail. It will then be relatively easy to describe other types of apparatus in the later portions of this chapter. The apparatus will be presented as one way to measure both the longituB. Hartmann and J . Jarzynski, J . Acousr. Soc. A m . 56, 1469 (1974).

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dinal and shear speeds and absorptions in a polymer at about 1 MHz, over a range of temperatures around room temperature, at atmospheric pressure, to 1% accuracy in sound speed. As with any ultrasonic measurement, the first step is to generate ultrasonic waves. This is done using a material that converts an oscillating electric field to a mechanical oscillation. Such a material is referred to as a piezoelectric transducer. When a transducer is subjected to an oscillating electric field, its thickness will alternately increase and decrease with the same frequency as the electric field. Similarly, when a mechanical oscillation is applied to such a transducer, it produces an oscillating electric field. Thus, transducers are used both to generate and to detect ultrasonic waves. Depending on its use, a transducer is called a transmitting transducer or a receiving transducer. Measurements are made by sending pulses of ultrasound through the specimen. An ultrasonic pulse is just a high-frequency sine wave that begins abruptly and ends abruptly, usually with a total duration of less than 1 msec. Pulses, rather than continuous waves, are generally used to make ultrasonic measurements because it is easy to determine the time it takes for a pulse to go through a specimen by looking for the beginning of the pulse. In the immersion technique, to be described here, the specimen, transmitting transducer, and receiving transducer are all immersed in a liquid. Ultrasonic pulses are sent from one transducer to the other, both with and without the specimen in the path of the sound beam. From the changes in the detected signal observed when the specimen is removed, the speed and absorption can be obtained. 12.1.2.1. Description of Apparatus. When an unbounded specimen (lateral dimension at least several times the ultrasonic wavelength) is held with its face perpendicular to the path of the sound beam, longitudinal waves are developed in the specimen. This technique has been used by several A variation of this method is to hold the specimen at an angle to the sound beam. In this way, both longitudinal and shear waves are generated in the specimen. If the angle at which the specimen is held is greater than the critical angle, the longitudinal wave is totally internally reflected and only the shear wave is propagated. This technique has also been used by several worker^.^-^

* D. G . h e y , G . A.

Mrowca, and E. Guth,J. Appl. Phys. 20, 486 (1949). A. Zosel, Kolloid-Z. 213, 121 (1966). H. J. McSkimin and P. Andreatch, Jr., J . Acoust. SOC.Am. 49, 713 (1971). a Y. Maeda, J . Polym. Sci. 18,87 (1955). R . Kono, J . Phys. SOC.Jpn. 15,718 (1960). H. A . Waterman, Kolloid-Z. 192, 1 (1963). R. E. Smith, J . Appl. Phys. 43, 2555 (1972).

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(b)

FIG.1. Schematic drawings of immersion devices. (a) Specimen rotated, (b) transducers rotated.

In a typical immersion apparatus, when making shear measurements, the specimen is rotated with respect to the fixed transducers to obtain shear waves. In the apparatus described here, the transducers are rotated to obtain shear waves. Schematic drawings of these two arrangements are shown in Fig. 1. A photograph of the apparatus is shown in Fig. 2. The entire device is immersed in a liquid. Measurements are made with different specimens or no specimen by turning the selector shaft. An immediate advantage of this arrangement is that more than one specimen can be used at one time. Especially when covering a temperature range, this can result in a major saving of time. Four major advantages of the immersion technique are: (1) The intimate contact of the liquid with the transducers and specimen provides good transfer of ultrasonic energy (good coupling) between the transducer and the specimen. (2) The coupling is very reproducible. (3) Shear waves can easily be generated with the same transducer that generates longitudinal waves. (4) Multiple specimens can be quickly moved into and out of position.

The transducers used in this case are made of PZT, a polycrystalline ceramic of lead zirconate titanate polarized in a strong electrostatic field.

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ULTRASONIC MEASUREMENTS

63

FIG.2. Photograph of immersion apparatus.

The diameter of the transducers is 2.5 cm. It has been showne that the larger the transducer diameter at a given frequency, the better collimated will be the ultrasonic beam. In this case, at 1 MHz, the diffraction attenuation is negligible and specimen alignment is not critical. The thickness of the transducers is 0.25 cm. The resonant frequency of such transducers is proportional to their thickness and in this case is 0.75 MHz. The output of the transducer is maximum at its resonant frequency, but it can be operated below its resonant frequency without too much loss in output. It can also be operated at harmonics of its resonant frequency to get higher frequencies. In this way, the apparatus has been used to make measurements over the frequency range 0.1- 10 MHz. The central wire of a miniature coaxial cable is soldered to the back of the transducer for one electrical lead. The ground lead is that part of the immersion device which presses against the front face of the transducers. The brass transducer holders are open to the atmosphere, thus providing air backing for the transducers, which improves their operation. One other feature is built into the device to accommodate its use with shear waves. When a shear wave goes through a specimen, the emerging H. Seki, A. Granato, and R. Truell, J . Acoust. Soc. A m . 28, 230 (1956).

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FIG.3. Geometry of shear wave propagation.

beam is displaced from the direction of the incident beam, as shown in Fig. 3. If the transducers are directly facing each other, the displacement will cause some of the sound energy to miss the receiving transducer, causing a reduced amplitude and thus giving a false contribution to the measured absorption. To correct for this, the transducers can be moved until the observed signal is at a maximum. This has been found to give only a small correction. A silicone fluid, poly(dimethy1 siloxane), is used as the immersion liquid. This liquid has a room temperature viscosity of 100 centistoke. To prevent swelling, all rubber gaskets in contact with the liquid are made of a chlorinated silicone rubber. Most of the immersion device is aluminum and all these parts are anodized. Specimen temperature control is obtained by using two 1 kW continuous heaters immersed in the tank holding the immersion device. The current to these heaters is controlled by a variable resistor, and in this manner the temperature can be controlled to about ? 03°C. A stirrer is run continuously to promote good mixing and approximate isothermal conditions. The highest temperature that can be attained is determined by the gelation point of the silicone liquid and is about 150°C. The lowest temperature that can be reached is set by the freezing point of the silicone liquid and is about - 50°C. Measurements can be made below room temperature by cooling the silicone liquid with dry ice. Since the tank dimensions are approximately 70 x 50 x 35 cm, there is a relatively large amount of silicone liquid so that temperature fluctuations are not great. The broad temperature range available is one reason why the silicone liquid is preferable to water as the immersion liquid. The final major topic to be considered in the description of the appa-

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65

PULSED OSCILLATOR Puke Outpul

Triqqw

OSClLLOSCOPE

Tronimillinq Transducer

IMMERSION APPARATUS Receivinq Troruducer O --t

Input

ratus is the electronics that generates and detects the electrical oscillations. There is commercial equipment on the market to readily perform these functions. In the simplest case, only two pieces of equipment are needed: a pulsed oscillator and an oscilloscope. A block diagram of the electronic setup is shown in Fig. 4. The output of the pulsed oscillator is an ultrasonic pulse of the desired frequency. The pulse length is variable but is usually 10-20 cycles, which at 1 MHz gives a pulse length of 10 psec or longer. The Fourier transform of such a pulse does not contain many different frequencies. The group velocity will therefore equal the phase velocity of a continuous wave of the same frequency. This is important for a dispersive material. However, the pulse should not be too long. The distance between transducers must be large compared to the length of the pulse so that standing waves are not formed. In this apparatus, the distance between transducers is about 20 cm while a wavelength in the liquid at 1 MHz is about 0.1 cm. In this case, therefore, a pulse of 20 cycles would be only 2 cm long. The pulses are repeated at a rate of about 60 per second. The output pulses from the pulse generator are applied to the transmitting transducer of the immersion apparatus. The output from the receiving transducer is then displyaed on an oscilloscope whose sweep is triggered by the start of the pulse. The oscilloscope should have a variable time delay incorporated in it. This feature allows convenient positioning on an expanded time scale. As discussed below, measurements are made utilizing the sweep calibration of the oscilloscope. Much more accurate measurements can be made by the addition of one more piece of equipment: a precise time mark generator. This equipment gives a mark on the oscilloscope at precisely controlled intervals. S U M M A R Y . In the immersion apparatus described here, the specimens,

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PZT transmitting transducer, and PZT receiving transducer are all immersed in a silicone liquid. Ultrasonic pulses are sent from one transducer to the other both with and without the specimen in the path of the sound beam. With the specimen perpendicular to the beam, longitudinal waves are developed. With the specimen at an angle, shear waves are also developed. Two pieces of electronic equipment are required: a megahertz pulsed oscillator and an oscilloscope. Using this apparatus, measurements can be made over the frequency range from 0.1 to 10 MHz and the temperature range - 5O-15O0C, at ambient pressure. 12.1.2.2. Measurement Technique. Speed measurements are made in the following manner. With the specimen between the transducers, the detected pulse is displayed on the oscilloscope and the position of, for example, the first peak of the pulse is noted. The specimen is then removed from the path of the sound beam. Since the speed in the immersion liquid is less than the speed in the polymer, the signal will take a longer time to reach the receiving transducer, and the pulse will move to the right on the screen when the specimen is removed, as shown in Fig. 5 . Note that to make identification of the first peak easier, the amplitude scale on the oscilloscope is changed in order to make the height of the peak approximately the same with and without the specimen in place. The difference in transmit times, At, is determined from the change in position and the sweep calibration of the oscilloscope. This difference in transit times is due to the difference in the times to traverse the specimen and to traverse an equal thickness of liquid. For a specimen of thickness L , the transit time is L/ul, where u1 is the longitudinal speed in the specimen. The transit time in the liquid is L/u(liq), where u(liq) is the speed in the liquid. (Since a liquid does not transmit shear waves, there is only one mode of propagation in a liquid.) Therefore, A t = L/u(liq) - L / u , .

(12.1.1)

By measuring A t (and L ) , Eq. (12.1.1)can be used to find u1 provided u(liq) is known. Conversely, if ul is known, Eq. (12.1.1) can be used to find u(liq). For example, uI for aluminum is reportedlo to be u,(Al) = 6428 - 0.91T m/sec,

(12.1.2)

with temperature Tin "C. Using this known value of ul in Eq. (12.1.1), the speed in the silicone liquid can be found, with the experimental result v(liq) = 976 - 2.5(T - 25), which is in good agreement with other measurements." lo I'

J. R. Asay and A. H.Guenther, J . Appl. Phys. 38,4086 (1967). A. Weisler. J . Am. Chem. Soc. 71, 93 (1949).

(12.1.3) Using this result

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67

FIG.5 . Photographs of detected signals with (a) (amplitude scale 2 V/cm) and without (b) (amplitude scale 5 V/cm) a specimen in place (time scale 2 psec/cm, frequency 2 MHz).

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in Eq. (12.1.I), uI for a polymer can be determined as a function of temperature. Since the speed of the silicone liquid changes with age, heat exposure, impurities, etc., it is a good idea to “recalibrate” the liquid with the aluminum standard each time it is used. For a poly(methy1 methacrylate) (PMMA) specimen of 1.27 cm thickness at room temperature, At = 8.3 psec and uz = 2690 m/sec. Shear speed is measured by rotating the transducers so that the sound beam strikes the specimen at an angle (see Fig. 1). At any off-normal angle, there will be induced both a shear and a longitudinal component. When both shear and longitudinal waves are generated, they overlap in the received signal and are difficult to separate. However, if and only if u(1iq) < u I , there is a critical angle beyond which there is total internal reflection of the longitudinal wave and only the shear wave propagates through the specimen. For this reason, an immersion liquid, such as silicone liquid, with a low sound speed is desirable. At the far end of the specimen, the shear wave is converted to the only type of propagation possible in a liquid, with speed u(1iq). The procedure for shear measurements is to start with the transducer faces parallel to the specimen face. Then as the transducers are rotated, the received signal due to longitudinal waves in the specimen will decrease in amplitude until it disappears and the signal due to shear waves becomes apparent. The shear signal is much lower in amplitude and is observed at a later time than the longitudinal signal. This behavior is illustrated in Fig. 6. Shear speed us is calculated in a very similar manner to longitudinal speed u I . The procedure is a little more involved only because the path length through the specimen is not just equal to the specimen thickness. According to Snell’s law, sin B/sin 4

=

u(liq)/u,,

(12.1.4)

where 8 is the angle of incidence and the angle of refraction (Fig. 3). The path length x through the specimen is x = L{ 1 - [us sin B/~(liq)]~}-~m.

(12.1.5)

Replacing L with x and uI with u, in Eq. (12.1.1) gives us = u(liqX[cos 0 - u(liq) At/LI2

+ sin2 O}-112.

(12.1.6)

For a PMMA specimen of thickness 1.27 cm at room temperature, e = 30°, At = 4.4 psec. Then us = 1340 m/sec and x / L = 1.4. Before describing the details of absorption measurements, it is worthwhile to discuss the units of absorption. The most common unit is a logarithmic measure of the ratio of the initial and final intensities, Ii and If,

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69

FIG.6. Photographs of detected signals (a) before (0 = 90") and (b) (0 = 30") rotating the transducers (time scale 2 psec/cm, amplitude scale 2 V/cm, frequency 2 MHz).

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of a sound wave that has gone through a thickness L of the specimen, 11 10 a = - log -

L

(12.1.7)

If’

where a then has units of decibels per centimeter. Generally, signal amplitudes rather than intensities are measured. Since the intensity is proportional to the square of the amplitude 20 A! a = - log L Af’

(12.1.8)

where A, and Af are the initial and final amplitudes. Instead of using the common logarithm in Eqs. (12.1.7) and (12.1.8), natural logarithm6 are sometimes used. The units of a are then nepers per centimeter, where 1 Np = 8.686 dB. Longitudinal absorption is found by comparing, on the oscilloscope screen, the peak of the received signal with no specimen in place to that of the signal with a specimen held perpendicular to the sound beam. The amplitude will be less with the specimen in place both because the absorption in the polymer is greater than in the liquid and because some of the sound energy traveling through the liquid is reflected when it strikes the specimen. The change in amplitude due to reflection can be approximated12 by (Yr

=

(2O/L) lOg[(Z, + Zd2/4zlZ~I,

(1 2.1.9)

where Z = pu = density x speed = acoustic impedance, and subscript 1 refers to the immersion liquid and subscript 2 to the specimen. Since the measured transmission attenuation represents the difference in absorption between the specimen and the liquid, al = (20/L) log(A+/A-) - ar + a(liq),

(12.1.10)

where a1is the longitudinal absorption in the specimen, A+ and A- the measured amplitudes with and without the specimen in the path of the sound beam, and CwOiq) the absorption in the liquid. For most liquids, the ultrasonic absorption in the low-megahertz region is proportional to the square of the frequency.12 Temperature dependence of the silicone liquid can be expressed by an empirical curve fit to the data of McSkimin.Is The result is cr(liq)/f2 = 5.91 x lX

- 4.30 X lO-‘T

+ 3.72 X

10”TZ, (12.1.11)

J. Blitz, “Fundamentals of Ultrasonics,” 2nd ed. Plenum, New York. 1967. H. J. McSkimin,J. Acousr. SOC.Am. 29, 1185 (1957).

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ULTRASONIC MEASUREMENTS

71

for a(liq) in dB/cm, f i n MHz, and T in "C. For example, at 2 MHz and room temperature, a(1iq) = 0.2 dB/cm. Reflection loss is estimated from Eq. (12.1.9). For a 1.27 cm thick specimen of PMMA at room temperature, p = 1.19 g/cm3 and v[ = 2690 m/sec, while for the silicone liquid p = 0.960 g/cm3 and u(liq) = 976 m/sec. Using these values in Eq. (12.1.9), the reflection loss is 2.4 dB/cm. For PMMA at 2 MHz and room temperature, al = 1.4 dB/cm. Viewing the last two terms in Eq. (12. I . 10) as corrections to the basic measurement, it can be seen that a, represents a significant correction, larger in fact than the quantity being measured, while aoiq) is negligible. One can avoid using the large reflection correction if one uses two specimens of different thickness. Since the amount of reflection can be assumed to be the same in both cases, the difference in amplitude is due solely to differences in absorption between the specimen and the immersion liquid occurring in a thickness equal to the difference in thickness of the two specimens. Therefore, =

2O(L2

-

LI)-' lOg(A,/A,),

(12.1.12)

where A, and A2 are the amplitudes measured for the specimens of thickness L , and L 2 . Greater accuracy is, of course, obtained if more than two thicknesses are used. Reasonably good agreement between the values calculated from Eqs. (12.1.10) and (12.1.12)have been found. Shear absorption can be calculated using Eqs. (12.I . 10) and (12. I . 12) by substituting x, Eq. (l2.1.5), for L in these equations. For PMMA at 2 MHz and room temperature, a, = 4.3 dB/cm. Using the simple method discussed above, an accuracy of 1% or better in sound speed can be obtained. With the addition of a time-mark generator, an accuracy of 0.1% is achievable. Absorption measurements using this simple method are more difficult and are accurate to about 10%. SUMMARY. As with most ultrasonic methods, the immersion measurement technique is based on measuring differences between two acoustic paths. Longitudinal speed is found from the difference in transit times with and without the specimen held perpendicular to the sound beam. Shear speed is found similarly after rotating the transducers to the critical angle, where the longitudinal wave is totally internally reflected. Longitudinal and shear absorptions are calculated from the signal amplitude received with and without the specimen with corrections for reflection and absorption in the liquid. An alternative method is to use two specimens of different thickness. Reflection should be the same in both cases and need not be calculated. 12.1.2.3. Experimental Results. Some representative experimental results are given in this section. Most of these results were obtained with

72

12. FURTHER MECHANICAL TECHNIQUES

TABLEI. Sound Speeds for Various Polymers‘

Polymer Phenolicb Polyepoxide Poly(hexamethy1ene adipamide) Polycaprolactam Poly(methy1 methacrylate) Polypropylene Poly@henyl quinoxaline? Polyoxymethylene Polyethylene, high density Polystyrene Poly(viny1butyral) Poly(ethy1ene oxide) Poly (acrylonitrile-co-butadiene-co-styrene) Polyurethane, polyether based’ Poly(isobuty1ene-co-isoprene),filled Polyethylene, low density Poly(viny1idene fluoride) Polychloroprene, filled Polyurethane, polybutadiene basedC I ,2-Polybutadiene, filled cis-l,4-Polyisoprene, filled Poly(carborane siloxane) Polytetrafluoroethylene Poly(dimethy1siloxane)

Density @/cma) 1.22 I .205 1.147 1. I46 1.191 0.913 1.209 1.425 0.957 1.052 1.107 1.208 1.041 1.104 1.13 0.922 1.779 1.42 1.008 1.10

1.12 1.041 2.177 1.045

Longitudinal speed (m/sec)

Shear speed (m/sec)

2840 2820 2710 2700 2690 2650 2460 2440 2430 2400 2350 2250 2160 2130 1990 1970 1930 1730 1660 1570 1550 1450 1380 1020

1320 1230 1120 1120 1340 1300 1130 lo00 950 1150

-

930

-

Measurements made at room temperature, at ambient pressure, and a frequency of 2 MHz. Taken from Hartmann and Jarzynski,’ except as noted. From hart man^'^ Previously unpublished results.

the immersion apparatus described above, but some measurements were made using techniques to be discussed later. Sound speeds at room temperature, ambient pressure, and a frequency of 2 MHz are given in Table I. Longitudinal speeds vary from about 1500 to 3000 m/sec, a range intermediate between that covered by most rnetalsI4(3000-6000 m/sec) and that covered by most liquids14(900- 1500 m/sec). Shear speeds vary from about 700 to 1400 m/sec compared to 1600-3300 m/sec for most metals.14 (Liquids, of course, do not normally propagate shear waves.) In some cases, the absorption was so high that I4 R. C. Weast, ed., “Handbook of Chemistry and Physics,” 56th ed. CRC Press, Cleveland, Ohio, 1975.

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ULTRASONIC MEASUREMENTS

TABLE 11. Absorptions for Various Polymers"

Polymer

Longitudinal absorption (dB/cm)

Shear absorption (dB/cm)

Poly(methy1 methacrylate)b Polyethylene, high densityb Poly@henyl quinoxaline)c Phenolicc Poly(ethy1ene oxide? Polyurethane, polyether basedd Polyurethane, polybutadiene basedd

I .4 3.3 3.5 4.1 7.1 7.5 9.1

4.3 25.0 15.0 19.0 -

-

Measurements made at room temperature, at ambient pressure, and a frequency of 2 MHz. From Hartmann and J a r ~ y n s k i . ~ ~ From Hartmar~n.'~ Previously unpublished results.

shear measurements could not be made. Densities are also given in Table I because the speed in a given polymer varies with density. Due to the usual variations from batch to batch of polymer, the sound speeds given in Table I can easily differ by 1% or more for the same polymer. For this reason, measurements to an accuracy better than 1% are often not warranted. Absorptions at room temperature, ambient pressure, and a frequency of 2 MHz are given in Table 11. Longitudinal absorption varies from 1 to 10 dB/cm while shear absorption ranges from 4 to 25 dB/cm. These values are higher than those observed in metals. The temperature dependence of sound speeds over a narrow temperature range near room temperature, at ambient pressure, and at a frequency of 2 MHz is given in Table 111. Qualitatively, the speeds decrease as the temperature increases. The decrease is usually linear. The magnitude of the decrease is greater than that in Eq. (12.1.2) for aluminum, a typical metal, and generally greater than that for 1iq~ids.I~ Over a broad temperature range, the ultrasonic properties of polymers typi~ally'~ exhibit the behavior shown in Fig. 7. As can be seen there, the sound speed always decreases as the temperature increases, but in a certain temperature range it decreases much more rapidly than above or below this range. Also in this range the absorption goes through a peak. Thus, in making measurements over a narrow temperature range, the absorption might increase, decrease, or remain approximately constant as a Is

B. Hartmann and J. Jarzynski, J . Polym. Sci., Part A-2 9, 763 (1971).

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74

FURTHER MECHANICAL TECHNIQUES

TABLE111. Temperature Dependence of Sound Speedsa Polymer Polystyreneb Poly(methy1 methacrylate)c Poly@henyl quinoxa1ine)d Pol y (acrylonitrile-ro-butadiene-cu-st yrene)c Phenolicd Polyethylene, high densityC Polypropylenec a

-4.4 -2.0 -1.3 -0.7 -4.0 -6.8 -6.7

-1.5 -2.5 -3.0 -3.5 -7. I -9.6 - 15.0

Measurements near room temperature, at ambient pressure, and a frequency from 2 to

6 MHz.

From Lamberson el From Hartmann and Jarzynski.' * From Hart~nann.'~

function of temperature. Because of the strong temperature dependence in this region, close temperature control is important. The frequency dependence of sound speed in polymers has been determined in several cases.16-18 Qualitatively, the sound speeds increase as the frequency increases. Over a frequency range of a factor of 10, the

1000

'

-60

I

- 40

-20 Temperature

0 ("C)

20

40

FIG.7. Ultrasonic properties of a poly(carborane siloxane) at 0.6 MHz. 1E

I7

J. R. Asay, D. L. Lamberson, and A. H. Guenther, J . Appl. Phys. 40, 1768 (1%9). H. J. Sutherland and R. Lingle, J. Appl. Phys. 43,4022 (1972). M. P. Felix, J . Compos. Marer. 8 , 275 (1974).

12.1. ULTRASONIC MEASUREMENTS

75

speeds increase only about 1%. In the limited number of cases for which data are available, absorptions usually increase either linearly with frequency or as the square of the f r e q u e n ~ y . ~A~linear , ~ ~ increase is called hysteresis absorption. It can be expressed as a / f = const or, more commonly, a h = const. Note that a h has units of decibels. More exactly, a h has units of decibels per wavelength. Since measurements as a function of frequency are more difficult to make than measurements as a function of temperature, they are less common. Over a very broad frequency range, the ultrasonic properties of a polymer should mimic the temperature dependence shown in Fig. 7, but with frequency increasing from right to left. Plots of this type are not available, however, due to the experimental difficulty in making measurements over the required range, perhaps five decades of frequency. The pressure dependence of the sound speed has been determined in a number of cases.3,16,1e-23Qualitatively, the speed increases as the pressure increases. For PMMA, the rate of change of the longitudinal sound speed was foundlg to be 0.3 m/sec-bar. In the two cases331 where absorption measurements were made as a function of pressure, the qualitative response was a reduction in absorption as pressure increased. S U M M A R Y . Ultrasonic sound speeds in polymers are intermediate between those in metals and liquids. Absorptions are high. Sound speeds increase with decreasing temperature, increasing frequency, and increasing pressure. Over a broad temperature range absorptions go through a temperature peak, and, in the few cases where such measurements have been made, usually increase linearly with increasing frequency or as the square of the frequency, and decrease with increasing pressure. 12.1.3. Other Experimental Techniques

In Section 12.1.2, the immersion technique was described in detail. Several other techniques useful for polymers are described qualitatively in this section. One of these techniques substitutes solid rods for the immersion liquid, while the other techniques both make use of multiple echoes in the specimen. 12.1.3.1. Delay-Rod Techniques. In the delay-rod apparatus,24 the acoustic path of liquid between the transducers and the specimen in the l8 2o

24

J . Gielessen and J . Koppelmann, Kolloicl-Z. 172, 162 (1960). H. Singh and A. W. Nolle, J . Appl. Phys. 30, 337 (1959). J . E. McKinney, H . V. Belcher. and R. S. Marvin, Trons. Soc. Rheol. 4, 347 (1960). D. L. Lamberson, J. R. Asay, and A . H. Guenther, J . Appl. Phys. 43, 976 (1972). Y. Wada, A. Stani, T. Nishi. and S. Nagai, J . Po/ym. Sci., Parf A-2 7, 201 (1%9). A. W. Nolle and P. W. Sieck, J . Appl. Phys. 23, 888 (1952).

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Tronsmitting Tronsducer

FIG.8. Schematic drawing of a delay-rod apparatus.

immersion apparatus is replaced with solid rods of quartz or metal, as shown in Fig. 8. The purpose of the delay rods (also called buffer rods) is to give a sufficiently long time separation between ultrasonic pulses that the complete pulse can be sent by the transmitting crystal before the beginning of the pulse arrives at the receiving crystal. This technique eliminates stray rf pickup by the receiving crystal. Since there is no liquid to freeze, this technique has been ~ s e ddown ~ ~ * ~ ~ to liquid helium temperature, 4.2 K. As this temperature is approached, the speed levels off and becomes temperature independent. Maximum speed values depend on chemical structure but can be as high as uI = 3700 m/sec and us = 2000 m/sec. There are two major problems introduced with the delay-rod technique. First, there must be a good bond between the transducers and the delay rods and also between the delay rods and the specimen. Second, shear measurements cannot be made by rotating the transducers used for longitudinal measurements but require separate transducers. Various bonding agents have been used. For longitudinal waves, silicone liquid, silicone stopcock grease, and glycerin are all acceptable. For shear waves, which are more highly damped, the problem is more critical. A low-molecular-weight poly(a-methyl styrene) liquid and mixtures of phthalic anhydride and glycerin have both been found to be effective. All of these bonding agents have the advantage of being relatively easy to remove so that the transducer can be recovered intact. An epoxy bond can give good coupling but is difficult to remove. Longitudinal and shear measurements are made separately with different sets of transducers for each mode. Quartz transducers are often used because it is relatively easy to excite these crystals into harmonics of their resonant frequency, thus giving a range of measurement frequencies from the same transducers. Depending on how the quartz crystal is cut, it produces different types of vibrations. An X-cut quartz crystal is used for generating longitudinal waves. Y-cut or AC-cut quartz crystals are I . 1. Perepechko and V. E. Sorokin, Sov. Phys. -Acousi. (Engl. Trans/.)18,485 (1973). P. D. Golub and 1. I . Perepechko, Sov. Phys.-Acoust. (Engl. Trans/.) 20, 22 (1974).

*I *O

12.1.

ULTRASONIC MEASUREMENTS

77

used to generate shear waves. While the Y cut has a bigger voltage output for a given mechanical stress, the AC cut is almost pure shear and the parasitic longitudinal wave also generated is 50 dB lower than the shear wave.22 In addition to single-crystal quartz transducers, transducers of polycrystalline ceramics that have been polarized in a strong electric field as they cool down from above their Curie point are commonly used. The two most common examples are barium titanate and lead zirconate titanate. Speed and absorption measurements can be made in the same manner as in the immersion technique or by using a null method.2s In this method the pulse input to the transducer is split in two. The pulse that has traversed the specimen is then mixed with an out-of-phase signal directly from the pulsed oscillator. By properly delaying and attenuating the direct signal, a null is obtained. Using this technique an accuracy of 0.5% in speed can be obtained. 12.1.3.2. Multiple-Echo Techniques. In multiple-echo techniques, pulses of sound that traverse the specimen are partially reflected at the face of the specimen and bounce back and forth with continually diminished amplitude. From the time it takes for the echo to travel from one face to the other and the reduction in its amplitude during this trip, the speed and absorption can be calculated. Since the pulse must make several trips through the specimen, this technique cannot be used for highly absorbing materials. On the other hand, the technique is very accurate. In one version of this technique2' that has been successfully applied to polymers,16two transducers are bonded directly onto the specimen. Two independently controlled pulses are applied to the transmitting transducer. The first applied pulse generates a series of signals at the receiving transducer corresponding to the directly transmitted signal and its subsequent echoes. The second applied pulse generates a similar set of signals. The envelope of each of these signals is essentially flat-topped, as indicated in Fig. 9. The second applied pulse is delayed and attenuated so that the envelope of the directly transmitted signal coincides with that of the first echo from the first pulse. By adjusting the frequency of the measurement, the coincident signals can be made to destructively interfere and cancel each other out. The required frequency depends on the transit time through the specimenls and allows accurate determination of the transit time. As shown in Fig. 9, complete cancellation of the two signals is seldom achieved. That is, generally there are residues left at the beginning and end of the cancelled peaks in the superimposed output. These are caused by dispersion in the specimen and because of dif-

*'

J. Williams and J. Lamb, J. Acousr. SOC. Am. 30, 308 (1958).

78

12.

FURTHER MECHANICAL TECHNIQUES

Transmitting Transducer,

Specimen

Receiving Transducer

I-

Pulse I Summed Outpul Pulse 2

(a) Schematic Drawing

-l-LhLL (b) Received Echoes Due To Pulse I

(c) Received Echoes Due To Pulse 2

n

(d) Received Signal Due To Summing b B. c When They Are Out Of Phase

FIG.9. Multiple-echo technique utilizing two transducers.

ferences in the pulse widths of the two applied signals. However, this effect does not influence the velocity calculations, since cancellation is generally obtained for the middle portions of the pulse. Absorption is obtained from the attenuation required to match the amplitude of the direct pulse with the first echo. Sound speed measurements can be made to an accuracy of 0.1% and absorption to 2 1 dB. Another version of the multiple-echo techniquez8that has been used22 to make very accurate speed measurements in polymers makes use of a single transducer bonded directly to the specimen. Pulses are reflected from the opposite face and return as echoes to the front face. By critically adjusting the pulse repetition rate, an echo from a later pulse can be made to overlap a multiple echo from an earlier pulse, as shown in Fig. 10. When the repetition rate is adjusted to obtain an in-phase condition, constructive interference will result in a maximum amplitude in the superimposed signals. A value of sound speed accurate to 0.05% can be obtained by observing several repetition rates that yield an in-phase condition at the transducer resonant frequency as well as some other frequency, say H. J. McSkimin, J . Acoust. SOC. Am. 33, 12 (1961).

12.1.

ULTRASONIC MEASUREMENTS

79

Tronsducer Pulse I Pulse 2 Summed Output

(01 Schemotic Drowing

m

(b) Received Echoes Due To Pulse I

n

(c) Received Echoes Due To Pulse 2

2 (d) Received Echoes Due To Pulse 3

(el Received Signal Due To Summing b , c , 8 d When They Are In Phase

FIG. 10. Multiple-echo technique utilizing one transducer.

10% below resonance, and repeating the measurements on other specimens that differ only in thickness. S U M M A R Y . Delay-rod and multiple-echo techniques can be used to measure speeds with an accuracy of up to 0.05% by obtaining either constructive or destructive interference between pulses. These techniques can also be used all the way down to liquid helium temperature. The measurement procedures are, however, more involved than those used in the immersion technique and require more elaborate electronics and different transducers for longitudinal and shear measurements. 12.1.4. Molecular Interpretation

In this section, sound speeds are used to calculate elastic constants, the equation of state is discussed, and ultrasonic measurements are related directly to the molecular structure of the polymer. 12.1.4.1. Elastic Constants. One of the purposes of measuring sound speeds is to be able to calculate elastic constants. Longitudinal and shear

80

12.

FURTHER MECHANICAL TECHNIQUES

sound speeds are related to elastic constants by the relationsz0 u1 = [ ( K

+ 4G/3)/plln,

(12.1.13) ( 1 2.1.14)

where K is the bulk modulus (equal to the reciprocal of the compressibility), G the shear modulus, and p the density. For an isotropic solid, there are only two independent elastic constants. These two can be taken to be K and G, but it is sometimes convenient to use other elastic constants, such as Young's modulus E and Poisson's ratio u. These constants can be calculated from K and G using the standard relations E

=

3G/(1

(T

=

4

-

+ G/3K),

E/6K.

(12.1.15) (12.1.16)

When the lateral dimensions of the specimen are much greater than the wavelength, a longitudinal wave is propagated, as in Eq. (12.1.13). When the lateral dimensions are much less than the wavelength, an extensional wave is propagated, where an extensional wave can be defined as a mode of propagation for which the sound speed is (12.1.17) This type of propagation is most often encountered in the lowerfrequency end of the ultrasonic spectrum. Alternative definitions of elastic constants are occasionally The Lame constants A and p are related to K and G in the following manner: p = G,

(12.1.18)

A = K - 3G.

(12.1.19)

Other elastic constants are defined starting from the generalized Hooke's law (linear stress-strain relation), (12.1.20) where i,j = 1 , . . . , 6 and the ctjare the elastic stiffness constants. In the most anisotropic material, there are 2 1 independent elastic stiffness constants or elastic moduli. As the structure of the solid becomes more symmetric, the number of independent constants reduces. For a cubic crystal structure, there are three independent constants and for an isoJ . Blitz, "Fundamentals of Ultrasonics," 2nd ed., p. 150. Plenum, New York, 1%7. W. P. Mason, "Physical Acoustics and the Properties of Solids," p. 357. Van Nostrand-Reinhold, Princeton, New Jersey, 1958. *@

30

12.1. ULTRASONIC MEASUREMENTS

81

tropic solid there are only two, ell and c12. The 6 x 6 matrix for cYjis given by ~ 1 = 1 ~ 2 = 3 ~ 3 = 3 h + 2~ = K + gG, (12.1.21) ~ 1 = 2 ~ 1 = 3 ~ 2 = 3 cgl

C44

=

C55

=

c66

=

(C11

=

~ 3 = 1 ~ 3 = 2

- C12)/2 =

/A

A = K - $G, =

(12.1.22) (12.1.23)

G,

all other terms being zero. For all those polymers listed in Table I for which both longitudinal and shear sound speeds are given, the elastic constants have been calculated at room temperature and pressure and 2 MHz and are given in Table IV. The moduli values are approximately one order of magnitude lower than those for metals. For example, using literature1*values for aluminum of uI = 6420 m/sec, us = 3040 m/sec, and p = 2.7 g/cm3, the bulk modulus is 7.81 x 10" dynes/cm2 and the shear modulus is 2.49 x 10" dynes/cm2. On the other hand, the range of Poisson's ratios for polymers tends t o be slightly higher than, but overlapping with, the range for metals. For aluminum, u = 0.35. By making speed measurements at different temperatures, the elastic constants can be calculated as a function of temperature. Illustrative results for sound speeds in polypropylene' are shown in Fig. 11. (The TABLEIV. Elastic Constants for Various Polymersa Bulk modulus

Shear modulus

Young's modulus

(10'0

(10'0

( 10'0

Polymer

dynes/cmz)

dynes/cm*)

dynes/cmz)

Poisson's ratio

Phenolicb Pol yepoxide Poly(hexamethy1ene adipamide) Polycaprolac tam Poly(methy1 methacrylate) Polypropylene Poly(pheny1 quinoxalinep Polyox ymethylene Polyethylene, high density Polystyrene Poly (acrylonitrile-cobutadiene-co-styrene) Poly(viny1idene fluoride)

7.02 7.13 6.53 6.45 6.49 4.37 5.21 6.59 4.54 4.21 3.64

2.13 1.83 1.43 1.43 2.33 1.54 1.54 1.43 0.91 1.39 0.90

5.79 5.05 3.99 4.00 6.24 4.13 4.20 4.01 2.55 3.76 2.49

0.36 0.38 0.40 0.40 0.34 0.34 0.37 0.40 0.41 0.35 0.39

5.18

I .07

.3.00

0.40

Measurements made at room temperature, at ambient pressure, and a frequency of 2 MHz. Taken from Hartmann and Jarzynski,' except as noted.

From Hartmann."

12.

82

FURTHER MECHANICAL TECHNIQUES

2000.

2700 ; i

-

2600 -

In 0

-2

2500

-

U

2400c

ul D

5

1300

-

1200

-

0

ul

I100

15

20

25

Temperature

30

35

PC)

FIG. 1 I . Sound speeds vs. temperature at 2 MHz for polypropylene.

slopes of such graphs are given in Table 111.) Calculated elastic constants for polypropylene are given in Fig. 12. Accurate measurements of elastic constants as a function of temperature require a knowledge of the thermal expansion coefficient,

There are two reasons why p is needed. First, in calculating sound speed, the specimen thickness depends on temperature. Second, in cal-

0.38

50 0.34 5 I 0.36

Poisson's Rotio

In

2

1

12.1. ULTRASONIC MEASUREMENTS

83

culating elastic constants, the density depends on temperature. Note that from Eq. (12.1.13) or (12.1.14), a density decrease produces a speed increase. Since it is experimentally observed that speeds decrease as temperature increases, it is clear that the modulus decrease with temperature is more significant than the density decrease. In comparing elastic constants measured ultrasonically with those obtained in a "static" (very low frequency) measurement, note that ultrasonic values are adiabatic while static values are isothermal. The two types of measurements are related3I by the standard thermodynamic equation (12.1.24)

where K s and KT are the adiabatic and isothermal bulk moduli and C, and Cy the specific heats at constant pressure and volume. For the shear modulus3* Gs = GT.

(12.1.25)

(This equation follows from the fact that shear deformation occurs at constant volume.) The magnitude of K s / K T is about 1.1 for polymers and increases as the temperature is raised above the glass transition temperat ~ r e . ~For * PMMA at room temperature, K s / K T = 1.07. Another reason why ultrasonically measured elastic constants of polymers differ from those measured in other ways is that the moduli of viscoelastic materials depend on frequency. Physically, the amount of deformation that is produced in a polymer by a given stress depends on the length of time that the stress is applied. During the short time that the stress of a sound wave is applied in one direction, only relatively small portions of the polymer can move; hence not as much strain is induced as in a typical static measurement, and the ultrasonic modulus is higher than the static modulus. This effect is not too pronounced for the bulk modulus (on the order of 20%) but can be significant for shear and Young's modulus (a factor of 10 or more).3334 Because of the above dependence on frequency, ultrasonic waves represent a mechanical probe for particular molecular motions, namely, those that can occur in the period of the sound wave. Viewed as one technique for making mechanical measurements on polymers, ultrasonic J. D. Ferry, "Viscoelastic Properties of Polymers," 2nd ed. Wiley. New York, 1970. R . W. Warfield. D. J . Pastine, and M. C. Petree, Appl. Phys. Leu. 25, 638 (1974). 33 D. J . Pastine, J . Chem. Phys. 49, 3012 (1968). 34 J. Schuyer, J . Po/ym. Sci. 36, 475 (1959). 31

32

84

12. FURTHER MECHANICAL TECHNIQUES

measurements probe motions of small parts of the polymer while torsional pendulum measurements probe larger-scale motions and creep measurements very large-scale motions. (See Chapter 1 1 , this volume.) As is well known for viscoelastic material^,^^ the frequency and temperature dependencies of polymer properties are related. From the ultrasonic point of view, the frequency dependence of polymer properties results from a distribution of relaxation times in the material. The relaxation times in turn depend on temperature. Thus, measurements made at high frequency are equivalent to those made at low temperature and vice versa. This behavior is described by an Arrhenius-type behavior below the glass transition and by the Williams -Landel-Ferry (WLF) equation3I above the glass transition. Strain dependence should also be considered when comparing ultrasonically measured elastic constants with statically measured values. As an example,35for polyethylene at room temperature, the modulus is independent of strain up to a strain of about Beyond this point, the modulus decreases as the strain increases. Typically,36 ultrasonic measurements are made in the strain range from 10" to lo+, where the moduli are strain independent, but static measurements may exceed the linear strain limit. SUMMARY. Polymer moduli can be calculated from the ultrasonic speeds and are an order of magnitude lower than metal moduli. Polymer moduli decrease as the temperature is raised, and this decrease is more significant in determining sound speed than is the density decrease. U1trasonic moduli differ from static moduli not only because they are adiabatic rather than isothermal but also because polymer moduli are frequency and strain dependent. 12.1.4.2. Equation of State. An equation of state, in this context, is a relation between p, V, and Tin a polymer, or a relation between any two of these variables holding the third constant. Frequently, these equations involve the bulk modulus and perhaps its temperature or pressure dependence, so that this section is a special topic related to the previous section since ultrasonic measurements are often used to determine the bulk modulus. Given the modulus and an equation of state, one can correlate much experimental data and also extrapolate to high pressure where measurements are difficult to make. From the second law of thermodynamics, dE=TdS-pdV,

( 12.1.26)

35 L. E. Nielsen, "Mechanical Properties of Polymers," p. 64. Van Nostrand-Reinhold, Princeton. New Jersey, 1962. B . Hartmann and J. Jarzynski, J . Appl. Phys. 43, 4304 (1972).

12.1.

ULTRASONIC MEASUREMENTS

85

and the definition of the bulk modulus, (1 2.1.27)

it follows that (1 2.1.28)

In this form, the bulk modulus is seen to depend on an energy (E) term and an entropy (S)term. For polymers, the second term is usually neglibible compared to the first3' so (1 2.1.29)

This relation is contrasted to that for ideal rubber elasticity, where the energy term is small compared with the entropy term.38 According to Eq. (12.1.29), KT depends on the volume dependence of the intermolecular energy. Making reasonable assumptions about the form of the intermolecular potential, bulk modulus can be calculated as an analytic function of volume3' and these results are in fairly good agreement with experiment, verifying our assumption that the intermolecular potential is the primary factor determining speed. The above calculation was done taking the special nature of polymers into account. The bulk modulus along the (covalently bonded) chain of a polymer is much higher than the modulus between the (van der Waals bonded) chains. Therefore, compression occurs primarily in the two dimensions perpendicular to the chain and very little along the chain. One of the most successful equations of state for polymers is the Gruneisen equation of state,33 in which the key quantity is the Gruneisen parameter y, which can be expressed in several ways. One expression for y is 1dlnK Y"TdlnV'

(1 2.1.30)

Experimentally, the volume of a solid may be changed by changing either the pressure or the temperature. In the first case, Eq. (12.1.30) can be rewritten as (12.1.31) 37

M. C. Broadhurst and F. 1. Mopsik, J . Chem. Phys. 52, 3634 (1970). A. V. Tobolsky, "Properties and Structure of Polymers," p. 20. Wiley, New York,

1960.

86

12.

FURTHER MECHANICAL TECHNIQUES

In the second case, Eq. (12.1.30) can be rewritten as 1

=

zp

(71. aInK

( 1 2.1.32)

For polymers, Eqs. (12.1.31) and (12.1.32) give approximately the same value.39 An alternative expres~ion'~ for y is Y = PKTV/C".

(12.1.33)

Values calculated from Eq. (12.1.33) are lower than those calculated from ~ Eq. (12.1.31) or (12.1.32) by a factor of about 5 . For e ~ a m p l e , 3for PMMA using Eq. (12.1.31) or (12.1.32), y = 6 while y = 0.8 using Eq. (12.1.33). This difference occurs the specific heat in Eq. (12.1.33) sums up all possible vibrations in the polymer, while in Eq. (12.1.30), only the interchain vibrations contribute. A comparison of Eqs. (12.1.33) and (12.1.30) gives a way of separating the interchain contributions to the specific heat from the intrachain contributions. For PMMA, interchain vibrations contribute about 13% of the total specific heat. Thus, of the total specific heat of 0.266 cal/g deg, interchain vibrations contribute 0.043 cal/g deg. In addition to the above relation between the Gruneisen parameter and bulk modulus, it has been that there is a relation between the Gruneisen parameter and ultrasonic absorption: the higher y is, the higher the absorption will be. S U M M A R Y . Sound speed measurements can be used to determine the Gruneisen parameter for use in equation of state calculations and also for determining the ratio of interchain to intrachain vibrations. The Gruneisen parameter is also related to absorption. 12.1.4.3. Direct Relations. A number of empirical relations have been observed between longitudinal sound speed and such polymer properties as crystallinity 34*41*42 degree of cr~sslinking,'~ and plasticizer loading. It is important to keep these relations in mind when comparing measurements made by different workers on presumably the same polymer. In addition, it is possible to use these relations to measure or at least monitor any of the above polymer properties. A molecular interpretation of such relations is based on the assumption B. Hartrnann, Acusrica 36, 24 (1976). J. C. Slater, "Introduction to Chemical Physics," p. 219. McGraw-Hill, New York, 1939. 'I P. D. Davidse, H. I. Waterman, and J. B. Westerdijk, J . Polym. Sci. 59, 389 (1962). A. Levene, W. J . Pullen, and J. Roberts, J . Polym. Sci.. Parr A 3, 697 (1965). Is B. Hartrnann, J . Appl. Polym. Sci. 19, 3241 (1975). 40

12.1.

ULTRASONIC MEASUREMENTS

87

that the dominant factor governing the sound speed of a given polymer is the specific volume of the polymer. This volume dependence arises in the following manner. As the specific volume of a polymer increases, the average separation distance of the molecules gets larger. The forces between the molecules depend on the separation between them and are most conveniently described in terms of an intermolecular potential that depends only on the separation distance. As the average separation increases, the intermolecular forces get weaker so that the modulus decreases and the sound speed decreases. The qualitative description is quantified in Eq. (1 2.1.29). It is worth pointing out that the only assumption made here is that sound speed is governed primarily by specific volume because the intermolecular potential depends on molecular separation and hence volume. It is not necessary to use the concept of free volume, as has been suggested.44 It is not contradictory to do so, but it is not required. There are some immediate consequences of the above assumption. Since sound speed depends on specific volume, anything that changes the specific volume should change the sound speed. Thus, increasing the temperature will increase the volume and decrease the sound speed. Similarly, increasing the pressure will decrease the volume and increase the sound speed. This is the qualitative explanation for the temperature and pressure dependence outlined in Section 12.1.2.3. Volume dependence also explains the relations between sound speed and polymer properties. As the degree of crystallinity of a given polymer increases, its specific volume decreases (with rare exceptions) and its sound speed increases. Likewise, cross linking decreases specific volume and increases sound speed. Plasticization increases specific volume and decreases sound speed. The above comments are believed to hold generally for variations in a given polymer. There have been attempts to plot different polymers on the same sound speed vs. density graph,44but the results are rather scattered. This is also evidenced in Table I. Higher densities tend to be correlated with higher sound speeds, but there are numerous exceptions. For sound absorption, the relation to volume is not as clear. However, it has been pointed O U ~ that ~ absorption ~ B ~ ~ also depends on volume, with the absorption decreasing as the volume increases. This fact has been used to follow the curing of several polymer ~ y s t e m s . ~ ~ * ~ ~ , ~ ~ Is I6

Y. Wada and K. Yamamoto, J . Phys. Soc. Jpn. 11, 887 (1956). R. Kono and H . Yoshizaki, J . Appl. Phys. 47, 531 (1976). A. G. H. Dietz, E. A. Houser, F. J . McGarry, and G . A. Sofer, Ind. Eng. Chem. 48,75

(1956). "

E. P. Papadakis, J . Appl. Phys. 45, 1218 (1974).

88

12.

FURTHER MECHANICAL TECHNIQUES

By itself, the above description df ultrasonic properties is incomplete. This can generally be seen when making measurements over a broad range of temperature as in Fig. 7. As can be seen there, the ultrasonic properties are very different at the lower temperatures than at the higher temperatures and make the transition between the two sets of properties over a fairly wide temperature range. In this case, the particular transition exhibited is the glass transition of the polymer and is the transition most often studied in polymers. Transition behavior can be explained in the following manner. There are two contributions to the sound speed: molecular separation and structural rearrangement. The molecular separation contribution was discussed above and gives rise to an approximately linear temperature dependence as shown in Fig. 7. Structural rearrangements in a polymer take a finite time, called the relaxation time ‘T, to occur and therefore this contribution to the speed is frequency dependent. But since T depends on the temperature of the polymer, the sound speed also has a temperature dependence due to structural rearrangement. This is the source of the rapid drop in sound speed on Fig. 7 between -30 and 0°C. There are similarly two contributions to the absorption. The peak in absorption occurs when W‘T

=

1,

(12.1.34)

where w = 2rfandfis the measurement frequency. Therefore, the absorption peak for the glass transition occurs at a temperature higher than the glass transition temperature Tg by an amount that varies with frequency. It is generally assumed that the temperature dependence of the relaxation time is given by an Arrhenius relation ‘T

=

T~

exp(AH/RT),

(12.1.35)

where ‘ T ~is a constant and AH the activation energy for the structural rearrangement. Using Eqs. (12.1.34) and (12.1.35) the activation energy can be determined by measuring the temperature of the absorption peak at several frequencies. The slope of a plot of logfvs. 1/T (using the temperature of the absorption peak) yields the activation energy. By making measurements similar to those in Fig. 7 at different frequencies, the activation energy for the glass transition in the polycarborane siloxane polymer was found15to be 20 kcal/mole. This would be exact if the glass transition of a polymer was a structural rearrangement involving a single relaxation time. Since, however, it is known31that theglass transition involves a broad distribution of relaxation times the activation energy determined above is an average value. In studying polymer transitions, it has been found very convenient to

12.1. ULTRASONIC MEASUREMENTS

89

correlate ultrasonic measurements with lower frequency measurements using Arrhenius plots (log f vs. 1 Valuable collections of such data are a ~ a i l a b l e . ~ * * ~ ~ In addition to the ultrasonic measurements at the glass transition, there have been some measurements at the melting point of polymer^.^^*^^*^^ As temperature is increased at constant frequency, there is a decrease in the sound speed and a peak in absorption. The frequency dependence has not been determined, however. S U M M A R Y . The ultrasonic properties of a given polymer depend primarily on the intermolecular potential, which is a function of intermolecular separation and hence volume. Thus, changes in ultrasonic properties can be used to follow changes in any physical property that affects volume, such as crystallinity, cross linking, and plasticization. In a transition region, structural rearrangements occur. From the temperature and frequency dependence of the properties in this region, the activation energy for the rearrangement can be calculated.

/n.

12.1.5. Conclusions

Using the particular immersion apparatus described here, both longitudinal and shear sound speeds and absorptions can be measured from 0.1 to 10 MHz and from - 50 to 150°C, at ambient pressure. Measurements accurate to within 1% in sound speed and 10% in absorption can be made very easily. More accurate measurements and other frequency, temperature, and pressure ranges are possible using other equipment. Polymer sound speeds are intermediate between those in metals and liquids. Absorptions are high. Sound speeds increase with decreasing temperature, increasing frequency, and increasing pressure. Over a broad temperature range, absorptions generally go through a peak. There are limited absorption data available vs. frequency, but the absorption usually increases linearly with frequency, or as the square of the frequency, and decreases with increasing pressure. From the measured sound speeds, one can calculate the elastic moduli of the polymer. These moduli are an order of magnitude lower than metal moduli. Polymer moduli decrease as the temperature is raised and this decrease is more significant in determining sound speed than is the density decrease. The bulk modulus determined ultrasonically can be used Y. Wada, J . Phys. SOC.J p n . 16, 1226 (1961). N. G. McCrum, B. E. Read, and G . Williams, “Anelastic and Dielectric Effects in Polymeric Solids.’’ Wiley, New York, 1967. R. K . Eby, J . Acousr. SOC.Am. 36, 1485 (1964). m N . A. Bordelius and V. K. Semenchenko, Sov. Phys. -Acoust. (Engl. Trans!.) 16,519 48

49

( 197 1 ).

90

12.

FURTHER MECHANICAL TECHNIQUES

in an equation of state to extrapolate to high pressure. A convenient equation of state makes use of the Gruneisen parameter, which is also useful in determining the ratio of interchain to intrachain vibrations. The ultrasonic properties of a given polymer depend primarily on volume. Thus, changes in ultrasonic properties can be related to changes in any polymer property that affects volume, including crystallinity, cross linking, and plasticization. In a transition region, structural rearrangements occur. The activation energy for the rearrangement can be calculated from ultrasonic measurements in this region.

12.2 Static High-pressure Measurements on Polymers?

By R. W. Warfield 12.2.1. Introduction In the past two decades the use of static high-pressure measurements as a tool for studying the structure, morphology, and properties of bulk polymers has led to a considerable advance in the understanding of the solid-state physics of these synthetic materials. Whereas Bridgman' studied the compressibility of only a few polymers at room temperature, more recent work2 has shown that pressure is a pertinent variable when studying polymers over broad pressure and temperature ranges. Recent investigations have greatly increased the pressure and temperature ranges, and data on a relatively large number of polymers are available. Among the noteworthy investigations are those of Weir and co-workers3 at the National Bureau of Standards, Wunderlich: Matsuoka and Maxell,^.^ Simha and c o - ~ o r k e r sSauer, ,~ Pae and co-workers,*BSBaer and co-workers,*Om and Pistorius. There appear to be almost as many experimental devices as there are researchers, due perhaps to each worker developing a device to best suit t Work supported by the Naval Surface Weapons Center Independent Research Fund.

' P. W. Bridgman, Proc. Am. Acad. Arrs Sci. 76, No. 3,71 (1948);76, No. 1, 9 (1945). J . M. O'Reilly. in "Modem Aspects of the Vitreous State" (J. D. MacKenzie, ed.,) Vol. 3. Butterworth, London, 1964. C. E. Weir.J. Res. Narl. Bur. Stand. 46,207 (1951);50,95, 153,311. and 321 (1953);53, 245 ( 1954). ' B. Wunderlich and T. Arakawa, J . Polym. Sci., Part A 2, 3697 (1964). B. Maxwell and S. Matsuoka, J . Polym. Sci. 32, 131 (1958). S. Matsuoka, J . Polym. Sci. 42, 51 1 (1960);57, 569 (1962). ' A. Quach and R. Simha,J. Appl. Phys. 42,4592 (1971);Macromolecules 4, 268 (1971). J. A. Sauer, D.R. Mears, and K. D. Pae, Eur. Polym. J . 6, I015 (1970). J . A. Sauer, and K. D. Pae, Colloid Polym. Sci. 252, 680 (1974). lo J. L. Kardos, and E. Baer, J . Polym. Sci., Part A 2827 (1%5). l o p E. Baer and J . L. Kardos, J . Polym. Sci., Part A 3, 2827 (1%5). l 1 W.W. Doll, and J . B. Lando, J . Macromol. Sci. Phys. 2, 219 (I%@. I2 C. W. F. T. Pistorius, Polymer 5, 315 (1964). 91 METHODS OF EXPERIMENTAL PHYSICS, VoL.

16c

Copyright @ 1980 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-475958-0

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FURTHER MECHANICAL TECHNIQUES

his or her own research interests. While many specialized types exist, most of the devices used by polymer scientists can be grouped in the following categories: piston-cylinder a p p a r a t ~ s ,various ~ types of dilat o m e r ~ , ’ ~“belt” J ~ ~ apparatus,14 and the recently developed diamond anvil.l5 Many variations exist on each of these basic types. Fortunately, the pressure range of greatest interest to the polymer scientist is between atmospheric pressure and about 10 kbar and for measurements in this range experimental techniques may be employed involving the direct measurement of the applied force and the area over which it acts. Also, the temperature to which most polymers can be heated without degradation is approximately 250°C. Thus, both the pressure and temperature ranges ofgreatest interest are accessible to devices that, for the most part, are relatively easy to construct, use, and maintain. At pressures above 10 kbar the devices in use until recently were usually massive, complex, difficult to use and maintain, and expensive. This set a limitation on any study of the effects of very high pressure on polymers. However, recent advances in the technology of high-pressure generation and its measurement have opened up new areas to research. In particular, the development of the diamond anvil high-pressure cell1s has been noteworthy. Presented in this chapter are experimental procedures for determining the static response of bulk polymers to pressure under a variety of experimental conditions. Examples of the various types of data that are usually obtained are drawn from experimental results employing a pistoncylinder device. 12.2.2. Types of Equipment

The measurement of the response of bulk polymers to high static pressure presents many problems not usually encountered in high-pressure studies. Perhaps the most important factor that must be considered by the investigator is the time-dependent nature of the relaxation process in polymers. This time dependence, in effect, prevents “instantaneous” measurements on polymers. Rather, one must carefully choose a rate of pressure loading so as to correspond to an equilibrium condition. This is often not easily done, particularly in the case of cross-linked polymers. We have noted several criteria by which one can judge if the experimental data are in at least approximate equilibrium.l6 Ideally one should meaG . M. Martin, and L. Mandlekern, J . Appl. Phys. 34, 2312 (1963). J. E. McKinney, and R. W. Penn, Rev. Sci. Instrum. 43, 121 1 (1972). l4 H. T. Hall, Rev. Sci. Instrum. 31, 125 (1960). C. E. Weir, E. R. Lippincott, A. Van Valkenburg, and E. W. Bonting, J . Res. Natl. Bur. Stand.. Sect. A 63, 55 (1959). R . W . Warfield, J. E. Cuevas, and F. R. Barnet, Rheol. Acra 9,439 (1970). ls

Isa

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STATIC HIGH-PRESSURE MEASUREMENTS ON POLYMERS

93

sure a stress relaxation modulus by first isothermally loading a specimen so as to produce a certain volume change and then measuring the pressure as a function of time. However, in practice, provided a very slow rate of loading is employed, one can obtain a reasonable approximation of an equilibrium measurement. Another problem is estimating thermal equilibrium. When one is conducting a series of isothermal experiments sufficient time must be allowed for the pressure device and enclosed polymer specimen to come to thermal equilibrium. The device itself usually has a high heat capacity so that heating with electrical resistance bands is easily accomplished. Once the desired temperature has been obtained, as indicated by the thermocouples embedded in the device near the polymer specimen, one should wait at least one hour before obtaining the next isotherm to ensure thermal equilibrium. Ideally one should have a device in which a pure hydrostatic environment can be established. Unfortunately, such apparatus is complex and expensive. Also, polymers do not lend themselves to the usual hydrostatic measurement. Matsushige et al." have recently shown that polymers interact physically with many of the common pressuretransmitting fluids to produce swelling, stress-crazing, and cracking of the specimen. Solids such as pyrophyllite, talc, silver chloride, and boron nitride, which plastically transmit pressure to solid materials, are frequently employed in high-pressure investigations and can be used in contact with most polymers. The choice of just which experimental technique to use depends to a large extent on what type of information is required. For example, if the change in volume with pressure (-AV) up to 10 kbar or the pressure dependence of the isothermal bulk modulus (BT) to 10 kbar is required, one should use a device in which a large sample volume can be employed. A piston-cylinder arrangement could be used to advantage here. On the other hand, if the response of a polymer to very high pressure is of interest, the diamond anvil deviceI5 should be very satisfactory. Also, if one needs to study the effects of pressure by means of various optical or x-ray diffraction techniques the diamond anvil device should be employed. If only relatively low pressures are of interest but the highest possible accuracy is necessary then some form of dilatometer should be employed. Sample size can also dictate the choice of the experimental technique. If only a very small amount of sample is available, again the diamond anvil device should be used, whereas if a few cubic centimeters of sample are available, a piston-cylinder device or conventional dilatometer may K . Matsushige, S. V. Radcliffe, and E. Baer, J . Mater. Sci. 10, 833 (1975); J . Mull, 565 (1975); J . Polym. Sci., Polym. Phys. Ed. 14, 703 (1976).

cromol. Sci.. Phys.

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be employed. Large samples can be used in the belt apparatus“ or in the dilatometer developed by Corsaro and co-workers.lB Often, bulk modulus or change-in-volume measurements at low pressures can be made in a conventional dilatometer. A number of highly specialized devices have been developed for determining the effect of pressure under very specific conditions. For example, Balchan and DrickamerIg developed a high-pressure electrical resistance cell that permits one to measure the electrical resistance of a material to 500 kbar. Wunderlichzohas developed an apparatus to crystallize a polymer under pressure as high as 5 kbar. Uhlmann and coworkersZ1have adapted the Bridgman anvil apparatus to study the densification of polymers under high pressures. A number of other devices have been developed that would appear to be suitable for use with polymers, including those of Kennedy and La Griggs and Kennedy,z3 and Griskey and In the sections that follow, the four principal devices currently most widely used to study the pressure response of polymers are described. 12.2.2.1. Piston-Cylinder Device The difficulties, complexities, and expense arising from the construction, maintenance, and operation of hydrostatic equipment have caused many workers to adopt a simpler type of device. A very widely used type is the piston-cylinder apparatus, which exists with many variations. Here, when studying rigid polymers the pressure is quasihydrostati~,~~ while with rubbery polymers and molten systems the pressure is truly hydrostatic. The piston-cylinder type of experimental high-pressure device was initially developed by Parsonszs and greatly improved by Bridgman.z7 While Bridgman studied the compressibility of a number of polymeric substances, it was not until the investigations of Spencer and Gilmore,2e Weir,3 O’Reilly,29Matsuoka and M a ~ w e l land , ~ Hellwege, Knappe, and Lehmann30that the use of piston-cylinder devices in polymer studies became relatively common. D. Corsaro, J. Jarzynski, and C. M. Davis, J . Appl. Phys. 45, 7 (1974). S. Balchan and H. G . Drickamer. Rev. Sci. Instrum. 32, 308 (1961). Wunderlich, Rev. Sci Insfrum. 32, 1424 (l%l). M. Kimmel and D. R. Uhlmann, J . Appl. Phys. 41, 2917 (1970). a G. C. Kennedy and P. N. La Mori, in “Progress in Very High Pressure Research” (F. P. Bundy, W. R. Hibbard, and H. M. Strong, eds.), p. 304. Wiley, New York, 1961. D. T. Griggs and G . C. Kennedy, Am. J . Sci. 254,722 (1956). z4 G. N . Foster, and R . G.Griskey, J . Sci. Instrum. 41, 759 (1964). *I D. Sardar, S . , V. Radcliffe, and E. Baer, Polyin. Eng. Sci. 8, No. 4, 290 (1968). A. Parsons, Proc. R . Soc. London 44, 320 (1888). 27 P. W. Bridgman, “The physics of High Pressures.” G. Bell, London, 1958. R. S. Spencer, and G . D. Gilmore, J . Appl. Phys. 20,502 (1949); 21, 523 (1950). 2B J . M. O’Reilly, J . Polym. Sci. 57, 429 (1%2). K. H. Hellwege, W. Knappe, and P. Lehmann, Kolloid-Z. & Z . Polym. 183, 110 (1962). R. A. zo B. * I R. Is

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95

Basically the device consists of a hardened steel piston driven by a hydraulic ram into a tightly confined cylindrical bore. The specimen under study is confined in the bore. Ideally, the pressure P on the specimen is given by (12.2.1)

P = F/A,

where P is the pressure, F the force on the piston, and A the crosssectional area. Generally, for solid polymers in a piston-cylinder device, no sealing plug of any type is required if the clearance between cylinder and piston is very small, i.e., about 0.0005 cm. If the polymer is very soft with a very low shear modulus, such as is the case with solid polytetrafluoroethylene or polyethylene oxide, there will be a tendency of some polymer to extrude between the piston and cylinder. When this happens it will bind the piston, thus preventing further measurements. Extrusion will also occur when studying molten polymers. We have employed a slightly modified Matsuoka-Maxwell-type device5 in our studies and this is shown in Fig. 1 . The device consists of two cylinders, which are held together by four bolts and aligned with four guide pins. The polymer specimen also acts as a guide when the two sections are brought together. The inner bushing of both cylinders is made

35cm DIA x 4.127 cm LONG HEA TlNG BAND HEATING BAND-

THERM(X O U P L E THERMOCOUPLE WELL

\

SPECIMEN

-

FIG. 1. Piston-cylinder apparatus.

96

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FURTHER MECHANICAL TECHNIQUES

of hardened steel and the inner surface of the bore is lapped and highly polished to an inside diameter of 0.635 ( + 0.0025, -0.0000) cm. This inner bushing is tightly fitted into an outer bushing, which is made of softer steel. It is necessary to have the device built as two cylinders so as to be able to remove the polymer specimen easily after study. The usual procedure is to remove the bolts and guide pins. It is usually possible then to remove the specimen from the bore of the device by tapping it gently. Heating coils can be placed around the cylinders to heat the apparatus to a predetermined temperature. Once the desired temperature is reached one must wait approximately one hour or longer to ensure thermal equilibrium. Two thermocouple wells are located in the inner bushing near the specimen. In preparation for an experimental run, a polymer specimen 7.62 cm long and 0.635 ( + 0.0000, - 0.0025) cm in diameter is placed in the bore of the device, and the two halves are bolted together. For the usual compressibility experiment the specimen must fit tightly in the bore. However, in a later section, we describe experiments in which this condition is not necessary. Once the specimen is in position, two case-hardened steel plungers 4.128 cm long and 0.635 cm in diameter are inserted in the open ends of the bore above and below the specimen. The device is mounted on its base and the entire assembly placed in a testing machine. The specimen is loaded in compression by pressing down on the steel plungers. By measuring the plunger travel and the load applied, a stress-strain plot is obtained. Strain is defined as A//lo and is assumed to be approximately equal to AV/Vo. The rate of loading chosen must be slow enough to correspond to or closely approximate equilibrium conditions. Experience has indicated that rates of loading corresponding to strain rates of less than 0.008/min approximate equilibrium, whereas more rapid loading does not approximate equilibrium. Ideally, one should conduct an experiment by imposing a stepwire reduction in volume on the specimen and measuring the accompanying pressure as a function of time. The stress-strain plot obtained from the recorder of the testing machine is analyzed depending upon the type of experimental measurement that has been made. When the compressibility characteristics of a new polymer are being studied, usually the first measurement made is a stress vs. strain measurement. An example of this type of plot is given in Fig. 2. If the compressibility is determined as a function of temperature the existence of any transitions, melting, etc., will be noted on the isotherms. However, much of the interest in compressibility experiments is directed toward obtaining data that can be employed in equation-of-state

12.2

STATIC HIGH-PRESSURE MEASUREMENTS ON POLYMERS

97

fn v)

W

L Iv)

FIG.2. Typical stress-strain plot for poly(ethy1ene oxide) above the melting point.

Usually one obtains the volume and/or the bulk modulus as a function of p-essure by analytical treatment of the stress-strain plot. Bulk modulus B T is found by differentiating the stress-strain plot BT =

d6/dc,

(12.2.2)

where 6 is the stress (pressure) and E the volume strain. The derivative is found as follows. At five closely spaced equidistant strains the value of the pressure is recorded. A quadratic curve is fitted to these points using the method of least squares. This introduces some smoothing of the data. The derivative of the quadratic curve so found is then evaluated at the midpoint of the five data points. This gives a value of the bulk modulus at the pressure corresponding to the midpoint and the procedure is repeated s1

See, for example, D. J. Pastine, J . Chem. Phys. 49, 3012 (1968).

98

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FURTHER MECHANICAL T E C H N I Q U E S

throughout the pressure range of the measurement. The resultant plot of bulk modulus vs. pressure can be extrapolated to zero pressure to obtain BT and the slope yields the Griineisen Parameter32.33(see Section 12.2.3.7). The use of these data in equation of state studies has been considered by P a ~ t i n e . ~ ' Several methods exist for calibrating the piston-cylinder device. Many workers, using high-pressure equipment, prefer to check their apparatus against published values of transitions that occur in many solids. Many polymers exhibit various types of transitions. For example, both Weir3 and Pistorius12have studied the pressure/temperature behavior of polytetrafluoroethylene and have prepared pressure/temperature phase diagrams for this polymer. We checked our experimental procedure against these transitions. We found that the crystal- crystal transitions reported by these workers could be detected at or very close to the reported pressure and temperature range of the transition. The transitions were evidenced by inflection points on the stress-strain plot. The crystal-crystal transitions were very sluggish. Even at the usual slow rate of loading we doubt that the change was complete. Furthermore, they appeared to be diffuse, occurring over a range of pressure. With our equipment we could only measure the transition pressure with increasing pressure, and so we could not average the onset pressures with increasing and decreasing pressures as Pistorius did. However, the results indicated that the calculated pressures on the specimen, as calculated by Eq. (12.2. I ) , are essentially correct. The errors that enter into these measurements have been considered34 and it has been observed that duplicate determinations agree very closely.35 Compressibility measurements are subject to errors arising chiefly from the fact that no method has been devised to measure directly the static compressibility of a solid. Unfortunately, the volume of the containing vessel, pistons, testing apparatus, and pressure-transmitting media will change in volume with pressure. Also, some of the applied pressure may be used to overcome frictional forces, which could occur between the polymer sample and wall of the bore. To check this possibility a series of experiments, suggested by the work of Tydings and Giardini,3s were These workers pointed out that, with cylinR. W. Warfield and B. Hartmann, J . Appl. Phys. 44, 708 (1973). R. W. Warfield, Makromol. Chem. 175, 3285 (1974). 51 R. W. Warfield, J . Appl. Chem. 17, 263 (1967). s5 R . W. Warfield, J. E. Cuevas, and F. R . Barnet, Rheol. Acra 9, 439 (1970). 36 J. E. Tydings and A. A . Giardini, in "High Pressure Measurements'' (A. A. Giardini and E. Lloyd, eds.), p. 220. Butterworth. London, 1963. I2 33

12.2

STATIC HIGH-PRESSURE MEASUREMENTS ON POLYMERS

99

drical samples having a height-to-width ratio of one or less, frictional forces are minor. We first determined the compressibility isotherm for a sample of polystyrene 7.62 cm long and 0.635 cm in diameter. We next prepared and measured the isotherms of specimens starting with a length of 0.635 cm and increased the length of each sample by 0.635 cm until the standard length of 7.62 cm was achieved. In each case the stress-strain plots were identical in shape. Since the height-to-width ratio varied from 1 to 12 we concluded that friction is not a significant source of error. Similar results were obtained with various polyepoxides. O’ReillyPewho has used a similar apparatus has noted that changes in the volume of the apparatus due to temperature and pressure are negligible. We estimate that the values of AV obtained by the above-outlined procedures are accurate to +2%. 12.2.2.2. Dilatometer. For many applications in which hydrostatic pressures from atmospheric up to about 1000 bars are required, it is advantageous to use a dilatometer. In principle, this method is very simple. The sample under study is placed in a calibrated tubular vessel of known volume. A confining fluid, usually mercury, is placed in the vessel surrounding the sample and extending partway up the tube. The level of the mercury in the tube is carefully measured at room temperature and atmospheric pressure.37 The filled dilatometer is then placed in a suitable oil bath contained in a system designed to accurately apply pressure to the contents of the dilatometer. The temperature of the oil bath must be accurately controlled. The PVT properties of the solid specimen and confining fluid are then carefully measured and the volume of the polymer specimen calculated as a function of pressure and temperature. Again there are many different types of dilatometers. For low and moderate pressures two interesting devices have been developed at the National Bureau of Standards. Martin and Mandelkern13have used a dilatometer to study the crystallization kinetics of natural rubber. Another interesting device has been developed by McKinney and Penn. 13’ For investigations at pressures between 1000 and 2000 bars, Quach and Simha7 have developed a high-pressure dilatometer using the bellows technique, which was originally developed by Bridgman.3B In principle, the idea is simple: The solid sample with confining fluid is sealed in a flexible metal bellows, which is then subjected to hydrostatic pressure. Contraction of the bellows occurs until the pressure within them balances the applied pressure. The volume change is obtained from measurement of 37 3B

N . Bekkedahl. J . Res. Nail. Bur. Stand. 43, 145 (1949). P. W. Bridgman. Proc. A m . Acud. Aris Sci. 66, 185 (1931).

100

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FURTHER MECHANICAL TECHNIQUES

the change in length of the bellows with the assumption that the crosssectional area remains constant under isothermal conditions. Additional details and operating instructions of this device have been given.' Baer and Kardosloa have described a high-pressure dilatometer that used a standard 2000 psi cylinder of nitrogen as a source of pressure. An interesting device has been developed by Corsaro, Jarzynski, and Davis1*that, in principle, is a form of dilatometer. These workers have developed a very accurate acoustic densitometer for studying the PVT behavior of polymers and other solids. The device consists of two sections, the upper of which contains a 23.7 ml sample chamber and a precision bore 6.35 cm high and 1.110 cm in diameter. Sample chamber and bore are connected by a small diameter passageway. The lower section contains a 5 MHz quartz transducer placed at the bottom of the bore. The material under study is placed in the sample chamber, which is then filled under vacuum with mercury. The excess mercury fills the passageway and part of the bore. A float I. 10 cm in diameter and 3.30 cm long is placed on the mercury. A pulse of sound is emitted by the quartz transducer and passes up the mercury column to be reflected at the mercury-float interface and returned to the transducer. The total time of flight (to0 is measured, and knowing the speed of sound in mercury the distance between transducer and mercury float interface can be calculated. Changes in temperature and pressure produce changes in the volume of the bore. Pressure changes are made by placing the device in a conventional pressure chamber, while for temperature changes, the entire assembly is placed in an oil bath. A drawing of the densitometer and of the electronics has been given together with a detailed description of the device.l6 Typical results with polyethylene oxide (PEO) are also presented. It is of interest to note that the results obtained with PEO agree well with data obtained by Warfield and H a ~ t m a n nusing ~ ~ the piston-cylinder device described in this section. 12.2.2.3. Belt Apparatus. One of the more significant high pressure devices is the belt apparatus, which was developed by Hall.14 This apparatus can generate pressures as high as 150 kbar with temperatures as high as 2000°C. While it was originally developed for diamond synthesis,3Bit has proven useful in polymer investigations. The device consists of a supported conical vessel in which the solid specimen is placed. The specimen is then compressed between two (top and bottom) conical pistons, which are driven by a hydraulic press. The F. P.Bundy, J .

Chem. Phys. 38, 631 (1%3).

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101

pistons have steep sides of varying slope as does the containing vessel, the vessel being convex so that contact is initially made with the middle of the conical face of the piston. Pressure from the pistons is transmitted to the specimen in the containing vessel by pyrophyllite, a hydrous aluminum silicate that also serves as electrical and thermal insulation. As the piston advances the pyrophyllite is compressed and partly extruded. The sample can be heated electrically if necessary. The supporting rings around the sample chamber form a torus around the sample vessel and give this device its name. Anderson and B a c k ~ t r o mhave ~ ~ employed this apparatus to develop an interesting technique to determine the thermal conductivity of various polymers as a function of pressure. A cylindrical polymer sample is contained in a pyrophyllite chamber and is heated along its axis by means of an embedded spiral resistance wire. Power is externally supplied to the resistance wire. Temperature differences between two points at different radii are measured by embedded thermocouples and the thermal conductivity, as a function of the applied pressure, calculated. Corrections are applied to the data to account for the compressibility of the pyrophyllite discs at the top and bottom of the sample chamber. Additional details of the experimental procedures and data treatment have been given.4o 12.2.2.4. Diamond Anvil Cell. The diamond anvil high-pressure cell is making a very large impact on the study of materials at high pressure. Until recently, the generation of extremely high pressures in the laboratory was difficult. The equipment required was massive and very expensive, and the operation was very time consuming and often inaccurate. In addition, the manner of incorporating the material under study in the apparatus limited the use of such associated measurements as x-ray diffraction, various optical techniques, and sonic measurements. Thus, the data obtained with the conventional apparatus were often limited. The development of the diamond anvil device by workers at the National Bureau of Standardsz5has made the generation of high static pressures much more convenient. Furthermore, recent studies have led to a method of accurately measuring the static pressure exerted by this device.41 The method, which is based on measuring the pressure dependence of the shift of the R, ruby fluorescence line (ruby laser line), is used to calibrate the device. Briefly, the device, which is shown diagramatically in Fig. 3, consists of two brilliant-cut opposed diamond anvils with the material under study between. The anvils are maintained in position in holders, and are 40 41

P. Anderson and G . Backstrom, J . Appl. Phys. 44, 705 (1973). G . Piermarini and S. Block, Rev. Sci. Insfrum. 46,973 (1975).

102

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FURTHER MECHANICAL TECHNIQUES

DETAIL OF DIAMOND CELL

SIDE VIEW

FRONT VIEW

FIG.3. Diamond anvil high-pressure optical cell

ground and highly polished to form flats parallel to the cell. Alignment of the anvils is important. To facilitate such alignment, the anvils are ground with different surface areas and fitted into the holders. A gasket is employed to retain the specimen under study, equalize the pressure, and maintain isotropic conditions throughout the sample. Screws are

12.2

STATIC HIGH-PRESSURE MEASUREMENTS ON POLYMERS

I03

provided for alignment of the anvils and an optical "window" permits the passage of radiation. The pistons are inserted into a steel piece, which carries the pressure-generating system. One piston is maintained against a narrow flange at one end of the steel piece while the other is driven by a pressure plate, which is in turn connected to a lever pivoted about a steel block. This gives a 3 : 1 mechanical advantage. This steel block is actuated by a calibrated spring, which is compressed by a screw. Thus, a relatively small force applied to the faces of the diamonds results in a high pressure between the faces of the anvils. The entire assembly can easily be mounted in many optical devices such as spectrophotometers, microscopes, and x-ray diffraction apparatuses, and the device is inexpensive and easy to operate and maintain. For many years after the initial development of this device the problem remained of accurately measuring the pressure exerted. Various methods were employed: transitions, change in lattice parameters of crystalline solids, change in electrical resistivity of a material at a pressureinduced transition point, etc. Warfield and P a ~ t i n studied e ~ ~ polymorphic changes in phosphorus and were able to estimate the pressure exerted by following visually the conversion of red to black phophorus. However, all these methods were time consuming and in many cases not particularly accurate. In 1972 the ruby fluorescence method was developed. Calibration using this technique is not difficult. The fluorescence is strong and easily excited, and the wavelength shift with pressure is linear and is relatively large, about 0.036 mm kbar-l. As previously noted most of the measurements of conventional interest on polymers have been made at pressures below 10 kbar. Few measurements have been made at higher pressures and it would be of great interest to ascertain the effect of very high pressures on these materials. For example, one might expect to find new polymorphic transitions occurring at high pressure (some of which might well be irreversible thus yielding a somewhat different material), different types of crystalline forms, and possible new pressure-induced polymeric materials. Recent work with the diamond anvil has shown that pressures as high as I Mbar can be generated. However, the very small-sized sample sets a limit on the accuracy of many of the measurements. For example, bulk modulus and change-in-volume measurements made in this device will not be as accurate as those made in a conventional piston-cylinder device. Other areas of interest in which this device has been employed are in R . W. Warfield and D. J. Pastine, unpublished results.

I04

12.

FURTHER MECHANICAL TECHNIQUES

polymerization reactions under pressure (Zhulin et showed how complex polymerization reactions can be followed) and transition behavior. Van Valkenburg and Powers44studied pressure-induced transitions by optical detection methods. A general review of the early work with the diamond anvil has been given by Whatley and Van V a l k e n b ~ r g . ~ ~ 12.2.3. Response of Polymers to Static High Pressure

In this section we note many of the effects of static high pressure on polymers. The specific examples cited are taken from our experimental investigations employing a piston-cylinder apparatus but could, at least in principle, be obtained from many of the experimental techniques described in the experimental section. Many additional details have been previously g i ~ e n . ~ . ~ ~ , ~ ' 12.2.3.1. Change of Volume (-AV) with Applied Pressure. One of the first measurements one makes with static high-pressure equipment is the decrease in volume that a polymer undergoes when subjected to an applied pressure. Figure 4 presents data, obtained by a piston-cylinder device, on the bulk compressibility of polystyrene in the form of applied pressure vs. the percentage change in volume at 22°C (295 K). Several isotherms are shown. The 22°C isotherm is typical for a glassy amorphous polymer. This polymer remains in the glassy state over the entire pressure range and the isotherm is a smooth curve. The change in volume is about 12% at a pressure of 10 kbar, which is a typical value for an amorphous polymer. The second isotherm shown in Fig. 4 is that of the same polymer but at a temperature of 140°C (413 K), which is above the glass transition temperature of the polymer. At this temperature the polymer is initially rubbery and as a result can be seen to be more compressible. As the pressure is increased the polymer may change over a narrow range from a rubbery solid to a glassy solid. At this point the now glassy solid exhibits the reduced compressibility characteristics of the glassy state. Thus, the application of pressure has induced a glass transition in a rubbery polymer. A greater pressure will be required at higher temperatures. The pressure dependence of the glass transition is considered later. The compressive stress-strain curve of polymers is nonlinear and the application of pressure will increase the glass transition temperature. V. M. Zhulin, E. R. Lippincott, and W. J . Bailey, Science 153, 649 (1966). A . Van Valkenburg, and J. Powers, J . App. Phys. 34, 2433 (1963). 4a L. Whatley and A. Van Valkenburg, Adv. High Pressure Res. 1, 327 (1966). 46 R. W. Warfield, Polym. Eng. Sci. 6,41 (1966). R. W. Warfield, Makromol. Chem. 116, 78 (1968). 41.

44

12.2

STATIC HIGH-PRESSURE MEASUREMENTS ON POLYMERS

105

20,000 18,000

AV/V,,.G

%

FIG.4. Compressibility of polystyrene.

Matsuoka and Maxwell5were among the first to point out that the pressure shift in the glass transition temperature occurs with greater ease at higher rates of compression and, as a result, the effects of time and temperature are interchangeable in the rubbery state. 12.2.3.2.Secondary Transitions T,,,. Many polymers exhibit small secondary transitions at temperatures well below that of the glass transition temperature. In many cases these small transitions can be detected and studied by compressibility measurements. For example, a small secondary transition was first detected4*in a highly cross-linked polyepoxide by compressibility measurements made as a function of temperature. The transition was evidenced by small inflection points on the stressstrain isotherms and a plot of temperature vs. pressure for this polyepoxide is shown in Fig. 5 . Knowing the sensitivity of these measurements to small transitions we next studied a very highly cross-linked phenolic polymer. Close examination of a series of stress-strain isotherms revealed small temperature-dependent inflection points, which when plotted vs. pressure indicated a small transition at about 120°C.

'*

R. W. Warfield, unpublished results.

12.

I06

170 IE0t

'

:I, 0

1000

FURTHER MECHANICAL TECHNIQUES

,

'

'

,

,

,

2000

3000

I ,

,

4ooo 5000

PRESSURE,

kg

6Ooo

,

,

7000 8ooo

,

1

9000 10@0

/em'

FIG.5. Secondary transition temperature T,, vs. pressure for a polyepoxide polymer.

The existence of this small secondary transition was confirmed by supplementary mea~urernents.~~ Additional studies by Fava and ChaneyJO and by Kosfeld and BrandtS1also show the transition. It is of interest to note that the pressure dependence of secondary transitions is only 4 to f that of the glass transitions, T g , or thermodynamic melting point T,. This reduced dependence can be attributed to the small number of polymer segments taking part in a secondary transition. Note that we have extended the plot in Fig. 5 to atmospheric pressure, which yields a value of T,,,. 12.2.3.3.Glass Transition (Tg). Static compressibility measurements can be employed to detect and study the glass transition in polymers. This major transition is both pressure and temperature dependent. It is evidenced by abrupt changes in slope in the stress-strain isotherm and/or the AVIV,, vs. pressure plot, as is noted on the 140°C isotherm on Fig. 4. Here at a pressure of 1200 bar and a volume change of -6%, the rubbery polymer undergoes a pressure-induced glass transition and the polymer reverts to the glassy state. With increasing temperature, higher pressure is required for the glass transition to occur. Once a series of isotherms has been obtained each isotherm is carefully examined for any changes in slope or inflection points. Usually the evi-

-

R. W. Warfield and G . F. Lee, J . Appl. Pulym. Sci. 21, 123 (1977). R. A. Fava and C. E. Chaney, J . Appl. Pulym. Sci. 21,791 (1977). 51 R. Kosfeld and J. Brandt, Rheul. Acfa IS, 64 (1976).

*@

12.2

STATIC HIGH-PRESSURE MEASUREMENTS ON POLYMERS

107

dence for a glass transition will be unequivocal. The observed changes are then plotted as a function of the temperature of the isotherm and of the applied pressure at which the change in slope occurs. The resultant Tg vs. pressure plot should, if a transition has occurred, be a reasonably good straight line. Again we must note that Tg is a time-dependent process and that the isotherms must be made at a slow rate. If doubt exists as to the exact nature of the observed transition one can easily calculate the pressure dependence of the transition from a series of isotherms. Secondary transitions have dT,,/dP values of about O.Ol"/bar, whereas dTg/dP and dT,/dP have values of about 0.02-0.03"/bar. 12.2.3.4. Thermodynamic Melting Point (T,). It is generally not feasible to directly determine the melting point T, of crystalline and semicrystalline polymers in a piston cylinder device. Upon melting many polymers tend to flow and/or extrude between the bore and cylinder and thus "bind" the device. Also, when pressure is applied to a melt, freezing occurs, yielding a specimen that usually does not fill the bore of the device and the measurements then become difficult to interpret. In the case of polyethylene oxide, a pressure-temperature cycle was devised for determining the melting point of this polymer.32 While we have not applied this cycle to any other polymer, it is assumed that it could be applied to any polymer that exhibits T,. The direct determination of T , as a function of pressure and temperature was first made by Wood et U I . , who ~ ~ observed that for a crystalline, natural rubber the average value of dT,,,/dP was 0.029"C/bar. Baer and Kardos,loa using a high-pressure dilatometer, determined the effect of pressure on the melting behavior of a number of polymers. A diamond anvil device with associated optical equipment also can be employed to determine T , and its pressure dependence. 12.2.3.5. Freezing Point of Polymers (Tf). It is well known that the freezing point Tfof polymers is less than the melting point T,, as a result of supercooling. Usually, the two points differ by a few degrees at atmospheric pressure. The freezing point can easily be determined in a piston-cylinder device and the procedure we have employed32 is described briefly below. The polymer under study is placed in the bore of the apparatus and slowly heated to a temperature above the atmospheric melting point of the polymer. Once melting occurs and the temperature has stabilized, the melt is held at constant temperature for one hour to ensure completeness of melting. Next the polymer is compressed, at the usual slow rate; and, 52

L. A. Wood, N . Bekkedahl, and R. E. Gibson, J . Chem. Phys. 13,475 (1945).

108

12.

FURTHER MECHANICAL TECHNIQUES

with increasing pressure, the polymer crystallizes, i.e., freezes. This is evidenced by a marked change in the stress-strain isotherm, as shown in Fig. 2 for the case of polyethylene oxide. To determine Tfas a function of pressure, the following procedure should be followed. A series of isotherms is obtained as noted above, with each isotherm being at a temperature about 3-5°C above the temperature of the previous isotherm. With increasing temperature the pressure required to induce crystallization will be greater. A discussion of the effect of pressure upon crystallization3**”has been given. 12.2.3.6. Elastic Constants of Polymers. It has been recognized by many workers” that experimental techniques are needed that will determine the elastic constants of an isotropic solid using a single specimen. This is necessary to eliminate interspecimen variability .55 Also, a common practice is to determine one constant by an isothermal method and another constant by a adiabatic method. The combination of using different specimens of materials and interchanging isothermal and adiabatic methods has created considerable confusion in the values of these constants for polymers. Artemov and c o - w o r k e r ~were ~ ~ among the fist to note the importance of using the same specimen to obtain values of the two elastic constants employed to calculate accurate values of Poisson’s ratio. Likewise, NederveenS7developed an apparatus in which a long thin rod of a polymer was employed to determine both Young’s modulus and shear modulus at low frequencies. Bonnin and c o - ~ o r k e r sattributed ~~ the poor results that had previously been obtained in determining Poisson’s ratio to a combinations of poor experimental techniques and the use of different samples and different test conditions for determining the moduli. Bonnin et al. 55 also noted the importance of eliminating interspecimen variability and variable anisotropy. These workers presented a method, based upon the determination of torsional and flexural deformations on the same specimen, for obtaining accurate values of Poisson’s ratio. All three of these methods satisfy the conditions, noted by Koster and Franz,“ needed to obtain accurate values of the moduli to be used in calculating Poisson’s ratio. These conditions are: a. The same specimen is used to determine each modulus. b. Analogous, i.e., either static (isothermal) or dynamic (adiabatic), techniques are employed. c. The specimens are in a quasi-isotropic state. a3 R. W. Warfield, Naval Ordnance Laboratory Technical Report 73-23 (1973). 51 W. Koster and W. Franz, Met. Rev. 6, 1 (1961). ss M. J . Bonnin, C. M. R. Dunn, and S. Turner, Plast. & Po/ym. 37, 517 (1959). w

P. G. Artemov, G. V. Shpak, and V. V. Simankov, Plast. Massy No. 5 , p. 58 (1962). C. I. Nederveen, Rheol. Acta 3, 2 (1%3).

12.2

STATIC HIGH-PRESSURE MEASUREMENTS ON POLYMERS

109

With these conditions in mind we developed a multimodulus method for determining Young’s modulus and bulk modulus consecutively on the same polymer specimen under identical experimental c ~ n d i t i o n s . ~ ~ The experimental apparatus employed consists of the standard Matsuoka- Maxwell5 compressibility tester. One major change is made in the specimen diameter. Instead of having the initial diameter of the specimen made equal to the bore of the tester it is made slightly smaller. In preparation for a determination of Young’s modulus and bulk modulus, a specimen 7.62 cm long and 0.630 cm in diameter is prepared and then inserted into the bore of the tester. It is very important that the specimen be 0.005 cm smaller in diameter than the bore of the tester. Two casehardened steel plungers 4.128 cm long and 0.635 cm in diameter are inserted into the open ends of the bore and the entire assembly is placed in the testing machine. The specimen is then loaded by pressing down on the steel plungers at a rate of 0.635 cm/min. Usually a load of 1000 kg is applied and a stress-strain plot is recorded. If the polymer specimen exactly fills the bore of the tester the application of pressure will only result in a decrease in volume of the specimen and only the bulk modulus can be determined. However, when the diameter of the specimen is slightly smaller than the bore of the containing vessel, the application of pressure will initially result in an increase in diameter (dilation) and a decrease in length. This is, of course, Young’s modulus, which can be determined either by compression or extens i ~ n . The ~ . condition ~ ~ for Young’s modulus measurement will continue until the bore of the tester is filled. Additional pressure will then result in a decrease of the volume of the specimen, which is the condition for the bulk modulus measurement. A typical stress-strain plot obtained with a standard undersize polyepoxide specimen is shown in Fig. 6. Two stages will be noted. From the first stage Young’s modulus E can be calculated. This is done as follows. A tangent T, is drawn to the first stage. A normal N1 is then dropped from a point on T , to the abscissa. The distance between its intersection and that of T , describes a characteristic specimen deformation A/, for the first stage, for a force equivalent to the height of N , . E is then calculated by using

(12.2.3) where I,, is the original specimen length, F, the applied force, and A the area of the plunger that transmits the load to the specimen. JB L. E. Nielsen, ”Mechanical Properties of Polymers,’’ p. 3. Van Nostrand-Reinhold, Princeton. New Jersey, 1962. ’’ R. C. Novak, and C. W. Bert, J . C’ompos. Muter. 2, 506 (1969).

110

12. FURTHER MECHANICAL TECHNIQUES

I

\

11'2 DEFORMATION

FIG.6. Stress-strain plot for a polyepoxide polymer.

The second stage develops at a time when the specimen is no longer capable of dilating in the radial direction due to the constriction of the bore. Under these conditions, the following relations are valid for bulk modulus B: (12.2.4)

Thus, in a similar manner, B is obtained by drawing tangent T,, dropping normal N, to the abscissa determining Al,, and then using Eq. (12.2.4) to calculate B. The moduli can easily be determined as a function of temperature by placing heating coils around the tester. After heating and equilibration at a preselected temperature, the moduli are determined as noted above.

12.2

STATIC HIGH-PRESSURE MEASUREMENTS ON POLYMERS

111

A BULK MODULUS 0 YOUNG’S MODULUS

p,

1 25

0 POISSON’S RATIO

50

75

125

100

TEMPERATURE, FIG. 7. Elastic constants of polyimide as

150

O C

a function of temperature.

Two embedded thermocouples enable the temperature to be measured very accurately. When determining E at elevated temperatures, the diameter must be carefully chosen so that even at the highest temperature it will be smaller than the bore of the tester by 0.005 cm. A plot of the elastic constants of a polyimide as a function of temperature is shown in Fig. 7. Once the values of E and B have been determined, Poisson’s ratio p is calculated by means of Eq. (12.2.5):

1 E p = T - z .

(1 2.2.5)

Results obtained by this procedure have excellent reprod~cibility.~~ The accuracy is somewhat more difficult to assess. Satisfactory agreement with other static values of E , B, and p has been repeatedly obtained.35*s0Hartmanns’ has recently completed an analysis of the differences in E between values determined in tension and those determined in compression and found that for polymethylmethacrylate when Young’s 6o

R. W. Warfield and F. R . Barnet, Angew. Makromol. Chem. 27, 215 (1972); 44, 181

(1975).

‘’ B. Hartrnann. Proc. Int. Conf. High Pressure, 4th 1974 p. 110 (1975).

12.

112

0

.n

FURTHER MECHANICAL TECHNIQUES

1.0

1.0

2.0

2.5

S.0

s.n

4.0

*.n

PRESSURE, I O ~ O Y N E / C H ~

FIG.8. Bulk modulus of poly(ethy1ene oxide) vs. pressure at 25°C.

modulus is determined at a strain E = Young's modulus determined in tension will be 6% less than-the same modulus determined in compression. 12.2.3.7. Bulk Modulus of Polymers (BT). 12.2.3.7.1. B T vs. PRESSURE. The isothermal bulk modulus BTof a polymer is found by differentiating the stress-strain plot; if this is done over a broad pressure range, one obtains the pressure dependence of BT. A typical plot of this type is shown in Fig. 8. Below 1.3 x lo9 dynes/cm2 the polymer does not completely fill the bore of the tester and reliable data cannot be obtained for BT. In addition, the analytical method employed for calculating B T does not yield data points below a pressure of about 2 x lo8 dynes/cm2. P)~. The slope of Fig. 8 is the dimensionless quantity ( c ~ B ~ / C ~ This quantity is of interest since one can calculate the Griineisen parameter yL from this slope. 12.2.3.7.2. GRUNEISEN PARAMETER OF POLYMERS.The Griineisen parameter y is an important quantity in characterizing the polymeric solid state. This parameter first appeared in the Griineisen equation of state,62 and it relates the thermal and mechanical properties of a solid and deterE. Gruneisen, in "Handbuch der Physik" ( S . Flugge. ed.), Vol. 10, p. 7. Springer Verlag, Berlin and New York, 1926.

12.2

STATIC HIGH-PRESSURE MEASUREMENTS ON POLYMERS

1 13

mines the pressure response of a solid to energy deposition. Also, it is a measure of the anharmonicity of a solid. Values of y have only recently become available and the details of the relationship(s) between y and polymer structure are still obscure. The experimental method that has been developed to determine y is based upon the measurement of the pressure dependence of the bulk modulus BTas has been previously described. From the plot of bulk modulus vs. pressure the Griineisen parameter yL is then calculated by (12.2.6) where (dBT/aP)Tis the slope of the BT vs. P plot. Note that when we determine y using the pressure dependence of the bulk modulus, we obtain yL, the lattice Griineisen parameter. This yLrelates to the polymer chains moving in relation to each other, i.e., interchain. On the other hand, when y is determined by various dynamic techniques one obtains the thermodynamic Gruneisen parameter yT, which is an average over all the vibrations. For metals and ionic crystals yL = yT,but for polymers this is not the case. Intrachain vibrations (high frequency and harmonic) involve covalent bonds and have low Griineisen parameters, while interchain vibrations (low frequency and anharmonic) involve van der Waals bonds and have high y values. In the experimental method employed here for determining y, the pressure dependence is determined primarily by the interchain component so that we obtain yLby this method. Generally values of yL range between 4 and 7 and since yL depends only upon the specific volume of the solid, which changes only slightly with temperature, it follows that it will be almost independent of temperature. On the other hand, changes in structure will be reflected in changes in yL. For example, we have noted that changes in the crystal structure of polytetrafluoroethylene are evidenced by changes in yL.47 12.2.3.8. Pressure Dependence of the Electrical Resistivity. The determination of the pressure and temperature dependence of the electrical volume resistivity is one approach to studying the free-volume concept of motion in amorphous polymers. In addition, as pointed out by Seanor,s3information on the nature of the electrical conduction process in polymers can be obtained from data on the pressure dependence of the electrical resistivity. Experimental procedures for determining the electrical properties of polymers under pressure are limited. Since some of the devices prea D. A. Seanor, Adv. Polym. Sci. 4, 317 (1965).

I14

12. FURTHER MECHANICAL TECHNIQUES

viously employed tend to become rather complex, a need existed for a relatively simple device and experimental procedure. Toward this end we succeeded4*in adapting the basic piston-cylinder device so as to measure the electrical resistivity of polymers to pressures of 10 kbar and temperatures of 220°C (493 K). This was accomplished by suitably insulating the basic device. An inner bushing of a dielectric, beryllium oxide, was substituted for the usual steel inner bushing. The dimensions of the ceramic inner bushing are the same as those of the steel one. The polymer sample was usually about 1.25 cm long and 0.635 cm in diameter. It was inserted into the inner bore of the inner bushing. One hardened-steel plunger 4.129 cm long is inserted below the specimen, while another one 10 cm long is inserted above the specimen. To properly insulate the piston-cylinder apparatus from the testing machine, a thin sheet of aluminum oxide is placed under the base of the apparatus. Another smaller sheet is placed between the piston and the ram of the testing machine. Insulated leads are then attached to each of the steel plungers and are in turn connected to a current monitoring device." Close contact with the plungers, which are now serving as electrodes, is maintained by the application of pressure. To accurately calculate the pressure dependence of the volume resistivity one must know how the volume of the polymer decreases with pressure. This is necessary so that one can continuously change with the applied pressure the area to length ratio A I L , which is used to convert the measured values of resistance into resistivity. It has been found that this pressure dependence (and temperature dependence if need be) can be accurately established by measuring the compressibility of the specimen before its resistivity is measured, either in the basic piston-cylinder device or in the device as modified for electrical measurements. Typical data obtained by means of this device are shown in Fig. 9. Previous studies in this area are limited. However, Saito and co-workers6s have described a experimental technique and presented data on the electrical conduction of polymers as a function of pressure and temperature. Likewise, Sasabe and Saitos6have determined the electrical conductivity of various acrylates under pressure. 12.2.3.9. Effects of Pressure on the Mechanical Properties of Polymers. Another area of interest is the pressure dependence of many of the mechanical properties of polymers. Such properties as the elastic constants, crystallization, compressive and tensile stress-strain behavior, &1

sa 88

R. W. Warfield and M. C. Petree, Makrumol. Chem. 58, 139 (1962). S . Saito, H. Sasabe, T. Nakajima. and K . Yada, J . Pu/ym. Sci., Purr A-2 6, 1297 (1968). H . Sasabe and S . Saito, J . Po/ym. Sci.. Part A-2 6, 1401 (1%8).

12.2

STATIC HIGH-PRESSURE MEASUREMENTS ON POLYMERS

115

10 I2

10

10'

I'

t 0

1 1400

2800

4200 5600 PRESSURE

7000 (Atmospheres)

FIG.9. Resistivity of a polysulfide-epoxide copolymer as a function of pressure.

and flow all exhibit variations in their response to pressure. Knowledge of the pressure dependence of these properties is of technological importance particularly in the fabrication of these materials. Earlier, we showed a method for determining the pressure dependence of the bulk modulus and the parameters that can be obtained from such data. Likewise, we showed the effect of pressure on the transitions T,,, and Tf. Wunderlich and c o - ~ o r k e rshowed s ~ ~ ~ that, ~ because the effect of pressure is to increase T,, it is possible to conduct crystallization at a higher temperature than could be possible at atmospheric pressure. Under these conditions extended-chain lamellae can be obtained with linear polyethylene and polychlorotrifluoroethylene. Specimens crystallized under these conditions exhibit higher densities and melting points and sometimes new crystal structures and morphologies. For instance, the less common triclinic crystal structure of polypropylene has been produceds7 by crystallization under pressures up to 4 kbar. Sauer, Mears, and Pae8 have developed an apparatus for studying the 67

K . D. Pae, D. R . Morrow, and J . A. Sauer. Nature (London) 211,514 (1%6),

I I6

12. FURTHER MECHANICAL TECHNIQUES

effects of hydrostatic pressure on the compressive and tensile stress of a polymer to a pressure of 10 kbar. Initially these workers found that the yield strength and elastic modulus of polymers increased as a function of pressure. The mechanism of these increases was also studied. Compressive stress-strain behavior was also studied and the rise in elastic modulus with pressure was observed. Also observed was the shift with pressure of the glass and secondary transition temperatures. The values of the shift were very similar to those obtained by measurements made with the piston-cylinder device. The effect of pressure on tensile properties has thrown light on the mechanisms of crazing and shear-banding as modes of mechanical yielding in glassy po1ymers.t Pugh et a1.,68for instance, showed that crazing in polystyrene could be suppressed at pressures greater than 3 kbar, where the material behaved in a ductile fashion. Sauer and Pae have also studied such properties as flow and deformation, fracture, yield, and e x t r u ~ i o nand , ~ have recently reviewed work of their own and others.se

t See also Part 15 (this volume). BB H. L. D. Pugh, E. F. Chandler, L. Holliday, and J. Mann, Po/ym. Eng. Sci. 11, 464 (1971). J. A. Sauer and K. D. Pae,Proc. I n t . Conf. High Pressure, 4th 1974 p. 17 (1975).

12.3. Stress-Strain Yield Testing of Solid Polymers

By John L. Rutherford and Norman Brown 12.3.1. lntroductiont

There are two broad reasons for measuring the stress-strain yield behavior of high-molecular-weight polymers. First, commercial users must know how the material will respond to loads so that the component can be designed for satisfactory performance. Second, stress-strain measurements are useful for a deeper understanding of the relationships between molecular structure and mechanical properties. Standard test methods have been developed to provide information needed for practical application of polymers. Since polymers exist in many forms, a variety of testing procedures is required. These are described briefly, emphasizing the features they have in common. Other test methods better suited to the study of structure/property associations are also discussed. The properties of polymers, to a greater extent than other solid materials, are strongly influenced by material and experimental variables. Thus, a list of property values should include the history of the specimen from fabrication to fracture as well as a description of the test method. The observation that the properties of polymers are sensitive to material and experimental variation points out the utility of mechanical property measurements in the study of molecular structure. 12.3.2. Defi nitions

One of the many excellent books that describe stress and strain should be consulted for definitions of those terms.'-' This discussion is limited t See also Volume 1 of this series, Chapter 3.7. S. Timoshenko, "Strength of Materials," 3rd ed., Part I . Van Nostrand-Reinhold, Princeton, New Jersey, 1955. * A. E. H. Love, "The Mathematical Theory of Elasticity." Dover, New York, 1944. L. D. Landau and E. M. Lifshitz, "Theory of Elasticity." Addison-Wesley, Reading, Massachusetts, 1959.

METHODS OF EXPERIMENTAL PHYSICS,

I I7 VOL. 16c

Copyright @ 1980 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-475958-0

I18

12.

FURTHER MECHANICAL TECHNIQUES

to the stress-strain behavior for four loading modes most commonly used with solid polymers: tension, compression, flexure, and shear. Stress is the force applied over a unit area and is commonly expressed as newtons per square meter, called pascals (Pa). The cross-sectional area on which the force is applied is used to calculate the stress. In uniaxial tensile testing the force tends to pull the specimen apart, while compression loading results in the specimen being squeezed together. For flexural testing a specimen is supported at the ends and a deflecting force applied at the center. In shear the force acts parallel to the surface. Strain is displacement per unit length and is dimensionless; it is often expressed as a percentage. Tensile strain is the fractional longitudinal increase in length of a premeasured portion of the test specimen called the gauge length. Compressive strain is determined in a similar fashion. Flexural strain is determined by the longitudinal strain in the outer fibers of the beam at midspan. When shear strain is determined from tests in which a torsional force is applied, the surface is circumferential displacement per unit length of the specimen. Modulus is the ratio of stress to strain and commonly takes the following forms: (1) initial modulus, the slope of the stress-strain curve at zero strain; (2) tangent modulus, the slope of the stress-strain curve at an arbitrary strain; and (3) secant modulus, the slope of a straight line drawn from the origin to a specific strain level on the stress-strain curve.

Linear and elastic behavior is often a small part of the total stressstrain curve. The exact point of departure from linear stress-strain behavior is sometimes difficult to determine and may be affected by slight imperfections in the specimen. A departure from linearity in the stress-strain curve is indicated by the fact that the unloading curve does not retrace the loading so that a hysteresis loop is generated (see Fig. 1). As the stress level is increased the area of the hysteresis loop increases. As the macroscopic yield point is approached, more and more nonclosure in the hysteresis loop occurs. The residual strain that is observed after immediate unloading will persist for various periods of time depending on the polymer, time, temperature, and magnitude of the prior strain. When A . E. Green and W. Zerna, "Theoretical Elasticity." Oxford Univ. Press (Clarendon), London and New York, 1954. R. G . Roark and W. C. Young, "Formulas for Stress and Strain." McGraw-Hill, New York, 1975. J. G . Williams, "Stress Analysis of Polymers." Wiley, New York, 1973. ' E. Gillam, "Materials Under Stress." CRC Press, Cleveland, Ohio, 1969.

12.3.

STRESS-STRAIN

g

microyield

E

precision elastic

,,, V)

Y I E L D TESTING OF SOLID POLYMERS

119

stress

limit

STRAIN

FIG.1. Stress-strain relationships in polymers.

high-sensitivity strain-measuring methods are used it is possible to determine deviations from elastic behavior at very low stresses: in some cases residual strain as small as can be measured. The stress level that produces a very small residual strain may be called a microyield point. 12.3.3. Methods for Measuring Strain

The choice of a method for measuring strain is made after a review of several factors: desired sensitivity, type of specimen, cost of test and necessary equipment, kind of information needed, time for testing, and compliance with standard test specification (if any). The six most common methods are discussed below. 12.3.3.1. Fiducial Marks. The least expensive and easiest way to determine strain is to measure the distance between two fiducial marks on the specimen at the beginning and at the end of a test. The original distance is the gauge length (I,) and the difference between the two readings is the extension (A& The strain is calculated by dividing lo into AI (dimensionless). With sharp fiducial marks and a good traveling microscope, strains of about can be measured. The fiducial marks may be made by applying pencil, ink, or crayon, which is then lightly scratched without damaging the specimen. 12.3.3.2. Mechanical Strain Gauges8 These gauges use the lever principle, simple or compound, or a rack and pinion to magnify the deformation of a specimen under load. Simple lever gauges can accommodate large strains, but the magnification is limited to about 20. Gauges using compound levers can provide magnifications up to 1000 with a more limited strain range. The advantages of lever gauges are that they are

* G . H. 1950.

Lee, "An Introduction to Experimental Stress Analysis." Wiley, New York,

I20

12.

FURTHER MECHANICAL TECHNIQUES

t

I fixed end

mirror

f,

rocking/ lozenge

gauge

lenath \

a I

\specimen FIG.2. Tuckerman-type mechanical-optical strain gauge.

self-contained, rugged, easy to use, and provide direct read-out. These mechanical gauges have several drawbacks: methods of attachment require holes, suction cups, clamps, notches, and rubber bands; they have limited sensitivity; they are large and bulky; and they are sensitive to vibrations. Rack and pinion dial gauges are convenient to use, but have the same advantages and disadvantages as the lever gauges. 12.3.3.3. Mechanical-Optical Strain Gauges.* These gauges, originally developed by Tuckerman, combine mechanical and optical levers to cm extension. A typical gauge provide sensitivities as great as 5 x consists of a mirror attached to a pivoting lever whose gauging end is in contact with the specimen (see Fig. 2). The fulcrum of that lever is connected to the second gauging end, which is attached to the specimen. The gauge length is established by the distance between the two gauging points. The output of this gauge is read through an autocollimator, which can be awkward for large strains. The gauge requires much space for the long optical paths. Notches must be cut in the surface of the specimen to seat the two gauging points, which are held on the specimen by clamps, springs, or rubber bands. 12.3.3.4. Bonded Resistance Strain Gauge.s The resistance-type gauge is widely used for strain measurements. It consists of fine wire grid, which is adhesive-bonded to the specimen. When the specimen is stretched or compressed, the strain gauge also deforms altering the electrical resistance of the grid. The change in resistance is a measure of the strain experienced by the specimen. The advantages are that it is direct reading, has small size, accommodates transient and rapidly changing strains, is not affected by vibration, has sensitivity of about 5 x cm, has good accuracy and reproducibility, and has a well-defined gauge

' C. C. Perry and H. R . Lissner. "The Strain Gauge Primer." McGraw-Hill. New York. 1962.

12.3.

STRESS-STRAIN YIELD TESTING OF SOLID POLYMERS

121

length. Resistance gauges require temperature compensation, generally must be bonded to the specimen surface, and the strain range is limited to elastic deformation of the wire grid. 12.3.3.5. Linear Variable Differential Transformer.lO The LVDT consists of a magnetic core suspended on the center line of a helical coil. The outer coil is divided into thirds to allow an excitation voltage of one polarity at the end coils and opposite polarity at the center coil. The coil assembly is attached to one end of the gauge length of the specimen while the magnetic core is fastened to the other end. When the test specimen is deformed, the core moves within the electric fields changing the voltages, which become a measure of the deformation. LVDTs have a sensitivity cm, with good accuracy and repeatability and a suitable of about 2 x deformation range. Mounting this extepsometer is difficult and often results in an indefinite gauge length. 12.3.3.6. Capacitance-Type Extensometer.11J2This instrument consists of two parallel plates that form an air gap capacitor (Section 12.3.5.2). Each plate is attached to the specimen; the distance between the mounting points establishes the gauge length. As the specimen is deformed, the distance between the plates changes, thus altering the capacitance. The extensometer is one leg of a capacitance bridge whose output is a direct voltage. One of the capacitor plates is mounted on a micrometer thread to provide for calibration. Extension sensitivities of cm are routinely obtained and the instrument has satisfactory accuracy and repeatability. The mounting problems are similar to those of the LVDT. In addition, special attention must be given to shielding the plates and the leads to eliminate stray capacitances. The extensometer is inherently nonlinear, but this can be compensated for by calibration or suitable electronic circuitry. 12.3.4. Test Method

The mechanical properties of polymers are dependent on temperature, humidity, rate of strain, and specimen dimensions to varying degrees. For this reason, standard test methods have been devised in which these parameters are kept constant and the values so obtained are quoted as standard tensile, flexural, and compressive strengths or moduli. Standard test methods have evolved at a national level so that each country has developed its own series of tests; there are ASTM standards in the U.S., Schaevitz Eng. Co., lo E. E. Hercey, "Handbook of Measurement and Control." Camden, New Jersey, 1976. 'I J . L. Rutherford, F. C. Bossler, and E. J. Hughes, Rev. Sci. Instrum., 39,666 (I%@. I* F. C. Bossler, M. C. Franzblau, and J. L. Rutherford, J . Phys. E 1, 829 (1968).

122

12.

FURTHER MECHANICAL TECHNIQUES

British standards in Britain, DIN standards in Germany, etc. The differences between the various standards are usually small and some efforts have been made to correlate the national test methods, for instance, by the American National Standards Institute (ANSI) in New York. For the purpose of this chapter we have chosen to describe the American standards as being typical. The American Society for Testing and Materials (ASTM) has developed and approved many standard test methods for evaluating the mechanical properties of polymers. Those tests most useful for studying stress-strain yield behavior are summarized here: additional information may be obtained from the published ASTM test methods,I3 which are updated annually. Other test procedures designed specifically for use with adhesives in bonded joints are also presented here. ASTM standards have evolved using units of pounds and inches but have more recently been converted to the SI system of units without changing any of the conditions such as specimen dimensions or testing speeds. For this reason there are such odd specimen dimensions as 12.7 mm (0.5 in.) and 177.8 mm (7 in.) and testing speeds of 0.13 cm/min (0.5 in./min). The pascal and megapascal are used here as the units of mechanical modulus and stress: 1 Pa = 1 N/mZ = 1.02 x kg/cm2 = 1.45 x Ib/in2. A number of precautions must be considered when measuring the mechanical properties of polymers. Gripping polymeric specimens presents difficulties because polymers have relatively low strength and, often, smooth surfaces. Slippage can be minimized by using grip surfaces that are serrated with a coarse cut for thermoplastics and a finer cut for thermosetting polymers. Smooth-faced grips can be used by sandwiching thin sheets of abrasive cloth or rubber between the specimen and the grips. The specimen can be damaged by excessive pressure in the grips. Beyond that, however, the grips tend to concentrate stress at local areas leading to premature failure at those points. Two steps can be taken to alleviate the condition. First, the specimen may be strengthened in the grip area by adhering strips of reinforcing material to both sides of the specimen. Second, as is standard practice in tensile testing, the specimen should have a reduced cross-sectional area to help ensure that fracture does not occur in the grips. The tensile strain is measured in this reduced cross-sectional area. The specimen must be carefully aligned while the load is being applied to avoid extraneous stresses that will modify the stress-strain behavior. Bending of the specimen is the most common consequence of misalignment. Universal joints at each end of l3

"ASTM Standards." Am. SOC. Test. Mater., Philadelphia, Pennsylvania, 1979.

12.3.

STRESS-STRAIN

YIELD TESTlNG OF SOLID POLYMERS

123

the tensile specimen are effective in minimizing bending; however, ball-socket joints are the best method for improving alignment. Problems in compression testing usually are caused by nonparallelism of the loading surfaces or the ends of the specimen. This condition can often be corrected by placing a steel ball between the loading ram and the plate that contacts the end of the compression specimen. Care should be taken when mounting the extension indicator to avoid introducing stress concentrations. For example, the notches that must be cut in the specimen for the gauging points of some indicators can lead to fracture at loads lower than normal. Very brittle polymers are often more notch sensitive than are ductile materials. Preparing test specimens with reduced cross-sectional areas usually involves machining, cutting, or molding operations. Any tool marks remaining after the final machining step should be as fine as possible and they should be parallel to the direction of the applied stress. Burrs should be removed after cutting the gauge section and all flash should be cut away from molded specimens before testing. Some polymers have properties that vary with direction in the material due to crystallinity, structural orientation, or reinforcing elements. The orientation of the specimen with respect to the direction of the applied force should be clearly identified. The rate at which the load is applied is an important experimental variable. Since polymers are viscoelastic, the value of the elastic modulus can be higher with rapid loading rates than with slower loading rates. The yield and fracture behavior may also be different depending upon the rate with which the load is applied. Another experimental factor that strongly influences polymer property values is the test temperature. At higher temperatures polymers usually have lower strength and greater ductility. When test results are reported, all the material and experimental conditions should be listed: composition, fabrication procedures, machining methods, specimen dimensions, type of test, test temperature and environment, thermal conditioning, loading rate, and type of strain indicator used. The substrate preparation is an important step when preparing adhesive-bonded specimens and should be completely described. 12.3.4.1. Tensile Properties of Polymers. ASTM method D 638 has been developed for rigid polymers having a thickness greater than 1 mm; another test is used for thinner sheet material. Figure 3 is a sketch of a common tensile specimen and includes dimensions that are keyed to the specimen thickness. The grips should be self-aligning so that the long axis of the specimen coincides with the direction of the applied pull through the center line of the grip assembly. A suitable extension indicator is used to monitor the

12.

124

FURTHER MECHANICAL TECHNIQUES

0 . 4 or

Over 0.71 to 1 .40 incl.

T -Thickness

under

W -W i d t h of narrow section

0.32

0.64

1.27

1.91

+ 0.05

L Length of narrow section

0.95

5.72

5.72

0.95

i0.05

W O - W i d t h over-all, min

0.95

1.91

1.91

2.87

+0.64

6.35

18.29

16.51

24.64

no mox

0.76

5.08

5.08

5.08

i 0.03

-

2.54

13.46

11.43

11.43

t 0.51

- Radius of fillet

1.27

7.62

7.62

7.62

t0.11

-

- Length over*lI,

LO

0.71 or under

Tolerances

min

G

- Gauge

Length

D Distance between grips

R

(b) FIG.

3. Specimen dimensions for tensile tests (cm).

deformation in the reduced gauge section. The usual speed of testing, i.e., the relative rate of motion of the grips, is 0.5 cm/min. Nonrigid polymers are usually tested at a speed of 5 cm/min. Polymer sheets less than 0.1 cm thick are tested in accord with ASTM D 882. The width of the specimens should be not less than 0.05 cm nor more than 2.5 cm. A width-thickness ratio of at least 8 should be used. Such thin specimens require special procedures during testing. Extension indicators cannot be mounted on the specimen; therefore, fiducial marks are used to calculate the deformation. Extreme care must be taken when cutting the specimens to produce straight, clean, parallel edges with no visible imperfections that could lead to premature failure. As there is no reduced gauge section with these specimens there is a greater tendency for failure at the grips. 12.3.4.2. Compression Testing of Rigid Polymers. The standard test method is described in ASTM D 695. Typically, the specimen is in the

12.3.

STRESS-STRAIN

YIELD TESTING OF SOLID POLYMERS

125

form of a right circular cylinder or prism whose length is twice its diameter or principal width. Preferred specimen sizes are 1.27 cm diam x 2.5 cm long, or for prisms 1.27 x 1.27 x 2.5 cm. Where elastic modulus and offset yield stress data for circular cross-section specimens are desired, the test specimen should have dimensions such that the slenderness ratio (length to least radius of gyration) is between 1 1 : 1 and 15: 1. The standard speed of testing is 0.13 cm/min. After the yield point has been reached, it is permissible to increase the speed to about 0.5 cm/min until the specimen breaks. Some materials do not fail by shattering, but continue to deform in compression until a flat disk is produced. The compressive strength is then an arbitrary value depending upon the degree of distortion that is regarded as indicating complete failure of the material. The loading faces should be flat to within 0.025 cm and parallel to each other in a plane normal to the vertical loading axis. The specimen should be machined to the same flatness and parallelism tolerances. 12.3.4.3. Flexural Test Method. The flexural properties of rigid polymers are measured using ASTM D 790 on rectangular bars with a three- or four-point loading system (see Fig. 4). In each case the specimen rests on two supports and is loaded by means of the one or two loading noses until the outer fibers break or until a maximum fiber strain of 5% is reached. The difference between the two methods is that the maximum axial fiber stresses occur on a line under the loading nose of the three-point method and over the area between the loading noses in the four-point system. The loading noses and supports are cylindrical in shape, having specified radii (shown in Fig. 4). For all tests the support span should be 14 to 20 times the thickness of the beam and the width should not be more than one-fourth the support span. The overhang at each end should be sufficient to prevent the specimen from slipping through the supports. The maximum fiber stress for the three-point system may be calculated using the following equation: S = 3PL/2db2,

(12.3.1)

where S is the stress in the outer fibers at midspan, P the load, L the support span, b the width of the specimen, and d the thickness of the specimen. The maximum stress in the four-point system occurs between the two loading noses and is given by Eq. (12.3.1) with the factor 312 omitted. The maximum strain for the three-point system is given as follows:

r = 6Dd/LZ,

(12.3.2)

where r is the maximum strain in the outer fibers, D the maximum deflec-

I26

12.

FURTHER MECHANICAL TECHNIQUES load

support span

(a) Radii for supports

min = 0.32 cm max = 1.5 x thickness

Radii for loading noses

L

min = 0.32 cm max = 4 x thickness

lood

L

7-7

Radii for supports and loading noses

min = mox =

0.32 cm 1.5 x thickness

(b) FIG.4. (a) Three-point loading flexure test. (b) Four-point loading flexure test.

tion at the center of the beam, L the support span, and d the depth. The deflection is usually measured with a dial gauge under the specimen in contact with it at the center of the span. For the four-point system, the maximum strain is given by Eq. (12.3.2) with the factor 6 replaced by 4.70. 12.3.4.4. Adhesives in Bonded Joints. The stress-strain relationships of polymeric adhesives should be observed in situ, that is, with the adhesive bonding two adherends together. Metallic adherends are useful

12.3.

STRESS-STRAIN Y I E L D TESTING OF SOLID POLYMERS

+; . :

p p . 3 5

' - 6 , 3 5 7

127

:

17.7 t L

FIG.5. Lap shear test specimen for adhesives (cm).

for transmitting a force to the adhesive layer. Free-standing, bulk test specimens can be prepared from adhesive materials; however, those results have little interest because the properties of adhesives are so strongly influenced by the constraints of the adherends that extrapolation from bulk to thin film behavior is risky. The epoxy adhesives discussed in this section are commercially available and are made up of bisphenol A diglycidyl ether coupled with amine or anhydride curing agents. The most common mechanical test for adhesives is the lap shear method described in ASTM D 1002. The adhesive to be tested is used to bond together two metal adherends having dimensions of 15 x 2.5 x 0.157 cm (see Fig. 5 ) . Usually the overlap length is 1.2 cm and bond-line thicknesses of 0.01-0.03 cm are common. A tensile load is applied with a rate of grip separation of 1.3 cm/min until failure occurs. The only property value given by this test is the fracture stress, which is obtained by dividing the fracture load by the overlap area. Due to the unsymmetrical nature of the lap shear specimen, the adherends are not fully aligned in the direction of the tensile force. When the load is applied the adherends bend at the overlap area. The resultant cleavage forces strongly affect the fracture stress. If the bending is minimized by using thicker adherends, the fracture stress is increased. When testing highstrength adhesives, the overlap area may have to be reduced to avoid failure in the adherends before the adhesive bond breaks. Although this test provides only the fracture stress and the results depend very strongly on the experimental conditions, it has gained wide acceptance as a standard for specifying adhesives and bonding procedures. Fracture in adhesive-bonded joints is characterized as cohesive or adhesive. Cohesive is the term used when fracture occurs within the adhesive (center-of-bond) so that each adherend remains coated with adhesive after the failure. If the adhesive material separates from the adherend when the bonded joint breaks, the failure is called adhesive. When both types of fracture are present, the major mode is quoted as a percentage of the entire fracture surface.

128

12.

FURTHER MECHANICAL TECHNIQUES

FIG.6. Cross section of microstrain adhesive tensile test specimen.

Alternative test methods have been developed for measuring the other mechanical properties of adhesives: modulus, yield point, plastic flow, creep and recovery, and fracture stress and strain. One of these new techniques has been named microstrain in recognition of the small deformation that must be measured in specimens having an effective gauge length of 0.02 cm, a typical adhesive bond-line thickness. Considering a strain of 5% in a gauge length of 0.02 cm the total deformation is only 0.001 cm (10 pm). A high-sensitivity capacitance-type extensometer is used to measure the deformation of bonded joints for tensile or shear loading. The butt-tensile specimen (see Fig. 6) consists of two rods, 1.2 cm diam, bonded together end to end. The specimen is loaded in tension (or compression) in the direction coincident with the longitudinal axis of the specimen. A capacitance-type extensometer is mounted as close as possible to the adherend-adhesive interfaces. The measured deformation includes a small amount of adherend extension, which must be subtracted from the overall reading. For a typical test, the adherend deformation is usually about 15% of the total measured extension. The load signal is fed to the X axis of a recorder and the output of the capacitance bridge to the Y axis. Stress-strain curves can be calculated from the recorded load-deformation data. With this system the stress-strain behavior of adhesives can be examined in great detail. To measure the shear properties of adhesives, a napkin-ring-type specimen has been developed. As shown in Fig. 7, it consists of two tubes, 7.5 cm outside diam. x 0.3 cm wall thickness, bonded together end to end. One adherend is held stationary while the other is rotated about an axis coincident with the longitudinal axis of the specimen. The capacitance-type extensometer is used to monitor the angle of twist as the adhesive is subjected to the shear forces. X- Y traces are obtained as in the butt-tensile tests. Great care must be exercised when preparing microstrain adhesive-bonded specimens to guarantee axiality and concentricity. Any misalignment during testing will invalidate the results.

12.3.

STRESS-STRAIN YIELD TESTING OF SOLID POLYMERS

plate

129

od hesive Ladherend rotates with torsion sprocket

FIG.7. Cross section of adhesive napkin-ring shear test specimen.

12.3.5. Significance of Results

12.3.5.1.Bulk Polymers. Test temperature has a strong influence on the mechanical properties of For example, the stress-strain curves of Fig. 8 illustrate the range of behavior that can be found as the test temperature is changed in a linear polymer. At very low temperatures the molecular segments are relatively rigid and cannot easily move past one another. As a consequence, the polymer behaves as a glassy solid-it has a high modulus, high fracture stress, and little ductility. When the temperature is raised, the atoms vibrate more vigorously, increasing the distance between them, making the structure less stiff, and allowing the molecular segments to deform more easily. There is a spectrum of micromodes of deformation in the polymer chain.I5 Each micromode is activated by its characteristic temperature. These characteristic temperatures are observed by means of mechanical spectroscopy (internal friction measurements). As the temperature is increased, more micromodes of the polymer chain are activated so that there is a decrease in the modulus and an increase in the non-linearity of the stress-strain curve as shown in Fig. 8. When a sufficient number of micromodes become active, then large-scale plastic deformation begins with the advent of a yield instability. The stress corresponding to the maximum in the stressstrain curve is a macroscopic yield point because it leads to large permaI' la

Y.Imai and N. Brown, Polymer 18, 298 (1977). S. Rabinowitz and N . Brown, J . Polym. Sci., Part A-2 5, 143 (1967).

130

12.

FURTHER MECHANICAL TECHNIQUES

OO

4

8 STRAIN

12 We)

FIG.8. Effect of temperature on stress-strain curves of a linear polymer (polychlorotrifluoroethylene). A, 77";B, 118": C. 150"; D, 169";E. 192";F, 220";G , 240";H, 275";I, 298"; J , 340"; K , 373'K.I'

nent strains and is the ordinary yield point that is usually tabulated in the literature. This maximum in the stress at the yield point can be observed in tension, compression, and in shear experiments. The maximum in the yield point and the subsequent yield drop is based on the concept of an intrinsic strain softening. l6 When sufficient micromodes of deformation become activated and lead to a large-scale yielding, there is an autocatalytic effect in that a general softening occurs, which leads to the drop in flow stress. The localized large-scale yielding propagates along the length of the specimen leading to a plateau region beyond the yield point. The process of long-range deformation leads to molecular orientation. In the case of tensile deformation there is a profound orientation strengthening. As the glass transition temperature is approached, the yield instability disappears possibly because strain softening is masked by the general softening caused by the higher temperature. There is a general observation1' in polymers that as the yield point of the polymer decreases (stress to produce a permanent strain), the modulus also decreases (see Fig. 9). IT

N. Brown and I. M. Ward, J . Polym. Sri., Part A-2 6, 607 (1968). N . Brown, Muter. Sri. Eng. 8, 69 (1971).

12.3.

STRESS-STRAIN YIELD TESTING OF SOLID POLYMERS

13 1

100

-

-8z

80

p"

60

w

a

I-

In D

-I

40

w>

20

'0

1000 2000 3000 4000 YOUNG'S MODULUS (MPa)

FIG.9. Yield stress vs. Young's modulus for a variety of linear polymers. CP, chlorinated polyether; PP, polypropylene; N6/6, nylon 6.6; Ng/lO, nylon 6,10; PC, polycarbonate; PA, polyacetal; PAC, poly(pheny1ene oxide).I7

The hydrostatic pressure also has an effect on both the modulus and yield point. A review of this effect has been presented by Sauer (Fig. 10).l**l@ The effect of hydrostatic pressure is to restrict chain mobility. As a consequence, the yield point in tension, compression, and shear have the following relationship: u,,(tensile) < u,,(shear) < u,,(compression), where all yield points are given in terms of the maximum shear stress in the specimen ( T , , , ~=~ 0.5 tensile stress). The differences in yield points between tension and compression are typically about 10%.

In

campmssive

OO

200 400 600 PRESSURE (MPa)

FIG. 10. Yield stress and elastic modulus vs. pressure for polychlorotrifluoroethylene.ls H. N. Yoon, J. A. Sauer, and K . D. Pae, J . Polym. Sci.. Polym. Phys. Ed. 14, 1611 (1976). IYJ . A. Sauer, Polym. Eng. Sci. 17, 157 (1977).

I32

12.

FURTHER MECHANICAL TECHNIQUES

TRUE STRAIN

(vi

Fic. 1 1 . Effect of strain rate on the stress-strain curve of oriented poly(ethy1ene terephthalate) at 300 K (%/rnin).*O

The effect of temperature on the stress-strain curves of highly crosslinked polymers such as epoxy adhesives is qualitatively the same as for the linear polymers (Fig. 8). However, the temperature ranges at which the various types of curves occur span the glass transition temperature. Extensive cross-linking severely restricts the mobility of the chain segments. Therefore, the low-temperature-type curve (A-D in Fig. 8) for linear polymers may occur nearly up to Tg for cross-linked polymers. Nonlinear behavior in the stress-strain curve becomes appreciable at Tg; however, large-scale yielding is not possible since it is restricted by cross-linking. As the temperature is raised so that cross links are broken, it is then possible to observe a drop in the load. This drop in the load is caused by the combination of stress and temperature, which breaks the cross links. At higher temperatures it is possible to observe curves of type I, J , K in shear testing. The stress-strain curve of polymers depends on the strain rateZoas regulated by the speed of the testing machine (see Fig. 11). The dependence of yield point on strain rate P is usually given by the following type of relationship: uYp = A + B In P , where A and B are material constants that also depend on temperature. The above equation arises because the process of deformation is thermally activated in accordance with the Eyring equation, P = C l v exp[-(Q - c r u ) / k a , 2o

M . Parish and N . Brown, J . Macrornol. Sci. Phys. 4, 649 (1970).

12.3.

STRESS-STRAIN YIELD TESTING OF SOLID POLYMERS

2

2.2

y

1.8

133

0 5

a z

1.4 BOND-LINE THICKNESS (IO-3cm)

FIG.12. Variation of effective tensile modulus with bond-line thickness for unfilled epoxy adhesive.

where Cl is a constant related to the density of units of deformation (called flow units), Y the vibrational frequency of the flow unit, Q an energy barrier that must be overcome to activate a flow unit (activation enenergy), u the activation volume related to the size and displacement of the flow unit, and c the stress. Generally the activation volume is a function of temperature and increases with increasing temperature. Q, the activation energy, is also a function of temperature. Thus, the change in the stress-strain curve due to increasing the strain rate is similar to the effect of decreasing the temperature. 12.3.5.2. Adhesive Joints. Bond-line thickness is an important experimental variable that affects the mechanical properties of adhesives.21 Figure 12 shows how the effective tensile modulus decreases in an unfilled epoxy adhesive as the bond line is made larger. If filler particles were added to enhance the electrical or thermal conductivity of the adhesive, the modulus values would be increased but the temperature dependence would be about the same. The minimum bond-line thickness that can be obtained with filled adhesives is established by the largest particle size. Adhesives that are supported by a carrier cloth have one bond-line thickness where the mechanical properties are optimized. Joints made with smaller bond lines are epoxy-starved and have lower strengths. Bond-line thicknesses greater than optimum often contain voids or have been cured with inadequate pressure leading to poor mechanical properties. The term “effective tensile modulus” is used for the results of the butt-tensile test (see Fig. 6) because the adhesive deformation is affected by the properties of the metallic adherends. That is, the adhesive material immediately adjacent to the adherend is forced to deform as the metal

** E. J . Hughes and J. L. Rutherford, “Study of Micromechanical Properties of Adhesive-Bonded Joints,” Tech. Rep. 3744. Singer Co., Little Falls, New Jersey, 1%8.

134

12.

FURTHER MECHANICAL TECHNIQUES

FIG. 13. Epoxy adhesive with carrier cloth tested with stainless steel adherends (upper curve) and aluminum adherends (lower curve).*'

dictates. Since Young's modulus is larger for metals than for polymers, the tensile modulus of the adhesive is increased. The magnitude of this effect is illustrated in Fig. 13, where the same epoxy adhesive was tested using stainless steel adherends (upper curve) with a modulus of 186 x 103 MPa and aluminum adherends (lower curve) having a modulus of 66 x lo3 MPa. Figure 13 also shows the temperature dependence of the modulus of this adhesive. The mechanical constraints due to the metal adherends penetrate the adhesive only a small distance from each interface. When the bond-line thickness (see Fig. 12) reaches about 0.02 cm, the interior, or unconstrained, layer predominates and the modulus value becomes constant as the bond line thickness is further increased. When the shear modulus of an adhesive is measured using the napkin ring test (see Fig. 7), the bond-line thickness effect is absent because the adherends do not deform when a torsional load is applied. There is evidence that the molecular structure of an adhesive suffers damage when a load is applied. For example, the shear modulus of an as-cured epoxy adhesive was 2.3 x lo3 MPa (Fig. 14). After applying a stress of 1.4 MPa the modulus dropped to 2.1 x 103 MPa. Reloading to 3 MPa caused a decrease to 1.9 x 103 MPa. The modulus continued to

n

2 v)

1.4 APPLIED STRESS (MPa)

FIG.14. Decrease in epoxy adhesive shear modulus caused by applied stress.

12.3.

STRESS-STRAIN

YIELD TESTING OF SOLID POLYMERS

135

drop after application of successively higher stresses until a limiting value of about 1.68 x lo3 MPa was reached at an applied stress of 10 MPa. It has been postulated that this phenomenon can be attributed to the breaking of bonds during the shear loading. A portion of the lower strength, stress-breakable bonds are broken during the first application of load, making the molecular structure less stiff. At higher applied stresses, another stronger set of bonds is broken, resulting in a further decay of the shear modulus. By the time the applied stress has reached 10 MPa all the breakable bonds have been exhausted and the elastic shear modulus becomes constant. The results reported in Fig. 14 were obtained from tests where no recovery time was allowed between subsequent load-unload cycles.

This Page Intentionally Left Blank

13. PRODUCTION AND MEASUREMENT OF ORIENTATION

By Ian L. Hay 13.1. Introduction One of the principal commercial uses of polymers is in the formation of fibers either for textile use or for fiber-reinforcing applications such as in tires or belts. These fibers, and also certain films, can exhibit very different property values when measured parallel and perpendicular to the fiber axis. This is true for such properties as, for example, thermal expansion, for which the value along the fiber axis is much lower than across it and may even be negative;' dimensional swelling agents; and the thermal conductivity, for which the value parallel to the fiber axis may be much higher than normal to it.2 Of greatest commercial interest is the high modulus exhibited parallel to the fiber axis compared with the bulk material. This anisotropy in properties arises because of the molecular orientation that exists in the fibers; the molecules are not randomly arranged but are roughly parallel to one another and to the fiber axis. It is, of course, a consequence of the intrinsic anisotropy of the molecular chain in respect to the different bonding parallel to the molecular axis and across it. In fact, theoretical estimates of the moduli of various polymers stretched parallel to their molecular axis have been made.3~4 It is only in recent years, however, that these moduli have been approached either by use of orientation techniques designed to maximize the degree of alignment of the m01ecules~-~or with polymers that are intrinsically stiff, such as poly (para-benzamide).' K. Nakamura, K. Imada, and M. Tdkayanagi, Polymer 15,446 (1974).

' C. L. Choy, W. H. Luk, and F. C. Chen, Polymer 19, 155 (1978).

F. C. Frank, Proc. R . Soc. (London). Ser. A , 319, 127 (1970). G. Capaccio and 1. M. Ward, Nature (London) 243, 130, 143 (1973). J . H . Southern and R. S. Porter, J . Macromol. Sci. (Phys.),B 4, 541 (1970). A. Zwijnenburg and A. J. Pennings, Colloid Polym. Sci. 254, 868 (1976). ' S . Kwolek. U.S.Patent 3,671.542. 137 METHODS OF EXPERIMENTAL PHYSICS, VOL. 16C

Copyright @ 1980 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-475958-0

138

13.

PRODUCTION A N D MEASUREMENT OF ORIENTATION

The properties of a fiber are therefore related to the degree of orientation that it exhibits, and in this part we discuss techniques that may be used in order to characterize the orientation. The techniques described are those most useful for the more highly oriented systems (such as fibers). Techniques mainly useful for lower degrees of deformation, e.g., light scattering, are not included.

13.2. The Production of Orientation It is well known that orientation is found in a number of naturally occurring polymer fibers. Typical examples are wool, cotton, jute, flax, silk, and spider threads. In most of these fibers, such as cotton, jute, and wool, the orientation is produced during the growth of the fiber, i.e., during the natural polymerization process. The degree of orientation may be quite high, as in flax, leading to a strong, stiff fiber or low, as in the case of fibers separated out from wood pulp. However, for silk and spider threads the orientation is produced by a spinning process. In the case of silk worms, it is the motion of their heads as they form their coccoon that pulls and orients the fiber, while spiders produce a well-oriented and strong fiber when they drop on their drag line. Orientation during polymerization has also been observed for manmade polymers. For example, fibrous crystals with chain axes parallel to the fiber direction have been observed in polyoxymethylene, polytetrafluoroethylene, polyethylene, and poly(viny1 chloride),s-lo where the crystals were either grown from the gas phase for polyoxymethylene; and polytetrafluoroethylene or from solution for polyethylene and poly(viny1 chloride). Another source of oriented materials from monomers is solid-state polymerization, in which oriented crystals of the monomers are polymerized to the polymer. Examples of this are polyoxymethylenell and polydiacetylene.12 To date, however, there is no commercial process based on producing fibers during the polymerization process. In general, polymers as prepared are not oriented, and only by subsequent processing can high degrees of orientation be developed. The most commonly employed technique is the so-called cold-drawing process, which despite its name is usually carried out at elevated temperatures so H. Staudinger, H. Johner, R. Signer, G . Mie, and J. Hengstenberg, Z . f h y s . Chem. 126, 425 (1927).

G. Buthenuth, Verh. Kolloid Ges. 18, 168 (1958). L. Melillo and B. Wunderlich, Kolloid Z . Z . folym. 250,417 (1972). I1 K . Hayashi, Y. Kitanashi, M. Nishi, and S. Okamura, Makromol. Chem. 47,237 (1967). lP R. H. Baughman and E. A. Turi, J . f o l y m . Sci.. Part A-2 11, 2453 (1973). @

lo

!3 2.

T H E PRODUCTION OF ORIENTATION

I39

as to be above the glass transition temperature but below the melting point of the polymer. It is characterized by being a solid-solid transformation in which unoriented material is converted to a material with a high degree of chain axis alignment by stretching or drawing, the deformation usually taking place in a highly localized zone, termed the neck region. The orientation, which is produced in the neck, occurs by a combination of crystal deformation or destruction, crystal rotation, and the extension of the polymer chains (Fig. 1). Cr ysta I deformation

Crystal rot a t ion

FIG. 1. Reorientation processes occurring in neck regions during the deformation of a polymer.

140

13.

PRODUCTION A N D MEASUREMENT OF ORIENTATION

Because of the large amount of strain hardening that occurs during orientation, the neck, once formed, is quite stable and this has made colddrawing the basis for most commercial fiber formation processes. In such processes an unoriented fiber is first spun from the polymer melt or solution and subsequently drawn-the degree of draw being controlled to give the desired properties for that fiber. Spin-draw processes of this type can be carried out at high speeds (a few thousand feet per minute), with the best properties normally achieved being moduli of about 10-20 GPa (for PET) and tenacities of 1.2 GPa. As mentioned in the Introduction, a great deal of attention has recently been focused on achieving the ultimate properties of polymers by process control. One of the most successful techniques has been by a more careful control of drawing than is achieved in standard fiber formation processes. Studies by Ward and C a p a c ~ i oClark,13 ,~ and Keller14 on the readily crystallizable polymers polyethylene, polypropylene, and polyoxymethylene have revealed the necessity for controlling the temperature and drawing speed as well as a marked dependence on the molecular weight, molecular weight distribution, and mode of crystallization of the ample.^ By control of these parameters, draw ratios as high as 40 have been achieved for polyethylene (- 10 is normally observed). Properties within a factor of two or three of theoretical have been r e p ~ r t e d . ~ A related process for production of orientation is that of solid-state extrusion5J5(Fig. 2). There are two basic techniques. In the first, the solid polymer is forced directly by a plunger through a die at temperatures close to the atmospheric melting temperature (Fig. 2a). Area reductions (which are equivalent to a nominal draw ratio) as high as 300 have been reported.Is However, the moduli measured are much lower than would be expected for a similar actual draw ratio. This process is characterized by high pressures and low extrusion speeds, e.g., 0.2 mm/min. The second technique is hydrostatic e x t r u ~ i o n . In ~ ~this the pressure is transmitted to the polymer via a hydrostatic fluid (Fig. 2b). which has the advantage of removing any friction at the walls of the pressure chamber and also lubricating the die. However, the useful range of area reductions that can be achieved is limited to about 10 for polyethylene and three for poly(ethy1ene tere~hthalate).'~Below these values rapid, uniform extrusion takes place at pressures of up to 1000 psi: above these values the pressures required are much higher and the extrudate generally is badly deformed. l5

-

E. S. Clark and L. S. Scott, Pol.vm. Eng. Sci. 14,682 (1974). P. J . Barham and A. Keller. J . Morer. S c i . 11, 27 (1976). la L. A. Davis. Po/ym. Eng. Sci. 14, 641 (1974). l s N . E. Weeks and R . S. Porter, J . PoI.vm. Sci., Pol.vm. Phys. Ed. 12,635 (1974). l3

I'

13.2.

tl%-

Heated barrel

(b) FIG.

141

THE PRODUCTION OF ORIENTATION

-rln.r

(0)

2. Extrusion processes for polymer orientation. (a) Ram extrusion and (b) hydro-

static extrusion.

Turning from fibers to films, orientation can also be produced by passing a sample between rollers (calendering). This has the action of uniaxially orienting the chain axes parallel to the rolling direction, but, in addition, can preferentially orient another crystal plane in the plane of the rollers. 17--20 Again, there are no well-known commercial applications for this process: however, it can be very useful in the laboratory for studying structure and deformation.'* On the other hand, there is a strong commercial interest in biaxially stretched films (e.g., polyester tapes) in order I'

*O

J . J. Point, Mem. et Soc. Sciences des Arts et des Lettres du Hainaut 71, 65 (1958). I . L. Hay and A. Keller, J . Mater. Sci. 1, 41 (1966). D. M. Gezovich and P. H. Geil, J . Mater. Sci. 6 , 509 (1971). J. J. Point, M. Dosiere, M. Gilliot, and A. Goffin-Gerin,J . Mater. Sci. 6, 479 (1971).

142

13.

PRODUCTION A N D MEASUREMENT OF ORIENTATION

FIG.3. Columnar crystals formed in polyethylene during melt extrusion. (Transmission electron micrograph of a stained thin section. Scale bar = loo0 A".)

to obtain films with balanced properties in the film plane. Such films are produced by initially forward stretching an extruded film, as with fiber drawing, and then transversely stretching using a tenter. In these films, the chain axis is oriented either randomly or with a greater or lesser degree in two preferential directions in the plane of the film leading to the desired better balance of mechanical properties. In some instances a particular plane may become oriented in the plane of the *l

C. J . Heffelfinger and P. G. Schmidt. J . Appl. Polyrn. Sri. 9, 2661 (I%%

13.2.

T H E PRODUCTION OF ORIENTATION

143

In contrast to the solid-solid transformation of cold-drawing, orientation can also be produced in polymers directly from the melt or solution by crystallization of an oriented melt, i.e., a melt-solid transformation. This type of orientation is produced most readily when a polymer melt or solution is subjected to a highly elongational flow field at temperatures close to the crystallization temperature of the polymer and then crystallized (an elongational flow field is one in which the velocity gradient is parallel to the direction of flow as opposed to shearing flow, in which the velocity gradient is normal to the direction of flow). The structures produced in this manner are characterized by a line nucleus parallel to the flow axis onto which the bulk of the polymer crystallizes as an oriented overgrowth leading to columnar crystal arrays parallel to the flow axis (Fig. 3). Depending on the number of line nuclei, the orientation may

FIG.4. Shish-kebob crystals formed frwn a stirred polyethylene solution. (Transmission electron micrograph. From Pennings el a / . , Kolloid. Z . Z . Polym. 236, 99,1970.)

144

13.

PRODUCTION A N D MEASUREMENT OF ORIENTATION

(b)

FIG.5. Highly oriented lamellar crystals in hard-elastic polypropylene. (a) Film sample held stretched 80%. Note voids and fibrils between lamellae. (b) Film sample after 80% stretch and relax. (Transmission electron micrographs of surface replicas. Emtrusion, stretch, and shadow directions are vertical.)

13.2.

THE PRODUCTION OF ORIENTATION

145

be low or maybe as high as that of a drawn fiber.22 In melts the structure has become known as row structure, from the parallel rows of nuclei, and in solutions as shishkebab (Fig. 4). In recent years this has been studied extensively, in particular by Mackley and KelleP3 for melts and by Pennings6 for solutions. This type of orientation is of interest commercially from several aspects. First, on the negative side, such crystallization can occur during molding, leading to undesirable properties of the fabricated part. Second, the hard-elastic fibers or films discussed by Clark,24Quynn et al. ,25 and more recently by a number of authors26-28can be formed. This material exhibits almost total elastic recovery from extensions as high as 100% (unoriented material forms a neck after about 50% extension; cold-drawn fibers have extensibilities of about 10-15%). Studies show that the oriented lamellae in fact separate on extension (Fig. Sa), forming voids that close up again on removing the stress (Fig. 5b). By a combination of extension and heat setting a stable microporous film can be achieved2e (Fig. 6). Third, it has been postulated that the line nuclei are fully extended polymer chains or at least should have mechanical properties close to theoretical value^.^ Pennings has reported polyethylene fiiaments with moduli and strengths better than those of the best highly drawn material^.^^ Thus, there exists the possibility of producing material with close to theoretical properties if the number of row nuclei can be enhanced. All of the examples in the above discussion of melt orientation processes have been for polyolefins; similar effects, however, have now been observed in poly(ethy1ene terephthalate) during high-speed spinning.31 In this case, however, the crystallinity of the as-spun samples is low and the range of orientations observed in the olefins is only produced on subsequent heat treatment of the fibers (Fig. 7). Thus, it is apparent that orientation in polymers can be produced by a number of distinct processes, either individually or in combination. These processes give rise to fibers having distinctly different properties and if an understanding of property structure relationships is to be achieved, then a detailed characterisation of the molecular orientation will be necessary. pp

25

30

A. Keller and M. J . Machin, J . Mucromol. Sci. (Phys. ), B 1, 41 (1%7). M. R. Mackley and A. Keller, Polymer 14, 16 (1973). C. A. Garber and E. S. Clark, J . Mucromol. Sci. (Phys.),B 4, 499 (1970). R. G. Quynn ef ul., J . Macromol. Sci. (Phys.), B 4 953 (1970). B . S. Sprague, J . Mucromol. Sci. (Phys.), B 8, 157 (1973). R . P. Wool, J . Polym. Sci. Polym. Phys. Ed. 14, 603 (1976). S. L. Cannon et a / . , Macromol. Res. 11, 209 (1976). M. Druin er d.,U.S.fatenf 3,679,538. A. Zwijnenberg and A. J . Pennings. J . Polym. Sci., Po/ym. Lett. Ed. 14, 339 (1976). E. Liska, KoNoid 2. Z. Polym. 251, 1028 (1973).

146

13.

PRODUCTION A N D MEASUREMENT OF ORIENTATION

FIG.6. Stable microporous polypropylene film. (Transmission electron micrograph of surface replica. Extrusion, stretch, and shadow directions are vertical. Scale bar = 10,OOO A". )

13.3. Description of Orientation By their very nature, the orientation processes just described lead to samples with a range of orientations of molecules or molecular segments. Ideally, one should be able to describe this distribution in terms of a unique function of 8, where, in the case of uniaxial orientation, 8 is the angle between the molecule or segment and the orientation axis (Fig. 8). In practice, there are several difficulties in achieving such a complete and analytical description of the orientation. In this chapter, we point out the reasons for this, show what can be achieved, and how this information may be used. The foremost difficulty to obtaining an analytical description of the orientation is that polymers are in general two-phase systems. One phase is crystalline and the other has traditionally been described as amorphous. This latter phase arises because, due to their length, individual molecules are not confined to one crystallite but can meander from one crystalline

13.3. Spinning Speed m/min.

1450

1800

DESCRIPTION O F ORIENTATION

2500

2000

3000

147

3500

C

d FIG. 7. Wide- and small-angle x-ray patterns from poly(ethy1ene terephthalate) spun at various speeds. Wide-angle patterns: (a) as-Spun, (b) annealed free to shrink, and (c) annealed at constant length. Low-angle patterns: (d) annealed at constant length. (From E. Liska, Kolloid Z. Z. Po/.vt?i. 251, 1028, 1973.) ation

Ori

5

I

- I

F I G .8. Diagram to show disorientation of the molecular axis with respect to the orientation direction.

148

13.

PRODUCTION A N D MEASUREMENT OF ORIENTATION

region to another. In these intercrystalline regions the molecules will be constrained and cannot pack in a regular lattice. However, when a sample is deformed, molecular segments in both crystalline and intercrystalline regions can become oriented, for example, as do the molecules in a wholly amorphous cross-linked rubber. Thus, it should be emphasized that in the case of polymers, the term “amorphous” is not synonymous with “unoriented” or “lacking structure.” The existence of two phases also leads to the concept of a supramolecular or morphological structure in polymers, i.e., the arrangement of the crystalline and noncrystalline regions. Since the basic morphological units are commonly lamellar, the supramolecular structure may also be oriented as was the case in the hard-elastic materials. Hence, for a complete characterization of the orientation all three aspects, crystalline, amorphous, and morphology, must be individually determined together with a measure of the relative amounts of crystalline and amorphous phases. In general, the determination of the distribution in orientation for even a single phase is at best complex. Take as an example what is probably the most straightforward case: that of the crystalline phase. Crystalline orientations are most readily determined from x-ray diffraction since the spread of an x-ray reflection shows directly the spread in orientation. A single wide-angle diffraction photograph, however, shows only the spread in orientation in a particular cross section of the sample. In order to obtain a complete characterization over the whole volume, a large number of photographs taken at various sample orientations is required. In practice, certain symmetries help to reduce this number. In particular, if the sample possesses fiber symmetry with the molecular axes well aligned, then one photograph is sufficient to characterize the orientation. In the case of the amorphous orientation there are no techniques, com,parable to that of x rays for the crystalline phase, which lead to a direct analytical evaluation of the amorphous component. In the first place, the property being measured, e.g., birefringence, may be a combination of the amorphous and crystalline contributions requiring a prior knowledge of the crystalline distribution in order to evaluate that of the amorphous. Secctid, a single value of the property is obtained that is an average of the contiibutions from all orientations. 5ince an analytical description of the amorphous orientation cannot be obtained, it must be expressed in terms of an orientation function. The concept of orietation functions was initially introduced by her man^^^ in studies gf cellulose fibers but is now used more generally. It is most 32 P. H. Hermans, “Contributions to the physics of Cellulose Fibres.” Elsevier, New York, 1946.

13.3.

DESCRIPTION OF ORIENTATION

149

useful when comparing the orientations of different samples from different polymers measured by the same technique, but also can be used to calculate the value of orientation-dependent properties that cannot be measured directly. The orientation function f for any anisotropic property P is defined by

f

PI1 = p10-

P, P,O'

(13.3.1)

where PI,and P, are the values of P measured parallel and perpendicular to the orientation axis, respectively. PI: and Pl0 are these values for the case of fully aligned molecules. One physical interpretation offis that in a system containing only molecules that are either fully aligned or randomly oriented, then: J' is the fraction of fully aligned molecules, and 1 - f the fraction of random molecules.33 Although a composite interpretation in these terms can help to give a picture of the meaning of the orientation function, it is only useful when making comparisons between measurements that have the same dependence on orientation since, under such circumstances, it is not necessary to know the actual distribution with respect to 8. In general, however, when a measured property is being used to predict the value of another property that cannot be measured directly, the orientation dependence of each will be different. To be of general use, therefore, it is necessary to evaluate the actual 8 dependence of the property being measured. This was done by her man^^^ for birefringence, leading to the well-known Hermans optical orientation function:

f

3 cOs26 - I - -An = 2 Ano'

(13.3.2)

i.e., the ratio of measured birefringence to ultimate birefringence gives a mean value of cos26 for the distribution. Nuclear magnetic resonance and - laser Raman spectroscopy, on the other hand, lead to a value for cos48 of the distribution. Orientation functions evaluated in this way have three uses. First, if the analytical 8 distribution is known for the crystalline phase from, for example, x-ray diffraction then its f value can be evaluated and used to determine the separate amorphous and crystalline distributions. Second, one may have a model for the drawing or deformation process that predicts 8 and hence can calculate f i n order to make comparisons with observed values. Third, they can be used to evaluate coS28 and cOS'8 for 33

R. D.B. Fraser,J. Chem. Phys. 21, 1511 (1953).

13.

150

PRODUCTION A N D MEASUREMENT OF ORIENTATION

substitution in models used to predict the mechanical properties of oriented systems.34 As already noted, determination of the orientation functions from a measured property requires knowledge of the relative amounts of the amorphous and crystalline phases. The most direct method for evaluating this is from the density of the sample p , using (13.3.3) where xcris the volume fraction of the crystalline phase, pcr the density of the crystalline phase, and pam the density of the amorphous phase. However, materials that are either 100% crystalline or 100% amorphous are not usually available and hence pcrand Pam have to be estimated. per can be calculated from the crystalline structure determined from x rays. pammay be estimated by extrapolation from the density of the melt. An added complication in oriented systems is that the density of the oriented amorphous material may differ from that of the unoriented. This possibility has been discussed in some detail for PET by several author^.^^^^

13.4. Measurement of Orientation 13.4.1. Wide-Angle X-Ray Diffraction 13.4.1.1. Principles and Procedures. The use of x rays to examine preferred orientation, such as exists in oriented polymers, differs from that often encountered in other applications of x rays, for example, crystal structure determinations. In the latter case, x rays are used as a tool to probe the distribution of atoms that exists in the crystal and, to achieve this, a detailed knowledge of the interaction of the x-ray beam with an atom is required. In the analysis of preferred orientation, the crystal structure is already known and the x-ray beam is used as a probe to examine the distribution of crystallites within a sample. Thus, only knowledge of the general conditions for a crystal to give rise to a diffracted beam is required and these can be deduced from three basic relationships: ( I ) the Bragg equation, (2) the reciprocal lattice, and (3) the Ewald construction. A crystal is a regular array of atoms formed by the repetition in three diI. M. Ward, “Mechanical Properties of Solid Polymers.” Wiley, New York, 1971. E. W. Fischer and S . Fakirov. J . Muter. Sri. 11, 1041 (1976). 36 W. L. Lindner. Polymer. 14, 9 (1973). s4 35

13.4. MEASUREMENT OF ORIENTATION

151

FIG.9. Crystal structure of polyethylene. (From A. Keller, Kolloid Z . Z . Polym. 197,98, 1964.)

mensions of a basic building block-the unit cell of the crystal (Fig. 9). Certain planes will be very dense in atoms and, in fact, these planes form the familiar facets of the crystal. At a certain relative orientation of a particular set of planes and an incident x-ray beam, the x rays will be reflected as shown in Fig. 10. The condition that has to be satisfied for a reflection to occur is the Bragg equation: (13.4.1)

2d sin 8 = nA,

where A is the wavelength of the incident radiation, d the interplane Incident beam

Diffracted beam

t FIG.10. Diagram to illustrate the Bragg diffracting condition. Difference in path length between successive rays is 2d sin 8.

152

13.

PRODUCTION A N D MEASUREMENT OF ORIENTATION

FIG.11. Diffraction pattern from a highly oriented polyethylene fiber.

spacing, and 8 is defined in Fig. 10. Figure 11 is a diffraction pattern from a polymer fiber, and the interpretation of such patterns is facilitated by the other two relationships referred to above. The importance of the reciprocal lattice is that it represents the relative orientations and spacings of sets of planes in the real lattice by single points. It is constructed by drawing from an origin vectors r of length l/d, in the direction of the normal to the set of planes and placing a point at the end of each vector. For proof that the set of points generated in this way gives rise to another lattice, i.e., the reciprocal lattice, the reader is referred to the standard texts.37 Figure 12 shows the reciprocal lattice for the real lattice in Fig. 9. The planes in a real lattice are referred to by their Miller indices, which are defined in terms of their intercepts on the axes of the unit cell. Again it can be that the coordinates of a 37

R. W. James, The Optical Principles of the Diffraction of X-rays.” G . Bell, London,

1958.

13.4.

MEASUREMENT OF ORIENTATION

153

point in the reciprocal lattice are the same as the Miller indices of the plane it represents. The x-ray beam can be represented in the reciprocal lattice by a vector of length l/h, i.e., a reciprocal length, and to determine whether a particular set of planes will diffract use is made of the Ewald construction. This construction (Fig. 13) consists of drawing a sphere of radius 1/A with one diameter ( Q O )parajlel to the x-ray beam and one end at the origin of the reciprocal lattice. If a lattice point P is lying on the sphere, then the LOOP is the angle between the incident x-ray beam and the normal to the set of planes represented by P:

LQOP = 90 - 9,

(13.4.2)

L O Q P = 29.

(13.4.3)

OP = 2 0 Q sin 9,

(13.4.4)

It follows that

and substituting for OP and OQ

2 sin 9 -1 = d

h

FIG. 12. Rccipwcal lattice for an orthorhokbic unit cell.

(13.4.5)

154

13.

PRODUCTION AND MEASUREMENT OF ORIENTATION

I

(a)

(b)

FIG.13. (a) Geometric construction of the Ewald sphere of reflection. (b) Corresponding x-ray pattern for (a).

and rearranging,

2d sin 8 = A,

(13.4.6)

i.e., for a point touching the Ewald sphere, the Bragg relationship is satisfied. The diffracted beam is in the direction of the radius QP. The Ewald sphere is often referred to as the sphere of reflection. In general, the chance that a reciprocal lattice point will lie on the sphere of reflection is small and to overcome this in single-crystal studies, for example, the crystal is rotated about an axis. Hence the lattice points are swept through the sphere of reflection. A sample composed of small crystallites with random orientation is equivalent to completely randomizing the reciprocal lattice by rotation about three perpendicular axes. Thus, each point in the lattice will sweep out a sphere in reciprocal space centered on the origin of the lattice. This is called the sphere of position and will intersect the sphere of reflection in a circle (Fig. 14), giving rise to the familiar diffraction rings for a randomly oriented sample. It follows that that the diffraction pattern that a particular nonrandomly oriented polycrystalline sample will exhibit can be deduced by first representing the sample in reciprocal space from consideration of the effect of the disorientations on the reciprocal lattice, and then examining the inter-

13.4.

I55

MEASUREMENT OF ORIENTATION

(a)

(hl

FIG. 14. (a) Intersection of the Ewald sphere and the sphere of position in reciprocal space for a randomly oriented sample. (b) Corresponding x-ray pattern.

section of this representation with the sphere of reflection. Particular examples of structures found in oriented polymers are discussed below. The most important example is that of a fiber. In this case, one axis of each polymer crystal is parallel to the others, forming the fiber axis, and the other axes are randomly distributed about this. The distribution in reciprocal space will be generated by rotating the reciprocal lattice about the fiber axis, i.e., each point will describe a circle centered on the fiber axis. In practice, the crystal axis shows some disorientation about the fiber axis and hence the rings become bands (like the layers of a sliced onion), as shown in Fig. 15. With the x-ray beam normal to the fiber axis, the sphere of reflection will cross these bands, giving rise to the arcs in the diffraction pattern as shown in Fig. 11. By convention, reflections that lie on the normal to the fiber axis, through the origin, are called equatorial and are due to planes parallel to the fiber axis. Reflections that lie on the fiber axis are due to planes perpendicular to the fiber axis and are called meridional. If the x-ray beam is parallel to the fiber axis, rings due to the equatorial bands will be observed. At intermediate orientations the reflections will be asymmetric along the fiber axis. Tilted fiber patterns are used in order to examine lattice points that lie on or close to the meridian. As Fig. 15

156

13.

PRODUCTION A N D MEASUREMENT OF ORIENTATION

c /

I (0)

(b)

FIG.IS. (a) Intersection of the Ewald sphere with the reciprocal space representation of a fiber. (b) Corresponding x-ray pattern.

shows, the sphere of reflection only touches the fiber axis at the origin. For high d spacings and low values of 8, the disorientation is sufficient to bring a lattice point on the meridian into reflecting position. However, at high 8 values, the disorientation is too low to give rise to a reflection and hence the fiber must be tilted, by the Bragg angle for the reflection, in order to observe it. Another commonly observed orientation is found in polymer films and is called the planar orientation. This structure is characterized by one crystal axis, e.g., the chain axis, usually lying in the film plane, with the other axes randomized about it. To develop the reciprocal space representation of this structure, it should be noted that it is equivalent to a random mat of oriented fibers in which the fiber axes all lie in the plane of the mat. In reciprocal space, the fiber pattern circle due to, e.g., an equatorial lattice point, will be rotated about a diameter sweeping out a sphere. However, the density close to the poles of the sphere will be greater than at the equator (cf. the lines of latitude on a globe) (Fig. 16). Thus, with the x-ray beam parallel to the film plane, the pattern will be more intense on the normal to the film plane; while with the x-ray beam normal to the film plane the pattern will show uniform circles. A related pattern is found in both fibers and fdms that exhibit the socalled row structure.2z In this instance, the sample is composed of co-

13.4.

157

MEASUREMENT OF ORIENTATION

(0)

(b)

16. (a) The reciprocal space representation of a planar orientation. sponding x-ray pattern. FIG.

(b) Corre-

lumnar spherulites (Fig. 3) parallel to the extrusion direction, and hence a cross section normal to this axis will have a planar orientation with the radial axis of the spherulite lying in the plane and the other axes random about it. Therefore, an x-ray pattern taken with the x-ray beam normal to the extrusion axis will exhibit the planar orientation just discussed. In polyethylene, for example (Fig. 17), this leads to an intensification of the 200 reflection on the meridian (extrusion axis) and the 020 reflections, defining the spherulite radius, on the equator. As already emphasized, two reflections are not in themselves sufficient to make an unambiguous determination of the orientation. The distribution of a third reflection, e.g., 002, has to be examined. For a row structure the 002 and 200 distributions will be the same. If, however, the a axis is truly preferentially oriented in the extrusion direction, then the 002 and 020 reflections would have similar distributions. 13.4.1.2. Pole Figures. Working with three-dimensional diagrams in reciprocal space presents many difficulties when one wishes to represent a particular distribution on paper. To overcome this, it is customary to work with a modified version of the pole figures used to describe the orientation of single crystals.34 In this application, the distribution of a

158

13.

PRODUCTION A N D MEASUREMENT OF ORIENTATION

FIG. 17. X-ray pattern of a polyethylene fiber with orientation equivalent to that illustrated in Fig. 16.

single pole on its sphere of position is considered and a diametrical plane drawn normal to a given axis of interest (e.g., the normal to the fiber axis in a fiber) (Fig. 18). This will be the plane of projection. Straight lines are then drawn from points on the sphere to the south pole and the points of intersection with the plane of projection are marked with the corresponding intensity of the particular pole of interest. Contour lines of equal intensity can then be mapped out and hence the pole distribution, for this particular projection, plotted. To fully describe any particular orientation, several pole figures are required. Figure 19 illustrates pole figures for some of the orientations already referred to for three different poles and for different projection axes. 13.4.1.3. Quantitative Evaluation of Wide-Angle X-Ray Orientation. To quantitatively define the distribution function it is satisfactory in some instances to utilize photographs and microdensitometry. Such an evaluation using single photographs is satisfactory, for example, for a well-oriented equatorial reflection in a fiber pattern or for a meridional re-

13.4.

159

MEASUREMENT OF ORIENTATION

Azimutha I

XI' \

,Plane of projection

South pole FIG.18. Construction of pole figure for an azimuthal reflection of a sample with fiber orientation.

flection in the same sample if the fiber axis has been tilted through the Bragg angle for that reflection. However, for less well-defined orientations, a number of selected photographs would have to be analyzed. An alternative technique is to use a pole iigure attachment mounted on a diffractometer. With this device a specimen can be rotated about a fixed axis N, e.g., the normal to the plane of the film. With the goniometer set for a particular reflecting angle (i.e., a particular pole) rotation about N will sample the intensity distribution of this pole around a section of the sphere of reflection normal to N (Fig. 20). By varying the orientation of N with respect to the plane of the goniometer, the intensity distribution over the whole of the sphere of position can be sampled. Due to geometrical considerations in the goniometer, it is more convenient to sample certain regions of the sphere in transmission and other regions in reflection. These two regions can be made to overlap and hence intensities normalized for the two cases. Finally, due to the small crystallite size, the reflections have a finite

13.

160

PRODUCTION A N D MEASUREMENT OF O R I E N T A T I O N

Or ienta t ion type

Projection axis

Random

Any

Fiber

Fiber axis ve rt ica I

Fiber

Fiber axis in plane of project ion

Projection pole

(010)

0

Planar

N vertical

Planar

N in plane of projection

W

FIG.19. Pole figures for various types of orientation of an orthorhombic unit cell.

width (i.e., 28 & A28). To correctly evaluate the intensities, the reflections should be scanned over this width and integrated. However, two approximations can be utilized. The first is that the intensity is proportional t o the peak height: this assumes that the linewidth is constant a t all points. A closer approximation is to use the peak height times the width at half the peak height (i.e., the half-width).

13.4.

MEASUREMENT OF ORIENTATION

161

Detector

II

X

FIG.20. Diagram to illustrate the geometry of a diffractometer pole figure attachment. As shown p-rotation samples the intensity on the locus of pole. By successive a-tilt and p-rotation the whole spherical locus of the pole can be sampled. 20 rotation selects other poles.

The pole figure can then be constructed by plotting the intensities determined on a Wolff net3' and constructing contours of equal intensity as described earlier. Alternatively, the measured intensity distribution can be used to determine cOs2e and the orientation function evaluated. 13.4.2. Birefringence 13.4.2.t. Description of Birefringence. The measurement of birefringence is probably one of the most widely used techniques for studying and comparing orientation in similarly processed polymers. It may even

162

13.

PRODUCTION A N D MEASUREMENT OF ORIENTATION

be used for process control where a particular level of birefringence is desired. Moreover, in contrast to x-ray diffraction, which determines principally the orientation of crystalline regions, birefringence is a measure of the total molecular orientation. Hence, in two-phase systems it can give additional information. It is, however, handicapped by the difficulties in translation of birefringence into orientation values, the reasons for which will become apparent in the following discussions. The velocity of light (c) in a medium depends on its polarizability (P) and is given by the Lorentz equation n2--

1 n2+1

4 3

- - ITP,

where n = co/c, n is the refractive index and co the velocity of light in vacuo. Due to the anisotropic nature of the bonding of polymer molecules, the polarizability parallel to the chain will differ from that perpendicular to it. Hence, the refractive index with respect to a light wave with its electric vector parallel to the chain will differ from that for a wave with its electric vector perpendicular to it, i.e., the polymer chains are intrinsically birefringent, the birefringence being the difference in refractive indices. The degree of birefringence for a particular molecule will depend on the molecular structure. In poly(ethy1ene terephthalate) the polarizability along the molecule is much higher than that perpendicular to it and the intrinsic birefringence is large (approximately 0.25). For polypropylene, the difference is much smaller and the birefringence is only about 0.02. If the refractive index is largest along the molecule, the molecule is said to have a positive birefringence [e.g., poly(ethy1ene terephthalate) or polyethylene]. In some instances, e.g., polystyrene the polarizability is largest perpendicular to the chain. In this case the birefringence is intrinsically negative. For randomly oriented polymer molecules, differences in refractive index will average out and the sample will not show any residual birefringence. If, however, all of the chains are fully aligned, the refractive indices perpendicular and parallel to the alignment axis will have the Values associated with the individual molecules and the birefringence will be maximum. If the molecules are partially aligned, the sample will exhibit a birefringence between these extremes and its value can be used as a measure of orientation. A uniaxially oriented sample such as a fiber, for example, can be characterized by two principal refractive indices: one parallel to the axis and the other in the plane normal to it. Hence, the orientation can be characterized by a single value of birefringence, i.e., the difference of these two principal refractive indices. In the general case, however, there are three

13.4.

I I I I

EP

y I I

I

I

*\d I

Light source

\ I

,

, , ,

I63

MEASUREMENT OF ORIENTATION

I


Detector

principal refractive indices along mutually perpendicular directions, and complete characterization of the orientation requires measurement of two independent values of birefringence. This second case is the biaxial case, which may exist, for example, in films drawn in two directions. We do not consider this type of orientation further here and refer the reader to papers by Stein38for further information. 13.4.2.2. Measurement of Birefringence. The direct measurement of refractive indices parallel and perpendicular to the axis is not very satisfactory for determination of birefringence since it involves calculating a small difference between two large numbers. However, such techniques can on occasion be useful and the interested reader can find details of relevant immersion techniques in books on optical crystall~graphy.~~ It is more convenient to measure the birefringence directly, and this can be achieved utilizing polarized light in a system such as that shown diagrammatically in Fig. 21. Light from the source L passes through a polarizer and an analyzer in crossed orientation so that no light is transmitted. The sample is placed between the polarizer and analyzer so that its principal axes are at 45" to the axis of the polarizer. A polarized beam of light incident at an angle 4 to the major axis of a specimen, i.e., the larger of the two principal refractive indices, can be considered to be split into two beams, one vibrating parallel to the major axis and the other parallel to the minor axis. Due to the different velocities in these directions, the two beams will develop a phase difference, the value of which is dependent on the difference in refractive indices and the thickness of the sample. This in turn modifies the polarization of the transmitted light, and when the two beams are recombined using the ana38

R. S. Stein, J . P d y m . Sci. 24, 383 (1957). E. E. Wahlstrom, "Optical Crystallography.'' Wiley, New York, 1960.

164

13.

PRODUCTION A N D MEASUREMENT OF ORIENTATION

lyzer in the crossed position then in general there will be some light transmitted. It can be shown that the transmitted intensity It is a maximum when 4 = 45” and is given by It = sinZ(7rR), -

(13.4.7)

I0

where R is the phase difference, given by R = - t- - t A,, hi’

(13.4.8)

where t is the sample thickness and A,,and hi are the wavelengths parallel and perpendicular to the major axis in the sample. Using

n=cO=o c A’

(13.4.9)

then

Hence, (13.4.11)

Knowing t and Ao, the birefringence A n can be estimated from the transmitted intensity. This technique is of particular interest for dynamic studies.40 The most satisfactory technique for measuring A n , however, is to use a compensator. For this, a precalibrated wedge or rotating crystal is introduced between the polarizer and analyzer, in addition to the sample, to introduce a variable phase difference in the path. This phase difference is adjusted so as to compensate for the phase difference introduced by the sample and the retardation is read directly from the calibration for the wavelength used. Only the thickness is then required to calculate A n . Some care must be exercised if the value of the retardation is large. In this case, the transmitted light will go through several maxima and minima as the compensation is increased, and difficulty may be experienced in identifying the zero order, i.e., when the phase difference is completely compensated. If An is not too large, then it is usually sufficient to initially compensate using white light, for which the higher order fringes are col‘O

R . S. Stein, S. Onogi. and D. A. Keedy, J . Po/ym. Sci. 57, 801 (1962).

13.4.

165

MEASUREMENT OF ORIENTATION

ored and only the zero order is dark, and then switch to a monochromatic source. However, for high birefringences, differences in dispersion (the variation of refractive index with wavelength) between the compensator and the sample make identification of the zero order fringe difficult. In this case, a wedge can be cut at the edge or end of the sample and the fringes actually counted since one is going from zero thickness to the full thickness of the specimen. The reader is referred to various references" for further details of this technique. 13.4.2.3. Estimating the Orientation. In order to derive information on the molecular orientation from the measured birefringence, it must be taken into account that there are several possible contributions to birefringence. These are additive and can be written An

=

Anform+ Andel + Ancr + Anam

(13.4.12)

where Anformis the contribution from the form birefringence, Andef the deformation birefringence, Ancr the crystalline contribution, and Anam the amorphous contribution. Only the latter two are related to orientation. The form birefringence is a consequence in two-phase systems of the dispersion of one phase in the other. The general conditions for it to occur are that the two phases have different refractive indices, that at least one dimension of the dispersion is of the order of the wavelength of light, and that the structure, as opposed to the individual phases, is anisotropic. A recent observation of the occurrence of this type of birefringence was by Folkes and Keller42 in extruded cylinders of polystyrene -polybutadiene block copolymers. The structure of the cylinders was of parallel polystyrene rods about 300 A in diameter dispersed in a polybutadiene matrix. All of the observed birefringence in these cylinders was accounted for by form effects, there being no residual birefringence in the individual phases. Semicrystalline polymers are twophase systems and Stein has pointed out that form effects can contribute as much as 5-10% of the observed birefringence in polyethylene. However, Samuel~'~ showed that the contribution of form birefringence in polypropylene was negligible. Steinu also discusses means of determining this contribution by swelling the material with a liquid so as to match the refractive indices of the two phases. Under these conditions, the form birefringence goes to zero and the observed birefringence will be a maximum or minimum depending on whether the form contribution is positive or negative. M. A. Sieminski, Microscope 23, 35 (1975). M. J . Folkes and A. Keller. Polymer 12, 222 (1971). I.3 R. S. Samuels, "Structured Polymer Properties." Wiley, New York 1974. F. A. Bettelheim and R. S. Stein, J . Polym. Sci. 27, 567 (1958).

"

166

13. PRODUCTION

A N D MEASUREMENT OF ORIENTATION

Deformation birefringence does not usually have a significant value in oriented polymers and is mentioned for completeness. It can arise when a sample is compressed or dilated without change in orientation. Changes in bond angles and/or interchain spacings can induce changes in polarizability and lead to the development of this type of birefringence. Having allowed for these two contributions, the remaining birefringence is due to orientation effects in the crystalline and amorphous regions. By definition, the orientation function for birefringence [see Eq. (13.3. l)] is rill -

nL

= n, - n,

- An

(13.4.13)

Ano'

where Ano is the intrinsic birefringence for the system. Obtaining a quantitative value for the orientation in the sample requires two steps: derivation of a function relating the ratio An/Ano to the average orientation of the birefringent units, and evaluation of the intrinsic birefringence Ano. For a semicrystalline polymer, values of Ano for both crystalline and amorphous phases must be known. The relationship between f and orientation was derived by her man^.^^ He considers a fiber as a single-phase system composed of birefringent units with principal polarizabilities aland uz. He then calculates the contribution to polarizability parallel to the fiber axis (q) of one of these units oriented at an angle 8 to the fiber axis. He shows that ull = f(ul + 2az) +

3(al - uz)(l- 8 sinz8).

(13.4.14)

Since polarizabilities are additive, for No units oriented over a range of values of 8 this becomes PI,= No[(al + az)+ 3(al - uz)(l- Q sin28)].

(13.4.15) ,

Similarly

pL =

+ 2az) - +(a,- uz)(i

-

Q Sinze)].

(13.4.16)

Subtracting (13.4.15) from (13.4.16) and rearranging, this becomes

where P1and Pz are equal to NOuland N o a z , respectively. As long as rill - nL is small compared with nl or n 2 , then this can be rewritten as ( 13.4.18)

This is the Hermans orientation function referred to earlier [Eq. (13.3.3)].

13.4. MEASUREMENT

OF ORIENTATION

I67

Alternatively, it can also be shown that

(13.4.19) where n is the mean value of refractive index for a sample. The greatest barrier to the use of birefringence in evaluating orientation functions has been in the determination of values for intrinsic birefringence, since samples that are 100% crystalline or totally amorphous and fully oriented cannot be produced and only the average refractive index is observed in unoriented samples. Calculation of these values has been attempted using bond polarizabilities. This has been done for the amorphous phases of polyethylene and polypropylene by Stein45 using the bond polarizabilities derived by Denbigh46from gas phase measurements, i.e., assuming no interaction effects between the molecules. This has given results in good agreement with experimentally determined values. One of the few estimates of crystalline birefringence was for polyethylene by B ~ n n . ~In ' this case, from consideration of paraffin crystals, it was shown that the Denbigh values could not be used and polarizabilities calculated from the oligomers were employed. The reason for this is that in the solid the molecule will be affected by the polarization field of the surrounding molecules. In the amorphous phase, the internal field will be essentially isotropic but in the crystalline regions the more ordered structure will lead to anisotropy of this field. Stein has examined the effect of internal field on birefringence in polymer crystals and accounted for the observed values in polyethylene. An alternative technique for evaluating amorphous birefringence and crystalline birefringence has been used by Samuels for p o l y p r ~ p y l e n e ~ ~ and Dumbleton for poly(ethy1ene terephthalaterg from sonic velocity measurements (see Section 13.4.3).

13.4.3.Sonic Modulust 13.4.3.1.Principles. The velocity of sound in a medium depends on the modulus in the direction of propagation of the sound. It can readily be seen therefore that the velocity of a sound wave propagated parallel to a polymer chain (i.e., the high modulus direction) will be greater than that 45 46

D. A . Keedy, J . Powers, and R. S . Stein, J . Appl. Phys. 31, 1911 (1960). K . G. Denbigh. Trttrrs. F~trcrdoySoc. 36, 936 (1940). C. W . Bunn and R. deP. Daubeny, Trcrris. Fctrctdny Soc. 50, 1173 (1954). D. R. Holmes and R. P. Palmer, J . Po/yru. Sci. 31, 345 (1958). R. J . Samuels, J . Po/yr?r. S c i . . Pcrrt A-2 3, 1741 (1965). J . H . Dumbleton J . Po/yrn. S r i . . Port A-2 6, 795 (1968).

47h

49

See also Chapter 12.1 (this volume).

13.

168

PRODUCTION A N D MEASUREMENT OF ORIENTATION

0

so

H. M . Morgan, Textile R t s . J . 32, 866 (1962). J . W . Ballou and S. Silverman, Textile R e s . J . 14, 282 (1944).

c _ _

13.4.

MEASUREMENT OF ORIENTATION

169

distance between maxima will be half a wavelength (A/2). The velocity is given by c

=

nA.

(13.4.20)

Typically, A lies in the range 5-20 cm. The alternative technique is, essentially, to measure the velocity directly by measuring the transit time of a pulse between two points along the fiber. In order to avoid excessively large sample lengths, however, the technique used in this case5'; is to modulate the amplitude of a high-frequency carrier wave (e.g., 10 kHz) with a much lower frequency pulse (e.g., 100-200 Hz). The phase difference between the amplitude at the transmitter and detector is then measured and translated into a transit time between these two points. With the modulation frequencies quoted above, phase differences can be adequately detected for sample lengths greater than a few centimeters. Again, end effects due to the transmitter or detector can be eliminated by measuring the transit time for several sample lengths. This technique lends itself to the continuous measurement of the sonic modulus. Moseleys2in fact used it to monitor changes in sonic modulus during stress-strain tests, and Fig. 23 is a diagram of his apparatus. He also used it to examine variations in orientation along a drawn fiber by running the fiber slowly over the transmitter and detector. 13.4.3.2.Estimating the Orientation. The calculation of the orientation dependence of the sonic velocity has been treated by M ~ s e l e and y~~ Ward.= Both of these authors start with similar assumptions to those used by Hermans in his treatment of birefringence. They consider a fiber to be a single-phase system composed of units with a range of orientation angles (0) with respect to the fiber axis, each unit having the properties that a fully oriented fiber would exhibit. Implicit in this assumption is that changes in orientation do not affect the properties of the individual units. M ~ s e l e ythen ~ ~ makes the assumption that the force constant for deforming this unit depends on the cosine of the angle (0) that it makes to the fiber axis. He considers two cases. The first corresponds to assuming the addition of deformations (equivalent to a uniform stress distribution or a Reuss- or Maxwell-type averaging)." The second assumes the addition of forces (equivalent to assuming uniform strain or a Voigt-type averagehs5 By comparison of experimental results with those predicted from his W. W. Moseley Jr., J. Appl. Polyrn. Sci. 3, 266 (1960). I. M. Ward, Textile Res. J. 34, 806 (1964). s4 See. e.g., Ref. 34 p. 89. I5 See, e.g., Ref. 34 p . 90.

170

13.

PRODUCTION A N D MEASUREMENT OF ORlENTATlON

Stress transducer

r L

Double pen recorder

Transmitter

PuIs e transit time meter

crystals

E'ongoting device

11

FIG.23. Diagram of apparatus used by Moseley (52) for the measurement of sonic modulus.

equations, Moseley concludes that only the first case leads to physically meaningful results. The equation corresponding to this case is (13.4.21) where E is the sonic modulus of the fiber and Et and Ep are the intrinsic sonic moduli for propagation transverse or parallel to the molecular axis, respectively. Since E, will be very much greater than Et , this reduces to (13.4.22) In applying the aggregate model to predict the orientation dependence of the mechanical anisotropy of crystalline polymers, Wardm derived an expression that shows the modulus of a fiber has a more complex orienta-

13.4.

MEASUREMENT OF ORIENTATION

171

tion dependence than cos26. Thus, it is probable that Moseley's expression involves some hidden approximations. Ward's expression for the extensional or Young's modulus is

where E, Et, and E, are as defined above, u is Poisson's ratio, and G is the torsional modulus for the perfectly oriented fiber. Ward notes that since u is not very different from 0.5 and since E, is much greater than El and G ,

1 sin46 E - E,

+

sin26 COSZe G .

( 13.4.24)

Furthermore, if one assumes that E, = G, then Eq. (13.4.24)reduces to Moseley's expression,

_1 - sin!% -= E-

I

El

-

cos26 E*

( 1 3.4.25)

Ward further notes that experimental values for poly(ethy1ene terephthalate), nylon, and polypropylene were in good agreement with this assumption, whereas values for polyethylene were not. As already pointed out, both of these techniques treat the fiber as a single-phase system. S a m ~ e l extended s~~ the treatment to include twophase systems by making use of (1) the experimentally observed additivity of the bulk compressibilities of the individual phases in a mixture,"*57 i.e., K

= XcrKcr

+

(1 -

Xcr)Karn,

(13.4.26)

where K is the bulk compressibility, the subscripts cr and am refer to crystalline and amorphous phases, respectively, and xcr is the volume fraction of the crystalline phase, and (2) the relation between compressibility and Young's modulus, K = 3(1 If we now assume that

u is

-

2u)/E.

( 13.4.27)

the same for both phases, then we can write (13.4.28)

56

R. J . Urick, J . Appl.

57

H . A . Waterman, Kolloiti. Z . 2. Pol-vm. 182, 9 (1963).

Phys. 18,983 (1947).

172

13.

PRODUCTION AND MEASUREMENT OF ORIENTATION

and hence

_E1 -- Xcr (1

- cos2e)

+

(1

- Xcr)(l -

EIcr

EIam

cos2B) 9

(13.4.29)

where again the subscripts refer to the intrinsic transverse sonic modulus for the crystalline and amorphous regions. For an unoriented sample, coS2fl = 6 and the modulus will be given by

1

(13.4.30)

Substituting (13.4.30) into (13.4.29),

where hr and jam are identical to Hermans orientation function 3 (cos28 - 1) for the crystalline and amorphous phases, respectively. The advantage of this technique over that of birefringence for fundamental orientation studies lies in Eq. (13.4.30). For most crystalline polymers, the crystallinity xcrcan be varied over a reasonable range and, as already indicated, can be estimated from values of the crystalline and amorphous densities. Thus, by measuring Eufor two values of xcr, one can determine El cr and El a m . In practice, one would carry out this determination for a range of values of xcrand average the results. Although, as the foregoing analysis suggests, the soundness of the theoretical basis of sonic modulus leaves something to be desired, its determination is a relatively straightforward process, particularly in the case of fibers or films. The equations for the velocity of sound in a material reduce to a very simple form when one considers its propagation in a specimen in which one dimension is much larger than the other For propagation of longitudinal (extensional)waves in a rod (cf. a fiber or strip of fdm) the velocity (c) is given by c = (E/p)'l2,

( 13.4.32)

where p is the density. For comparison purposes between fibers, this means that the specific moduli, usually quoted in grams/denier ( 1 denier is the weight in grams of 9000 m of the fiber), are directly proportional to the velocity squared. On the other hand, the density of polymer fibers is readily determined via density gradient column techniques, and hence the true Young's modulus can be calculated for use in Eq. (13.4.22), (13.4.28), or (13.4.29). [Note that p is also required for the calculation of xcrin Eqs. (13.4.28) and ( 13.4.29).]

13.4. MEASUREMENT

OF ORIENTATION

I73

13.4.4.Infrared Dichroism Dichroism can be measured over a wide range of the electromagnetic spectrum from ultraviolet to infrared. The principles are the same for each case, and since infrared is the most commonly used technique we concern ourselves principally with this technique here. When a broad spectrum of infrared radiation passes through a polymer sample, the transmitted beam will show a number of absorption bands. Such absorptions occur when there is a resonance between a normal vibrational mode of certain groups of atomic nuclei in the molecule and a frequency in the incident radiation. Activation of the vibrational mode gives rise to a change in the dipole moment associated with the group of nuclei. This change is called the transition moment of the absorption. The transition moment is a vector quantity and therefore the absorption due to a particular transition will depend on its orientation relative to that of the electric vector of the incident beam. If the electric vector is parallel to the transition moment, the absorption will be strong: if it is normal to it there will be no absorption. Thus, by illuminating the sample with a polarized beam parallel or perpendicular to the orientation axis, the orientation in the sample can be characterized. Moreover, due to differences in the local fields experienced by a group of atomic nuclei in the crystalline and amorphous regions, certain absorptions may be associated solely with the crystalline regions or solely with the amorphous regions. This gives rise to the possibility of determining the individual orientations of each of the two phases. A drawback inherent in these studies is that the transition moment will not in general be parallel to the molecular axis, but will lie at an angle a,to it. Hence, in a fully oriented sample it will not be aligned with the molecular axis but will lie on a cone of half-angle a, and the absorptions observed with the polarization axis either parallel or perpendicular to the orientation axis will be a function of this angle. If the specific molecular motion associated with a given transition is known, then a,can be determined. The correlation of absorption bands with transitions is discussed in detail in Chapter 3.1 (this volume, Part A). In the discussion on birefringence it was shown that in general a beam of polarized light will have its plane of polarization rotated on traversing an oriented specimen. This rotation does not occur if the incident beam is polarized parallel or perpendicular to the orientation axis. Hence, to avoid such effects, the absorption is measured with the beam in these two directions. The results are expressed as the dichroic ratio R defined by

174

13.

PRODUCTION A N D MEASUREMENT OF ORIENTATION

where A,, and A L are the respective absorption coefficients parallel and perpendicular to the orientation axis. The dichroic ratio R , with perfect axial alignment has been estimated theoretically by F r a ~ e rwho , ~ ~showed that (13.4.34)

There will be no dichroism (R, = 1) for a = 54.74". For larger angles, R , > 1 and in particular for the transition moment parallel to the molecular axes a,,= 0 and R, will be infinite. For smaller angles R, < 1 , and if av = 90",then R, will be zero. It is apparent therefore that the sensitivity of the measurement will be dependent on a,,. For a real sample in which the molecular axis has a distribution about the fiber axis use is made of the concept that the orientation can be considered as a fractionfof fully oriented molecules and a fraction 1 - fthat are completely random.33 The dichroic ratio is then given by R='

cos2a

+ +(l - f)] + HI - f)]'

[jcf sin%)

(13.4.35)

On rearranging, this becomes R=

1 + i(R0 - 1)(1 + 2 f ) 1 + +(Ro - 1)(1 -f)

( 1 3.4.36)

or ( 1 3.4.37)

Note that this is independent of a",as would be expected, since it refers to the molecular disorientation, and if Ro is known thenfcan be calculated. In orientation studies, one is basically interested in how f is related to the disorientation 8. Fraser calculated the expression for R supposing that all of the molecules were oriented at an angle 8 to the fiber axis. By then allowing for a distribution in 8, the expression he obtained was 1

R = I

+ (R, - 1)cosze + )(Ro - 1 ) G Z 8

( 13.4.38)

or

i.e., a relationship identical to the Hermans orientation function. This expression can either be used to determine R , , and hence a,,, given cOs20

13.4.

MEASUREMENT OF ORIENTATION

175

from x-ray data, for example, or if R , is known, to find coS28. The infrared technique has been used extensively for studies on polyethylene,% p~lypropylene,~~ and poly(ethy1ene terephthalate).sO 13.4.5. Small-Angle X-Ray Scattering As stated earlier, polymer crystals have a very characteristic asymmetric shape. In particular, if grown from solution, they are lathlike with the molecular axis nearly normal to the surface of the lath. Similar lathlike crystals are observed in melt-crystallized material either forming spherulite radii (Fig. 24) or stacks of lamellae in oriented materials (Fig. 25). In a melt-crystallized material, these lamellar structures represent alternating crystalline and amorphous phases and can therefore affect its mechanical properties. Hence, a full characterization of the orientation of a sample must include a characterization of the size, shape, and orientation of these structures, i.e., of the sample morphology. The crystal morphology can be obtained from the broadening of the crystalline x-ray reflections,61but this gives no information regarding the amorphous material. However, the thickness of the lamellae is generally of the order of one to a few hundred Bngstrom units. Structures of this order of magnitude will scatter x rays at angles very close to the x-ray beam, e.g., 10-3-10-2 rad. This is small-angle x-ray scattering (SAXS), sometimes termed low-angle x-ray scattering. There are two main types of scattering at small angles: continuous and discrete. Continuous scatter is by far the most common type of scattering encountered. It arises from scattering by a system of noninteracting particles and is analogous to the scattering of light by small parIf the particles are identical in size, then they give rise to a series of rings around the strong central peak (Fig. 26) with the fist minimum given by E = 1.22 h / d , where d is the diameter of the particle. In general, the particles are not identical and the rings become smeared out to form a continuous distribution in diffracted intensity, giving a characteristic exponential fall-off in intensity from the center. The exact distribution depends on the particle shape and size. This type of scatter has been

W. Glenz and A . Peterlin. J . P o / ~ l mSci., . Purr A-2 9, 119 (1971). A . Cunningham. A . J . Manuel and I . M. Ward, Po/ymer. 17, 125 (1976). @O 1. M. Ward, J . Po/jrwr S c i . . Po/yrn. Symp. 58, 1 (1977). 61 See. e.g.. H . P. Klug and L. E. Alexander "X-ray Diffraction Procedures." Wiley. New York. 1974. O2 See, e . g . , F. A. Jenkins and H. E. White, "Fundamentals of Optics." McGraw-Hill, New York. 1953. 58

5B

13.

176

PRODUCTION A N D MEASUREMENT OF ORIENTATION

FIG. 24. Lamellae on the surface of a melt-crystallized polyethylene film. (From E. W. Fischer, Disc. Furcrdoy Soc. 25, 205 1958.)

extensively studied by Guinier and FourneP3 and KratkyM and his coworkers. The differences in the scattering curves for different particle shapes and sizes in a random distribution are usually small and require a (o

A . Guinier and G. Fournet, “Small Angle Scattering of X-rays.” Wiley, New York,

1955.

0. Kratky, Progress Biophys. 13, 105 (1963).

13.4.

MEASUREMENT OF ORIENTATION

177

FIG.25. Almost parallel lamellae in a melt-crystallized oriented polyethylene film. (Surface replica. Shadow and chain orientation directions are vertical.)

very careful plot of the variation of intensity as close to the central peak as is feasible. For oriented particles it should be noted that, as with all diffraction, the width in radians of the central peak is inverse to the size of the particle [Eq. (13.4.39)]. Hence, the shape of the scattering from a system of oriented particles is reciprocal to the shape of the particles. In particular, a streak on the equator of a pattern from a drawn fiber or film may be related to the presence of fibrils parallel to the draw direction. Care must be taken, however, when using a bundle of fibers to eliminate

13.

178

PRODUCTION A N D MEASUREMENT OF O R I E N T A T I O N

Cent ra I

peak

I

x

c .In aCl

c

C

0

1.22 X/d

2.23 X/d 3.24X/d Diffraction angle

-€

FIG.26. Intensity distribution due to diffraction by a spherical particle. Central scatter is about 100 times as intense as the first peak.

interfiber effects. This may be achieved by using an immersion medium that has an electron density close to that of the fiber.s5 Discrete low-angle reflections arise when there is a periodic structure within the material and can be interpreted according to the Bragg relationship, which, at small diffraction angles, can be written 26 =

A/d,

(13.4.40)

where 2r is the diffraction angle in radians. Diffracting conditions and the interpretation of patterns follow the same rules as described for wide-angle diffraction. However, in view of the fact that the diffracting angles are so small and the maxima broad, no approximation is involved in assuming that the diffraction patterns represent the distribution of poles in a reciprocal lattice plane normal to the beam. Thus the distance from the center represents the interplanar spacing and the position of the spot represents the orientation (Fig. 27). In general, with polymers only one or possibly two reflections are observed. This may arise from the fact that there is a distribution in repeat distances that, as with a distorted crystal, will lead to a smearing out of the higher order reflections. A second effect that may lead to the absence of higher order reflections is the nature of the repeat unit itself. It is a crystal -amorphous periodicity and, since the crystallinity is usually of 65

H. D. Noether and I . L. Hay, J . Appl. Crysr. 11, 546 (1978).

13.4.

MEASUREMENT OF ORIENTATION

I79

Orientation axis

r

c

F I G . 27. Schematic of low-angle diffraction. is orientation of lamellar normal to the orientation axis. OP is proportional to the reciprocal of the lamellar spacing.

the order of 40-70%, the two phases are of a similar size. Scattering theory indicates that under these conditions very little intensity is scattered to the higher orders3' and hence only a limited number of orders will be visible. Two basic low-angle patterns arise in well-oriented systems: (a) twopoint patterns, in which the diffraction is concentrated on the meridian

FIG.28. Typical two-point low-angle pattern for polypropylene.

180

13.

PRODUCTION AND MEASUREMENT OF ORIENTATION

FIG.29. Typical four-point low-angle pattern for polyethylene.

(Fig. 28), and (b) four-point patterns, in which the diffracted intensity lies approximately on a layer line, perpendicular to the meridian, but with a minimum in intensity on the meridian and maxima in the four offmeridional quadrants (Fig. 29). In keeping with the Bragg interpretation of these patterns the meridional, two-point patterns imply lamellar planes perpendicular to the orientation axis. Such an interpretation is borne out by studies on hard elastic polypropylene, for example, in which the lamellar spacings obOrientation axis

t

FIG.30. Lamellar normal distribution giving rise to four-point patterns.

13.4.

MEASUREMENT OF ORIENTATION

181

Draw Direct i o n

t

Normal to Rolling Plane

(a) FIG.31. Low-angle patterns from a drawn and rolled polyethylene film. (a) Definition of axes (X). X-ray beam parallel to X, (Y). X-ray beam parallel to Y.

served in transmission electron micrographs (Fig. 5a) and the low-angle patterns (Fig. 28) coincide. Similarly, four-point patterns indicate lamellae with planes inclined to the orientation axis and with the plane normals lying on a cone of half-angle a as shown in Fig. 30. Patterns obtained from fibers indicate only the orientation of the lamellar surfaces with respect to the molecular or c axis of the crystal. In order to obtain their relationship to the other crystal axes, films in which the three crystal axes have been separated must be studied. Such films may arise during processing as with poly(ethy1ene terephthalate) films2*or be obtained by callendering as discussed ear lie^.'^,^^ In such films the small-angle x-ray patterns obtained with the x-ray beam parallel to the three principal axes will differ. This is illustrated for polyethylene (Fig. 31), which has been drawn, callendered, and lightly annealed. Three axes can be defined for such a film, as shown in Fig. 32. With the x-ray beam normal to the draw direction and the film plane, i.e., parallel toX in Fig. 32, a weak two-point pattern is observed (Fig. 31X), while parallel to the film plane and still normal to the draw axis (i.e,, parallel to Y) a four-point pattern is found

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PRODUCTION A N D MEASUREMENT OF ORIENTATION

/020\

Z

X

Y

FIG.32. Wide-angle patterns from same sample as Fig. 31 with x-ray beam parallel to Z. X, and Y. Note the presence of 020 reflections in X and their absence in Y.

(C)

(d)

FIG. 33. Different types of low-angle patterns. (a) Polyethylene, (b) poly(ethylene terephthalate), (c) polypropylene as spun, and (d) polypropylene annealed.

13.4.

MEASUREMENT OF ORIENTATION

183

FIG.34. Diagram of four-point pattern shown in Fig. 33(a). y is layer line separation and 0 is equal to tilt of the molecular axis.

(Fig. 31 Y). These patterns indicate that the diffraction poles lie in the X Z plane and hence the lamellar surfaces are parallel to Y and inclined to both Z and X. Wide-angle x-ray patterns show (Fig. 32) that the b crystal axis is also parallel to Y and hence the lamallae surface contains the b crystal axis. Confirmation of this structure for polyethylene samples of this type has also been obtained by electron microscopy.ss The small-angle patterns also exhibit characteristic shapes, and some examples are illustrated in Fig. 33. Two point patterns on the meridian may appear fan- or pear-shaped (Fig. 33a), as a distinct spot, or droplet-shaped (Fig. 33b). The four-point patterns may not lie on exact layer lines but may be convex (Fig. 33c) or concave toward the origin 66

D. Grubb. J . Dlugosz. and A . Keller, J . Mate,. Sci. 10, 1826 (1975).

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(Fig. 33d). Such distinctive shapes can give further information on the morphology of the material. The fan shape (Fig. 33a) observed with two-point patterns is found, for example, in as-spun hard-elastic polypropylene fibers and, on annealing, changes to the droplet pattern of Fig. 33b. The annealing also gives rise to an improvement in orientation of the wide-angle patterns. This pattern therefore arises from a distribution of spacings, caused by the rapid quenching of the fiber, and a twisting of the lamellae. On annealing, the layer thickness becomes quite uniform (e.g., Fig. 5a) and hence the droplet corresponds to a sample with uniform layer thickness. The broadening of the reflections along layer lines typically found in the four-point patterns probably arises from the limited lateral dimensions of the lamellae. The curved nature of the layer lines indicate the presence of two sets of lamella that have rotated in opposite directions, giving rise to two sets of layer lines inclined to one another as shown in Fig. 34. For further discussion on the shapes of small angles and their relationship to morphology, the reader is referred to the and also to Chapter 6.2 (this volume, Part B).

G. W. Groves and P. B. Hirsch, J . Mrrrer. Sci. 4, 929 (1969). V . I . Gerasimov et al., Kolloid X. X . Polym. 250, 578 (1972). 68 M . Kakudo and N . Kasai "X-Ray Diffraction by Polymers." Elsevier. New York, O7

1972.

14. ESR STUDY OF POLYMER FRACTURE

By Toshihiko Nagamura 14.1. Introduction One of the important themes in polymer science is the search for molecular models from which macroscopic properties can be properly inferred. Polymeric materials often show complex responses to mechanical excitations and it is well known that macroscopic strength is generally much lower than that calculated on the basis of atomic bond strength. A number of useful phenomenological models have been proposed to explain such mechanical properties of polymers based on the information obtained from studies of the elastic, anelastic, and plastic deformation of the material at various temperatures and under various modes of mechanical excitations. However, it is difficult to choose a “real” model among them without information on molecular events. A new tool for investigation of polymer deformation and fracture has been added with the development of microwave resonance absorption technique. This is called electron spin resonance (ESR) or electron paramagnetic resonance (EPR). With this technique, the number of ruptured chains can be determined as a function of time or other parameters and information as to the location and environment of the mechanically generated radicals can be obtained. ESR gives valuable information whenever formation, trapping, or reactions of radicals are involved. Therefore, ESR has been used widely in the fields of polymer chemistry and physics to study radicals produced by high-energy radiation, by thermal degradation, or during polymerization processes. Only in the last two decades has it become one of the most useful methods for investigating molecular events associated with macroscopic deformation or fracture of polymers. Various mechanical actions have been found to generate radicals. They are roughly divided into two groups from the viewpoint of ESR investigation. In the first group, polymer samples are mechanically fractured by various methods outside the ESR spectrometer and all molecular fracture processes are completed before the ESR spectrum is recorded. I85 METHODS OF EXPERIMENTAL PHYSICS, VOL. 1 6 ~

Copyright @ 1980 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-4759584

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In the second group, various modes of tensile load or strain are applied to the polymer sample placed in the ESR cavity assembly and the radical formation behavior is observed as a function of time and other experimental parameters during deformation and fracture. Some review articles have appeared on the application of ESR for the study of polymer deformation and fracture. Kausch' first comprehensively reviewed the literature through 1969: DeVries and Roylance2 reviewed mainly their own works up to 1972; DeVries3 reviewed ESR studies of rubber fracture: Kausch and DeVries* reviewed more recent works; and Sohma and Sakaguchi5 reviewed their own works through 1975. Andrews and Reed5" published a review article on molecular fracture in polymers as studied mainly by the ESR method. In this part various applications of the ESR method to studies of deformation and fracture of polymers and interpretations of experimental results are described, together with limitations of this method and comparison with results obtained from other methods.

14.2. Basic Theory and Experimental Techniques? 14.2.1. Principle of ESR Method

The theory of ESR and details of experimental and instrumental techniques are found in a number of excellent texts and publications,6-12and H. H. Kausch-Blecken von Schrneling, J . Mucromol. Sci.. Rev. Mrrc,roniol. C h r m . 4, 243 (1970). K. L. DeVries and D. K . Roylance. Prog. Solid Stcctc Chetn. 8, 283 (1973). K. L. DeVries, Rubber Chcm. Trchnol. 48, 445 (1975). ' H. H. Kausch and K. L. DeVries, Int. J . Frrrcr. 11, 727 (1975). J. Sohrna and M. Sakaguchi. A h , . P o / y m . Sci. 20, 109 (1976). sa E. H . Andrews and P. E. Reed. A h . Po/ym. Sci. 27, 1 (1978). D. J. E. Ingram, "Free Radicals as Studied by Electron Spin Resonance." Academic Press, New York, 1958. ' G. E. Pake, "Paramagnetic Resonance." Benjamin, New York, 1962. R. S. Anderson, Methods Exp. Phys. 3, 441 (1962). C. P. Slichter, "Principles of Magnetic Resonance." Harper, New York, 1964. M. Bersohn and J. C. Baird, "An Introduction to Electron Paramagnetic Resonance." Benjamin, New York, 1966. l 1 A. Carrington and A. D. McLachlan, "Introduction to Magnetic Resonance with Application to Chemistry and Chemical Physics." Harper, New York, 1967. ** R. S . Alger, "Electron Paramagnetic Resonance: Techniques and Applications." Wiley (Interscience), New York. 1968. t See also Volume 2B (Electronic Methods) of this series, Chapter 9.7.

14.2.

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187

also in Chapter 5.3 (this volume, Part A). Therefore only an outline of principles is presented here. ESR spectroscopy is based on microwave absorption between energy levels of a Zeeman-type splitting of unpaired electrons in the presence of a magnetic field. ESR can be used, in principle, to study any system containing unpaired electrons with spin S = 4 and an associated magnetic moment p = - g p S , where g is a dimensionless constant called the spectroscopic splitting constant or electron g factor and p is electronic Bohr magneton. According to the detailed quantum theory of ESR, the resonance condition for unpaired electrons in the presence of a magnetic field of strength H can be formulated as hv = AE = g p H , where h is Planck’s constant and v is microwave frequency. If either the frequency v or, as is generally the case, the magnetic field H is swept through the resonance condition, one or more absorption lines are observed. The population of unpaired electrons in each energy level is given by Boltzmann’s law and at thermal equilibrium at ordinary temperature the number in the lower level is greater than that in the upper level. If the incident radiation power is sufficiently high, the absorption will disappear owing to equalization of populations in two levels. There are relaxation processes that enable electron spin in the higher energy level to dissipate the excess energy and to relax down to the lower level. However, one must be very careful to avoid this so-called power saturation effect, especially in the study of radical concentration. In most organic free radicals the energy levels of the unpaired electrons are split further into 21 + 1 sublevels by an interaction with neighboring nuclei having nuclear spin 1 # 0. This phenomenon is called hyperfine splitting and it gives a hyperfine structure to the ESR spectrum. The most common and important situation is an interaction with the hydrogen atom with 1 = 3. This hyperfine structure together with the g value can give most important information on identification of free radicals and their environment. 14.2.2. Radical Concentration

Radical concentration is an important experimental quantity in the study of polymer fracture. The number of free radicals contributing to an observed spectrum is given by an area under the absorption curve. In practice, the double integration of the observed first-derivative spectrum gives the number of radicals. The absolute radical concentration is determined by two alternative methods: an absolute and a comparison method. In the absolute method, the unknown radical concentration is calculated from the relationship between the radical concentration and the magnetic

188

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STUDY OF POLYMER FRACTURE

s~sceptibility.'~In the comparison method, the unknown radical concentration is measured relative to a standard reference such as DPPH (1,ldiphenyl-2-picrylhydrazyl,having one unpaired electron per molecule) or other calibrated substance having stable radicals. Usually this comparison method is used to determine radical concentration by evaluating the ratio of areas under two integral (absorption) spectra recorded from the reference and the unknown sample under "identical" experimental conditions. ESR signal intensity (area under the absorption curve) is affected by many factors, such as temperature, sample size, sample position in the resonant cavity, filling factor and dielectric loss, and the width and shape of the spectrum other than spectrometer parameters such as the power saturation effect and modulation width. Therefore, it is very difficult to determine the absolute radical concentration with a high degree of accuracy, and the experimental error in radical concentration measurements is said to be f 25% or more.loJ4 However, the relative concentration using the same material, such as in the case of investigations of time dependence, is much more accurate, within f5%. The double cavity developed by Kohnlein and Miiller,15which enables the study of two samples at the same time, is most appropriate for determination of radical concentration. The Mn2+ signal of six lines with equal intensity is used as an auxiliary standard to monitor small changes of experimental conditions. A small amount of nonmagnetic powder containing Mn2+ ion as impurities can be inserted into the cavity and measured with the unknown sample or the standard reference as long as its six lines do not overlap seriously with the object spectrum. Some care should be taken in the preparation of the standard sample. DPPH must be recrystallized from solution for purification and the solvent must be completely removed before weighing. A standard DPPH sample is usually used as a solution in benzene by dissolving the carefully weighed DPPH solid into a given volume of solvent. Although solid DPPH is very stable over a wide temperature range, DPPH in solution is rather unstable, and so the solution must be kept in a cool, dark place. Powdered carbon pitch is also frequently used as a standard.

14.2.3.System for Observing Mechanically Generated Radicals

14.2.3.1.ESR Spectrometer. Since detailed descriptions of the ESR spectrometer are given in many books and also in Chapter 5.3 (this voll3 J. A. McMillan, "Electron Paramagnetism." Van Nostrand-Reinhold, Princeton, New Jersey, 1W. J. W. Boag, Radiur. Eff. Phys. Chem. Biol., Proc. Inr. Congr.. 2nd. 1962 p. 194 (1963). *a W. Kohnlein and A. Muller, Z . Naturforsch., Teil B 15, 138 (1W).

14.2.

BASIC THEORY A N D EXPERIMENTAL TECHNIQUES

189

ume, Part A), they are not repeated here. When chain rupture of a polymer molecule takes place, two electrons making up a covalent bond may be uncoupled, forming two polymer free radicals. These radicals can be detected and identified by ESR as long as they are produced in sufficient numbers and are stable enough to remain at a high concentration in the ESR cavity during observation. ESR spectrometers are divided into X band (v 9.5 GHz), K band (v 24 GHz), Q band ( v 35 GHz), etc., according to the frequency of the microwave. Most of the work to date has been done in the X-band region due mainly to the large cavity volume in this type of ESR spectrometer. The ESR spectrum is usually recorded as a first derivative of the absorption curve for increased sensitivity. For most current highresolution ESR spectrometers of this type, the threshold sensitivity is about 1012spins in the cavity (which is usually 3 cm long and I cm diam.). Some of the spectrometers are equipped with a variable-temperature accessory that facilitates investigations between about - 180 and 300°C. 14.2.3.2. Methods for Obtaining Free Radicals from Mechanical Fracture of Polymers. Free radicals have been known to be generated in polymers during macroscopic fracture by various methods such as cutting, ball-milling, crushing, grinding, slicing, or filing. Most mechanical fracture experiments on polymers and subsequent ESR measurements must be done at low temperature and/or in an inert atmosphere, because most mechanically generated radicals are very reactive and easily disappear or are converted to other species before observation. The simplest method for obtaining mechanical radicals is to file, drill, or cut the polymer sample in liquid nitrogen and to transfer the fracture remnants, without exposing to air, to an ESR sample tube kept at liquid nitrogen temperature. The liquid nitrogen in the ESR sample tube can be removed by using a vacuum pump or by very slight warming. In some cases, however, these methods cannot prevent reactions of the generated radicals with small amounts of oxygen dissolved in the liquid nitrogen. Special devices are needed to fracture polymer samples in vucuo and at low temperature. Zhurkov et d . 1 6 used a vacuum mechanical manipulator in which a folded film of polymer, placed on the inner wall of a quartz tube that was placed in the ESR cavity, was shaved by two miniature steel cutters. The shavings thus formed were collected in the tube and the ESR spectrum was recorded. Abagyan and Butyagin" used a vibration mill with a glass working chamber of 25-30 ml volume containing 18-25 glass beads of 8-10 mm diam. After dispersion, the material was poured into the narrow tube in the upper part of the glass chamber and the

-

-

-

l o S. N . Zhurkov, E. E. Tornashevskii, and V . A. Zakrevskii, Sov. Phys.-Solid (Engl. Trans/.) 3, 2074 (1%2). '' G.V. Abagyan and P. Yu. Butyagin, Biophysics (Engl. Trans/.)9, 188 (1964).

State

190

14.

ESR STUDY OF POLYMER FRACTURE

(b) FIG.I . (a) Apparatus for slicing polymer samples. (b) Environmental chamber for slicing apparatus.'O

14.2.

BASlC THEORY A N D EXPERIMENTAL T E C H N I Q U E S

1.1

L------J

f

191

A, "

"$8

FIG. 2. Schematic sketch of a ball-milling apparatus: ( A ) glass ampule; ( B ) ESR sample tube; (C) connector to a vacuum system; (D) glass balls for milling; (S)holder for ampule; (M) motor; (N) belt; (P) pulley; ( R ) crank; (V) Dewar flask containing coolant.*0

tube was transferred into the ESR cavity. Lazar and SZOCS'~ used a special small drilling apparatus in which the powder so formed fell into a cooled sample tube, which was later sealed off and used for measurements. Backman and DeVrieslg used a slicing apparatus that could slice a sample in a controlled environment and at various temperatures. The slicing device consisted of a plane driven by a small dc electric motor as shown in Fig. la. A jet of dry pure nitrogen gas carried the slices through a tube where they were quenched in liquid nitrogen within 0.5 sec. Figure Ib shows the environmental chamber. Sakaguchi and Sohmazoused a ball-milling apparatus similar to that of Abagyan and Butyagin as shown in Fig. 2. They used polymer flakes cast from solution and the powder was collected into a sample tube after dispersion. 14.2.3.3. Methods for Observing Radical Generation during Tensile Deformation and Fracture. Although it is experimentally easy to produce mechanical radicals outside the ESR spectrometer, the ESR spectrum is always recorded after completion of fracture in such experiments and the fracture mechanism is expected to be very complex. Therefore, it is not appropriate to study in this way the relation between molecular events and macroscopic mechanical properties. Some workers have constructed a loading frame around the cavity and magnet assembly of the ESR spectrometer to try to observe simultaneously radical formation behavior, strain, and stress while stretching the sample in situ.

Is *O

M . Lazar and F. Szocs, J . folyrn. Sci.. f u r l C 16, 461 (1967). D. K . Backman and K . L. DeVries, J . f d y r n . Sci., Purr A - / 7, 2125 (1969). M. Sakaguchi and J . Sohma, J . f d y t n . Sci.. f d y r n . f h y s . Ed. 13, 1233 (1975).

I92

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ESR STUDY OF POLYMER FRACTURE

FIG.3. Schematic representation of tensile loading apparatus with environmental controlling chamber.zz

1 4 . 2 . 3 . 3 . 1 . LOADING SYSTEM.The loading system can be mechanically controlled or hydraulically servo-controlled. Becht and Fischerzl used a loading apparatus with a screw-and-lever loading mechanism, which can make constant-strain or constant-stress experiments at temperatures hetween 02s Nauamiira and ---..---_ -50 - - and hn°C -- - in nitrnucn ___-_I._- Tn*o--. -.-o-*---_kayanagiz2 constructed a similar but somewhat improved loading appa-

___

I__-

*I 22

0---

J . Becht and H. Fischer, Kolloid-2. Z . Polyrn. 240, 766 (1970). T. Nagamura and M. Takayanagi, J . Polyrn. Sci.. Polyrn. Phys. Ed. 12, 2019 (1974).

14.2.

BASIC THEORY A N D EXPERIMENTAL TECHNIQUES

193

Fic. 4. ESR spectrometer with servo-controller on the left, hydraulic system in the center, and loading frame on the right.

ratus with a screw-and-lever loading mechanism driven by an electric motor, as shown schematically in Fig. 3. With this apparatus, constant strain, constant stress, constant strain rate, or cyclic strain can be applied to the sample in a temperature range from about - 80 to 160°C in nitrogen gas or at room temperature in wicuo. The load is recorded by a stress transducer attached at one end of the loading frame and the displacement is directly read by a cathetometer or a dial gauge. Williams and DeVriesZ3constructed a servo-controlled hydraulic loading apparatus as shown in Fig. 4. This apparatus makes it possible to do almost every kind of programmed deformation mode, either strain- or stresscommanded, in the temperature range between about - 160 and 200°C, while simultaneously recording ESR signal, stress, and strain as a function of time. The load is detected by a stress transducer attached at one end of the loading frame and the displacement is detected by a linear potentiometer. It is necessary to make special grips to hold a polymer film or a bundle of fibers in these experiments. There are some requirements for these grips: 23 M. L. Williams and K . L. DeVries, Proc. Int. Congr. Soc. Rheol.. 5111, 1968 Vol. 5 . p. 139 (1970).

194

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ESR STUDY OF POLYMER FRACTURE

(a) They must be as compact as possible to be used in the limited space of the cavity-magnet assembly of the ESR spectrometer and, in some cases, in a small environmental chamber. (b) They must be nonmagnetic so as not to affect the homogeneity of the magnetic field. (c) They must be strong enough to hold a sample for high load of up to about 500 kg. (d) They must assure uniform deformation of the sample. Figure 5 shows some types of grips developed. The material generally used is brass, stainless steel, or high-strength aluminum.

14.2.3.3.2. ENVIRONMENTAL C O N T R O L . In some cases mechanically generated radicals during tensile deformation react very easily with oxygen (air) to form peroxy radicals or peroxides. To prevent these reactions a polymer sample must be stretched under high vacuum or in a purified nitrogen gas. The glass tube and 0 ring system shown schematically in Fig. 3 is designed to maintain high vacuum (1.5 X Torr) during stretching experiments. The central part of the glass tube consists of a double-walled and evacuated quartz tube. Small amounts of oxygen and water in commercial nitrogen gas can be removed by passing the gas successively through an aqueous potassium hydroxide solution of pyrogallol, an activated copper column, and silica gel columns.

FIG.5. Examples of grips for tensile loading of polymers in the ESR spectrometer.

14.3. MECHANICAL FRACTURE OF POLYMERS

I95

14.3. Radical Formation by Mechanical Fracture of Polymers 14.3.1. Radical Species

The formation of free radicals in mechanically fractured polymers was first reported in 1959 by Bresler et u/.24,25and by Butyagin et Since then, many natural and synthetic polymer materials have been investigated with respect to the formation of free radicals by mechanical acAll available information is listed in Table I concerning tion.1*5J6-20*27-55 24 S. E. Bresler. S. N. Zhurkov, E. N. Kasbekov, E. M. Saminskii, and E. E. Tomashevskii, Sol'.Phys.-Tech. Phvs. (Engl. Transl.) 4, 321 (1959). 2s S. E. Bresler, E. N. Kazbekov. and E. M.Saminskii, Polym. Sci. U S S R (Engl. Transl.) 1, 540 ( 1 959). 2E P. Yu. Butyagin, A. A. Berlin, A. E. Kalmanson, and L. A. Blyumenfeld, Vysokomol. Soedin. 1, 865 (1959). 27 K. Ulbert, Nirtrtre (London) 195, 175 (1962). 28 T. N. Kleinert and J. R. Morton, Nutirre (London) 196, 334 (1962). R. L. Ott. 3. Polym. Sci.. Purl A 2, 973 (1964). 30 J. J. Windle and A. K. Wiersema, J . Appl. Polym. Sci. 8, 1531 (1964). 31 P. Yu. Butyagin, I. V. Kolbanev, and V. A. Radtsig, SOP. Phys.-Solid State (Engl. Trunsl.) 5, 1642 (1964). 32 S . N. Zhurkov, V. A. Zakrevskii, and E. E. Tomashevskii, Sov. Phys.-Solid Slate (Engl. Trunsl.) 6, 1508 (IW). 33 V. A. Radtsig and P. Yu. Butyagin, Polym. Sci. U S S R (Engl. Transl.) 7 , 1018 (1%5). 34 G. V. Abagyan and P. Yu. Buiyagin, Polym. Sci. U S S R (Engl. Transl.) 7 , 1563 (1965). "P. Yu. Butyagin V. F. Drozdovskii, D. R. Razgon, and 1. V. Kolbanev, Sov. Phys. -Solid State (Eng/. Transl. ) 7 , 757 (1965). 38 V. A. Zakrevskii and E. E. Tomashevskii, Polym. Sci. U S S R (Engl. Transl.) 8, 1424 (1%6). 3' P. Yu. Butyagin, Polym. Sci. U S S R (Engl. Transl.) 9, 149 (1967). 38 V. A. Radtsig and P. Yu. Butyagin, Polym. Sci. U S S R (Engl. Trunsl.) 9, 2883 (1967). 38 V. A. Zakrevskii, E. E. Tomashevskii, and V. V. Baptizmanskii, Sov. Phys.-Solid Stute (Engl. Trunsl.) 9, I 1 18 (1%7). 40 T. Urbanski, Nuture (London) 216, 577 (1967). 4 1 V. A. Zakrevskii, V. V. Baptizmanskii, and E. E. Tomashevskii, Sov. Phys. -Solid Stute (Engl. Trunsl.) 10, 1341 (1968). I2 A. M. Dubinskaya and P. Yu. Butyagin, Polvm. Sci. U S S R (Engl. Trunsl.) 10, 283 (1968). " A. M. Dubinskaya, P. Yu. Butyagin. R. R. Odintsova, and A. A. Berlin, Polym. Sei. U S S R (Engl. Trunsl.) 10, 478 (1968). V . A. Zakrevskii, E. E. Tomashevskii. and V. V. Baptizmanskii. Vysokomol. Soedin. Ser. B 10, 193 (1968). 4s M. L. Williams and K. L. DeVries, Surf. Interfaces 2: Proc. Sagumore Army Muter. Res. Conf., 14rh.~1967p. 139 (1968). J. Tiho, M. Capla. and F. Szocs, Eur. Pulym. J . 6, 397 (1970).

I96

14. ESR STUDY OF POLYMER FRACTURE

the identification of primary and secondary radicals and the experimental conditions. Kausch' first published a similar table covering data through 1969. The direct products from polymer main-chain rupture are called primary radicals; secondary radicals are those formed from primary ones by various mechanisms such as reactions with oxygen or other polymer chains, or migration of unpaired electrons to more stable position in the polymer chain. These radicals have been identified by comparison with well-established results on radicals caused by high-energy radiation and with theoretical calculations. As shown in Table I, primary radicals have been observed for many polymers, which indicate that various mechanical actions actually break the polymer main chain. These results also show that mechanical action does not randomly break any type of bond in a polymer molecule. In other words, it is suggested that there are one or more weak bonds in some polymers. For example, it has been established for nylon-6 that molecular fracture leads to three types of primary radicals: -CH,-CH, (I), -NH-CH, (11), and -CO-CH, (111). None of the other radicals that may be produced by fracture of other main-chain bonds or by removal of side atoms correspond with the observed spectrum. Using aand emethyl-substituted polycaprolactam, Zakrevskii et al.38concluded that the two CH2-CH2 bonds adjacent to the amide group are broken with equal probability. They estimated the delocalization energy by the Huckel molecular orbital method and showed that the energy required to break these bonds is about 15 kcal/mole lower than that need to break an ordinary C-C bond. Rupture of these bonds thus leads to the formation of free radicals in the proportions (1) 50%, (11) 25%, and (111) 25%. Some primary radicals, however, that are assumed to be formed during mechanical fracture have not been detected even at very low temperature. For example, poly(2,6-dimethyl-p-phenyleneoxide) gives a very stable primary 2,6-dimethyl-substituted phenoxy-type radical upon frac-

'' K. L. DeVries. D. K. Roylance, and M. L. Williams, J . Pulym. Sci.. Purr A - / 8, 237 '' K. L. DeVries, D. K. Roylance, and M. L. Williams, J . Polym. Sci.. Purr B 9, 605

(1970).

(197 1 ).

K. L. DeVries, D. K. Roylance. and M. L. Williams, J . Pulym. Sci.. Purr A-2 10, 599 (1972).

K . A. Akhmed-Zade, V. V. Baptizmanskii, V. A. Zakrevskii. S. Misrov, and E. E. Tomashevskii. Polym. Sci. USSR (Engl. Trunsl.) 14, 1524 (1972). R . J. Salloum and R. E. Eckert, J . Appl. Pulym. Sci. 17, 509 (1973). T. Kawashima, S. Shimada. H. Kashiwabara, and J. Sohma, Pulym. J . 5, 135 (1973). T. Nagamura and M. Takayanagi, J . Polym. Sci.. Pulym. Phys. Ed. 13, 567 (1975). J. Tiho, J. PlaEek, and F. Szocs, Eur. Pulyrn. J . 11, 609 (1975). J. Pilaf and K . Ulbert, Pulymer 16, 730 (1975).

14.3.

MECHANICAL FRACTURE OF POLYMERS

I97

turing in liquid nitrogen or even at room temperature in air.53 This is the most stable primary radical observed so far, with half-life of about 20 hr at room temperature. Meanwhile, another type of primary radical (phenyl-type radical) that is expected to be formed from the molecular structure of this polymer could not be detected even by fracturing in liquid nitrogen. Nagamura and TakayanagiJ3observed a weak peroxy radical signal overlapped on the strong phenoxy radical spectrum at very low temperatures and found that it disappeared upon annealing to -30°C. Essentially the same signal was observed in UV-irradiated poIy(2,6dimethyl-p-phenylene oxide) at - 196°C by Tsuji and Seiki.= They discovered that this signal was not found in the sample irradiated in vucuo but appeared after maintaining the sample overnight at -196°C in the presence of air. From these results it was concluded that the phenyl-type radical formed by rupture of the main chain at an ether bond reacted with small amounts of oxygen dissolved in liquid nitrogen or that some of them decayed rapidly by recombination or reaction with other polymer chains owing to their very high rea~tivity.~' 14.3.2. Reaction and Location of Radicals

It is well known that most mechanically generated radicals are less stable than those caused by high-energy radiation. Chemical reactions initiated by mechanical action are not only scientifically interesting in the field of mechanochemistry, but also they have been practically important, long before the ESR method was applied, in fields such as the mastication of r ~ b b e r s . ~ *These ~ ~ - reactions ~~ are generally based on the high reactivity of mechanically generated radicals. Most primary radicals convert into less unstable secondary radicals, upon annealing, by radical migration in the polymer molecule or proton subtraction from other polymer molecules. Such a conversion might occur even at - 196 "C during long-time fracture e x p e r i m e n t ~ . ~ ~ ~' the effect of temperature on the ESR spectra Zakrevskii et ~ 1 . studied of mechanically generated radicals in helium gas at - 1% "C in polyethylene, nylon 6, and natural silk. In all cases they observed spectral

* K . Tsuji and T. Seiki, fo/.vm. J . 4, 589 (1973). J . E. Bennett, B. Mile, and A. Thomas, f r o c . R . Soc. London, Ser. A 293,246 (1966). K. Baramboim, in "Mechanochemistry of Polymers" (W. F. Watson, ed.), p. 1 . Rubber and Plastic Research Association of Great Britain, Maclaren. 1964. 59 W. F. Watson, Rubber Chem. Techno/. 33, 80 (1960). Bo W. F. Watson, in "Chemical Reactions of Polymers" (E. M. Fettes, ed.). p. 1085. Wiley (Interscience), New York, 1964. A. Casale, R. S. Porter, and J . F. Johnson, Rubber Chem. Techno/. 44, 534 (1971). A. Casale and R. S. Porter, Adv. f o l y m . Sci. 17, 1 (1975). 57

* N.

TABLE I. Free Radicals Observed Conditions of fracture (“KY

Polymer Polyethylene -CHZ-CHZ-

LN LA LN vc.77 VC,80-100 VC,80-100 HE.77 HE,77 HE,77 HE.77 LN LN LN VC,80-100 VC,80-100 LN VC,77

Polypropylene -CHZ-CH(CHs)-

Polyisobuty lene -CHX-C(CHa)Z-

LN LNd LNd LN LN vc,77 vc.77 vc ,77 LNd LNd LN VC,77 LN

Polystyrene -CHZ-CH(C6Hs)Pol y (a-deuterostyrene) -CHa-CD(CeH5)Poly(a-methylstyrene) -CHS-C(CHs)(C6H5)Poly(viny1 acetate) -CHZ-CH(OCOCHS)Poly(methy1 acrylate) -CHZ-CH(COOCHs)Poly(methy1 methacrylate) -CHZ-C(CHj)(COOCHs)-

LN VC,RT vc,77 AR.77 LN

Pol ycarbonate -O-CIH~-C(CH~~-C~H,-O-COPolyoxy methylene -CHS-OPoly(ethy1ene oxide) -CHZ-CHZ-O-

VC.80 VC.80 VC,80 VC.80 VC,8W VC.80 VC,80

Poly(propy1ene oxide) -CHx-CH(CHs)-O-

198

Conditions of measurement of ESR (oKY

vc,77 vc,77 vc.77 VC,150 HE,77 HEJ35,

in Mechanically Fractured Polymers Assignment of observed ESR spectra Primary radicals

Secondary radicalsC

-CH,-CH~

-cH,-CH-CH,-CH~-CH=CH-~H-CH~-CH,-~H-CH~ -CH,-tH-CHI, R06

R06

-cH,-C(CH,)-CH~ROO

-

ROO

-

-cH,-c(cH,)(~H,)-~ RO6 ROO

R06

R-CH-R~ RO6

-O-CH-O--, -0-CH-OH -O-CH~-~H-O-0-CH-O-

-

-O-C(CH3)-CH*-O-0-C(CH3)-CH2-0-

References 24 25 39.52 39 38 38 41 41 41 41 52 52 52 38 38 52 20 24,25 31 31 24 16.25 46 46 46 31 31 25 16 25 24,52 16 16.20.31 16 44

25 25 25 25 25 25 25 (continued)

I99

TABLEI. Conditions of fracture Polymer

(“KY

Poly(2.6-dimethyl-p-phenylene oxide) -C6Ha(CH3h-OPoly(viny1 alcohol) -CHZ-CH(OH)Pol ycaprolac tam -CO(CHt)s-NH-

Poly (a-methylcaprolactam) -CO-CH(CH3)-(CH,)4-NHPoly(r-methyl caprolactam) -CO-(CHz)4-CH(CH,)-NHPolytetrafluorethy lene -CFS-CFZ-

Organosilicone polymers Pol yisoprene -CHz-CH=C(CH3)-CHzPolyurethane

Poly(styrene-co-butadiene) Pol ybutadiene -CH2-CH=CH-CHZPoly(ethylene glycolmethacrylate) (crosslinked) Poly(ethy1ene sulfide) Other polymers containing sulfide bond in the main chain Natural silk

Conditions of measurement of ESR (“K)b

LN LN AR,RT VC.80 VC.80d

v c . 7 7 (77-227) VC.77 (243-RT) VC,RT (RT-400) vc.77 vc.77

HE, VC ,77

HE, VC ,77

HE.77 HE.77 HE.77 HE,77

HE.125 HE.240 HE.295

vc,77 VC.77

VC,RT vc.77

vc.77

vc.77

VC,77 VC.77 vc,77 vc,77 LN LN

vc.77

LN AR,RT LN vc.77

AR. I23 AR.RT AR.123 VC.77

VC,77 LN VC,RT vc.123 LN LN VC,RT LN HE,17

VC.77 (243) AR.123

HE.77

HE,RT

VC.243 VC.77 1n,77

vc.123 VC.77 VC.77 VC,RT HE.77

Other natural polymers a LN, liquid nitrogen; LA, liquid ammonia; VC, in vac‘uo: HE, helium gas: AR, air: RT, room temperature. Temperature shown in parentheses is an annealing temperature.

200

(Conrinued )

Assignment of observed ESR spectra Primary radicals

Secondary radicalsC

References

53 53 53 25 25 39,4134

41 41 41 41 -CO -t H(CH3), -NH-~H, -CO-tHZ, -NH - ~ H ( c H ~ ) -

-CF,

- t ~ ~

-R,Si - t H , -

-

C H ~ - CH=CH -C H ~ -

-

54 39 39

ROO

24,52 52 20 20 50 24

ROO -

ROO ROO

ROO, others ROO, others ROO. others -CH~-~H-CH=CH-CH,-, others others

-CHz-t(CH3)(COOR)

-R-~H-R--c

- c H , - s - ~ H ~(80%)

-CH~--S-~H-CH~-

(20%)

-CHZ-$ R--S

-

-NH-CHz, -CO-CHZ, -N H-QCH,) -

-N H -CH -CO-

~

ROO: Peroxy radicals. e

Fracture products were prepared from the solid solution of polymers. Identification of radical is not complete. 20 I

48,49 48.49 49 20 20 49 55 55 36 25.27.30 26 30 41 41 17.28.29, 34.40

202

14.

ESR STUDY OF POLYMER FRACTURE

changes upon increasing temperature. The sextuplet (-CH2-CH2)plus-triplet(-NH-CH, and -CO-CH,) spectrum observed at - 196°C in nylon 6 changed to the triplet alone at - 158°C. A quintuplet appeared at -148°C on the triplet background and its intensity increased with increasing temperature. At - 33 "C the observed spectrum consisted almost entirely of the quintuplet corresponding to the secondary radical -NH-CH-CH2-. The spectrum at room temperature consisted of the quintuplet and a small amount of the singlet -NH-CRO-CH2--. Most primary and some secondary radicals react readily with oxygen to form peroxy radicals. The readiness of this reaction suggests that mechanically generated radicals are located on or near the newly formed surfaces. In contrast, it is well known that the radicals formed by highenergy radiation do not react readily with oxygen but preferentially decay by annealing in the presence of oxygen, suggesting a rather random distribution of radicals throughout the whole volume of the sample. The reaction of mechanically generated radicals with oxygen causes serious problems in identifying the original radical species in some polymers, but it also serves as a tool to investigate newly formed fracture surfaces. Backman and DeVrieslS determined the depth of the damaged layer by observing the change of ESR signals with oxygen diffusing into the sample. Their results indicate that the radical concentration decreased exponentially with distance from the fracture surface, with 85% of all radicals being within 2 Fm from the surface. Photoreactions of mechanically generated radicals are also used to study the location of radicals. Sohma and SakaguchP studied the effect of UV irradiation on the ESR spectra of peroxy radicals originating from mechanical fracture and y irradiation of polypropylene. They found that the conversion of peroxy radicals by UV irradiation was almost complete for the former but incomplete for the latter. From these results they concluded that almost all mechanically generated radicals were trapped on the fracture surface while the radicals produced by y irradiation were distributed more inside the sample than near the surface. Mechanically generated radicals are known to initiate polymerization of added monomers at very low t e m p e r a t ~ r e . ~These ~ ~ ~ ~results , ~ ~ also suggest that radicals are trapped on or near the fracture surface. 14.3.3. Radical Concentration and Fracture Surface

Radical concentration might be used as a measure of the fracture path. The number of chains cut by a cross-sectional plane in fully oriented polymers is on the order of lOI4/cm2. The number of chains per unit area in unoriented bulk polymers is assumed to be about 4 of this value. The

14.4. OR1ENTED CRYSTALLINE POLYMERS

203

observed radical concentration upon fracture at low temperature is one or two orders of magnitude smaller than this These results suggest that the actual fracture occurs along the selective path of least resistance such as crystalline boundary regions. The energy necessary to break such a number of bonds is much smaller than the observed fracture surface energy of actual polymers. It is thus indicated that energy dissipation by anelastic and plastic deformation of a sample prior to fracture consumes the major portion of the fracture energy.

14.4. Radical Formation during Tensile Deformation and Fracture of Oriented Crystalline Polymers 14.4.1. Radical Species

It has been found that chain rupture occurs not only at fracture but also during tensile deformation long before macroscopic fracture of the sample. The formation of polymer radicals in mechanically deformed polymers was first reported in 1964 by Zhurkov et a/." for polycaprolactam fibers. Since then, radical formation during tensile deformation and fracture has been observed for many p o l y m e r ~ . ~ ~Table ~ - ~ I1~ ~ - ~ ~ shows all available data on radical species and experimental conditions. Kausch' first compiled data published by 1969 into a similar table. At very low temperatures below about - 100°C most polymer fibers or films

* S. N. Zhurkov, A. Ya. Savostin, and E. E. Tomashevskii, Dokl. Akud. Nuuk SSSR 159, 303 (1%); Sol'.Phys. -D(ikl. (Enel. T ~ ~ n s 9, l . 986 ) (1965). S. N . Zhurkov, Int. J . Fruct. Mcch. 1, 311 (1965). a S. N . Zhurkov and E. E. Tomashevskii, Prcic. Conf. Phys. Basis Yield Frucr. 1966 p. 200 (1966). D. Campbell and A. Peterlin, J . Polym. Sci., Part B 6,481 (1968). D. K. Roylance, K. L. DeVries, and M. L. Williams, Fruct.. Proc. I n t . Conf.,Znd, 1969 p. 551 (1%9). O8 J . Becht and H. Fischer, Kolloid-Z. Z . Polym. 229, 167 (1969). G . S . P. Verma and A. Peterlin, J . Mucromol. Sci., Phys. 4, 589 (1970). G. S. P. Verma and A. Peterlin, Kolloid-Z. Z . Polym. 236, 111 (1970). 'I K. L. DeVries, J . Polym. Sci. Purr C 32, 325 (1971). '* B . Ya. Levin, A. V. Savitskii, A. Ya. Savostin, and E. E. Tomashevskii, Polym. Sci. USSR (Enel. Trunsl.) 13, 1061 (1971). A. Ya. Savostin and E. E. Tomashevskii, Sov. Phys.-Solid State (Engl. Trunsl.) 12, 2307 (1971). T. C. Chiang and J. P. Sibilia, J . Polym. Sci.. Polym. Phys. Ed. 10, 2249 (1972). 'I L.A. Davis, C. A. Pampillo, and T. C. Chiang,J. Polym. Sci.. Polym. Phys. Ed. 11,841 (1973). U . Johnsen and D. Klinkenberg, Kolloid-Z. Z . Polym. 251,843 (1973). @'

''

@ '

TABLE11. Free Radicals Observed during Tensile Deformation o r after Tensile Fracture

Polymer Polyethylene -CHZ-CHzNylon 6 -CO- (CH, )S-N H -

Nylon 6.6 -CO-(CHZ)I-CO-NH-(CHz)g-NHNylon 12 -CO-(CHZ)Il-NHPoly(ethy1ene terephthalate) -(CHZh-O-CO-C6H4-CO-OPoly[p-(2-hydroxyethoxy)-benzoicacid] -(CHzk-O-CgH4-CO-OPolystyrene Polypropylene Poly(viny1 chloride) Poly(methy1 methacrylate) Natural silk a

PolYlners stretched

ESR spectra recorded

(“K).

(“K)”

AR,RT HX.243 NG,RT AR,RT AR,RT

AR,RT LN.77 NG,RT AR, 103 AR,RT

HX,243 VC,RT NG,RT 10.178-293 AR,RT NG.206-290 AR,RT VC,RT NG,RT

LN.77 VC,RT NG,RT AR,RT AR.113 NG,206-290 AR,RT VC,RT NG,RT

HX.243 AR,RT VC,RT NG,RT HX.243 AR,RT NG,RT AR,RT HX.243 AR,RT HX.243

LN,77 AR, 103 VC.RT NG,RT-345 LN.77 AR,RT NG,RT AR,RT LN ,77 AR,RT LN.77

Assignment of observed ESR spectra Primary radicals

-

-1 -1

-

-

-O-CO-C6H4-0 -O-CO-C~H,-O b

Secondary radicals ROO ROO ROO

-NH--~H-CH~-

R06 -NH-CH-CH~-

-NH-~H-cH*c

-NH-~H-cH~-

References 47.71 65 21 75 45,47,63, 61,67,71 65 66.70.74 21.68 73 74 76 47.69 66 21

ROO ROO -

b

b

b

b

b

b

b

b

b

b

b

-

-NH-CH-CH~-

RT, room temperature; AR, in air; HX, in hexane: NG, in nitrogen gas; VC, in vacuo: 10, in isooctane; LN. in liquid nitrogen. The number of radicals at fracture was below the sensitivity of ESR spectrometer. The observed spectrum was due to impurities in the sample.

65 74 22 22 65 47 21 47 65 47 65

14.4.

O R I E N T E D CRYSTALLINE POLYMERS

205

break without producing sufficient number of radicals, and so the temperature of ESR observation is limited to a relatively high region. As mentioned in Section 14.3.1, most primary radicals are so highly reactive in this temperature range that all except one radical species observed during tensile deformation are secondary or peroxy radicals as shown in Table 11. The most extensively studied material is nylon 6 fiber, which gives a large number of fairly stable radicals during stretching at or near room temperature. The spectrum observed has a quintuplet pattern and is identified to be due to -NH-CH-CH2radical. Although this is a secondary radical formed by proton subtraction from nylon 6 main chain, it has been assumed to be an indirect indicator of main-chain rupture caused by stretching for the following reasons: (a) this spectrum is observed only after stretching, and (b) as mentioned in Section 14.3.2,primary radicals in fractured nylon 6 are transformed to secondary radicals of this type upon heating to room temperature. Therefore, it has been assumed that the primary radicals formed by main chain rupture of nylon 6 are nearly instantaneously transformed to secondary ones during tensile deformation at room temperature. The assumption of main-chain rupture by stretching polymers has been proved directly by Nagamura and Takayanagi22 by observing primary radicals. These authors stretched poly[p-(2-hydroxyethoxy)-benzoic acid] (abbreviated as PEOB in the following) fibers at room temperature in vucuo or in a nitrogen atmosphere and detected fairly stable primary phenoxy radicals that are generated by main-chain rupture at an ether bond. They identified this radical from a detailed analysis of the. ESR spectrum and a comparison with spectra of a model compound and theoretical calculations taking into account the anisotropy of the g tensor. They attributed its fairly high stability to the high glass transition temperature of PEOB (80°C) and to the considerable delocalization of unpaired electron density in the benzene ring. Meanwhile, another type of primary radical (alkyl end radical) expected to be generated from the molecular structure of PEOB is not detected in these experiments. The alkyl end radical is known to be very reactive and is reported to be detected only below about - 138"C.38 Therefore, it is assumed that these radicals react with each other or with other polymer chains to form phenoxy radicals.22

14.4.2.Reactivity and Location of Radicals The reactivity of radicals generated during tensile deformation is high, as is the case for radicals from mechanical fracture mentioned in Section 14.3.2. Since oxygen reacts so readily with these radicals, only peroxy

206

14. ESR STUDY OF POLYMER FRACTURE

radicals can be detected in some polymers, as shown in Table 11. In some cases almost no radicals can be detected by stretching in air, such as in a PEOB sample, where mechanically generated radicals are believed to react with oxygen to form peroxides.22 If the peroxy radicals formed are relatively stable, this reaction can be used to convert primary radicals to less unstable ones as a convenient way to study radical formation kinetics, as reported by Chiang et u1.74*75 Verma and Peterlin'O studied the reaction of radicals with oxygen as a function of time. They found that mechanically generated radicals reacted with oxygen readily and reversibly while those formed by y irradiation did not react at all. They also studied the anisotropy of ESR spectra of mechanically stretched and y-irradiated nylon samples. They reported that the spectrum in the former case did not show any anisotropy, whereas the one in the latter case showed clear a n i s o t r ~ p y . ~ ~ Fischer and stretched nylon samples immersed in methacrylic acid, which penetrates the amorphous region but not the crystalline region of nylon. The spectrum observed during stretching was only that of polymerization radicals of methacrylic acid. All these results indicate that radicals observed during tensile deformation are generated and trapped in the amorphous regions of the oriented polymer, whereas those formed by y irradiation are present predominantly in the crystalline region. 14.4.3. Radical Concentration

Table 111 shows radical concentration observed during tensile deformation or after fracture. Kausch' compiled data published by 1969 to a similar table. If a fracture occurs in a single plane perpendicular to the fiber axis of a drawn polymer, about 5 x 10" free radicals/cm3 are expected to be generated.66 The observed value of radical concentration in nylon 6, nylon 6,6, and PEOB is much higher than this value. Therefore it is concluded that the generation of radicals in mechanically deformed or fractured samples of these polymers is not localized at a single plane but is distributed throughout the whole volume of the sample. Some polymers like polypropylene, polystyrene, and poly(methy1 methacrylate) do not give detectable radicals during deformation and fracture. This result suggests that fracture occurs by breakage of secondary (intermolecular) bonds rather than that of primary (main-chain) bonds. In polyethylene, poly(ethy1ene terephthalate), and nylon 12, radicals can be detected, but the concentration is relatively small. The interpretation of these results is that those polymers without any strong intermolecular bonds will break I7

H.Fischer, Adv. Polym. Sci. 5, 463 (1%8).

207

14.4. ORIENTED CRYSTALLINE POLYMERS TABLE111. Radical Concentration Observed during Tensile Deformation or after Tensile Fracture Polymer Polyethylene

Radical concentration (spins/cma) 5 x 10'6a 5 x 1015

2.5 x Nylon 6

1.3 7.6 I 1.2 5 8

9 7

1015

x x IO1O x 1017 x 1017 x 10"a x IOl7 x 1017 x 1017 x 1018

1 1.5 x loLa

Nylon 6.6 Nylon I2 Poly(ethy1eneterephthalate)

Poly[p-(2-hydroxyethoxy)benzoicacid] Natural silk

I x 5 x

1017

1015

b

8 x 1.5 x 1017 6 x lotB 7 x 1017a

References 65 21

71 75 74 66 67 65 68 73 80,81

21 76 66 21 21 65 74 1 I8 65

a Radical concentration was measured after tensile fracture. Other values are the maximum radical concentration during tensile deformation. * Radical concentration was below the sensitivity of ESR spectrometer.

along the m i ~ r o f i b r i l sor ~ ~by* ~the ~ pulling out of chains from the crystalline region2I with a small amount of generated radicals. But using the method of converting mechanically generated radicals to peroxy radicals, Chiang er u/.74.75observed many more radicals in polyethylene and poly(ethy1ene terephthalate) and suggested that the relatively small amount of radicals reported before might be due to the decay of original radicals during observation. 14.4.4. Constant-Rate and Stepwise Stretching

The kinetics of radical formation during constant-rate and stepwise stretching have been extensively studied in connection with macroscopic deformation mechanisms and the strength of polymers. It is not appropriate and is practically impossible to record the whole ESR spectrum during deformation in a study of the time dependence of 78

79

A. Peterlin, Tex. Res. J. 42, 20 (1972). A. Peterlin, J . Macrornol. Sci., f h y s . 6, 583 (1972).

208

14.

ESR S T U D Y OF POLYMER FRACTURE

3q-

'Ot

FIG.6. Strain, stress, and radical concentration vs. time for a constant-load-rate test of PEOB fibers at room temperature in nitrogen gas.76a

radical formation. If the shape of the spectrum does not change with increasing intensities, the peak height of any peak of the spectrum should be linearly correlated with the area under the absorption spectrum. This is experimentally confirmed for mechanically generated radicals in nylon 6, PEOB, and polyethylene. Experimentally, the magnetic field is set at the position of the desired peak and its change is recorded as a function of time during deformation. It is then converted to radical concentration using the calibration curve. Figure 6 shows the result of a constant-load-rate test in PEOB fibers at room temperature in nitrogen gas. It clearly indicates a very rapid increase of main-chain rupture after about 6% strain. Figure 7 shows the result of a constant-load test (creep) in PEOB fibers at room temperature in nitrogen gas. The radical concentration first increases rapidly and then slows down to a fairly steady rate of increase. This suggests a correlation between creep behavior and the main-chain rupture. T. Nagamura and K . L. DeVries, Polyrn. Eng. Sci. 19, 89 (1979).

14.4. ORIENTED CRYSTALLINE POLYMERS

209

Figure 8 shows the result of a constant-strain test (stress relaxation) in PEOB fibers at room temperature in nitrogen gas. In this case the rate of radical generation decreases to almost zero after an initial high value. This means that the number of chains broken under a constant strain reaches a constant equilibrium value. Figure 9 shows the result of successive stretching tests of PEOB fibers in vuciio at room temperature. In this case, the sample was first stretched by 14.9% and then the strain was removed. The radicals still remaining after relaxation were killed by introducing air. Then the system was evacuated and the sample was stretched again in vucuo. No radicals were detected until the effective strain in the second run exceeded the maximum strain in the first run. After that, “new” radicals were detected and increased with stretching. These results suggest that there is a distribution of load-bearing chains, which are successively broken during tensile deformation. In a step-strain test the radical concentration increases stepwise after reaching an equilibrium value at each step as shown in Fig. 10. The increment of radical concentration in each step was found to vary with strain level, suggesting the existence of a distribution of lengths of load-bearing chains. This will be discussed in more detail in Section

I

I

1

I

3ot

Time

(min)

FIG. 7. Strain, stress, and radical concentration vs. time for a constant-stress test of PEOB fibers at room temperature in nitrogen gas.7ea

14. ESR STUDY OF POLYMER FRACTURE

2 10

Time

(min)

FIG. 8. Strain, stress, and radical concentration vs. time for a constant-strain test of

PEOB fibers at mom temperature in nitrogen 14.6.2 in connection with molecular mechanisms of deformation and frac-

ture. 14.4.5. Effects of Temperature and Heat-Treatment

Temperature has strong effects on the radical formation behavior as well as on other mechanical properties of polymers. Figure 1 1 shows the histogram of radical formation obtained from step-strain tests of nylon 6 fibers at -25, 22, 50, and 100°C.*o The changes of the shape and the width of histograms with temperature were interpreted as an apparent (effective) distribution of load-bearing chains varying with temperature by mechanisms such as partial pulling out of chains from crystalline region and/or unfolding. The distribution parameters derived from these histograms were found to correlate with the strength of nylon 6 fibers and films.4g0g1 The tensile strength of drawn nylon 6 samples increased as B. A. Lloyd, K. L. DeVries. and M. L. Williams, J . Polym. Sci., Part A-2 10, 1415 (1972).

K. L. DeVries, B. A. Lloyd, and M. L. Williams, J . Appl. Phys. 42, 4644 (1971).

14.4.

ORIENTED CRYSTALLINE POLYMERS

21 I

the width of the histogram of radical formation decreased. This was interpreted as a result of increased homogeneity in a distribution of molecular loads. The morphological changes during heat-treatments of drawn polymers have significant effects on the molecular stress distribution and chain rupture behavior as revealed by the ESR method4B0as well as on the mechanical properties and others. The histogram of radical formation from step-strain tests of slack-annealed nylon 6 fibers was broader and shifted to a higher strain region compared with that of the original sample. The fracture strength and the number of radicals at fracture were decreased in

ezl9.72 (eA4.981

after air introduction

20 gauss

FIG.9. ESR spectra observed in two successive stretching experimentsof PEOB fibers at room temperature in vacua. l l 8

212

14.

:I

L

ESR STUDY OF POLYMER FRACTURE

1

I

1.i

G

f

" 0.8

Time

(nin)

FIG.10. Strain, stress, and radical concentration vs. time for a successive step-straining of PEOB fibers at room temperature in nitrogen gas.7aa

slack-annealing and slightly increased in tension-annealing with increasing annealing temperature. These results were interpreted as a loss of extended-tie molecules due to chain folding and/or defect migration in crystallites in slack-annealing. In tension-annealing, a slight increase of extended tie molecules due to a relaxation of local molecular strains was deduced from these results. Johnsen and Klinkenberg7s studied radical formation behavior in step-strained nylon 6 fibers at temperatures increased stepwise in the region from 206 to 290°K. Their result showed a stepwise increase of radical concentration and stress relaxation at each temperature. Thus it has been proved that the temperature and the strain have similar effects on the kinetics of radical formation. 14.4.6. Effects of Strain Rate and Cyclic Loading

The mechanical response of a polymer sample during tensile testing can be attained by various viscoelastic mechanisms other than main-chain rupture. Therefore, rapid loading is expected to generate more radicals than slow loading. This is experimentally confirmed for nylon 623and polycarbonate.82 But the effect seems to be not so large; the radical cone* Yu. B. Zaks, M. L. Lebedinskaya, and V. N . Chalidze, folyrn. Sci. USSR (Engl. T r w s l . ) 12, 3025 (1971).

14.5.

213

FRACTURE I N ELASTOMERS

STRAIN

(%)

a.

STRAIN

STRAIN (2)

b.

(%)

C.

STRAIN

(%)

d.

Fic. 1 1 . Histograms of radical formation from step-strain tests of nylon 6 fibers at various temperatures in nitrogen gas; (a) -25"C, (b) 22°C. (c) 50°C. (d) 100°C.80

centration increased by about 60% with a tenfold increase of the strain rate in nylon 6 and by about 220% with a 100-fold increase of the strain rate in polycarbonate. There are a few experimental data on radical formation during cyclic loading.23*47*81*83 It is indicated that radical concentration increases stepwise at each cycle up to the fracture point. But the radical concentration at fracture is found to be significantly less under alternating than under monotonically increasing loads. These results reflect a redistribution of stress on the molecular level that is substantially widened by an alternating load.

14.5. Fracture in Elastomers The ESR method has been also found to be useful in studying deformation and fracture in elastomers. These studies include ozone-stress Iu

B . A. Lloyd, K . L. DeVries, and M. L. Williams. Rheol. Acia 13, 352 (1974).

214

14.

ESR STUDY OF POLYMER FRACTURE

cracking of rubber^,^-^' low-temperature deformation and fracture of preoriented r ~ b b e r s , ~ ~fracture - ~ l in granular-filled and mechanical fracture in block copolymers at low t e m p e r a t ~ r e . ~ ~ . ~ ~

14.5.1. Ozone-Stress Cracking

It is well known that a combination of ozone and stress causes extensive crazing of rubbers. It is a problem of practical importance in strength and serviceability. There have been extensive studies on this phenomenon mainly from macroscopic viewpoints."-" DeVries ef u1.84-87tried to study ozone-stress cracking on a molecular scale by using ESR and its correlation with results from macroscopic experiments. Figure 12 shows the ESR spectra of acrylonitrile-butadiene rubbers during ozone attack with 25% strain in an ozone concentration of 2.8 mg/liter, where the lowest curve is the residual signal and each spectrum was recorded 5 minutes after the preceding These strong and fairly stable ESR signals were detected before any microscopic evidence of cracks could be obtained. This result suggests that the ESR method has the potential of being one of the most sensitive means of detecting the very initial stage of ozone attack in rubbers. The rate of radical forma-

K. L. DeVries, E. R. Simonson, and M. L. Williams, J . Basic Eng. 91, 587 (1%9). K. L. DeVries, E. R. Simonson, and M. L. Williams, J. Appl. Polym. Sci. 14, 3049 ( 1970).

K. L. DeVries, E. R. Simonson, and M. L. Williams, J . Macromol. Sci.. Phys. 4,671 (1970).

K. L.DeVries, N. B. Moore, and M. L. Williams, J. Appl. Pulym. Sci. 16, 1377 (1972). R. Brown, K. L. DeVries, and M. L. Williams, in "Polymer Networks. Structural and Mechanical Properties" (A. J. Chornpff and S. Newrnan, eds.), p. 409. Plenum, New York, 87

BB

1971.

R. T. Brown, K. L. DeVries, and M.L. Williams, J. Polym. Sci., Part B 10,327 (1972). R. Natarajan and P. E. Reed, J. Polym. Sci., Part A-2 10, 585 (1972). W. T. Mead and P. E. Reed, Polym. Eng. Sci. 14, 22 (1974). K. L. DeVries, T. B. Wilde, and M. L. Williams, J . Macromol. Sci., Phys. 7 , 633 (1973). era M. Jamroz, K. Kozlowski, M. Sieniakowski, and B. Jachym, J . Polym. Sci. Polym. Chem. Ed. 15, 1359 (1977). wb K. Kozlowski, M. Jamroz, T. Shpkowski, and B. Jachym, Polymer 19, 709 (1978). W. T. Mead, R. S. Porter, and P . E. Reed, Macromolecules 11,56 (1978). e3 R. G. Newton, J . Rubber Res. 14, 27 and 41 (1945). M. Braden and A. N. Gent, J. Appl. Polym. Sci. 3, 90 (1960). OS M. Braden and A. N. Gent, J . Appl. Polym. Sci. 6 , 449 (1962). Od A. N. Gent and J. E. McGrath, J . Polym. Sci., Part A 3, 1473 (1%5). @' E. H.Andrews, J. Appl. Polym. Sci. 10,47 (1966).

14.5. FRACTURE IN ELASTOMERS

215

FIG.12. Increase of radical concentration in acrylonitrile-butadienerubber during ozone attack at constant stress. The lowest spectrum was the residual signal and each spectrum was recorded 5 min after the preceding one.B*

tion was found to be directly proportional to the strain above a threshold (about 3% in natural rubber and 13% in acrylonitrile-butadiene rubber) and to the ozone concentration. In each type of rubber studied, the kinetics of radical formation were found to correspond well with the extent of decrease in mechanical properties (stress relaxation). By using the Griffith-type energy balance approach, the rate of radical formation was successfully correlated to the strain-energy release rate during ozone cracking. Very accurate predictions of expected behavior in creep and cyclic loading tests were obtained by this approach using the surface energy density determined from stress relaxation tests. This approach was confirmed also to be effective for torsionally loaded rubbers.8T These results indicate a good correlation between molecular phenomena and macroscopic fracture mechanisms; cracks propagate only if bonds are ruptured and at rates depending on the strain energy.

216

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ESR STUDY OF POLYMER FRACTURE

14.5.2. Low-Temperature Deformation of Preoriented Rubbers and Granular Filled Rubbers At low temperatures, below the glass transition temperature, rubbers are usually very brittle. However, ductility was found by Andrews and Reeds8to be greatly increased by prestraining the rubber before reducing the temperature. Similar results were obtained for natural polybutadiene rubber,g1and silicone rubber.*@ Such treatments change the fracture mechanism of rubbers at low temperatures from a brittle one producing very few radicals to a ductile one with large numbers of radicals after the yield point. ESR results showed a rather general bond rupture throughout the loaded sample volume during deformation, leading to fracture. The increase of ductility has been attributed to the increase in crystallinity and preferred orientation induced by the strain and low temperature. Large numbers of small ordered regions in the prestrained rubbers could arrest the growth of cracks to cause ductile deformation and a large amount of radical formation. Ductility of rubbers at low temperature can be achieved in another way. DeVries et al.@2reported that certain granular fillers produced similar effects on the bond rupture and mechanical properties of rubbers. They used ESR and a scanning electron microscope (SEM) to study radical formation and filler-matrix interaction in (a) ethylene -propylene -diene terpolymer(EPDM) filled with 29 pm glass beads, (b) NaC1-filled polyisoprene, and (c) silica-filled polyisoprene. They could observe an ESR signal increase almost immediately upon loading in NaCI- and silica-filled samples, but they could not detect any increase in the number of radicals in the glass-filled sample. The NaCI- and silica-filled samples showed similar ductility at low temperature as did the unfilled and prestrained rubbers. SEM photographs showed that the glass beads were easily dewetted while silica particles were embedded in the matrix at the fracture surface with the rubber adhering completely to their surfaces. The analysis of ESR results showed that a much larger number of radicals per unit filler surface area was present in the silica system than in the NaCl system. From these results it appears that ESR could also be useful in investigating the effectiveness of fillers. Jamroi ef ~ f . studied ~ ~ the ~ free * ~radical formation during rubber mastication and tilling of rubber with various carbon blacks. ESR spectra of a complex character were obtained at room temperature, a narrow component line superimposing on a much broader signal of carbon black. ESR data indicated that the shearing forces caused degradation of rubber 98

E. H . Andrews and P. E. Reed, J . Pdyrn. Sci., Purr E 5, 317 (1967).

14.6.

DEFORMATION A N D FRACTURE OF POLYMERS

217

chains and that interactions between the rubber molecules and carbon blacks existed. Mead et u/.gzcstudied oxidation in uniaxially deformed polybutadiene and polyisoprene during initial processing, low-temperature mechanical degradation, and subsequent warming. Two or more radical species resulted from mechanical deformation at 83°K when oxygen was present. Two of which were an ally1 radical resulting from main-chain rupture and a peroxy radical. Experiments in oxygen-free environment indicated that the rupture site was at the weakest bond, between the a-methylene groups. The ESR was shown to be a sensitive method of observing the oxidation of diene rubbers.

14.6. Molecular Mechanism of Deformation and Fracture of Polymers It has been proved by the ESR method that macroscopic fracture of bulk polymers and tensile deformation and fracture of oriented crystalline polymers are accompanied by the rupture of the polymer main chain. It has been also found that the observed radical concentration during tensile deformation depends on the temperature, the strain, and the nature of the polymer. These results should be incorporated into the “correct” model together with other experimental results. 14.6.1. Some Models of Polymer Fracture and Polymer Morphology

Although many macroscopic and microscopic models have been proposed, there seems to be no fully satisfactory model that can explain the very complex phenomena of deformation and fracture of polymers. Statistical approaches are intended to explain the variability of experimental values such as strength, time to failure (lifetime), or radical concentration by the following arguments: (a) the occurrence of fracture is described statistically by a probability law, (b) the material contains large numbers of flaws of different size or severity, and (c) the fracture is caused as the result of many molecular processes. Detailed statistical arguments of polymer fracture have been found in many publications. 99-103 T. Yokobori, Kalloid-Z. Z . Polym. 166, 20 (1959).

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ESR STUDY OF POLYMER FRACTURE

One of the most commonly used macroscopic approaches for fracture is that based on creating some type of fracture envelope. This method was developed by Smith for viscoelastic materials,’@’ and was found to work well for a large number of rubbers and vulcanizates. In this approach, stress, strain, and temperature (or time) are reduced to construct a master curve called a fracture envelope. Fracture is then predicted whenever one of these parameters lies beyond the boundary. Another type of macroscopic approach is based on an extension of Griffith’s brittle-fracture theory,lo5which treats a material as a continuum. The Griffith theory assumes the existence of a small crack within the material. In this approach the total energy is expressed as the sum of the elastic energy and the surface energy of the crack. Fracture is then predicted as the result of crack instability, which occurs when the stress field can provide more energy than can be dissipated by the growth of the crack. In viscoelastic materials, a modificatioc of Griffith’s theory is necessary, as pioneered by Rivlin and Thomas106and Williams.’07 Microscopic models for fracture in polymers have led to theoretical formulas that could be fitted to a set of macroscopic experimental data with the “correct” choice of parameters. There has been, however, no experimental verification of such models on a molecular level, and so it has been difficult to choose among them. Most microscopic theories of fracture have been kinetic theories that intend to explain time-dependent changes within the material that will lead to the fracture. These timedependent changes are related to and depend on the fiuctuating thermal energy of the atoms, molecules, or segments. Reviews of some recent kinetic theories are given by Kauschlo2and Henderson et al. lo8 The morphology of polymers is one of the most important factors affecting the physical properties of polymers. However it is not the purpose of this part to discuss it in detail, and so only a brief description of the morphological model of highly oriented crystalline polymers is presented. Although models proposed by many workers differ in some details, they have a basic common element in their microstructure4*78~10s; all include a “sandwich”-type structure of alternating crystalline (ordered) regions and amorphous (disordered) regions, with adjacent crystalline D. Prevorsek and W. J . Lyons, J . Appl. Phys. 35, 3152 (1964). S. Kawabata and P. J. Blatz, Rubber C h m . Techno/. 39, 923 (1966). H. H. Kausch von Schmeling, Kolloid-Z. Z. Polym. 236,48 (1970). H. H. Kausch,J. Polym. Sci., Part C 32, 1 (1971). lo( T. L. Smith, J . Po/ym. Sci. 32,99 (1958). loo A. A. Griffith, Philos. Trans. R . SOC. London, Ser. A 221, 163 (1921). R. S. Rivlin and A. G. Thomas, J . Polym. Sci. 10, 291 (1953). lo’ M. L. Williams, Int. J . Fract. Mech. 1, 292 (1965). C. B. Henderson, P. H. Graham, and C. N. Robinson,lnt. J . Fract. Mech. 6,33 (1970). ‘09 A. Peterlin, Macromol. Chem. 8, 277 (1973). loo

Io1

14.6. DEFORMATION AND FRACTURE OF POLYMERS

219

end of microfibril-

Jbierofibril+ interfibrillar mol~cule

FIG. 13. Morphological model of highly oriented crystalline polymer.'

regions that are connected by so-called tie molecules existing in the amorphous regions, as shown schematically in Fig. 13.' Amorphous regions also contain chain ends and folded chains at the crystalline surface. Various conformational changes of all these chains in the amorphous regions can occur under an applied stress, and thus lead to local stress relaxation. However, only two modes of stress relaxation are possible in the fully extended state of segments: chain rupture or chain slippage from crystalline regions. Chain rupture generates free radicals that can be detected by ESR. This is the reason why ESR investigations of deformation and fracture in situ have been made using highly oriented crystalline polymers. 14.6.2. Molecular Models of Deformation and Fracture Mainly Based on ESR Results

Zhurkov et a1.64*110 studied the lifetime of polymers and metals under constant tensile load over ten orders of magnitude in time ( lo7 sec) at various temperatures and found that their results can be described by fb

= to

exp[(UO - ')'a)/kT],

(14.1)

110 V. R. Regel, A. I . Slutsker, and E. E. Tomashevskii, Sov. Phys.-Usp. (Enel. Trans/.) IS, 45 (1972).

220

14. ESR

S T U D Y OF POLYMER FRACTURE

where t b is lifetime, k Boltzmann's constant, T absolute temperature, cr applied stress, and t o , y, and Uo are material constants. This is a first approximation of the kinetic theory proposed by Tobolsky and Eyring."' sec for all materials studied, of the They found that to was about order of the atomic vibration period; that Uowas equal to the binding energy of atoms for polymers and to the heat of sublimation for metals: and that y was a coefficient taking into account the overstress on a bond and depended on the orientation of polymer chains or the amount of disorder in metals. Then they tried to confirm this theory experimentally on a molecular scale. They first detected free radicals during tensile deformation and fracture of nylon 6 fibers and reported the existence of a direct correlation between the rate of radical formation and the lifetime of polymers under load expressed by Eq. (14.1). From these results they proposed that stress-aided thermal bond rupture is the controlling factor in fracture. Their model was found to explain the exponential increase in the rate of radical formation under constant load rate.67 But it could not explain the complex nature of the time-dependent radical formation in other types of stretching experiments such as the decreasing rate of radical formation at a constant load shown in Fig. 7. The essential difficulty in applying a kinetic theory to polymer fracture is due to the following reasons. First, Eq. (14.1) was derived from the macroscopic observation of lifetime over a limited range of stress or time. Therefore, there is no assurance that this equation can be used for molecular events such as main-chain rupture or radical formation. There are some arguments that this theory does not hold for very short or very long time fracture experiment^.'^*.^^*.^^^ Second, in the equation for molecular events, the atomic stress should be used instead of macroscopic stress. The expression for atomic stress is a complex function of the macroscopic stress, temperature, morphology of polymers, and time. Zhurkov ct a / . further studied the deformation and fracture of polymers by other experimental methods, as discussed in Chapter 14.7. From these results they concluded that macroscopic fracture occurs when the number of microcracks that are assumed to be initiated by mechanically generated radicals reaches a critical value.114-116 found from extensive studies on radical formation DeVries er d.80*81 A. Tobolsky and H. Eyring,J. Chrtn. Phys. 11, 125 (1943). D. R. Curran, L. Seaman, and D. A. Shockey, Phys. Toduy 30,46 (1977). 113 A. Peterlin, J . Mrrcromol. Sci., Phys. 7, 705 (1973). M S . N . Zhurkov, V . A. Zakrevskii, V . E. Korsukov, and V. S. Kuksenko, J . f o l y m . 'I*

Sci.. frrrr A-2 10, 1509 (1972). IIS lle

S. N . Zhurkov and V . E. Korsuk0v.J. Polytn. Sci.. Polym. Phys. Ed. 12, 385 (1974). S. N . Zhurkov and V. S. Kuksenko, I n t . J . Frrrcr. 11, 629 (1975).

14.6.

DEFORMATION A N D FRACTURE OF POLYMERS

22 I

under various loading conditions and temperatures that the strength of nylon 6 fibers is correlated with the region where radicals were formed. They obtained histograms of radical formation for nylon 6 fibers from step-strain tests at various temperatures and found that histograms could be well described by a Gaussian distribution. They interpreted these results to reflect the effective distribution of chain lengths of tie molecules in some critical amorphous regions. They developed a deformation model based on the length distribution of load-bearing tie molecules and the reaction rate theory using the Zhurkov-type equation for the lifetime of stressed tie molecules taking into account the atomic stress on a bond.80*81s3Their equation for the rate of chain rupture, dC,/dr, is expressed by (14.2)

where CUiis the number of unbroken tie molecules in the ith group of length I( exposed to an atomic stress of q , and w,, U,,and y are the kinetic parameters. In their model it is assumed that the polymer fiber has a sandwich structure of crystalline and amorphous regions connected by tie molecules, and that at least some of these amorphous regions are critical regions in which fracture initiates. They assumed that only fully extended tie molecules bear the load and established a relationship between the macroscopic stress and the microscopic strain. The input parameters in their model include S (standard deviation of the distribution of tie molecule lengths in one critical region), RC (the ratio of the number of polymer chains in the crystalline region to the number of tie molecules in the amorphous region), RL (the ratio of the original length of the crystalline region to that of the amorphous region), Eb (the elastic modulus of a single polymer chain), oo(the collision parameter), U,(the activation energy), y (the activation volume), ut or (macroscopic stress or strain as a function of time), and T (temperature). They determined best fit values of these parameters from a comparison of theoretical calculations with the experimental curves for constant-strain-rate and constant-stress tests using the experimental value of S from ESR results and the best-known theoretical values of E b and o,. After determining these parameters at room temperature, they used them to predict mechanical and radical formation behavior in tests at other strain rates and during creep, constant-load-rate stretching, and cyclic stress fatigue. They obtained a fairly satisfactory correlation between experimental results and theoretical predictions from their model. They also successfully predicted and explained the effect of temperature on deformation and fracture behavior and on radical formation. They observed a similar and even slightly wider histogram of radical formation for nylon 6 drawn film

222

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ESR STUDY OF POLYMER FRACTURE

and confirmed that such observed distribution is not caused by macroscopic artifacts such as an uneven distribution of strains due to differences in the lengths of the many individual filament^.^ Although their model can give fairly good predictions at various loading conditions and temperatures, it still contains some problems to be solved. First, their data depend on the secondary radical formation behavior, introducing uncertainties concerning the transformation process from the primary radicals “originally formed” to the secondary radicals “experimentally observed” and the variation of radical concentrations and polymer microstructure during such process. Second, their theoretical and experimental curves for a constant-strain-rate test, which is one of the experiments they used to get “best fit” parameter values for their subsequent theoretical calculations, showed a maximum and subsequent drop of stress at higher strain regions. A monofilament does not show such behavior, as reported by Crist and Peterlin117and Nagamura et al. 118 Their results at higher strains are presumed to reflect mostly the macroscopic differences in lengths of many filaments. Their predictions show better fit at’higher strains rather than at smaller strains, where a monofilament and a bundle of many filaments show an identical stress-strain curve. Therefore, some best fit parameter values will need to be revised. Third, their model cannot explain the mechanical and radical formation behavior during a repeat stretching of previously strained or broken material. Peterlin and Verma80J10~1*0 initially suggested that radical formation during tensile deformation reflects the distribution of lengths of tie molecules. However, from some problems encountered in this concept such as the third problem mentioned above and the fact that the radical concentration is l(r-105 times larger than the number of tie molecules in a single amorphous region and 10-102 times smaller than the total number of tie molecules in a sample, they introduced a defect distribution of the microfibrillar superlattice of the fiber structure. They proposed that the basic structural element is an extremeley strong microfibril that contains alternating crystalline and amorphous regions bridged by a large number of taut tie molecules. The ends of microfibrils form a vacancy defect or point dislocation as schematically shown in Fig. 14.121 They proposed that radical formation reflects the increase of strain in microfibrils adjacent to these defects rather than the distribution of average strain in tie B. Crist and A. Peterlin, Makromol. Chem. 171, 21 1 (1973). T. Nagamura, K . Fukitani, and M. Takayanagi, J . Polym. Sci., Polym. Phys. Ed. 13, 1515 (1975). ‘I9 A. Peterlin, J . Polym. Sci., Part A-2 7 , 1151 (1969). A. Peterlin. J . Polym. Sci.. Part C 32, 297 (1971). A. Peterlin, i n t . J . Fracf. Mech. 7 , 4% (1971).

14.6.

DEFORMATION A N D FRACTURE OF POLYMERS

223

b

J

1 D' Section A A

D' Section BB'

FIG.14. Microfibrillar superlattice model with (a) vacancy defect and (b) point dislocation in two perpendicular planes CC' and DD' through the point dislocation and two planes AA' and BE' perpendicular to the fiber axis.IP1

molecules .79J09J17J21-123 To account for the high initial Young's modulus of fibrous material, Peterlin proposed a new model, which includes very taut tie molecules having lengths less than the average thickness of the amorphous layer.123 They concluded that their model can explain almost all mechanical properties and radical formation behavior during the stretching of highly oriented crystalline polymers. But this model still does not seem to be perfect by the following reason. First, a marked increase of stress or strain should not be presumed to exist in the absence of a detailed quantitative consideration of tensile and shear stresses in a system of aligned microfibrils. Even if the stress or strain enhancement is sufficient to cause adjacent microfibrils to break, there remains the question of the stability of defect regions at the ends of each microfibril that are six or eight times as large as a microfibril.lU These would cause further development of microcracks and a decrease of strength. Second, this model seems not to explain the variation of the number of microcracks reported by Zhurkov et al.lza; the microcracks observed by small-angle x-ray scattering varied in number but those expected from this model would vary in size. Nagamura et al. 11* proposed another model for explaining mechanical properties and radical formation behavior during tensile deformation of PEOB fibers. The PEOB sample is the only polymer to give fairly stable A. Peterlin, J . Macrumul. Sci., Phys. 8, 83 (1973). A. Peterlin, I n t . J . Fract. 11, 761 (1975). H. H. Kausch and J. Becht, Kulloid-Z. 2. Polym. 250, 1048 (1972). Ips S. N . Zhurkov, V. S. Kuksenko, and A. I . Slutsker, Suv. Phys.-Solid la

Transl.) 11, 238 (1969).

State (Engl.

224

14.

ESR STUDY OF POLYMER FRACTURE

0.0

-,

'7

FIG.15. Schematic representation of the deformation mechanism of a PEOB fiber. For simplicity, only tie molecules are shown in the amorphous region (A). The shaded area in the crystalline region (C) represents the p-form (planar zigzag) crystal.118

primary radicals, as described in Section 14.4.1. Therefore, the deformation mechanism can be discussed more directly in connection with main-chain rupture. After confirming that the stress-strain curve of the bundle of multifilaments used for ESR study was identical with that of a single monofilament up to about 21% strain, they investigated the dependence of radical formation on strain, taking into account radical decay during stretching. They found that the dependence of radical formation on strain was well described by a Gaussian distribution in accordance with results for nylon 6 fibers. This was attributed to a distribution of effective lengths of tie molecules, as in the structural model of DeVries et a / . In addition to the rupture of overstressed tie molecules, the effect of crystal transformation in the crystalline region connected to tie molecules during stretching was included in their model, as shown schematically in Fig. 15. The crystal transformation from a helix to pform (planar zigzag) was found to be caused by tie molecules in highly oriented crystalline PEOB films by Kuroishi et U I . ' * ~ The lifetime of stressed tie molecules was evaluated as the maximum strain from energetic considerations. In their model, contributions to the load from all chains in the amorphous region such as chain ends, chain folds, and taut and loose tie molecules lz8

M. Kuroishi, M. Fujisaki, and M. Takayanagi, Nippon Kagaku Kaishi 7, 1281 (1972).

14.7. LIMITATIONS OF ESR METHOD

225

were evaluated. The calculation showed fairly satisfactory predictions of the stress -strain curves and radical formation behavior during single or repeated stretching experiments. However, this model still has problems to be included such as the time effect, the pulling out of chains from the crystalline region, the statistical aspects of the criticality of fiber microstructure, and the effect of molecular weight distribution. In summary, the deformation and fracture phenomena of polymer is very complex and at present there is no perfect model that can satisfactorily correlate the microscopic events to all macroscopic properties.

14.7. Limitations of ESR Method and Comparison with Associated Studies Although the ESR method has been proved to be very effective in studying molecular mechanisms of deformation and fracture of polymers and has provided much important information, it still has inherent problems, as have many other experimental methods. In recent years several analytical methods have been used to study molecular events during deformation and fracture of polymers. These methods may provide information to corroborate or modify the molecular mechanisms based on ESR results. 14.7.1. Problems in ESR Investigations

The most serious problem in the ESR investigation of polymer fracture is the fact that this method is effective only when a sufficient number of fairly stable free radicals are generated. Many polymers do not suffer enough chain rupture during tensile deformation and fracture, and so the applicability of this method is limited to a few polymers such as nylon 6, nylon 6,6, PEOB, and polyethylene. Another problem is the uncertainty in radical concentrations due to the reactivity of radicals and/or to many factors in measurement as discussed in Section 14.2.2. The relatively high reactivity of most mechanically generated radicals could cause large discrepancies between the numbers of “originally generated” and “actually observed” radicals. Therefore, decay reactions of radicals and the effect of the environment must be very carefully studied before interpreting experimental results. The ESR method cannot tell exactly the locations of main-chain rupture in the supermolecular structure of polymers, although ESR has proved that mechanically generated radicals are located in the amorphous regions of fiber structure.

226

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STUDY OF POLYMER FRACTURE

14.7.2. Other Methods for Studying Micromechanism of Polymer Deformation and Fracture 14.7.2.1. Infrared Spectrosc0py.t Infrared (IR) spectroscopy has been used to study overstressed chemical bonds and newly formed end groups during deformation. The phenomenon of frequency shifting in the IR spectrum of stressed polymers was first reported for the hydrogen-bonded -OH and -NH stretching bands of some polymers under high pressure.1z7 Since then it has been demonstrated both theoretically and experimentally that the frequency and shape of skeletal vibration bands depend on an applied stress. Experimental observations in tensile deformation are reported for highly oriented polypropylene, nylon 6, poly(ethy1ene terephthalate), polyacrylonitrile and p~ly(phenyl-p-sulfide).~~~-~~~ The common features of these results are (a) the peaks of these bands shift to lower frequency and (b) a “tail” appears at lower-frequency side of these bands upon loading. The peak shift has been found to be proportional to the applied stress. The tail is interpreted as being caused by the nonuniformly stressed bonds. The distribution function of molecular stress among bonds can be obtained from the analysis of this tail. The result shows that there is a wide distribution of molecular stress with maximum stress of about 1000-2000 kg/mm2, while most (50-95%) of the bonds experience stress close to the applied stress. In constant-stress tests, the distribution of molecular stress has been found to vary with time in such a way that the number of strongly overstressed bonds increases with time at the expense of lightly stressed bonds.1zD*134 The number of ultimately stressed to near the maximum stress decreased with time and the amount of this decrease was found to be linearly correlated with the number of ruptured bonds determined by the end-group a n a 1 y ~ i s . l In~~ creases in temperature or applied stress are found to have the same effect J. Reynolds and S. S. Sternstein, J . Chem. Phys. 41, 47 (1964). S. N . Zhurkov, V. 1. Vettegren. I . 1. Novak, and K. N . Kashincheva, Dokl. Aknd. Nauk S S S R 176,623 (l%7). I** S . N . Zhurkov, V. I. Vettegren, V. E. Korsukov, and I . I . Novak, Fracr., Proc. Inr. Conf., 2nd. 1969 p. 545 (1%9). S . N . Zhurkov, V. I. Vettegren, V. E. Korsukov, and I. I. Novak. Sov. Phys.-Solid Stare (Engl. Transl.) 11, 233 (1969). D. K. Roylance and K. L. DeVries. J . Polym. Sci.. Parr B 9,443 (1971). Is* V. I. Vettegren and I . 1. Novak. J . Polym. Sci., Polym. Phys. Ed. 11, 2135 (1973). V. A. Kosobukin, Sov. Phys.-Solid Stare (Engl. Transl.) 14, 2246 (1973). R. P. Wool and W. 0. Statton, J . Polym. Sci.. Polym. Phys. Ed. 12, 1575 (1974). Is6 R. P. Wool, J . Polym. Sci.. Polym. Phys. Ed. 13, 1795 (1975). 13(1 V. I. Vettegren, I. I . Novak, and K. J. Friedland, Int. J . Fracr. 11, 789 (1975).

t See also Volume 10 (Far Infrared) of this series, as well as Volume 13B (Spectroscopy), Chapters 4.1 and 4.2.

14.7

LIMITATIONSOF ESR METHOD

227

as time on the stress distribution.129 Overstressed bonds are found to lie in the amorphous region and to be oriented in the direction of the applied stress. These results correspond well with ESR observations such as the distribution of lengths of tie molecules and the time-dependent main-chain rupture of overstressed chains. As to the detection of main-chain rupture, the end-group analysis by IR appears to have significant advantages over the ESR method in that end groups once formed by chemical reactions of mechanically generated radicals should be relatively stable, which would allow the study of materials over long times. In the end-group studies, specimens of the same polymer are placed in each light beam of a double-beam spectrometer and one of the specimens is stretched during observation. In this way the spectrometer records the difference in absorption due to the original and the deformed specimen. In principle, this differential method should cancel the end groups originally present and indicate only those newly formed. But IR is also sensitive to changes of orientation, sample thickness, and many other factors that may occur during deformation of the polymer. Therefore, in some cases it is very difficult to detect the changes of end groups alone. If the absorbance of end groups is determined and the extinction coefficient is known, the concentration of end groups can be estimated from the Beer-Lambert law. The intensities of the IR bands of end groups such as -C = CH,, -OH, -CH3, -COOH, and -CHO are reported to increase upon tensile loading in polyethylene, polypropylene, poly(viny1 alcohol), or polyoxymethylene. 114-118~137-1c0 The concentration of newly formed end groups was found to increase with applied stress and time,138and to be much larger than the radical concentration detected by ESR. These results are discussed in Section 14.7.2.4 in connection with results from other methods. 14.7.2.2. Mass Spectroscopy. The application of mass spectroscopy to polymer fracture was pioneered and has been used to detect the volatile products from polymers deformed or fractured in the spectrometer by Regel et c ~ / . ~They ~ ~ observed * ~ ~ ~ some small molecular fragments in 137 S. N. Zhurkov, I . I . Novak, and V. 1. Vettegren, Dokl. Akad. Nard S S S R 157, 1431 (1964). Is8 S. I. Veliev, V. I . Vettegren, and I. I. Novak, Polym. Mech. (Engl. Transl.) 6, 369 (1970). S . I . Veliev, V. E. Korsukov, V. I . Vettegren, L. F. Shalaeva, and I. I. Novak, Polym. Mech. (Engl. Trans/.)7 , 347 (1971). I4O U. G. Gafurov and 1. 1. Novak, Polym. Mcch. (EngI. Trunsl.) 9, 517 (1973). 141 V. R. Regel, T. M. Muinov, and 0. F. Pozdnyakov, Sov. Phvs.-Solid Stute (EngI. Transl.) 4, 1809 (1963). 14* V . R. Regel, 0. F. Pozdnyakov, and A. V. Arnelin, Polym. Mech. (Engl. Trrrnsl.) 11, 13 (1975). Their preceding works are cited in this review article.

228

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ESR STUDY OF POLYMER FRACTURE

stretched poly(methy1 methacrylate) and polystyrene and assumed that these products were caused by chemical reactions initiated by mechanically generated radicals. The activation energy of mechanical fracture was estimated to be equal to that of the initial-stage thermal degradation due to rupture of "weak" bonds. The rate of formation of these products was found to increase with applied stress. From these results it was concluded that the kinetic theory of polymer fracture developed by Zhurkov et (11. was confirmed. There are, however, some ambiguities about the origin of the lowmolecular-weight products of mechanical degradation. Grayson er a / .143 studied the mechanical degradation of well-characterized polystyrene samples. They observed large amounts of styrene evolved from both the as-received and vacuum-baked samples, but essentially none was observed from the sample dissolved and reprecipitated to remove residual monomers. From these results they concluded that previous reports on volatile products upon loading might be the result of residual monomers. The very small amount of styrene released during the mechanical fracture of purified polystyrene was interpreted as indicating that either the number of primary chains broken or the subsequent unzipping reaction chain length is relatively small. The ESR result that polystyrene or poly(methy1 methacrylate) yields very small numbers of radicals during tensile deformation and fracture seems to support the former interpretation. 14.7.2.3. Molecular Weight Measurements. The rupture of main chains detected by the ESR or IR method is expected to significantly reduce the molecular weight of the polymer sample. In other words, the determination of changes in molecular weight can yield information on the amount of bond rupture if the type of chain rupture and molecular weight distribution function are assumed. Although solution viscometry or gel permeation chromatography (GPC) is not as sensitive as ESR, such methods have some advantages over ESR or IR: (a) they are not sensitive to changes of various factors in the sample during deformation and fracture, (b) they are not affected by radical reactions and the environment, (c) all soluble polymers can be studied, and (d) GPC can provide information as to which chains among the molecular weight distribution are broken. There are a few reports on the changes of molecular weight caused by mechanical action, but they showed a reduction in average molecular M . A . Grayson. C. J . Wolf, R. L. Levy, and D. B. Mil1er.J. Po/yrn. Sci.. Polyrn. Phys. Ed. 14, 1601 (1976).

14.7

LIMITATIONS OF ESR METHOD

229

Molecular Weight FIG. 16. Molecular-weight distributions obtained from GPC before and after fracture in virgin.14E drawn nylon sample: (---) fractured, (-)

weight during deformation and Roylance and D e V r i e ~ obtained '~~ from GPC studies shown in Fig. 16 almost the same amount of chain scissions as the result from ESR in nylon fiber, i.e., about 101*/gm. Stoeckel et d.146a studied chain rupture and tensile deformation of semicrystalline polymers such as nylon 6, nylon 6,6, polypropylene, polyethylene, and polyethyleneneterephthalate by the viscometry and ESR. The chain rupture concentrations estimated from the changes in molecular weight assuming a random-scission scheme were shown to be 1018/cm3 for nylon and polyethyleneterephthalatefibers and 1018/cm3for polypropylene and isotropic samples of polyethylene, polypropylene, and nylon 6. Their results indicated that these values were about 10 times larger than those obtained from ESR. Some possible explanations were given for this discrepancy. The stretching environment and thermal and mechanical history were found to affect the bond rupture. The relation between the bond rupture and the tensile properties of polymers was considered. 1M A. M . Ar'ev, N . N . Zaslavskii, and N . G. Dolzhenkova, f o / y m . Mech. (Engl. Trunsl.) 4, 434 ( 1968). 145 R . E. Mehta. cited by K . L. DeVries and M . L. Williams,J. Mucromol. Sci.. f h y s . 8, 691 (1973). *" D. K . Roylance and K . L. DeVries, P o / ~ mP. r e p . . Am. Chem. Soc.. Div. folym. Chrm. 17(2), 720 (1976). 148a T. M. Stoeckel, J . Blasius, and B. Crist, J. folym. Sci., f o / y m . f h y s . Ed. 16, 485 (1978).

230

14.

ESR STUDY OF POLYMER FRACTURE

14.7.2.4. Small-Angle X-Ray Scattering. Small-angle x-ray scattering (SAXS) as a method of investigating deformation and fracture was pioneered by Zhurkov et (11. and has been used mainly in the USSR. 114-116~125~147-150 A small-angle light-scattering method was used together with SAXS in some cases. It was found that the intensity of the SAXS pattern of both crystalline and amorphous polymers increased and its peak shifted to smaller angles upon loading, These changes are attributed to the appearance of submicrocracks, which are assumed to be formed as a result of the coalescence of chain scissions detected by the ESR method. The dimension, shape, and number of these submicrocracks were evaluated as a function of time or applied stress from an analysis of these diffraction patterns by Guinier's formula.151 It was found that the number of submicrocracks increased rapidly upon loading and approached an equilibrium value under constant load, while the change? in dimension were relatively small in both crystalline and amorphous polymers. The rate of submicrocrack formation increased exponentially with the applied load. The average lateral dimension of submicrocracks was 250-600 8, and their final concentration just before fracture was of the order of 1015-1016 per cm3 depending on the polymer species. Zhurkov et al. compared the number of submicrocracks (N,) with that of radicals (N,) detected by ESR and that of end groups (N,) detected by IR.114*152The result showed that N, is roughly equal to N,, while N , is about 103 times larger than N,. Based on these results they proposed a submicrocrack formation mechanism such that the mechanically generated radicals react successively with adjacent polymer chains to form many end groups without increasing the radical concentration and that the region surrounded by these end groups becomes a submicrocrack. They assumed that macroscopic fracture occurs when a sufficient number of submicrocracks are formed so that the average distance between them becomes comparable with their dimensions and some of them are given 14'S. N . Zhurkov, V. A. Marikhin, and A. I. Slutsker, Sov. Phys.-Solid

Stare (Engl.

Trans/.) 1, 1060 (1960). 14* S . N. Zhurkov, A. 1. Slutsker, and V . A. Marikhin, Sov. Phys. -Solid Srure (Engl. Trans/.) 1, 1601 (1960). Ire S. N . Zhurkov, V. S. Kuksenko, and A. I. Slutsker, Fruct., Proc. Inr. Conf., 2nd. 1969 p. 531 (1969). Irn M. A. Gezalov, V. S. Kuksenko, and A. I. Slutsker, Sov. Phys. -Solid Srate (Engl. Trunsl.) 14, 344 (1972). A. Guinier and G . Fournet, "Small-angle Scattering of X-rays." Wiley, New York, 1955. IS1 E. E. Tomashevskii, V . A. Zakrevskii, 1. I . Novak. V . E. Korsukov, V . R. Regel, 0. F. Pozdnyakov, A. I. Slutsker, and V . S. Kuksenko, I n f . J . Frucr. 11, 803 (1975).

14.7

LIMITATIONS OF ESR METHOD

23 I

favorable conditions for coalescence and rapid growth. But this hypothesis has some problems with the unclarified termination mechanism of chain reactions and the strength of a once deformed or fractured sample. lZ4 A more detailed treatment of submicrocracks and crazes in glassy polymers is given in Part 15 of this volume. Many alternative methods of studying such phenomena are discussed there. Acknowledgments The author wishes to acknowledge Professor K. L. DeVries for reading the manuscript and making valuable comments. The author appreciates the National Science Foundation Grant DMR74-03271 for support during the preparationof this manuscript as well as support for some of the studies reported.

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15. METHODS OF STUDYING CRAZING

By Norman Brown 15.1. Introduction There are several reasons for studying crazing in polymers: (1) The fracture of a polymer is usually preceded by crazes and to understand fracture we must understand crazing. (2) Crazing may enhance the toughness of polymers as in the case of multiphase high-impact polystyrene. (3) There are two modes of plastic deformation in polymers below the glass transition temperature: shear flow and crazing.

In the simplest terms, shear flow is like the flow of a viscous liquid; the flow deformation may be homogeneous or be localized in the form of shear bands that are approximately parallel to the maximum shear strain. When the specimen yields in tension via shear flow, a neck usually forms in the specimen. Except for the existence of shear bands, it is difficult to detect an intrinsic change in the bulk properties of the polymer when it first begins to yield via shear flow. After yielding, the shear flow produces an orientation of the molecules. Crazing, on the other hand, is a form of plastic deformation (permanent deformation) caused by discrete entities called crazes. The existence of crazes in a specimen is readily observed with the naked eye if the specimen is properly tilted with respect to transmitted light. The crazes are visible because they consist of a thin platelike region whose index of refraction is less than that of the surrounding polymer; a silvery appearance is produced by the internal reflection of the light. A low-power optical micrograph of crazes is shown in Fig. 1. The crazes are generally perpendicular to the maximum tensile stress. An electron micrograph of the cross section of a typical craze is shown in Fig. 2. Crazes are typically about 1 p m thick. Their internal structure consists of (1) oriented fibrils, which are about 200 8, thick with most fibrils oriented in the direction of the tensile stress, and (2) pores with a porosity that may range from about 30 to 90% with typical values of 40-60%. When a specimen deforms by 233 METHODS OF EXPERIMENTAL PHYSICS, VOL.

16C

Copyright 0 1980 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12.475958-0

234

15. METHODS OF STUDYING CRAZING

FIG. 1 . Optical micrograph of crazes (from lmai and Brownlo) at surface of a polymer (2.5 X ) .

crazing there is no deformation transverse to the tensile stress as in shear yielding because the extension parallel to the stress is accommodated by internal dilatation. Poisson's ratio is effectively zero. A craze is not a crack although crazes and cracks look very much alike when they are each observed in the closed state. A literal translation of craze in Russian is "silver crack." Under stress there is only space between the surfaces of the crack, whereas in a craze these surfaces are bridged by oriented fibrils. The tensile stress normal to the surfaces of a crack is always zero, whereas the stress supported by the craze may be just as great as that in the uncrazed region. Crazes are usually initiated at the surface of a polymer at defects such

FIG.2. Electron micrograph of a craze (courtesy of KambourO).

15.1. INTRODUCTION

I

235

I

FIG.3. Schematic of a craze.

as notches, scratches, voids, or dirt particles, just as cracks are initiated at points of stress concentration. If there are points of sufficient stress concentration inside the polymer, as is the case of rubber-modified polystyrene, which consists of small rubber particles in a polystyrene (PS)matrix, then craze can be initiated internally. The shape of a craze is illustrated by Fig. 3. The length of the craze, I, as indicated by the dimension visible on the surface of the specimen, can be as large as the dimensions of the specimen or too small to see with a microscope, depending on the conditions under which the craze was produced. The dimension perpendicular to the surface p is approximately proportional to 1. The thickness a of the crazes, ranges from about 0.2 to 20 pm. The thickness of the crazes tapers throughout its length as shown in Fig. 2. Occasionally crazes have unusual shapes as observed by Imai and Brown' for polycarbonate in a C02 environment around 200 K (Fig. 4). Crazing is very general in that it can be produced in both amorphous and crystalline polymers.e In the past it was not thought to occur in crystalline polymers only because the crazes are more difficult to observe in nontransparent materials. Crazes have been produced in two of the most crystalline polymers: polytetrafluorethylene (PTFEY and high-density, high-molecular-weight p~lyethylene.~A basic question that is often asked is: Why under a certain set of conditions as prescribed by the stress, temperature, and environment does a particular polymer craze in-

' Y. Imai and N . Brown, M a w . Sci. Eng. 28, 209 (1977).

* A. van der Boogaart, "Basis of Yield and Fracture." Inst. Phys. and Phys. SOC., London, 1966. S. Fischer and N. Brown, J . Appl. Phys. 44, 4322 (1973). Research in progress.

236

15.

METHODS OF STUDYING CRAZING

FIG.4. Unusual-shaped crazes in polycarbonate formed in COz at 150 K*(400 x).

stead of deforming by shear? This question can only be answered in a very limited way at this time. One general statement can be made about the occurrence of crazing in relation to the stress field. If the stress field is compressive or there is a net hydrostatic pressure on the specimen, then crazing is more difficult; this is understandable since crazing involves pore formation. Another generality of crazing is the effect of molecular weight. If the molecules of a solid are very short then the entanglement is insufficient to support the drawn fibrils that constitute a craze, and a simple crack will form. The lower limits of molecular weight to form a craze have not been extensively studied, but it appears that such a limit definitely exists. For example, it is difficult to observe crazes below a number-average molecular weight of about 35,000 in polystyrene5 and 45,000 in poly(methy1 methacrylate) (PMMA).s Crazes can be classified into two types: ( I ) intrinsic crazes, which occur when the specimen is stressed in an inert environment, and (2) environmental crazes, which occur by a combination of stress and an active environment. Many polymers may not craze intrinsically, but practically every linear polymer that has been thus far investigated can be made to craze in a suitable environment. For example, practically every linear polymer will craze under a tensile stress when exposed to nitrogen or J. F. Fellers and B. F. Kee, J . Appl. Polym. Sci. 18, 2355 (1974). R. P. Kusy and D. T. Turner, Polymer 18, 391 (1977).

15.2.

STRUCTURE

237

argon gas at sufficiently low temperatures in the neighborhood of their dew p ~ i n t . ~ , ? ~ Research on crazing can be divided into the following areas: (a) the structure of crazes as observed by microscopy and interferometry, (b) the influence of stress on the initiation and growth of crazes, (c) the effects of environments, and (d) the relationship between the macroscopic mechanical behavior and the microproperties of the individual craze. Fracture will be discussed in area (d), since crazing and fracture are closely connected. The excellent review articles by Rabinowitz and Beardmore* and by KambourOfurnished much of the foundation for this work. The author has tried to distill the essential features of crazing as presented in these review articles and then to add the research findings of subsequent publications.

15.2. Structure In this chapter methods for observing the size, shape, and internal structure of crazes are presented. It is of interest to observe both static structure and the changes in structure as the craze develops. The stress field associated with the craze may also be considered part of the overall structure. The existence of the craze when the specimen is under stress leads to a distortion of the stress field as in the case of a crack. The stress concentration thus produced is an important factor in understanding the growth of crazes. The dimensions of crazes may vary from the submicron range to a macroscopic dimension that covers the entire cross section of a specimen. 15.2.1. Optical Methods

The optical microscope is useful for determining the number of crazes, and their size and shape. Crazes can be observed under reflected and transmitted light. For sheet specimens, transmitted light usually gives better contrast. If the specimen is observed after the stress that produced the crazes has been removed, then the crazes retract or heal. For some polymers it is found that the crazes are more visible just after they are formed, as compared to some later time. This is especially true for PTFE and polyethylene when the crazes are produced in liquid nitrogen. N. Brown, Mmrrr. S u . h t g . 25, 87 (1976). M . F. Parrish and N. Brown, J. Mucromol. Sci., Phys. 8, 665 (1973). S . Rabinowitz and P. Beardmore, Crii. Rev. Macromol. Sci. 1, 1 (1972). R . P. Karnbour, Macromol. R p i ~ 7, . 1 (1973).

238

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METHODS OF S T U D Y I N G C R A Z I N G

FIG.5. Crazes in a direction parallel to the surface of the specimen ( 1 1 0 ~ ) .

To determine the area and shape of crazes they should be viewed from two or three directions. The simplest type of observation is the observation of the length 1 (Fig. 3 and as shown in Fig. 5). The depth of penetration p is determined by longitudinally sectioning the specimen and metallographically polishing the exposed surface8as shown in Fig. 6. One can also observe crazes from a direction perpendicular to their large dimensions by cutting a thin slab parallel to the area of the crazes and metallographically polishing the parallel faces. The slab can be several millimeters thick, but should be as thin as possible if there are many overlapping crazes. The slab is then viewed in transmitted light and needs to be tilted slightly in order to obtain contrast. Figure 7 is an example of this technique. Crazes produce surface groovese whose height can be measured by optical interferometry. The height of the groove or step is on the order of 1 pm. The interferometers are generally available as standard accessories on most metallurgical microscopes. These surface grooves can also be measured by a profilometer for measuring surface roughness (Talysurf, for example). Sometimes shear bands form at the tips of crazes at an angle of approxi-

15.2.

STRUCTURE

239

FIG. 6. Crazes in polycarbonate in a direction perpendicular to the surface at 77 K (loox).

FIG.7. Crazes in PS viewed in a direction nearly parallel to the thickness of the crazes (ax).

240

15.

METHODS OF STUDYING CRAZING

FIG.8. Crazes and microshear bands in polycarbonate (1 10 x).

mately 45” D ~ l o(Fig. 8). It may be important to determine the relationship between the crazes and the shear bands. The conditions for optimum optical contrast are usually not the same for crazes as for the shear bands and it was foundlo that a series of through-focus photographs gave the best view of how the crazes and shear bands were related. The fractional porosity of crazes is an important parameter. Kambourll has measured the porosity by filling the craze with a liquid of known index of refraction. The effective index of refraction of the crazes is measured by determining the critical angle for total reflection. By using the Lorentz-Lorenz equation, the porous content of the craze can be calculated.ll If the craze is not filled with a liquid, the critical angle for total reflection is more difficult to measure. The profile of the craze, as determined by its variation in thickness, can be measured interferometrically. For crazes that were nucleated in precracked specimens, Kambour12and Brown and Ward13 observed the type of interference fringes shown in Fig. 9. Observations were made perpendicular to the plane of the crack and craze with an optical microscope Imai and N. Brown, J . Marer. Sci. 11, 425 (1976). R. P. Kambour, Polymer 5, 143 (1964). l* R. P. Kambour, J . Polym. Sci., Purr A-2 4, 349 (1%6). H. R. Brown and I. M. Ward, Polymer 14, 469 (1973). lo

l1

15.2.

STRUCTURE

24 1

FIG.9. Interference fringes at crack tip in PMMA. Large set from crack, small set from craze (courtesy of Kambours).

using reflected light. The fringes were observed in loaded and unloaded state, loading being produced by a razor blade wedge in the crack.13 From the ratio of the fringe in the loaded and unloaded state, Brown and WardI3 calculated the extension ratio A at each point in the craze and showed that the index or refraction in the loaded state was given by (15.2.1)

242

15.

METHODS OF STUDYING CRAZING

Pm

CRACK f

t

-I CRPLE

I

-2

\

FRACTURE SURFACE LAYER THICKNESS -.58Cm I

1 1

-70

-60

1

I

a

1

-40

-30

-20

I

I 1

-50

I

I

I

1

-10

1 1

0

10

I

20 pm

LEWH

FIG.10. Profile of crack tip and craze in PMMA as calculated from interference pattern in Fig. 9 (courtesy of KambouP).

where yois the index of refraction of the uncrazed polymer. The data indicated that p was constant along the length of the craze. Figure 10 shows the craze and crack profile.12 Since crazes usually precede fracture, optical observations of the fractured surface give information about the structure of crazes. This craze structure may produce interference colors that depend on the thickness and density of the craze material. That part of the fracture surface where the craze existed prior to rapid fracture is usually very smooth even if the interference colors are not observable. The existence of interference colors will also depend on how much healing of the craze matter has occurred after fracture. 15.2.2. Electron Microscopy?

The thickness of mature crazes often ranges from about 0.2 to 20 pm. Since the oriented fibrils and the pores are about 200 A in size, electron microscopy is the best tool for determining the internal structure. The simplest way to estimate the thickness of the craze at a point on the surface of the specimen is to coat the surface of the specimen with a conducting film and observe the surface in the scanning electron microscopy. Better resolution can be obtained by making a replica of the surface using polyacrylic acid and then coating the polyacrylic acid replica with t See also Part 7 in this volume, Part B.

15.2.

EXTENDED C RAZ E

ORIGINAL POLYMER

243

STRUCTURE

RETRACTED CRAZE

FIG. 1 1 . a, is the primordial thickness. b the craze-opening displacement, and u the thickness of the retracted craze.

carbon-platinum shadowing. The polyacrylic acid is dissolved in water and the carbon -platinum replica is viewed in the transmission electron microscope. The above observations have two drawbacks because the craze is being observed in the retracted state. The retraction obscures the true thickness and tends to obliterate any tapering in thickness. The measured retracted craze thickness may be simply related to the original displacement caused by the opening of the craze as follows1*: If we define b as the craze-opening displacement, a. as the primordial thickness of polymer, and N the measured retracted craze thickness (see Fig. 1 l ) , then the measurable ratio 9 of the macroscopic strain in retracted to extended states is given by 9 =

(U

-

ao)/b.

Defining p as the fractional porosity of the extended state, p = b / (b +

4,

then eliminating a, from these equations yields

b=a/

(

P )

( 15.2.2)

Crazes may retract almost completely when the stress is released. 9 values in the range of 0.1-0.01 have been measured and in this case, if p = 0.5, b = a , and the craze thickness in the extended state is -(b + a). The original thickness of the bulk polymer from which the craze evolved may vary by a factor of 3 within the same specimen. Probably this initial thickness depends on the heterogeneity from which the craze nucleated. In order to view the internal structure of a craze, it is necessary to maintain the craze in its extended state. Kambour and c o - w ~ r k e r s ~ ~ * ~ ~ impregnated the craze while under stress, with a liquid-sulfur-iodine euI'

Is

N. Brown and B. D. Metzger, J . Appl. Phys. 48, 4109 (1977). R. P. Kambour and A. S. Holik, J . Polym. Sci., Parr A-2 7 , 1393 (1969). R. P. Kambour and R. R. Russell, Polymer 12, 237 (1971).

244

15. METHODS

OF S T U D Y I N G C R A Z I N G

tectic and then solidified the impregnant under stress. The specimen was then microtomed and viewed in the transmission electron microscope (TEM). Under the vacuum the impregnant sublimed but the structure of the extended crazes seemed to remain intact. No doubt the impregnation and subsequent sublimation somewhat modified the extended structure, but the structure (Fig. 2) is not greatly different from that obtained by other techniques. Beahan ef ul. 17-1e microtomed bulk-crazed specimens of polystyrene. They mounted the 50- 150 mm thick slices between copper grids, and deformed the sandwich in a microstrain jig in order to open the retracted crazes. They also crazed microtomed slices in the same jig and found that both procedures gave about the same internal structure. Their work" indicated that crazing a thin film would give about the same internal structure as for crazes produced in bulk material. It is now more to study the structure of crazes produced in thin films because it is a much easier procedure than starting with bulk material. A factual representation of craze morphology as it changes along the length of the craze is shown in Fig. 12. The thin-film method^^^*^^*^^ start with a solution of the polymer in a suitable solvent at a concentration that may vary from 0.5 to 10%. The film may be solution-cast onto a clean glass slide and, after drying the film, it can be separated from the glass by allowing deionized water to penetrate the polymer-glass interface. The film can be placed on a soft copper grid and then the copper grid may be deformed and viewed in the TEM. The film will adhere to the copper grid if its bars have been previously coated with the polymer. Then both are heated to the softening temperature.21 Krenz et produced their solution-cast film on rock salt instead of glass. The solution-cast film can also be placed on a Mylar poly(ethy1ene terephthalate), substrate,'* which can be subsequently deformed to produce crazes. While under stress, crazes can be stabilized to prevent retraction by coating the specimen with a thin film of carbon in an evaporator.20 After removal from the Mylar by immersion in deionized water, the film is placed on a grid and viewed in the TEM. Wellinghoff and B a e P viewed crazes by deforming thin films of polymer on a Mylar substrate. They also evaporated gold on the surface prior to deforming the film. The gold forms a random distribution of P. Beahan, M. Bevis, and D. Hull, J . Muter. Sci. 8, 169 (1972). P. Beahan, M. Bevis, and D. Hull, Polymer 14,96 (1973). P. Beahan, M. Bevis, and D. Hull, Proc. R . Soc. London Ser. A 343, 527 (1975). zo E. L. Thomas and S . J . Israel, J . Muter. Sci. 10, 1603 (1975). z1 H. G. Krenz, E. J. Kramer, and D. G. Ast, J . Muter. Sci. 11, 221 I , (1976). IZ S . T. Wellinghoff and E. Baer, J . Mucromol. Sci., Phys. 11 (3). 367 (1975). I7

15.2.

STRUCTURE

245

FIG. 12. Factual diagram of the variation in micromorphology along the length of the crazes (courtesy of Beahan, ei d.'O).

small particles, which then function as markers during the deformation and thus permit localized nonhomogeneous strains to be observed as the crazes develop in the early stage of initiation. The results of staining the films with OsO, suggested to Wellinghoff and Baer that a microporosity, which was too fine to be resolved with the TEM, developed during the very initial stage of craze nucleation. Crazes usually extend perpendicular to the tensile stress. However, if shear bands that usually make an angle of about 55" with the tensile axis are produced, it is possible for the shear band to fibrillate and have an internal structure about the same as an ordinary craze, as shown by Thomas and IsraeI2Oand Wellinghoff and Baer.23 It is possible to fibrillate shear bands and at the same time to produce the usual type of craze in the same specimen. The structure of crazes has been investigated with small-angle x-ray The SAXS gives indirect inforscattering2, (SAXS) and laser mation on the size and shape of the pores and the number of pores per unit volume of specimen. The interpretation of the laser results is not definite at this time. Fracture surfaces have been viewed with the scanning electron microscope (SEM) and from replicas viewed with the TEM. The details of the fractured crazes are generally consistent with the structure that is observed in the extended crazes, as discussed above. Observations of the fractured surface are most helpful in understanding the relationship of crazing to the fracture process, but do not furnish much additional information about the morphology of the craze itself. 15.2.3. The Stress Field -

The stress field of a craze is defined as the distortion of an applied homogeneous stress field by the presence of a craze. As in the case of a crack it is expected that the craze will concentrate the applied stress field ZIS. T. Wellinghoff and E. Baer, "Microstructure and Its Relationship to Deformation Processes in Amorphous Polymers." T. R. No. 28. Case-Western Reserve University, Cleveland, Ohio, 1976. *' T. R. Steger and L. E. Nielsen "Microvoid Formation During Crazing of Styrene Polymers," APS Bull., Paper FG2. Monsanto Co., 1976. C. C. Hsiao, Appl. Phys. Lei/. 23, 20 (1973).

246

15.

METHODS OF STUDYING CRAZING

I

-0

t

POSITION

FIG. 13. Probable stress and displacement field surrounding a craze (courtesy of Verheulpen-Heymans and B a ~ w e n s ~ ~ ) .

at some point and this concentrated stress field may be important for the propagation of the craze. KnightZsfirst attempted to calculate this stress field based on an assumed shape of the extended craze. The basis of his calculation was the method proposed by Sneddonz7for determining the stress field of a crack. Knight's calculation predicted a concentration of the applied tensile stress at the craze tip. Narisawa and Kondoz8failed to observe a stress concentration by using a photoelastic method; possibly calculated their method was not sufficiently sensitive. Wilczynski et the stress field of a craze by assuming the craze consisted of a circular membrane with elastic restraints. Their results showed that the existence of a concentration in the stress depended strongly on the degree of porosity that was assumed. Verheulpen-Heymans and Bauwens30 calculated the stress field of a craze using Muskheli~hvili's~~ method. They used mixed boundary conditions by assuming discontinuous stresses or displacements over the tip and long portion of the craze. The resulting stress field (Fig. 13) was qualitatively similar to Knight's calculation and similar to experimental observations by Kramer and c o - w o r k e r ~ . ~ ~ Kramer and c o - w o r k e r ~ ~have ~ - ~developed ~ a holographic interferometric technique that measures the displacement field associated with a A. C. Knight, J . Polym. Sci.. Part A 3, 1845 (1965). 1. N. Sneddon, "Fourier Transforms," p. 426. McGraw-Hill, New York, 1951. 1. Narisawa and T. Kondo, J . Polym. Sci.. Part A-2 11, 223 (1973). *O A. P.Wilczynski, C. H. Liu, and C. C. Hsiao, J . Appl. Phys. 48, 1149 (1977). 3o N. Verheulpen-Heymans and J. C. Bauwens, J . Muter. Sci.. 11, 7 (1976). 31 N. I. Muskhelishvili, "Some Basic Problems of the Mathematical Theory of Elasticity," Noordhoff Int.. Groningen, Leyden, 1953. 32 E. J. Kramer, H. G. Krenz. and D. G. Ast, "Growth Criteria for Solvent Crazes,'' Rep. No. 2684. Mater. Sci. Cent., Comell University, Ithaca, New York, 1976. 33 H. G. Krenz, Ph.D. Thesis, Mater. Sci. Cent., Comell University, Ithaca, New York, p8

27

1977.

T. L. Peterson, D. G. Ast, and E . J. Kramer, J. Appl. Phys. 45, 4220 (1974). H. G. Krenz, D. G. Ast. and E. J. Kramer, J. Mater. Sci. 11, 2198 (1976). 38 E. J. Kramer, Proc. I n t . Conf. Mech. Environ. Sensitive Cracking Mater., I977 p. 249 35

(1977).

15.2.

L

STRUCTURE

247

Laser

FIG.14. Holographic system. M , mirror; BS, beam splitter: SF, spatial filter:, A , aperture: CL, collimating c:ollimatinglens: S, sensitivity vector: H , holographic plate (courtesy of' Krenz et

craze. The holographic technique is a double-exposure method that produces fringes related to the change in displacement field between exposures. The change in displacement can be produced by adding an increment of load or by nucleating a craze at the root of a crack in the specimen. Figure 14 shows the optical system. The changing profiles of the craze as represented by the craze opening displacement (as the crazes move away from a crack) are shown in Fig. 15.32 The stress field of the craze corresponding to the profiles in Fig. 15 are shown in Fig. 16.32 The stress field, which shows a maximum and a minimum, qualitatively agrees with the theoretical ~ a l ~ ~ l a t i The o n region ~ . ~ of ~ low ~ ~ stress ~ is in the tip region of the craze, where the polymer has begun to develop porosity so that its strength is less than that of the bulk and less than that of the oriented fibrils in the main body of the craze. The region of low stress at the tip was not observed during the initial stages of growth, possibly because the craze had not yet established its equilibrium tip shape and was still under the influence of the stress field of the starting crack.

248

15.

METHODS OF STUDYING C R A Z I N G

I 4

Z (rnm) FIG. 15. Craze-opening displacement vs. distance along craze as craze increases in length. PMMA craze in Methanol.32

I

3

4

d)

I

I

3

4

-

Z (rnrn) F I G .16. Stress field of crazes in Fig. I5 (courtesy of Kramer ef

a/.32).

15.3.

I N I T I A T I O NA N D GROWTH

249

15.3 Initiation and Growth 15.3.1. Stress Criteria for Initiation A central question is what critical combinations of the principal stresses initiate a craze. The basic procedure is to design a loading device such that more than one of the three principal stresses can be applied simultaneously. Then the craze initiation surface is mapped in stress space by determining various combinations of the magnitudes of the stresses that cause crazing. Sternstein and O n g ~ h i nused ~ ~ tension-tension, Sternstein and Myers38used tension -torsion, Oxborough and B ~ w d e used n~~ tension -compression, and Matsushige et a1.40*41used tensionhydrostatic pressure. In the last method it is important to shield the specimen from the liquid that transmits the hydrostatic pressure because an environmental effect will be superimposed on the stress effect. In order to compare the results from different investigators it is necessary to use the same definition for the stress at craze initiation. Sternstein ct ~ 1 . ~ ~ " ~ used the yield point, the maximum in the stress-strain curve, as the critical condition: others, however, used the stress for producing the first visible craze and this occurred appreciably below the yield point. Since the yield point associated with crazing depends on both the strain rate and the velocity of the crazes, as pointed out by it seems that the appearance of the first visible craze is the more desirable condition for establishing a stress criterion for crazing. The above experimental investigations have produced the following criteria for crazing: Criterion

X Y Z

01

-

= A -I-B / I , , I c, = C D / l 1 ,I u1= E 03

+

Stress field o

>o

tension-tension, tension-torsion tension-compression tension-hydrostatic pressure

Here, a1is the greatest principal stress and a3the smallest, el is the largest principal strain, I, is the hydrostatic component of the stress, the sum of the principal stresses, and A , B, C, D, and E are material constants that 37

S. S. Sternstein and L. Ongchin, fol.vm. P r e p . , Am. Chem. Soc., Div. Polym. Chem.

10, 1117 (1969).

S. S. Sternstein and F. A . Myers, J . Mricromol. Sci.. Phys. 8, 557 (1973). R. J. Oxborough and P. B. Bowden, fhilos. M u g . [8] 28, 547 (1973). 'O K. Matsushige, S. V. Radcliffe, and E. Baer, J . Muter. Sci. 10, 833 (1975). " K. Matsushige, S. V. Radcliffe, and E. Baer, J . Polym. Sci., Polym. Phys. Ed. 14,703 38 38

( I 976). IZ N . Brown, fhilos. Mug. [8] 32, 1041 (1975).

250

15. METHODS OF STUDYING CRAZING

depend on strain rate and temperature. Contrary to criteria X and Y,criterion Z permits crazing to occur when I < 0, as was observed in polystyrene.41 It was suggested that the sequence with which the stresses are applied may be a factor that produced the differences in the above criIt is also likely that there is no simple criterion for craze initiation which applies to all possible stress fields; such is the case for the fracture of metals. Wang et u I . ~ ' supported the critical strain criterion Y by experiments with hard and soft spheres embedded in polystyrene. The specimen was exposed to uniaxial stress and the location of the point where crazes initiated was observed. It was found that, in accordance with the nonhomogeneous stress field produced by the spheres, the crazes nucleated at the points where el was a maximum. For the soft spheres, u1 and el occur at 37" with respect to the stress axis, whereas other criteria predicted crazing at other points. Additional experiment^^^ using embedded rubber spheres with variable spacing and subjecting the polystyrene to a uniaxial stress supported the idea that crazing initiates at the points of maximum strain, in agreement with criteria Y and Z. For a given stress field, the magnitude of the stress at which the first craze appears depends on the polymer, the time, the temperature, the environment, and the defects such as notches, scratches, dirt particles, and voids, which concentrate the stress. Thus, the condition of the surface is important in determining the magnitude of the stress to initiate the first craze. Generally crazes first initiate at the s ~ r f a c e ~of ~ -the * ~specimen where defects usually exert the greatest stress concentration. It is possible for the first craze to initiate internally if the point of greatest stress concentration is internal. Thus, the number of crazes produced for a given level of stress depends on the number of defects that produce a certain level of stress concentration. All defects at which the local stress exceeds a certain critical value will be the source of a craze. Figure 17 shows how the number of crazes per unit area changes with stress. As pointed out by Brown and Fischer" and Argon ez the surface density of crazes is a map of the surface density of points of stress concentration. For a given stress, the number of crazes also changes with time. This also means that the stress or strain to initiate the first craze may be a funca

R. A. Bubeck, Ph.D. Thesis, Mater. Sci. Cent., Cornell University, Ithaca, New York,

1976.

T. T. Wang. M. Matsuo, and T. K . Kwei, J . Appl. Phvs. 42,4188 (1971). M. Matsuo. T. T. Wang, and T. K . Kwei, J . Po/.vm. Sci.. Part A-2 10, 1085 (1972). J. A. Sauer and C. C. Hsiao. Trans. ASME 75, 895 (1953). '' N . Brown and S. Fischer. J . Po/.vm. Sci.. Po/ym. Phys. Ed. 13, 1315 (1975). a A. S. Argon, J. G. Hanoosh, and M. M. Salama, Fracture 1, ICF 4. 445 (1977). "

15.3.

2

10

I N I T I A T I O N A N D GROWTH

25 1

t 1

I

STRESS ( K S I

I

1

FIG.17. Number of crazes per unit area vs. stress for PCTFE in liquid nitrogen (from Brown and Fischer").

tion of time. Ziegler and demonstrated the time effect by measuring the critical stress and the corresponding time to initiate the fist craze. They found that the critical stress approached a lower limit as the time became very large. This lower limit depends on temperature and the environment. Ziegler and Brown49found that a stress of 3600 psi would produce crazes immediately in polystyrene in air, and a stress of 1400 psi would nucleate a craze in about lo3 hours. When the stress is applied, it takes time for the craze to form and grow to a size where it can be observed. There is an arbitrary line of demarcation between the processes of nucleation and growth since it is determined by the resolution of the microscope that is used for the observation. Argon et ~ 7 1 visualized . ~ ~ nucleation as a two-stage process: (1) The applied shear stress is concentrated at a defect, and the resulting local shear is blocked by heterogeneities in the structure of the polymer so that voids are created whose nucleation rate is thermally activated. (2) The voids grow into a craze under the hydrostatic component of the applied stress.

'*

E. E. Ziegler and W. E. Brown, Plasr. Techno/. 1, 341 and 409 (1955).

252

15.

METHODS OF STUDYING CRAZING

= 17.92 MP6 I586 1337

0

Ilsl

I

1021 I

I

I

10

1

lo2

1

1

lo3

I

1

I o4

I

lo5

Croze Initiotion time, sec

Fic. 18. Increase in number of craze nuclei with time in polystyrene subjected to different combinations of global deviatoric stress and negative pressure at room temperature. Continuous curves were computed from a theory (courtesy of Argon ct

Thus, Argon’s theory shows that the surface density of crazes increases with time to a saturation value that depends on both the hydrostatic and shear components of the stress. The experimental observations of craze density vs. time along with the theoretical curves are shown in Fig. 18.

15.3.2.Growth of Crazes Crazes grow in a direction that is normal to the maximum tensile stress if the material is isotropic. If the direction of the maximum tensile stress changes, the direction of the craze will change accordingly. If the polymer molecules are oriented in one direction, the growth direction will depend on the angle between the direction of orientation and the maximum tensile stress as shown by Rabinowitz and Beardmore.8 The length of a craze depends on the time the specimen is under stress, the velocity (which depends on stress), and the environment. If fracture does not intervene, crazes can be grown over the entire cross section of the specimen. Kambour and KoppS0 grew isolated crazes completely across 3 x d in. specimens of polycarbonate by loading to a very low stress in ethanol and waiting a few hours. They were then able to measure the stress-strain behavior of single crazes. In order to produce a single craze, starter cracks are usually used. A general procedure is to R. P. Kambour and R. W. Kopp, Philos. Mag. [8] 7, 183 (1969).

15.3. INITIATION A N D GROWTH

253

cut a notch in the specimen and then to lightly stress the specimen in a crazing agent such as liquid n i t r ~ g e n ,n-heptane, ~' n-butyl methanol, or ethanol. The initial crack may be removed by machining the specimen if it is not wanted, but the machined edge must be carefully polished to prevent the nucleation of additional crazes.s1 The motion of the craze is usually observed with a microscope. Han00shS2placed a beam of light from a light pipe on one side of the thin specimen. The slanted tip of the light pipe and the microscope on the other side were so arranged that bright crazes appeared on a dark background. The lengths of the crazes were photographically recorded as a function of time. Marshall et al. 53 observed the velocity of single crazes immersed in methanol by viewing the craze in a direction perpendicular to its plane with a microscope. B u b e ~ measured k~~ the velocity of a single craze in polystyrene in an environmental chamber using an optical method (Fig. 19). Andrews and Levy" measured the velocity of crazes in PMMA immersed in various environmental liquids, using an ultrasonic detector to observe the position of the craze. The ultrasonic detector is useful for crazes or cracks that may not be visible optically if the specimen is opaque. If the stress field is homogeneous and constant, then the velocity of a craze is constant after a craze initiation period.46-48~s1~ss If a craze grows under the nonhomogeneous stress field of a crack, then its velocity will decrease as it moves further from the crack and will eventually stop if the stress at the craze tip becomes too low. If the craze grows in a liquid environment, then it may also stop if the flow of liquid through the craze becomes impeded by "craze clogging," as suggested by Williams concerning the solvent crazes observed by Kramer et Earlier Marshall et a / . = had observed that a craze in PMMA immersed in methanol stopped when it moved out of the stress field of the crack, but they attributed it to a decrease in the pressure gradient of the environmental fluid as it passed through the lengthening channel of pores in the craze. Kamhowever, suggested that the capillary pressure would always keep the craze tip full of liquid. At high values of stress intensity the craze will continue to grow even when its tip is far beyond the influence of the con-

'' E. H . Andrews and G. M. Levy, Po/ymer 15, 599 (1974). '' J . G. Hanoosh, Ph.D. Thesis, Dept. Mech. Eng., M.I.T., Cambridge, Massachusetts, 1975.

G. P. Marshall, L. E. Culver, and J . G. Williams, Proc. R. Sor. London. Ser. A 319, 165 ( 1970).

E. H . Andrews and G . M. Levy, J . Muter. Sci. 6 , 1093 (1971). M. I . Bessonov and E. V. Kuvshinsku, Fiz. Tverd. Telu 3, 1314 (1961); Sov. Phys. Solid State (Engl. Transl.) 3, 950 (1961). "

254

15.

METHODS OF STUDYING CRAZING

-Gas

lnlel Pipe

Stranded Wire

Inner Chornber for Vapor or Liquid

Specimen with Craze Circulcting Oil From Hoake K-2 Circulator (-15OC lo llO°C)

Two Layers of Oplical Gloss (sealed)

FIG.19. Apparatus for measuring change in length of a craze in a specimen being stressed in an envii-onmental chamber (courtesy of Bube~k'~).

centrated stress produced by the crack; it simply grows under the homogeneously applied stress. The remarkable feature is that the velocity still depends on the stress intensity of the starting crack.= Andrews and LevyS1showed that after the starting crack was machined off the specimen, the craze continued to move at the same velocity as before. This means that the starting crack influenced the geometry of the craze and this geometry was related to the craze velocity. Andrews and LevyS1 suggested that the craze velocity depended on the craze thickness and that this thickness was originally determined by the stress intensity of the crack from which it started. Since the crack opening displacement is related to the stress intensity, and since the thickness of the craze is related to the crack opening displacement, the craze thickness should depend on the stress intensity of the crack. The thicker craze, according to An-

15.4.

ENVIRONMENTAL EFFECTS I N LIQUIDS AND GASES

255

drews and Levy,51produces a more concentrated stress field before its tip so that its velocity should be greater. Various dependences of craze velocity u on stress u have been observed. Sauer and H ~ i a for o ~dry ~ polystyrene and Andrews and LevyS1 for PMMA in environmental fluids observed u (u - uo),where uois a starting stress. Brown and FischeF for polytrifluoroethylene in liquid N2found u eS' for low stresses. Argon et al.48observed that u e d / u for dry polystyrene. A detailed quantitative theory for the dependence of the velocity of dry crazes on stress is given by Argon et ~ 1 . ' ~Initially Argon56proposed that the rate of pore formation determined the velocity, but later observation^^^ showed that this process was too slow and did not account for the connectivity of the pores. A new was proposed based on the instability of a meniscus. Argon likened the growth of a craze to the flow of a film confined between two plates that are being spread apart. Verheulpen-Heymans and BauwensS7 and Satos8 found for polycarbonate in air at room temperature and above that the length of the craze varied logarithmically with time. Since homogeneous creep also occurs with the crazing of polycarbonate above room temperature, Kamboure suggested that the homogeneous deformation retarded the craze growth and gave the logarithmic behavior instead of a constant velocity. Verheulpen-Heymans and Bauwen30 proposed that the logarithmic behavior occurred because the deformation of the fibrils in the craze was retarded by their orientation strengthening and that the lengthening of the craze was proportional to its thickening. Since shear deformation prior to crazing affects the subsequent craze velocity, it is very likely that the shear flow that occurs simultaneously with crazing will also affect the craze velocity. For environmental crazing, the kinetics of craze growth will depend on the flow of liquid through the craze to its tip or, in the case of gaseous environmental crazing, on the diffusion of gas into the bulk polymer.

-

-

-

15.4. Environmental Effects in Liquids and Gases Whereas some polymers may craze in an inert environment only under certain conditions, almost all polymers can be made to craze by an environmental agent. For example, liquid nitrogen causes almost every linear polymer to ~ r a ~ e . ~In *general, ~ * ~ crazing ~ * ~ agents ~ lower the stress

* A. S. Argon, J . Mucromol Sci., Phys. 8 (3-4). 573 (1973). N. Verheulpen-Heymans and J . C. Bauwens, J . Murer. Sci. 11, 1 (1976).

* Y. Sato, Kohunshi Kugukic 23, 69 (1%6).

256

15.

METHODS O F S T U D Y I N G CRAZING

or strain to initiate crazing and also increase the craze velocity. The structure of the craze will also depend on the crazing agent.21 When measuring the critical strain for crazing for polymers that contain solute molecules, it is important to prevent evaporation. If the surface becomes depleted in solute relative to the interior, it will shrink to create tensile surface stresses that may even be high enough to produce crazes in the absence of an applied stress. Conversely if solute molecules have a higher concentration at the surface than in the interior, compressive stresses are produced at the surface, which makes crazing more difficult. Kamboure and co-workers have extensively studied the effects of different liquid environments on the strain to initiate crazing. They used the B e ~ g e elliptical n~~ jig. A sheet specimen about 1 mm thick is bent around the surface of an elliptical jig and thus is subjected to a different but known strain at each point. The surface of the polymer is then exposed to a wick that contains the environmental fluid while avoiding the rough cut edges of the specimen. The boundary between crazing and nocrazing determines the critical strain, which depends on the environment and the time of exposure. Kambour et ul.e*ao-a2found that the critical strain decreased as the solubility of the environmental agent increased. In order to determine whether the presence of free liquid made a difference, a series of polystyrene specimens with different amounts of a dichlorobenzene dissolved in them were tested in the “dry” state. The relationship between the critical strain and the solubility was about the same for the wet and dry crazes (Fig. 20). Kambour et ul.e-s2found that the critical strain decreased as the glass transition temperature Tg of the solvated polymer decreased. Andrews and Bevans3 measured the critical value of stress intensity at which craze or crack growth stopped. The polymer, PMMA, was immersed in different liquids at various temperatures in a double-walled container in which the specimen and loading assembly were placed. The specimen was observed through a window with a microscope to determine the stress at which growth stopped in the prenotched specimens. They found that the critical stress intensity decreased with increasing temperature and then became constant above a critical temperature. The critical temperature was associated with the Tgof the solvated polymer at the tip of the craze. The findings of Kambour et ul.n~s0-s2and Andrews and Bevans3 have R. L. Bergen, Jr., SPE J . 24, 77 (1968). G . A. Bernier and R. P. Kambour, Mucroniolecules 1, 393 (1968). R. P. Kambour, C. L. Gruner, and E. E. Romagosa, J . f o / y m . Sri. 11, 1879 (1973). R. P. Kambour, C. L. Gruner, and E. E. Romagosa, Mucromolecules 7, 248 (1974). as E. H. Andrews and L. Bevan, Polymer 13, 337 (1972). 59

Bo

15.4.

E N V I R O N M E N T A L EFFECTS I N LIQUIDS A N D GASES I

1

I

257

1

00-

-

40 -

-

20

-

0

-

0

c-

- 20

-

-

i

-40-

DIETHYL ETHER AT Sr ~0.75

-601

1

I

1

I

'Y* "DCSIPS

FIG.20. Dependence of critical strain to initiate crazing on concentration of small molecules dissolved in polystyrene: ( 0 )free liquid present,).( dry crazing (courtesy of Kambour et u / . ~ ' ) .

been brought together by Kambour.64 The environment weakens the polymer by weakening the intermolecular Van der Waals bond. The degree of weakening is related to the particular environment agent and amount in solution, the net effect being reflected in a lowering of the Tgof the polymer (Fig. 21). The temperature per se also determines the strength of the Van der Wads bond in a polymer as reflected by the decrease in yield strength with increasing temperature. Therefore the critical strain for crazing decreases as the difference between the test temperature and the Tg of the solvated polymer decreases, as was shown by Kambour.s4 Generally, gases at sufficiently low temperatures make almost all linear polymers ~ r a ~ e . ~ The * ~effect ~ , ~of ~the- partial ~ ~ pressure and temperature of various gases on stress-strain behavior has been ~ t u d i e d . ~ ~ - ~ ' The specimen is enclosed in an environmental chamber that is cooled on the outside by an appropriate liquid. A precooled mixture of the active a R. ( 1977).

P. Kambour, Proc. Int. Conf. Mech. Environ. Sensitive Cracking Mater., 1977 p. 213

N . Brown and Y. Imai, J . Appl. Phys. 46, 4130 (1975). Y. lmai and N . Brown, J . Muter. Sci. 11, 417 (1976). O7 Y. Imai and N . Brown, Po/ymer 18, 298 (1977). M. J. Parrish and N . Brown, Nuture (London), Phys. Sci. 237, 122 (1972). E8 J . R. Kastelic and E. Baer, J . Macromol. Sci., Phys. 7 (4), 676 (1973). lo H. G. Olf and A . Peterlin, J . Po/ym. Sci.. Polyrn. Phys. Ed. 12, 2209 (1974). 71 B. D. Metzger, M.S. Thesis, Dept. Metall. Mater. Sci., University of Pennsylvania, Philadelphia, 1977. 72 N. Brown, B. D. Metzger, and Y. Irnai, J . Polym. Sci., Phys. Ed. 16, 1085 (1978). BB

258

15. METHODS OF STUDYING CRAZING

+ KKVSULFONE

2.4

2.0

c

' \.

FIG.21. Change in critical strain to initiate crazing vs. T. of the solvated polymer (courtesy of Kambour er a/.=).

gas and an inert gas (helium)is passed over the specimen. The exact temperature of the specimen is determined by a heating element that surrounds the specimen. The concentration of the active gas is controlled to within a few percent by accurately controlling the flow of each gas prior to mixing by means of manometer-controlled flow meters. The environmental chamber is attached to a tensile- or creep-testing machine. Typical stress-strain curves for polycarbonate as a function of the partial pressure of the various active gases are shown in Fig. 22, while Fig. 23 shows the effect of temperature on the tensile strength of PMMA in various environments. The vapor of solid COOat a vapor pressure as low as 2 X atm can cause crazing in PMMA. Pressures as low as 0.05 atm of N2 at 78 K can make polycarbonate craze. Many gases such as N2, A, 02,C02, CO, HIS, CHI, and N 2 0 have been shown to cause crazing if the temperature is sufficiently low. At a pressure of one atmosphere, the temperature above which the crazing action of a particular gas ceases corresponds roughly to the critical temperature of the gas. Generally, the propensity for crazing as indicated by the density of crazes or by the craze velocity increases with the pressure of the gas and decreases with increasing temperature. It appears that the surface concentration of the adsorbed gas is a key factor in determining its effectiveness as a crazing agent.

15.4.

:[ 30

30

.Oatn

.-

20

u)

X Y

v) v)

15

15

10 PC

10

- N2

77 K

-

1

20

W LL I-

u-l

30

OOatn

25

25

-

259

E N V I R O N M E N T A L EFFECTS I N L I Q U I D S A N D GASES

5

5

0

0

PC - Ar

I

87 K

4

8

STRAIN

12

16

V

0

(%I

4

8 12 STRAIN (%)

16

FIG. 22. Tensile stress-strain curves of polycarbonate as a function of pressure of various gases.E5 200 [

I

I

I

I

I

I

I

I

1

E

0' 0

I

50

I

100

1

I

150

200

I

I

250

300

350

400

TEMPERATURE ( K )

FIG.23. Effect of temperature on tensile strength of PMMA in various environments for P = 1 atm. GH He; AB: N2 and Ar: CD: 0,; FH: C 0 2 gas; FE: COPvapor in equilibrium with solid; 1J: water.BB

15.

260

METHODS OF STUDYING CRAZING

The effect of hydrostatic pressure and the combined effect of an environmental fluid and hydrostatic pressure have been studied by Matsufor high pressures up to about 4 kbar. The pressureshige et a/.40*41,73 transmitting fluid, silicon oil, exerts an environmental effect that was prevented when the gauge section of the specimen was sealed with PTFE tape and covered with a transparent silicon rubber. In order to obtain a simple tensile stress plus a pure hydrostatic pressure, the specimen holders were specially designed and the pressure-compensated load cell was incorporated within the pressure chamber. The combined effects of environment and pressure on the craze initiation stress and on fracture shear yielding are shown in Fig. 24. With increasing hydrostatic pressure, crazing is more difficult, but as the pressure increases further, the environmental fluid is forced into the polymer so that the fracture stress decreases to such an extent that brittle fracture occurs before general crazing occurs. However, since the crack is always preceded by a craze, a stress-cracking agent should also be considered to be a craze-enhancing agent. There is a dynamic aspect of gaseous environmental crazing that should also be considered, as suggested by B ~ b e c k .During ~ ~ craze growth, it is necessary for the environmental agent to diffuse into the polymer and plasticize it in order for the environment to be effective. If the craze grows slowly, as under a low stress, then the craze velocity is limited by the rate of plastic deformation and not by the diffusion of the environmental agent. At higher craze velocities, as produced by a higher stress, the environmental agent cannot fully plasticize the craze because diffusion imposes a limitation. Therefore, the change in velocity with a change in stress will decrease as the process becomes diffusion limited. Experiments by Imai and Brown74at extremely high strain rates showed that there is a critical velocity above which general crazing ceases to be observed and only brittle fracture occurs, as in an inert atmosphere. Above the critical strain rate, the environmental agent diffuses too slowly into the specimen to be effective as a crazing agent. Recent experim e n t ~ 'on ~ the effect of the pressure of the gaseous crazing agent on craze velocity also show this effect. At low craze velocities, log velocity is proportional to pressure as can be explained by a deformation-controlled growth. However, at high velocities, the velocity is linearly related to pressure, which is in accordance with diffusion-controlled growth. The gaseous environment acts just like a liquid environment in that the polymer is plasticized at the tip of the craze and therefore the stress to ini73

K . Matsushige, E. Baer, and S. V. Radcliffe, J . Macrumol. Sci.. Phys. 11 (4). 565

(1975). 74

Y. Imai and N . Brown, J . Po/.vm. Sci.. Polym. Phys. Ed. 14, 723 (1976).

15.4.

E N V I R O N M E N T A L EFFECTS I N LIQUIDS A N D GASES

261

I-

V

U

a Y

UNSEALED

I

O

h

i

2

i

PRESSURE

4 (kb)

FIG.24. Fracture stress of sealed and unsealed PS as a function of pressure. Craze formation was observed from 0 to 3 kb for the unsealed specimen (courtesy of Matsushige e / ol.73).

tiate a craze is lower. One difference between gaseous and liquid crazing is in the transport of gas through the craze to its tip. In the case of the gas the transport is very rapid, so that the concentration of gas at the inner surface of the craze is the same as at the exterior surface of the specimen.

25 -

- -Relaxation - - - - -in-He -*

n

EFFECT OF N2 GAS ON THE STRESS RELAXATION BEHAVIOR OF PC

I

0

I

I

I

I

25

I

I

I

I

I

50

I

I

I

I

I

75

TIME (min 1

FIG.25. Effect of switching from a helium to nitrogen environment during a stress relaxation test on PC at 78 K.'a

262

15.

METHODS OF STUDYING CRAZING

301 25

20 R

'9

-x

15v)v)

w

rT

t7

l0-

STRAIN %

FIG.26. Effect of switching from nitrogen to helium during a stress-strain test on PC at 78 K.'a

In the case of a liquid, the craze velocity will be determined by the depth of the craze, as observed by Marshall et al." for the end-flow of methanol in PMMA. Whereas the inner surface of the craze is saturated with liquid molecules, the concentration of gaseous molecules can be controlled by the pressure P and temperature T . The surface concentration is essentially proportional to PeQIRT,where Q is the binding energy between the gas and the polymer. Thus, in gaseous crazing, dynamic experiment^'^ can be performed by switching from one gaseous environment to another. The quick response of the crazing behavior shown in Figs. 25 and 26 indicates that the gas molecules do not have to penetrate very deeply into the bulk polymer in order to become effective as a crazing agent.

15.5. Relationship of Crazing to Macroscopic Mechanical Behavior This chapter describes the relationship between the properties of the individual crazes and the macroscopic mechanical behavior of the material as exhibited by the stress-strain curve, creep curve, stressrelaxation curve, impact strength, and fracture. First, simple systems are described in which the only types of deformation are crazing and

15.5

MACROSCOPIC MECHANICAL BEHAVIOR

263

elastic strain. The complexities that result when crazing and shear deformation occur are discussed. Then the relationship between crazing and fracture is considered. Finally, the more complex systems such as rubber-modified polystyrene are presented. 15.5.1. The Stress-Strain Curve

If fracture does not intervene, then pure craze deformation in the absence of shear yielding produces the tensile stress-strain curves shown in Fig. 22. The following explanation for the shape of the macroscopic stress -strain curve is based on the original work of Hoare and and a subsequent investigation by The first craze is initiated at a critical stress that depends on the environment, temperature, rate of elongation, and condition of the surface. The first craze initiates at the point of highest stress concentration. As the stress continues to rise relative to the craze initiation stress, more crazes are nucleated. The resulting number of crazes depends on the stress level, the density and severity of points of stress concentration on the surface of the specimen, and time. Each craze begins to grow at a velocity that depends on the magnitude of the stress. Each craze contributes to the tensile strain by growing and thickening as indicated by 6, = py[b dA/dt

+ A db/dt],

(15.5.1)

where i,is the strain rate produced by the crazes, p the number of crazes per unit surface area of specimen, y the amount of surface per unit volume in the specimen, b the craze opening displacement, A the average area of the disk-shaped crazes (cf. Fig. 3), and t the time. This was derived on the assumption that the strain cc was equal to the fractional increase in volume of the tensile specimen resulting from the opening up of crazes. At all times, the total strain rate i.,.is fixed by the testing machine and consists of

iT = i,

+ iE= const,

(15.5.2)

where iEis the elastic contribution. We note that the stress rate is proportional to & by Hooke’s law. When crazing starts, i,continuously increases since all the parameters except y in Eq. (15.3) increase with stress or time. Finally the total strain rate is completely accommodated by crazing, so that iEbecomes zero and the stress becomes constant. However, i, tends to rise even though the stress-dependent terms p, dA/dr, and db/dt remain constant, because b and A are always increasing. Since J . Hoare and D. Hull, Philos. M u g . [8] 26, 443 (1972).

264

15.

METHODS OF STUDYING C R A Z I N G

the rate of elongation of the specimen cannot exceed the cross-head speed of the machine, the stress decreases and thereby decreases dA/df and dbldr. Thus, beyond the maximum in stress, which is called the yield point, CE becomes negative and ic increases. The stress continues to fall, so that kc + kE remains constant. This explanation of the yield point is a dynamic one in the sense first proposed by Johnston and Gilman7s for dislocation-controlled deformation in crystalline solids. By knowing the dependence of p, dA/df, and db/dr on stress it is possible to calculate the stress-strain curve from the properties of the individual crazes. This was done for the case where p depended only on stress and not on time and where it was assumed that db/dr was negligible.42 More recent experiments with creep curves indicate that db/dr makes an important contribution to the deformation. This dynamic picture of yielding explains the effect of strain rate on yield point but underestimates the magnitude of the drop in stress after yielding. Another explanation for the maximum in the stress-strain curve can be based on the nature of the deformation that occurs during crazing. The stress to initiate a craze should be greater than the stress to develop the subsequent f i b r i l I a t i ~ n . ~ *It, ~is~generally thought that the stress to initiate a craze is determined by the stress to form voids in the original materia1.48 After the voids are formed, the stress that produces fibrillation is lower than the initiation stress because a material with the voids is weaker. Probably the intrinsic drop in stress that is described here produces the major part of the drop after yielding in the stress-strain curve, which was not predicted by the dynamic theory as described above. The stress -strain behavior of individual crazes was measured by Kambour and KoppSoand Hoare and They first crazed the specimen and then measured the stress -strain response of the precrazed specimen. Kambour and Kopp could observe the strain in an individual craze with a microscope because the crazes, which were formed in methanol, were very thick. Hoare and worked with very thin, 0.1-0.4 pm crazes in polystyrene. They determined the average strain in an individual craze by determining the stress-strain curve of the entire precrazed specimen and then subtracted the elastic deformation in the bulk material between the crazes. Both Kambour and Kopp and Hoare and Hull obtained the same type of stress-strain behavior. The stress-strain behavior of the individual craze shows an initial modulus that is much less than the elastic modulus of the uncrazed polymer and a yield stress that is much less than the yield point of the virgin specimen, as shown in Fig. 27. The final part of the stress-strain curve of the individual craze shows an increasing 76

W.G . Johnston and J . J . Gilman, J .

Appl. Phys.

30, 129 (1959).

15.5

MACROSCOPIC MECHANICAL BEHAVIOR

265

0

1 4

8 12 16 20 24 28 32 36 40 44 48 52 STRAIN (o/ol

F I G . 27. Stress-strain behavior of an individual craze in precrazed polycarbonate (courtesy of Kambour and Kopp").

slope associated with the orientation strengthening of the fibrils. The lower modulus and lower yield stress of the retracted craze as compared to the virgin specimen are explained on the basis that the packing of the molecules in the retracted craze is looser than in the virgin specimen.

15.5.2.Creep

Creep has been extensively investigated in polymers under conditions where shear flow was the primary form of deformation. There are some creep studies where shear flow and crazing occurred together, but there only appears to be one creep investigation where only crazing occ ~ r r e d . ~Creep ~ , ~curves ~ that are produced by pure crazing are shown in Fig. 28. The creep equation has the form = B1 + B,t (Fig. 29). In our analyses, B , will be neglected because it corresponds to a small initial strain that is nearly equal to the experimental error. The analysis of the creep curve is easier than for a stress-strain curve, because with a constant stress the number of crazes is constant except for a short time at the beginning of the test. Also the creep velocity is usually c ~ n ~ t a n t . ~ ~ - ~ The analysis of the creep equation is based on the following model: (1) the craze grows proportionally in all directions, (2) the growth velocity o is constant. The strain is given by E =

pybA.

(1 5.5.3)

266

15. METHODS OF STUDYING CRAZING PCTFE N2 ENVIRONMENT

0.08- ( a )

0

2

T =78 K

4

6

10 12 TIME (MIN)

8

14

16

18

20

T=78K u = 120 6 MPa

0.50 ATM

0

2

4

6

10 12 TIME (MIN)

8

14

18

16

20

P = 10 ATM IT= 117 MPa

$5 K 1 0

10 15 TIME (MIN)

5

20

88 K 25

FIG.28. Creep curves from pure crazing at low temperatures for polychlorotrifluoroethylene.7P

If b = K,vt and A = K2v2t2,where K , and K 2 are constants, then E =

pyK,K,v3P.

(15.5.4)

Equations (15.5.3) and (15.5.4) have been compared with the experimental observations by measuring the microparameters of crazing, p , a, p , I , and v (Fig. 3). These microparameters are directly related to the total strain and the experimental constant B2 since I = vt, b comes from a through Eq. (15.2.2) and A = P I . The agreement between theory and experiments is very good.72 It is to be noted that in Fig. 28 the creep rate in a gaseous environment decreases as the temperature increases. As discussed earlier, the desorption of gas with increasing temperature decreases the environmental effect of the gas. If the temperature is increased further, the creep rate will

15.5

MACROSCOPIC MECHANICAL BEHAVIOR

267

T = 70K p = 0.5 ATM

T = 70K

u =117MPa

I OATM

0 5

I

l

I

I

I

06 -

I

I

(

l

P = I OATM

-

t (min)

FIG.29. Creep curves in Fig. 28; follows

=

B1 + B,r."

begin to increase with temperature because the rate of microplastic deformation associated with crazing is a thermally activated process and shear flow begins to occur. Figure 30 shows the transition from the lowtemperature environmental crazing behavior where the creep rate de-

268

15.

METHODS OF STUDYING CRAZING 1

I

I

I

I

1.0ATM N2

~

100

110

120

130

140

TEMPERATURE

150

( O K )

FIG.30. Shows creep rate as a function of temperature in range where the creep rate undergoes an inversion. On the low-temperature side, adsorption is controlling, and on the high-temperature side, thermally activated creep deformation mechanisms dominate (courtesy of Metzger").

creases with increasing temperature to the high-temperature regime where thermally activated deformation processes dominate. In a stress-relaxation test, the specimen is suddenly extended to a fixed amount and the stress is then observed to relax with time. The basic equation is i,

+iE

= 0.

(15.5.5)

Thus if the parameters that govern the craze strain rate e, are known in the basic equation given above, then the stress relaxation curve of u vs. t can be calculated. 15.5.3. The Size Effect

It is axiomatic that, if the unit of deformation nucleates at the surface and then propagates inward, the mechanical behavior depends on the

15.5

MACROSCOPIC MECHANICAL BEHAVlOR

200

I

I

I

I

I

I

I

I80

-

1

I

1

-

269

7

/--

160

I

i

,"

I

&-

140

0

a

2 I20 -

W

> loo[ 80

I

I

,

y.E;iM;NT

L ~ o - ~ ~ o - ~ ~ o - 10'~ ~1o0-~~ 1~104105 0o ~- ' ~ o ~ VOLUME SURFACE AREA

(mm)

FIG.31. Craze yield point vs. ratio of volume to surface area of polycarbonate test specimen (courtesy of Wu and Brown").

ratio of surface to volume in the specimen. This size effect has been designated as the parameter y in Eq. (15.5.3). Thus the creep, stressrelaxation, and stress-strain curves depend on y . Wu and Brown7' have demonstrated the existence of the size effect. The magnitude and functional form of the size effect are illustrated in Fig. 31. The lower bound for the yield point is the stress to nucleate the first craze and corresponds to a specimen of zero thickness; the upper bound is the yield point without crazing and corresponds to the specimen of infinite thickness. 15.5.4. Shear Flow and Crazing

For pure crazing, the macroscopic deformation behavior is easy to analyze if the parameters that determine kc are known. These parameters are density of crazes as a function of stress and time, the stress dependence of the thickening and growth rates, and finally the shape of the craze as it grows. However, if shear flow occurs with crazing, the situation becomes very complex. It is difficult to quantitatively describe pure shear flow because the unit of flow is not observable as is a craze. In addition, shear bands interact with crazes. It appears that, in general, prior shear flow makes crazing more difficult. It also leads to polymer orientation. The interaction between crazing and shear flow is only partially underJ . B. C. Wu and N. Brown, ./. Mater. Sci. 12, 1527 (1977).

270

15.

METHODS OF S T U D Y I N G CRAZING

stood even from a qualitative viewpoint. As mentioned previously, shear bands may degenerate into crazes, so that the distinction between craze deformation and shear flow may become blurred under certain conditions .20

15.5.5.Fracture Crazing and fractures are very closely related because, when crazing occurs, the crack initiates at a craze. The amount of ductility prior to fracture depends on the amount of crazing. For a completely brittle fracture, crazes will not be observed because the crack may initiate at the first craze very shortly after it forms. Therefore, in general, the fracture toughness of a polymer is related to the energy to form the craze or bundle of crazes that precede the crack. Since the fracture of polymers is a large area of research, this section will be restricted to a brief review of those fracture studies that emphasized the role of crazing. The important variables that control crazing and thus determine the fracture toughness are temperature, speed of applying the stress, molecular weight, and the environment. The most detailed microscopic description of the relationship between crazing and the fracture process has been presented by Beahan et ul. l 9 Their approach has been the use of optical and scanning electron microscope observations of the fractured surfaces and an optical examination of the growth of cracks within crazes. In addition, greater detail was obtained with high-resolution electron microscopy using replica techniques on the fractured surface and transmission methods on the crazes as discussed previously. Their observations indicate that fracture initiates within the well-developed fibrillar region; thus the initial part of the fracture surface is very smooth, the craze having broken essentially at midthickness. As the crack grows it subsequently follows a path that oscillates between the opposite boundaries of the craze. If crazing is generated during the rapid growth of the crack, then it is very likely that multiple crazing will occur and the fracture energy will increase. Using compact tensile specimens, Brown and Ward,13 measured the thickness profile of a craze that formed in front of a slowly moving crack. They found that the size and shape of the craze was in accord with the Dugdale model, which is the basis of a general theory for describing the plastic zone in front of a crack. The length of the plastic zone is given by R = (T/~)K~/UO~,

(15.5.6)

and the thickness of the plastic zone adjacent to the crack is 6 = K'//aoE,

(15.5.7)

15.5

MACROSCOPIC MECHANICAL BEHAVIOR

27 1

where K is the stress intensity factor, wo the stress to form the craze, and E Young’s modulus. K is proportional to the square root of the crack length times the applied stress. Morgan and Ward78found that agreement with the Dugdale model persisted from +45 to -30°C. It is interesting to compare the energy to form a unit area of crack with the energy to form a unit area of craze. Measurements of the energy to fracture PMMA have ranged from 2000 j/mz 79 to 400j/rns8Oat 77 K. The energy to form a craze at this temperature is 60 j/m2 according to Brown and Metzger.14 The energy to form a unit area of craze was determined by measuring the mechanical work per unit volume to form crazes and dividing this by the total area of crazes per unit volume of the specimen. The discrepancy between the energy to form a crack and the energy to form a craze cannot be explained on the basis of the surface energy involved in breaking the fibrils in the craze, because this energy has been estimated to be only about 1j/m.gJ4*7gThe explanation most probably involves two factors: (1) a bundle of crazes often follow a crack unless it moves very slowly and is very sharp, and (2) the energy of the crack is the stress to form a craze multiplied by the crack-opening displacement, whereas the energy to form an isolated craze is the stress to form a craze multiplied by the craze-opening displacement, and the crack-opening displacement is much larger than the opening displacement of a craze that is produced in a virgin specimen. Kramer et ~ 1 measured . ~ ~the energy to form crazes in methanol in PMMA that had been grown from a sharp crack at room temperature; they measured energies of 80-3 j/m2, demeapending on the length of the craze from the crack. Kramer er sured the craze energy by measuring the stress and the craze-opening displacement, with double-exposure holography, as a function of craze length. For comparison Morgan and Ward78found a value of 400 j/mz for the fracture energy of PMMA in air at room temperature. Since both of the above measurements involved single crazes grown from a crack and since the surface energy of the crack makes a minor contribution to the fracture energy, the difference between the Kramer et al. results and the Morgan- Ward results must be attributed to the methanol environment. 15.5.6. High-lmpact-Strength Polymers

Rubber-toughened plastics consist of rubber particles blended with polymers such as PS, PMMA, or polycarbonate. One of the most common is high-impact PS, which consists of micron-sized polybutadiene G . P. Morgan and 1. M. Ward, private communication. J . B. Berry, in “Fracture Processes in Polymeric Solids” (B. Rosen, ed.), Chapter IIB. Wiley (Interscience), New York, 1964. 8o G . P. Marshall, L. N. Coutts, and J. G. Williams, J . Mafer. Sci. 9, 1409 (1974). @ ’

212

15. METHODS

OF STUDYING CRAZING

FIG.32. Rubber-toughened polymer. Crazes are between rubber particles (courtesy of KambourO and Lawrences2).

15.5

MACROSCOPIC MECHANICAL BEHAVIOR

273

particles in a polystyrene matrix. This is most commonly achieved by polymerizing styrene in which the rubber is dissolved. Bucknall and Smiths1first pointed out that the toughness is caused by crazing occurring between the rubber particles, as shown in Fig. 32.9,82 The energy absorbed by crazing is equal to the number of crazes per unit volume times the average area per craze times the craze energy per unit area. Since crazes would make it easier for a crack to form, one might think that the material would fracture easily. However, it appears that the rubber particles arrest the growth of a crack by blunting its tip. For a given volume fraction of rubber, it would be most desirable to make the particle size very small in order to increase the number of particles. Each rubber particle generates crazes by functioning as a point of stress concentration. One advantage of a large particle over a small particle is that it is likely to produce a larger craze-opening displacement and thus increase the energy per unit area of the craze. A full description of the structure and properties of high-impact-strength polymers is given in the book by Manson and Sperling.83 A further discussion is also given in Part 16 of this volume. ACKNOWLEDGMENTS It is my pleasure to acknowledge that the research on gaseous crazing was done with the collaboration of Dr. Mark F. Parrish, Solomon Fischer, Professor Yasafumi Imai, Dr. J. B. C. Wu,and Bruce D. Metzger. The Army Research Office and the National Science Foundation MRL Program under Grant No. DMR 76-80994 supported the research.

C. B. Bucknall and R. R. Smith, Polymer 6, 437 (1965). K. Lawrence, British Petroleum Company Limited, Epsom, England (unpublished work). J . A. Manson and L. H. Sperling, "Polymer Blends and Composites." Plenum, New York, 1976. 82

This Page Intentionally Left Blank

16. POLYMERIC ALLOYS By J. Roovers 16.1. Introduction In this part experimental techniques and results obtained in the study of polymeric alloys are reviewed. Polymeric alloys are formed when two polymers are thoroughly mixed in one body.' This definition excludes composites, laminates, coatings etc., in which microscopic randomness is missing. In contrast to metallic alloys, covalent links between the constituting polymers are allowed as, for example, in block and graft copolymers. When two polymers are mixed together, deviations from absolute randomness will occur. Compatibility between two polymers will be defined as a ~ariablel-~ and measured by the average dimension of the heterogeneities in the polymer mixture. When this average dimension is larger than the average radius of gyration of a polymer molecule and each heterogeneity is composed of many polymer molecules, the two polymers are said to be nearly incompatible. If the average dimensions of the heterogeneity are smaller than the radius of gyration of a polymer molecule, the two polymers are highly compatible. When miscibility at the monomeric level occurs, the two polymers form one homogeneous phase. A somewhat special situation arises when two polymers are chemically linked. The maximum size of the heterogeneities cannot extend much beyond the end-to-end distance of each polymer chain.' Obviously, this part focuses on morphologies in polymer mixtures. While thermodynamics are used to determine the conditions of stability of polymer-polymer mixtures, kinetic aspects often prevent unstable or metastable mixtures from reaching the equilibrium state. This brings out, to a larger extent than for mixtures of low-molecular-weightcompounds, A . J . Yu,Adv. Chem. Ser. 99, 2 (1971). Stoelting, and F. E. Karasz, Adv. Chem. Ser. 99, 29 (1971). D. S. Kaplan, J . Appl. Polym. Sci. 20, 2615 (1976). ' D. J . Meier. J . Po/ym. Sci.. Parf C 26, 81 (1969).

* W. J . MacKnight, J .

275 METHODS OF EXPERIMENTAL PHYSICS, VOL.

16C

AU

Copyright @ 1980 by Academic Ress. lac. rights of reproduction in MY form reserved. ISBN 0-12-475958-0

16. POLYMERIC

276

ALLOYS

the importance of the thermal, pressure, and mechanical history of the samples that are studied. Methods for mixing polymers are either mechanical or Mechanical mixing includes blending, roll milling, and coextrusion. Sufficient mechanical and thermal energy can be put into a mixture of polymers to produce a fine dispersion that may be stable for kinetic reasons. Nevertheless, on the basis of microreversibility one would expect that during mixing more compatible polymer pairs produce finer dispersions. Mixing of polymers in the presence of a low-molecular-weight carrier includes such methods as latex blending followed by drying and/or coagulation, and dissolution of two polymers in a common solvent followed by solvent evaporation, freeze drying, or polymer coprecipitation. 1.1 all these cases, the final state of the polymer-polymer mixture will depend on the thermodynamic conditions prevailing at the time of the polymer immobilization. Examples of chemical blending of polymers are copolymerization, block copolymerization, the polymerization of a monomer in the presence of another polymer with the formation of graft copolymers, and interpenetrating networks. Various degrees of chemical mixing can be realized with all these methods. Chemical linking between two polymers is often observed during mechanical mixing at high shear rates and at high temperatures. Under such conditions radicals are produced that may lead to polymer degradation, oxidation, branching, and grafting.s Such changes in the chemical composition and structure of the polymers, although not easily investigated, may influence the morphology of the polymer-polymer blend.

16.2. Thermodynamics 16.2.1. Polymer Mixtures

The lattice model for liquid-polymer mixtures was extended to the mixturz of two polymer^.^ The mean free energy change per segment on mixing two polymers is (16.2.1)

where (&

and c $ are ~ the volume fractions of polymer 1 and polymer 2 I), and m,and m2their degree of polymerization expressed in

~ $ 1 ~

+ dbl

=

P. J . Corish and B. D. W. Powell, Rubber Chem. Techno/. 47, 481 (1974). A. Castle and R. S. Porter, Rubber Chem. Techno/. 44, 534 (1971). ' R. L. Scott, J . Chem. Phys. 17, 279 (1949).

a

16.2.

277

THERMODYNAMICS

a common lattice unk8 The purely enthalpic Van Laar type interaction parameter x12of two different polymer segments is defined by AH

=

kT~iz414z.

(16.2.2)

Because m,and m2are normally large, the two entropy-of-mixing terms in Eq. (16.2.1) are small, and any small positive xlz value (endothermic mixing) will prevent mutual solubilization of the two polymers. When a mixture of two polymers separates into two phases the compositions of the two coexisting phases lie on a binodul, which can be calculated by equating the chemical potentials of each component in the two It is usually easier to calculate the spinodul, which is the composition of absolute stability a2 AG,,,/a+," = 0. Then (16.2.3)

For the critical point at which two coexisting phases coincide and az AG,,x/a~,2 = a3 AG,,x/a4,3 = 0, 1

XlZ,

=2

(-m1y +*) 1

2

( 1 6.2.4)

*

Equations (16.2.3) and (16.2.4) indicate how phase separation is dependent on the degree of polymerization of the two polymers in the mixture. The higher m, and m,, the smaller xlzhas to be for the pair in order to observe mutual solubility. In order to better match experimental spinodals (cloud point curves) with curves calculated from Eq. (16.2. l ) , the requirement that xlzbe a constant is relaxed to1O XlZ

=

XlZ.0

+ x12,142 +

*

*

*

'

(16.2.5)

Since only xlz,ois considered dependent on TI, the higher terms are entropic correction terms. A theory based on a lattice model cannot account for volume changes on mixing. Flory has indicated how an equation-of-state approach takes changes in the local liquid structure into account and allows the calculation of an excess volume of mixing and the residual enthalpy and entropy of mixing connected with it.11J2 For the particular case of mixtures of S. Krause, A. L. Smith, and M. G.Duden,J. Chem. Phys. 43, 2144 (1965). H. Tompa. "Polymer Solutions." Butterworth, London, 1959. R. Koningsveld, L. A. Kleintjens, and H. M. Schoffeleers, Pure Appl. Chem. 39, 1 (1974).

P. J. Flory, R. A. Orwoll, and A. Vrij, J . Am. Chem. Soc. 86, 3507 and 3515 (1964). P. J. Flory,J. Am. Chem. SOC. 87, 1833 (1965).

278

16.

POLYMERIC ALLOYS

two nonpolar polymers, polyethylene and polyisobutylene, it was calculated that the positive free energy of mixing, AGmix,is caused not only by a small positive neighbor interaction parameter and an overall positive enthalpy of mixing but mainly by a larger negative entropy of mixing.13 Model calculations using realistic values for the characteristic equationof-state parameters have shown that the difference between the thermal expansion coefficients of two polymers is the main cause for polymerpolymer incompatibility: A difference in their thermal pressure coefficients and the presence of a low-molecular-weight polymer can extend the ~ ~ calcutemperature and concentration range of mutual s o l ~ b i l i t y .These lations also indicate that demixing of polymer-polymer solutions at higher temperatures (lower critical solution temperature) must be expected especially when the nearest-neighbor interaction parameter is negative.14 Up to now the use of the equation-of-state approach has been limited. 16.2.2. Block Copolymers A thermodynamic theory, which considers the free energy change on mixing two infinitely long completely separated blocks without consideration of domain morphology, gives for the mean free energy of mixing per segment l5

X

[41In 41 + & In & - 2(n - 1)

+ In(n

-

R

-

l)]}.

(16.2.6)

The first three terms in Eq. (16.2.6) are comparable with Eq. (16.2.1). AS,,,/R represents the change in entropy on disorienting the junction between the blocks that is frozen in the interface. Its value is about 1.0. The last term allows for the choice of surface placements of the junctions in multiblock or graft copolymers ( n is the number of blocks). From Eq. (16.2.6) the value xlZn,for which AG = 0, can be calculated for any block polymer when m,, m2, 41,4z, n, and ASdis/R are known: XlZ,

=

'' P.J . I'

Is

1

414dm1 + mz)

Flory, B. E. Eichinger. and R. A. Orwoll, Mucromolecules 1, 287 (1968). L. P. McMaster, Macromolecules 6, 760 (1973). S. Krause, J . Polyrn. Sci.9 Purr A-2 7, 249 (1%9); Mucromolecules 3, 84 (1970).

279

16.2. THERMODYNAMICS

1

FIG.I . Values of xlrer as a function of QI,the volume fraction of the second polymer. (a) rn,. (b-d) According to Eq. (16.2.3)for two homopolymers, rn, = 100 and rnt = Eq. (16.2.7)with AS,,,/R = 1.0. (b) Diblock copolymer with rn,, = 100. (c) Triblock copolymer with m A = 100. (d) graft copolymer with rn,, = 300 and n = 30.

If the calculated is larger than the real xlz for the polymer pair, the blocks of the block polymer can mutually dissolve and form a single phase. Comparison of critical conditions of mixing calculated from Eq. (16.2.7) with those for the mixture of the corresponding homopolymers according to Eq. (16.2.3) indicate that block copolymers will more easily form homogeneous mixtures than the corresponding homopolymers (Fig. 1). According to Eq. (16.2.7) mixing is favored for low-molecular-weight block copolymers at constant n , and in multiblock copolymers (n > 2) at constant molecular weight. The presence of one or both homopolymers in the block copolymer will reduce miscibility.16 If one block is crystalline, dissolution of the blocks will be opposed by the free energy of melting. A term equal to

(16.2.8) lo

S. Krause, in "Colloidal and Morphological Behavior of Block and Graft Copolymers"

(G.E. Molau, ed.), p. 223. Plenum, New York, 1971.

280

16.

POLYMERIC ALLOYS

will have to be added inside the brackets of Eq. (16.2.7), where T,,, is the equilibrium melting temperature, AH,,, the molar enthalpy of melting, and mi the number of crystalline m, segments per chain. This term easily dominates Eq. (16.2.7) when T < T,,,.” The domain dimensions in block copolymers can be calculated from thermodynamic considerations. In block copolymers, phase separation can occur only to the extent that A-B junctions must lie in or near the interface of the different domains and that the domains must be filled uniformly.‘ Therefore, the characteristic domain dimension of the discrete A domains can be expressed as radius, = KminlA = KaeqAmy2loA,

( 16.2.9)

where my2 I, and my210Astand for the root mean square and unperturbed root mean square end-to-end distance of an A block with m A segments. I, is the effective segment length and aeqA is the equilibrium expansion coefficient. For lamellae, thickness, (TA) replaces radius, in Eq. (16.2.9). Values for K for different morphologies are given in Table I.18,*s Two opposing tendencies determine (yep,. The domain tends to grow in order to reduce the surface free energy. But as the domain grows, the polymer chains that are anchored at one end in the interface have to expand more in order to fill the domain uniformly causing a reduction of elastic entropy (AS,,). The equilibrium expansion coefficient is given by (16.2.10)

where y is the interfacial tension between polymers A and B, MA the molecular weight of the A block, C a shape constant (see Table I), pAthe density of the A block, and @ is given by (16.2.11)

with n also given in Table I. It turns out that aeqaisa function of m~05-0.15, the exponent depending on y . The characteristic dimension according to Eq. (16.2.9) varies therefore as m~65-0.65.4 The free energy of domain formation per block copolymer molecule can S. Krause, in “Block and Graft Copolymers” (J. J. Burke and V. Weiss, eds.), p. 143. Syracuse Univ. Press, Syracuse, New York, 1973. D. J. Meier, Polym. Prepr. 11, 400 (1970). 1B D. J. Meier, in “Block and Graft Copolymers” (J. J. Burke and V. Weiss, eds.), p. 105. Syracuse Univ. Press, Syracuse, New York, 1973.

16.2

28 1

THERMODYNAMICS

TABLEI . Constants for the Evaluation of the Doman Size in Block Copolymers Type

Morphology

Dimension

AB ABA

spheres A spheres B spheres cylinders lamellae A lamellae B lamellae

radius

K Eq. (16.2.9)

~~

AB AB ABA

C Eq. (16.2.10)

Eq. (16.2.11)

2.25

0

2

d

1.4

t

:::: ] ] 1 I

radius thickness

n

1.2

be calculated using the equilibrium expansion coefficient NABk

T

=

-AS,, - AS,, +

4 -

(I In

+ 0') -

3-

Xlz(mA

+ mB)+A,

(16.2.12)

where ASp is the entropy associated with restricting the A-B junction to the interface and AS,,is the entropy associated with the restriction of each From the block to its domain. Values for these terms were cal~ulated.~ relative stabilities of the different morphologies it is calculated that for diblocks spherical domains will be stable when +A < 0.2, cylinders when 0.2 < +A < 0.25, lamellae when +A > 0.25.18 Triblocks would form < 0.33, cylinders when 0.33 < +A < 0.37.'O A difspheres when ferent theory predicts spheres when +A < 0.14 and cylinders for 0.14 < +A < 0.39." Equation (16.2.12) allows evaluation of the critical interaction parameter x12 for spherical domain formation. Alternatively, the minimum molecular weight for domain formation can be estimated when x12is known.4 In the original theory y was left a variable. However, y and x12 are interrelated.22 During casting from solution, the presence of solvent at the point of polymer immobilization may influence the type of morphology that is thermodynamically favored. The presence of a solvent preferential for one of the blocks tends to produce the morphology according to the volume fraction of the solvated block.192' The theory was further refined by taking into account the presence of an interphase of mixed polymer segR. T. LaFlair, Pure Appl. Chem., Suppl. 8, 195 (1971). T. Inoue, T. Soen, T. Hashimoto, and H. Kawai,J. Po/ym. Sci., Parr A-2 7,1283 (1969). 22 E. Helfand and A. M. Sapse, J . Chem. Phys. 62, 1327 (1975).

2o

21

282

16. POLYMERIC ALLOYS

ments between the domains.= The joint placement entropy AS, becomes then In(volume fraction interphase). 16.2.3. Polymer-Polymer lnterphase

It has since been realized that a sharp boundary between two different polymers is unlikely because it would create a region of reduced density.24 This is avoided when polymer 1 and polymer 2 segments interpenetrate slightly and form an interphase. The density profile in the interphase is (16.2.13)

when the properties of the two polymers are identical, i.e., poA = po, = po, 1, = lo, = lo and, at x = 0 in the center of the interphase, p A = pB = #po. A measure for the interphase thickness al is then a1 = 210/(6~12)"~,

(16.2.14)

i.e., the more compatible the two polymers (x12--* 0) the larger is their interphase. For example, when lo = 6.6 A, the interphase is 17 8, thick when x12= 0.1. A similar technique was used to calculate the interphase is small compared to the dofor a block c o p o l y m e ~ - . ~If~the * ~interphase ~ main size, the thickness of the interphase is given approximately by Eq. (16.2.14). However, the fact that A-B junctions exist in the interphase and domains have to be filled uniformly tends to cause the thickness of the interphase in block copolymers to increase beyond that in homopolymers. Accurate calculation shows that extensive interpenetration occurs when 4(mA + mB)X12I10. In those cases no pure block domains exist.26 The free energy per unit volume of forming lamellar domains for a diblock copolymer with a small interphase is approximated byz5

where d = T A + TB. The first term on the right-hand side of Eq. (16.2.15) represents the surface free energy of the polymer pair in terms of x12,the second term is the entropy loss AS, of placing the A-B joint in the interphase al given by Eq. (16.2.14), and the third term is the entropy of Oq

D. E. E. E.

I. Meier, Polym. f r e p r . 15, 171 (1974). Helfand and Y. Tagami. Polym. Lett. 9, 741 (1971). Helfand, Acc. Chem. Res. 8, 295 (1975). Helfand, Macromolecules 8, 552 (1975).

16.2 THERMODYNAMICS

283

confining the segments to their domain and filling the domain uniformly (AS"+ ASe& The value of d for which Eq. (16.2.15) becomes a minimum will represent the equilibrium domain dimensions of the block polymer. Equation (16.2.15) leads to an M i B 5dependence of the domain size. In principle, Eq. (16.2.15) will give the condition for which the homogeneous mixture is favored over a phase-separated structure as g(mA+ rnR)xlt < 8, but at this point the interphase is no longer small compared to the domain size and the theory not strictly applicable. Many other theories of phase separation in block copolymers have been

16.2.4. Segmental Polymer-Polymer Interaction Parameter

The segmental polymer-polymer interaction parameter x12plays a crucial role in all thermodynamic theories. From its definition [Eq. (16.2.2)] x12 can be obtained calorimetrically. Experimentally, two polymers cannot be mixed fast enough to generate the heat of mixing over a sufficiently short period. The use of low-molecular-weight oligomers may introduce an entropy contribution in the experimental heat of mixing.30 The measurement of any property of a binary polymer mixture, derivable from the Gibbs free energy of mixing, yields x12. The equilibrium composition of coexisting phases, the cloud point curve or spinodal [Eq. (16.2.3)], or the critical miscibility temperature with Eq. (16.2.4) yield x12 for that polymer pair.103031 Mixtures of oligomers of polystyrene and p o l y i ~ o p r e n e ~and ~,~~ of polyisobutene and poly(dimethylsil~xane)~~ exhibit upper critical solution behavior (Fig. 2a). Poly(viny1 methylether) -p o I y ~ t y r e n e ~ ~and ~ ~ - ~pol ~ y( ecaprolactone) -poly(styrene~o-acrylonitrile)~~ exhibit lower critical solubility behavior (Fig. 2b). Transitions from one to two phases have been measured for the block copolymers poly(a-methyl~tyrene-b-styrene)~~ and poly(styrene-b-iso-

'' D. J. Leary and M. C. Williams, J. Polym. Sci., Parr B 8, 335 (1970);J. Polymer Sci., Polym. Phys. Ed. 1 1 , 345 (1973). " W. R. Krigbaum, S. Yazgan, and W. R. Tolbert, J. Polym. Sci..Polvrn. Phys. Ed. 11, 511 (1973). 29 H. Kromer, M. Hoffmann. and G . Kampf,Ber. Bunsenges. Phys. Chem. 74,859(1970). 30 G. Allen, G. Gee. and J . P. Nicholson, Pol.ymer 2, 8 (1961). 31 D. McIntyre, N . Rounds, and E. Campos-Lopez, Polym. Prepr. 10, 531 (1969). 32 M . Bank, J . Leffingwell, and C. Thies. J. Polym. Sci.. Part A-2 10, 1097 (1972). 53 T. Nishi and T. K . Kwei, Polymer 16, 285 (1975). T. Nishi, T. T. Wang, and T. K. Kwei, Macromolecules 8, 227 (1975). 35 D. J . Dunn and S. Krause, Polym. t e r r . 12, 591 (1974).

16. POLYMERIC ALLOYS

284

I

1

0.2

35i

I

I

0.4

I 0.6

I

1 0.8

I

1.0

@POLYSTYRENE

8OL

I

I

I

1

I

I

1

I

I

$ POLYSTYRENE FIG.2. (a) Phase diagram with upper critical solution temperature for mixtures of polystyrene (MW, 2700) and polyisoprene (MW, 2000). Visual observation. Heating rate: 0, O.ZS"C/min; O,O.OYC/min (with the kind permission of the publisher and the authors, taken from McIntyre cr aLS1).(b) Phase diagram with lower critical solution temperature for mix-

16.2

THERMODYNAMICS

285

~ r e n e ) .However, ~~ the value of x12depends on the theory used to calculate the entropy of mixing of block polymers. The interaction parameter for two polymers is most often determined from the properties of a mixture of two polymers in a common solvent. Heats of mixing measurements of two polymer solutions have shown that most polymers mix end~thermally.~’The analysis of coexisting phases in a common solvent for polystyrene-p~lybutadiene,~~polystyrenep o l y i s o b ~ t e n e , and ~ ~ ~ ~polyisobutene-poly(dimethylsiloxane)38 were used to evaluate x12of those polymer pairs. Enormous progress has now been made in the accurate light-scattering measurement of spinodal compositions in ternary mixtures of a solvent and two p o l y m e ~ s . ~ @ - ~ ~ The measurement of the second virial coefficient A2 is used to determine xlZ4’:

(1 - xS, - xa + ~ 3 1 , (16.2.16) where is the molar volume of the solvent, c: = cl/(cl + c2), and C1 is the partial molar volume of polymer 1, etc. Since x12is usually small compared to the solvent-polymer interaction parameters xsl and xsz (which are typically 0 . 3 - 0 3 , x12is determined as the small difference between large numbers. Very accurate experimental data are therefore required and large uncertainty limits must be assigned to the x12values so determined. Generally, such measurements lead to low values of x12. For example, for polystyrene-poly(methy1 methacrylate), x12= 0.0144z; for polystyrene-polybutadiene, x12= 0.0138; for polystyrene-polyisobutene, x12= 0.02,380.016.3@ These values suggest rather unrealistically high compatibility among these polymers in the bulk state.

v,

X

38 P. M. Toporowski and J. E. L. Roovers, J . Polym. Sci.. Polym. Chem. Ed. 14, 2233 (1976). 37 G. L. Slonimskii, J . Polym. Sci. 30, 625 (1958). G. Allen, G. Gee, and J. P. Nicholson, Polymer 1, 56 (1960). M. W. J. Van den Esker and A. Vrij. J . Polym. Sci.,Polym. Phys. Ed. 14, 1943 (1976). 40 M . W. J. Van den Esker, J. Laven. A. Broeckman, and A. Vrij,J. Polym. Sci., Polym. Phys. Ed. 14, 1953 (1976). R. Kuhn and H. J. Cantow, Mukromol. Chem. 122,65 (1%9). W. H. Stockmayer and H.E. Stanley, J . Chem. Phys. 18, 153 (1950).

’*

tures of polystyrene (MW,200,000) and poly(viny1 methylether) (MW, 51,000). Heating rate: 0.2”Clmin. Solid line: temperature of initial increase in optical density. Dotted line: temperature at which the maximum optical density is reached. ( 0 )Composition and temperature for which phase separation by nucleation and growth has been observed. (0) Phase separation by spinodal decomposition. The dashed line represents the demarcation of the two mechanisms of phase separation (with the kind permission of the publisher and the authors, taken from Nishi er ~ 1 . ~ ‘ ) .

286

16. POLYMERIC

ALLOYS

Chemical potential measurements at high polymer concentrations were recently used to determine the negative x12parameter in films of the compatible polystyrene-poly(viny1 m e t h ~ l e t h e r ) . ~Similarly, ~ the meltingpoint depression of poly(viny1idene fluoride) in the presence of poly(methy1 methacrylate) and poly(ethy1 methacrylate) yields negative x12values for these polymer An inverse gas chromatography technique has also been described.46 The polymer-polymer interaction parameter is often estimated from the cohesive energy density differences of the two polymers (6, cal 1/2/cm39 (16.2.17)

in which V , is a reference molar volume, taken either as that of the small.~ est polymer segment or equal to 100 ~ m ~ / m o l eNotice that m, and m2, or m A and m Bof a block copolymer, are then degrees of polymerization in terms of that reference volume. In the case of low-molecular-weight compounds a2 = U / V , the molar energy of vaporization divided by the molar volume. For polymers, however, 6 cannot be determined directly, but may be found by equating 6 of the polymer to that of the solvent in which it expands most. This expansion can be measured from swelling, intrinsic viscosity, second virial coefficient, radius of gyration, or other experiments. Unfortunately, large variations in experimental 6 values have been ob~erved.~'It was shown that this is caused by the varying entropic contributions present in solvent-polymer interaction parameter^.^^ Finally, 6 values can be calculated from the additive molar attraction constants ( F J of the atoms and groups that constitute the polymer segment: (16.2.18)

Different sets of Fihave been c ~ m p i l e d . ~The ~ - ~same ~ tables can be found el sewhere .47*52,53 T. K . Kwei, T. Nishi, and R. F. Roberts, Macromolecules 7, 667 (1974). T. Nishi and T. T. Wang, Macromolecules 8,909 (1975). T. K. Kwei, G . D. Patterson, and T. T. Wang, Macromolecitles 9,780 (1976). 0 . Olabisi, Macromolecules 8, 316 (1975). H. Burell, in "Polymer Handbook" (J. Brandrup and E. H. Immergut, eds.) Sect. IV, p. 337. Wiley. New York, 1975. J . M. G. Cowie, R. J . Ranson, and W. Burchard, Br. Polym. J . 1, 187 (1%9). P. A. Small, J . Appl. Chem. 3, 71 (1953). i3

44

16.3.

DIRECT OBSERVATION

287

The 6 values for random copolymers can be obtained as 6=

61419

(16.2.19)

f

where 4i is the volume fraction of each c o m p ~ n e n t . Two ~ ~ random copolymers with different compositions have to be treated as two different polymers with regard to their phase behavior. The temperature variation of 6 is given by"

(51

= -ffp,

(16.2.20)

in which apis the thermal expansion coefficient at constant pressure. The use of the solubility parameter is restricted to polymer pairs whose interactions are determined solely by dispersion forces. It always leads to x12> 0. The number of nonpolar polymer pairs, for which compatibility can occur because 16, is small, is limited. At the same time, when x12+ 0, the interphase between two polymers becomes large. For polymers with polar groups, the solubility parameter concept is not applicable. Nevertheless, many pairs of compatible polar polymers have similar 6 values.s5 It is becoming increasingly clear that compatible polar polymers have negative x12.43-46Their mutual solubility is caused by specific interactions between the polar groups of both polymers. The density of such polymer mixtures is often higher than that calculated from the components by an additivity rule. Equation-of-state effects are important for the limits of stability of the mutual solubility of such polar polymers.

16.3.Direct Observation 16.3.1. Visual Observation

The size of heterogeneities that can be observed by visual inspection has a lower limit of about 0.2 mm. However, when a mixture of two polymers contains heterogeneities with dimensions larger than about 3 the D. W. Van Krevelen, Fuel 44, 236 (1965). K . L. Hoy, J . Paint Technol. 42, 76 (1970). sp S. Krause, J . Macromol. Sci.. Rev. Macromol. Chem. 7 , 251 (1972). D. W. Van Krevelen "Properties of Polymers." p. 135. Elsevier, Amsterdam, 1972. sI J. Biros, L. Zeman, and D. Patterson, Macromolecules 4, 30 (1971). 55 L. Bohn, Rubber Chem. Technol. 41,495 (1968); in "Polymer Handbook" (J. Brandrup and E. H. Immergut eds.), Sect. 111, p. 212. Wiley, New York, 1974. 51

288

16.

POLYMERIC ALLOYS

wavelength of visible light, i.e., about 0.1 pm, it appears opaque to the eye. When the heterogeneities are smaller, the mixture appears clear. Specific turbidities as a function of particle size and refractive index difference between the components of a polymer blend have been calculated for spheres in a continuous medium.58 Even if the size of the heterogeneities is large, a mixture will appear clear when the refractive indices of the two zones are matched. Several examples in which the composition of the components is tuned to give nearly identical refractive indices to two discrete phases are k n o ~ n . ~ ' -The ~ ~ resulting transparency of the blends occurs only in a narrow temperature range.5s*5B Lack of transparency has been used extensively as a criterion for incompatibility in polymer mixtures. The reverse of the argument does not necessarily hold. The phase diagram of Fig. 2a is based on the visual observation of the clear/turbid transition temperature; that of Fig. 2b is based on light transmission measurements. Phase diagrams based only on the criterion of film clarity give the composition limit for which, under the condition of sample preparation, small domains are formed.8o Furthermore, when it is claimed that block or graft copolymers cornpatibilize their parent homopolymers, it means only that they reduce the domain size to the point where sample clarity results, and not necessarily to where a homogeneous mixture is formed.81 Often, observation of phase separation of a polymer mixture in a common solvent has been used to establish incompatibility in the bulk mixture. Indeed, the addition of solvent terms in Eq. (16.2.1) almost always lowers the Gibbs free energy of mixing from that of the bulk polymers. Lists of compatible and incompatible polymer pairs are available8z-s4and the results based on many other visual observations have been c ~ m p i l e d . ~ * * ~ ~ 16.3.2. Optical Microscopy

The phase contrast microscope has a maximum resolving power of about 0.2 p m and accurate size measurements can be made down to about 0.5 pm. It has been used to examine the morphology of blends of B. F. Conaghan and S. L. Rosen, Polym. Eng. Sci. 12, 134 (1972). B. D. Gesner, J . Appl. Polym. Sci. 11, 2499 (1967). B. Baum, W. H. Holley, H. Stiskin, R. A. White, P, B. Willis, and A. F. Wilde, Adv. Chem. Ser. 154, 263 (1976). R. G. Bauer, R. M. Pierson, W. C. Mast, N. C. Bletso, and L. Shepherd, A d v . Chcm. Ser. 99, 237 and 251 (1971). M, J. Kohler, G. Riess, and A. Banderet, Eur. Polym. J . 4, 173 (1968). T. Inoue, T. Soen, T. Hashimoto, and H. Kawai, Macromolecules 3, 87 (1970). (pl A. Dobry and F. Boyer-Kawenoki, J . Polym. Sci. 2, 90 (1947). (p( R. J . Kern and R. J. Slocombe, J . Polym. Sci. 15, 183 (1955). K. Friese, Plasre Kaursch. 12, 90 (1%5); 13, 65 (1966). ST

16.3. DIRECT OBSERVATION

289

rubbers. One zone analysis used a television scanning device coupled with a computer to measure optical densities and compare them with a set threshold value.s5 The grey image is thereby converted to a contrast-rich black-and-white picture from which the number of zones, their size, and their size distribution may be calculated. The measured average diameter of rubber zones in a mechanical blend varies from 0.3 to 30 pm.65*66 There seems to be a crude correlation between zone size and the solubility parameter difference (6, - tip)of the pairs of rubbers. The discrete zones of A in a 25/75 mixture of A and B have about the same size as the ~ *50/50 ~ blends form either discrete B zones in the 75/25 m i ~ t u r e . ~The two meshlike continuous phasess5or discrete zones in the continuous matrix of the component with lower viscosity.gs The final mixing stage is reached within a few minutes even when the rubbers show a high degree of ~ompatibility.~' Mixtures of poly(styrene-co-butadiene) (SBR) and polybutadiene are sometimes described as h o m o g e n e o ~ s and ~ ~ * sometimes ~~ as heterogeneousss*won the basis of phase contrast observation. These opposing results are, to some extent, caused by poor phase contrast. Results from glass transition behavior studies indicate that a minimum 20% styrene content difference is required for the observation of heterogeneity in the rni~ture.~-~O Even then, the different zones may be too small for their detection under the microscope. Both natural rubber and high 1,Ccispolyisoprene form heterogeneous mixtures with c i s - p ~ l y b u t a d i e n e , ~ ~ ~ ~ ~ * ~ ~ while natural rubber forms homogeneous mixtures with polybutadiene ( T g = -48"C),which contains about 50% 1,2 segment^.^^"^ The microstructure of diene polymers is therefore a factor in their mutual compatibility, The phase contrast microscope is used extensively in the study of the morphology of high-impact polystyrene (HIPS), which consists of rubber ~ - ~ ~ the rubber particles dispersed in a polystyrene m a t r i ~ . ~Optimally, particles have diameters of 2-10 pm. The rubber particles in ABS, which consists of polybutadiene dispersed in poly(styrene-co-acrylo65

BB

J . E. Callan, W. M. Hess, and C . E. Scott, Rubber Chem. Techno/. 44, 814 (1971). P. A. Marsh, A. Voet. L. D. Price, and T. J . Mullens, Rubber Chem. Techno/. 41,344

(1968).

P. A . Marsh, A . Voet, and L . D. Price, Rubber Chem. Techno/. 40, 359 (1967). M. H. Walters and D. N . Keyte, Truns. Inst. Rubber Ind. 38, 40 (1962). '@ N . Yoshirnura and K . Fujirnoto, Rubber Chem. Techno/. 41,669 (1968); 42, 1009 (1%9). 70 D. I. Livingston and R. L. Rongone, Proc. Inr. Rubber Conf., Sth, 1967 p. 337 (1%8). G . M. Bartenev and G. S. Kongarov, Rubber Chem. Techno/. 36, 668 (1963). '* E. H. Merz, G. C. Claver, and M. Baer, J . Polym. Sci. 22, 325 (1956). 73 P. A . Traylor, Anul. Chem. 33, 1629 (1961). 74 G. E. Molau and H . KeskkulaJ. Pnlym. Sci., Purr A - / 4, 1595 (1966). 67

BB

290

16.

POLYMERIC ALLOYS

nitrile), are usually less than 1 pm.75 The fine structure of these blends is described in Section 16.3.3. The phase contrast microscope is used in the study of the morphology of polymer crystals. In the context of polymeric alloys, the crystals of poly(styrene-b-ethylene oxide) should be mentioned. Platelets and extremely regular single crystals, with up to 50 pm sides have been crystallized from solution.76 Careful recrystallization yields I mm long dendrites .76 Crystallization of one polymer dissolved in another polymer occurs with the formation of spherulites, as for example, when poly(rcaprolactone) crystallizes from its solution in poly(viny1 ~ h l o r i d e ) . ~ ~ These 1-5 pm spherulites can be studied between the crossed polarizers of a microscope. Spherulites are often formed when one block of a block copolymer crystallizes either on solvent evaporation or from the melt. Spherulites have been found in poly(styrene-b-ethylene ~ x i d e ) , ~ ~ * ~ ~ * ' ~ poly(ethy1ene oxide-b-ethyl m e t h a ~ r y l a t e ) ,poly(ethy1ene ~~ oxide-bisoprene),eo poly(styrene-6-tetramethylene oxide),*' segmented poly (tetramethylene oxide-b-terephthalate),82and segmented poly(ethet--burethanes).w Only limited crystallization occurs when the crystallizing block is the minor component in a glassy matrix.7g No superstructure is then observed between crossed polarizers. The first results on the mechanism of phase separation of two polymers from their mixture have now been published.34." In the metastable region between the binodal and spinodal, phase separation is expected to proceed by nucleation and growth. Although the small initially formed nuclei are not directly observed by optical or electron mi~roscopy,~~." their existence is inferred from their slow coarsening after thermodynamic equilibrium is attained.84 This type of phase separation has been found when a homogeneous 75/25 mixture of poly(styrene-coK. Kato, Polym. Eng. Sci. 8, 38 (1%7); Kolloid-Z. & Z . Polym. 220, 24 (1967). B. Lotz and A. J. Kovacs, Kolloid-2. & 2. Polym. 209,97 (1966); A. J. Kovacs, Chim. & Ind., Genie Chim. 97, 315 (1967). I7 F. B. Khambatta, F. Warner, T. Russell, and R. S . Stein,J. Polym. Sci. Polym. Phys. Ed. 14, 1391 (1976). 78 R. G . Crystal, P. F. Erhardt, and J. J . O'Malley, in "Block Copolymers" ( S . L. Agganval, ed.), p. 179. Plenum, New York, 1970. Is P. K. Seow, Y. Gallot, and A. Skoulios, Makromol. Chem. 177, 199 (1976). en E. Hirata, T. Ijitsu, T. Soen, T. Hashimoto, and H. Kawai, Adv. Chem. Ser. 142, 288 75

(1975).

A. Takahashi and Y. Yamashita. Adv. Chem. Ser. 142, 267 (1975). A. Lilaonitkul, J. C. West, and S.L. CooperJ. Macromol. Sci.. Phys. 12, 563 (1976). 83 S. L. Samuels and G . L. Wilkes, Polym. Lett. 9, 761 (1971). L. P. McMaster, Adv. Chem. Ser. 142, 43 (1975).

16.3 DIRECT OBSERVATION

29 1

acrylonitrile) with 28% acrylonitrile and poly(methy1methacrylate) is held at 265°C.84 The full circles in Fig. 2b represent conditions where polystyrene and poly(viny1 methylether) are observed to separate by nucleation and growth.34 Near the critical point and far from the binodal, spinodal decomposition is expected to dominate phase separationss (the open circles in Fig. 2b). It is characterized by the formation of concentration modulations (wavelength A,,) of a nearly constant size, in which the polymer composition changes fast and continuously until the equilibrium binodal compositions are reached. A, is given byee

where T, is the spinodal temperature, T the decomposition temperature, and L the range of molecular interaction, which is of the order of the radius of gyration of the polymer, 4 milZ 1 1 6 , according to McMaster.84 Microscopically, A,s have been estimated as 0.9 and 1.2 pm at 20 and 10°C above T,, re~pectively,~~ and as about 1 pm." There is evidence, however, that subsequent coarsening of these zones intervenes before thermodynamic equilibrium is established.M Results from an NMR pulse technique indicate that the exponential growth of the phases takes less than 5 min.34 Primarily, spinodal decomposition leads to interconnected zones that coarsen in order to reduce interfacial tension." This process stops when the interconnections break up and a disperse phase is formed. Above their glass transition temperature, therefore, two polymers can rapidly separate into two phases that produce fairly large zones. Phase separation appears then as a fast clear-to-opaque transition. 16.3.3. Electron Microscopy

The resolving power of the electron microscope as applied to phase morphology studies in polymeric alloys can reach about 30 8,. Under favorable conditions sizes down to about 50 8, can be measured accurately. This is well within the range of the average dimension of a polymer chain, but not yet the size of the individual monomer unit in the chain. Both surface (replica) and transmission viewing are used. Contrast usually is based on electron density difference, either natural or enhanced by staining, selective swelling, or radiation degradation. A full account of the techniques of electron microscopy as applied to polymers is described in Part 7 (this volume, Part B). J . W. Cahn, J . Chem. Phys. 42, 93 (1965). J . J. van Aartsen, Eur. Po/vm. J . 6, 919 (1970).

292

16. POLYMERIC

ALLOYS

16.3.3.1. Homopolymers The electron microscope can distinguish between HIPS that is a mechanical blend of rubber and polystyrene and graft-type HIPS. The former has structureless rubber particle^,^^*^^ while in the latter the rubber particles have the cellular s t r u c t ~ r e ~shown ~ * ~ ~in- Fig. ~ 3. The cellular structure is produced when styrene is polymerized in the presence of a rubber (usually polybutadiene). Incompatibility of polystyrene and polybutadiene causes the formation of polystyrene droplets in the early stages of the p o l y m e r i z a t i ~ n . ~Complete ~ ~ ~ ~ demixing is prevented by the poly(butadiene-g-styrene) species f ~ r m e d . However, ~ ~ ~ ~ ~ when the amount of polystyrene formed increases, phase inversion occurs and the rubber becomes the discrete phase in the polystyrene rich matrix.74*g1*g2 Without agitation during the phase inversion the rubber becomes a spongy network that is heavily grafted and cross-linked. Agitation during phase inversion allows the rubber to coagulate into spherical particles swollen with ~ t y r e n e . ~ ~ When ” ~ * ~the * ~latter polymerizes, a secondary phase separation inside the rubber particles occurs.74 The product of the stirred reaction is s o l ~ b l e . ~This ~ . ~mechanism of phase inversion occurs whenever the two polymers are i n c ~ m p a t i b l e .The ~ ~ rubber particle size and the fraction of polystyrene they occlude increase with decreasing agitatione8and with increasing viscosity of the polymerizing m e d i ~ m . ~ ~ . ~ Particles with 2-10 pm diam occluding about 20% of the polystyrene give HIPS its optimum proper tie^.^^^^^.^^ The product of the copolymerization of styrene and acrylonitrile in the presence of polybutadiene rubber (ABS) sometimes contains cellular part i c l e ~ , ~but ~ , ~most ~ often 0.1-0.5 pm structureless rubber part i ~ l e s . ~ This ~ * is~ the ~ ~direct ~ * ~result ~ of the emulsion copolymerization of styrene and acrylonitrile on rubber particle seeds.Bs The electron microscope has revealed how the acrylonitrile content of poly(butadiene-co-acrylonitrile)(NBR)influences the morphology of its blends with poly(viny1 chloride) (PVC).87,g7 Polybutadiene blended into M. Matsuo, Jpn. Plast. 2(3), 6 (1968). E. R. Wagner and L. M. Robeson, Rubber Chem. Techno/. 43, 1129 (1970). J. D. Moore, Polymer 12,478 (1971). R. J. Williams and R. W. A. Hudson, Polymer 8, 643 (1967). G. E. Molau, J . Polym. Sci., Part A 3, 1267 (1965). B. W. Bender, J . Appl. Polym. Sci. 9, 2887 (1965). OJ G. F. Freeguard, Polymer. 13, 366 (1972). o( M. R. Grancio, A. A. Bibeau, and G.C. Claver, Po/ym. Eng. Sci. 12, 450 (1972). 95 C. B. Bucknall and I. C. Drinkwater, J . Muter. Sci. 8, 1800 (1973). P. Keusch and D. J. Williams, J . Polym. Sci., Polym. Chem. Ed. 11, 143 (1973). cn M. Matsuo, C. Nozaki, and Y. Jyo, Polym. Eng. Sci. 9, 197 (1969).

16.3

DIRECT OBSERVATION

293

FIG.3. Transmission electron micrograph of graft-type high-impact polystyrene showing cellular structure of 6% polybutadiene (with kind permission of the publisher and the authors, taken from Wagner and Robesonan).

poly(viny1 chloride) forms rough aggregates of about 10 pm particles. When NBR contains 8% acrylonitrile the rubber particles are smaller than 1 pm. At 15% acrylonitrile the rubber appears as a continuous network and the PVC particles are about 1 p m in diameter. At 40% acrylonitrile the rubber phase is less than 10 nm thick around PVC particles of about 0.1 p m diam in which NMR is very finely dispersed. Obviously with 40% acrylonitrile in NBR the copolymer is highly compatible with PVC. Mixtures of poly(ethylene-co-vinylacetate) containing 45% vinyl acetate with PVC have morphologies similar to those of NBR-20 (containing 20% acrylonitrile) with PVC .w Optically clear impact-resistant plastic blends have their structure revealed under the electron microscope, even when optical clarity is obtained by refractive index matching. Such blends sometimes show large zones (= 1 pm).57,59 In other instances the zones are smaller (<0.25 pm) as a result of extensive grafting.58sge Particles of ABS or poly(methy1 Y. Jyo, C. Nozaki, and M. Matsuo, Marcomolecules 4, 517 (1971). R . G. Bauer and M. S. Guillod, Adv. Chem. Ser. 142.231 (1975).

294

16.

POLYMERIC ALLOYS

FIG. 4. Electron micrograph of film of poly(styrene-6-butadiene) containing 17 wt% polystyrene. Film annealed for 1 hr at 100°C. OsO, staining causes polybutadiene matrix to appear dark. The grains in which the polystyrene domains are coherent are about 1 p m (with the kind permission of the publisher and the authors, taken from Kampf er ul. Iol).

methacrylate-co-butadiene-co-styrene)(MBS) in poly(viny1 chloride) have 0.1 pm diam.e7 The zone reduction in blends of homopolymers in the presence of increasing amounts of graft copolymer was demonstrated in a series of electron micrographs of blends of polystyrene and polyethylene.'"'' Ad: dition of 7.5% graft copolymer reduces the size of the dispersed phase from more than 5 pm to about 0.5 p m . 16.3.3.2Block Copolymers The morphology of block polymers, especially the poly(styrene-bdienes), has been made visible by electron microscopy. The diene is either butadiene or isoprene. An example is shown in Fig. 4. The method is, however, not without experimental difficulties and possible pitfalls. Samples are either solvent-cast films or ultramicrotomed sections preferably less than 1000 8, thick. Details of sample preparation and data acquisition are described.20.28~101~10* Special attention has been drawn to artifacts caused by staining.lm Determination of the morpholW. M. Barentsen, D. Heikens, and P. Piet, Polymer 15, 119 (1974). G. Kampf, M. Hoffmann, and H. Kromer, Ber. Bunsenges. Phys. Chem. 74, 851 ( 1970). Ioz P. R. Lewis and C . Price, Polymer 13, 20 (1972). lW G. Kampf and H. Schuster, Angew. Mukromol. Chem. 14, 1 I 1 (1970). loo Io1

16.3

295

DIRECT OBSERVATION

TABLE11. Morphological Relations for Block Polymer Domains

Morphology

Ratio of Bragg spacings

Characteristic dimension of dispersed phase A

Single cubic Body centered cubic

1:4:4:4

Face centered cubic

1:4:4:G

RA= d

[E] 'lJ

Hexagonally packed cylinders Lamellae

I :b:f:t

ogy in block copolymer samples requires, ideally, observation of the sample in two perpendicular directions. An overview of the extensive literature data shows that polystyrene spheres in a rubber matrix are obtained when 0 < 4ps< 0.30. Polystyrene cylinders (hexagonally packed) are obtained when 0.15 < 4ps< 0.40 and polystyrene and polydiene lamellae when 0.35 < 4ps< 0.70. Polydiene cylinders are found when 0.60 < +ps < 0.85 and polydiene spheres when rpPs > 0.85.20*21.101.1~~105 It seems best to allow for overlapping volume fractions for the different morphologies. At 0.26 < 4ps< 0.32 it was shown that the three different morphologies-polystyrene spheres, cylinders, and lamellae-can coexist each with its own characteristic dimension.1m Obviously, the three morphologies are energetically nearly equal at certain volume fractions. This would also explain why some authors find polystyrene spheres in poly(styrene-b-butadiene-b-styrene) samples with c $ = ~ O.27,lo7while others find cylinders108 and lamellaelOefor very similar block polymers. In this regard it is important to realize that the morphology of solvent-cast block polymers is fixed when polymer mobility is severely reduced, but there is still solvent present. Variation of casting solvents and procedures have been found to induce differences in block polymer morphologies.21~110~111 M. Matsuo, S. Sagae, and H. Asai, Polymer 10, 79 (1969): H. Hendus, K.-H. Illers, and E. Ropte, Kolloid-Z. & Z . Polym. 216, 110 (l%7). '00 M . Hoffmann, G. Kampf, H. Kromer, and G. Pampus, Adv. Chem. Ser. 99,351 (1971). E. Campos-Lopez, D. McIntyre, and L. J. Fetters, Macromolecules 6,415 (1973). E. Pedemonte, G. Dondero, G. C. Alfonso, and F.de Candia, Polymer 16,531 (1975). loo E. Pedemonte and G. C. Alfonso, Macromolecules 8,85 (1975). 110J. F.Beecher, L. Marker, R. D. Bradford, and S. L. Agganv4.J. Polym. Sci., Purr C 26, 117 (1969). IM

lo5

296

16. 2.d

SPHERES

P O L Y M E R I C ALLOYS

-4

N 0 0

-I

2.8

- 2.2 2.6 -2.0

-

"5

t

v)

2.4

@

2.2

-J

LAMELLA€

0 0

3.6

4.0

I

I

I

4.5

5.0

5.5

2.0

Log MWPS block

FIG.5. Characteristic dimensions of polystyrene blocks in poly(styrene-b-dienes) and poly(styrene-b-diene-b-styrene). Full symbols: electron microscopy. Open symbols: small-angle X-rky scattering. 0:reference 28; 0: reference 106; 0: reference 107; A: references 112 and 156; V: reference 113. Broken line for spheres: Eqs. (16.2.9) and (16.2.10) with lo = 6.6 A and y = 1 dyne/cm. Broken line for lamellae: Eq. (16.2.15) with /, = 6.6 A and xlz = 0.142.

The characteristic dimension R of the dispersed domains and their spacing (center-to-center distance) d are determined from electron micrographs by a careful statistical sampling. The dimensions of the domains and their spacing in a particular pattern are related to the volume fraction of the blocks as given in column 3 of Table 11. Figure 5 shows the relation between domain diameter (thickness for lamellae) and the molecular weight of polystyrene blocks observed in poly(styrene-b-dienes) and poly(styrene-h-diene-b-styrenes).The data are selected from studies that explicitly dealt with this relation.28~108~107~"2."3 Some literature data do not agree with these data.21 The delicate energy balance struck in the formation of the domains, the experimental difficulties in obtaining the data, and the uncertainty in the molecular weights of the blocks explain the appreciable scatter of the data. There is a wealth of single precise measurements on polystyrene domains in the 10,000-20,000 molecular weight range when they form 111

112

lls

T. Uchida, T. Soen, T. Inoue, and H. Kawai, J . Polym. Sci., Part A-2 10, 101 (1972). A . Douy, R. Mayer, J. Rossi, and B. Gallot, Mol. Crysr. Liq. Crysf. 7, 103 (1969). E. B. Bradford and E. Vanzo, J . Polym. Sci., Part A - / 6 , 1661 (1968).

16.3

DIRECT OBSERVATION

297

when they appear as hexagonally packed cylind e r ~ , and ~ when ~ ~ they ~ ~form ~ lamellae.28*10sJ18 - ~ ~ ~ The lines in Fig. 5 indicate that the diameter of spheres and cylinders and the thickness of lamellae of polystyrene are in the ratios 2: 1.5: 1, in close agreement with the K factor of Eq. (16.2.9). Experimentally, it is found that a block polystyrene with molecular weight 12,600 forms 230 8, diam spheres, 170 8, cylinders, and 115 8, thick lamellae. The unperturbed rms endto-end distance for such polystyrene is 76 8, (based on ( s 2 ) , / M = 7.6 x 10-l8 cm2 mole/g). From Eq. (16.2.9) it follows that the polystyrene blocks are therefore expanded by a = 1.15-1.09 in their domains. The slope of the lines in Fig. 5 is drawn as 0.56, comparable to 0.58,29but is obviously not very well established. Theories predict a slope between 0.55 and 0.65.4.25 Domains of a characteristic dimension in a geometrical pattern can extend in an ordered fashion over a certain distance. The area over which a morphology is coherent is called grain.20*101*114 Grains can be seen in Fig. 4. The dimensions of the grain are even more influenced by the sample preparation and treatment than are the domain morphology. In solution-cast samples this long-range order is often absent but it can be promoted by slow evaporation of a solvent equally good for both blockslo1 and by subsequent annealing of the cast film above the glass transition temperature of the plastic component.20 Normal grain sizes are of the order of a few microns.114 Extremely well-ordered structures are produced by the slow extrusion of the sample above the higher glass transition temperature. The lattice order can extend over several millimeters and encompass the whole sample as in a single crystal. Hexagonally packed cylinders lie parallel to the extrusion direction."' Lamellae are extruded as concentric cylinders.118 Electron microscope studies have been made on poly(styrene-b-dimethylsiloxane),lle poly(amethylstyrene-b-dimethylsiloxane),120 poly(styrene-b-acrylonitrile),'21 poly(dimethylsiloxane-b-bisphenol-A-carbonate),122 and presumably many others. 'I4

'I5

P. R. Lewis and C. Price, Polymer 12, 258 (1971). C. Price, A . G. Watson, and M. T. Chow, Polymer 13, 333 (1972). C. Price, T. P. Lally, A. G. Watson, D. Woods, and M. T. Chow,&. Polym. J . 4,413

(1972).

"'J. Dlugosz, A . Keller, and E. Pedemonte, Kolloid-Z. & Z . Polym. 242, 1125 (1970). 'I8 J . Dlugosz, M. J. Folkes, and A. Keller, J . Polym. Sci.. Polym. Phys. Ed. 11, 929 ( 1973).

I*' Ia2

J. C. Saam,D. J. Gordon, and S. Lindsey, Macromolecules 3, 1 (1970). M. Morton, Y. Kesten, and L. J. Fetters, Appl. Polym. Symp. 26, 113 (1975). E. Perry, J . Appl. Polym. Sci. 8, 2605 (1964). R . P. Kambour, Polym. Left. 7, 573 (1969).

298

16.

POLYMERIC ALLOYS

Block copolymers with more complex molecular structures, e.g., multiblock and graft copolymers, form well-ordered morphologies with more d i f f i c ~ l t y . ~ ~ ~An * ~exception @ " . ~ ~ ~seems to be star-shaped block polymers of general formula (AB),X with n > 2. There is evidence that they take on the same morphology as the linear di- and triblock polymers of the same chemical c o m p o s i t i ~ n ,while ~ ~ ~ other * ~ ~ evidence suggests that they acquire a more disperse morphology than linear block copolymers (cylinders rather than lamellae, spheres rather than ~ y l i n d e r s ) .Important ~~~~~~~ are multiblock, also called segmented, poly(ether-6-urethanes) and poly(ester-b-urethanes), whose electron micrographs show small 30- 100 8, polyurethane domains randomly dispersed in the When triblock copolymers with chemically different A, B, and C blocks phase-separate, either two or three phases can form. The nature of the phases has not been established in the case of poly(styrene-bbutadiene-b-ethylene sulfide)128or poly(styrene-b-butadiene-b-a-methyl styrene).lZg In the latter case the glass transition temperature suggests that polystyrene and poly(a-methyl styrene) form an intimate mixture. There is electron-microscopic evidence that polystyrene and poly(viny12-pyridine) blocks form a single phase in poly(styrene-b-is0prene-b~inyl-2-pyridine).'~~ Electron microscopy of block copolymers in which one block has crystallized gives details of the spherulites and crystallite structure to the level of the individual monolayers. Lamellae of crystalline poly(ethy1ene oxide) sandwiched between layers of the amorphous polystyrene seem to dominate the morphology of poly(styrene-b-ethylene oxide). Since poly(ethy1ene oxide) is never 100% crystalline, a layer of amorphous poly(ethy1ene oxide) is assumed at the interface of the two b l o ~ k s . ~ ~ ~ ~ From replica and transmission electron micrographs the crystalline poly(ethy1ene oxide) layer is estimated to be either 95 or 180 A.132 Shadowed replicas of single-crystal layers yield poly(ethy1ene oxide) thicknesses of 60-100 A, depending on the crystallization conditions and a C. Price, R. Singleton, and D. Woods, Polymer 15, I17 (1974). R. Mayer, Polymer 15, 137 (1974). E. Pedemonte, G. Dondero, F. de Candia, and G. Romano, Polymer 17, 73 (1976). *Ie L.-K. Bi and L. J. Fetters, Macromolecules 8, 90 (1975). J . A. Koutsky, N. V. Hien, and S. L. Cooper, Polym. Len. 8, 353 (1970). I** W. Cooper, P. T. Hale, and J. S. Walker, Polymer 15, 175 (1974). Ire G.S. Fielding-Russell and P. S. Pillai, Polymer 15, 97 (1974). C. Price, T. P. Lally, and R. Stubbersfield, Polymer 15, 541 (1974). 131 B. Lotz, A. J. Kovacs, G . A. Bassett, and A. Keller, Kolloid-Z. & Z . Polym. 209, 115 (l%6). J. J. O'Malley, R. G . Crystal, and P. F. Erhardt, in "Block Copolymers" ( S . L. Agganval, ed.), p. 163. Plenum, New York, 1970.

16.4. SCATTERING TECHNIQUES

299

polystyrene layer of about 30 A.131 Globules of amorphous polymer have been observed in poly(styrene-b-ethylene and poly(is0prene-b-ethylene oxide).80 Spherical domains of poly(ethy1ene oxide) were observed when it was the minor component.80

16.4. Scattering Techniques 16.4.1. Small-Angle Light Scattering (SALS)

Typically, depolarized small-angle light scattering (e.g., from a laser source) of spherulites will produce a cloverleaf H, pattern and an elongated V, pattern. H, indicates that the light is horizontally polarized; the analyzer detects vertical polarized light. These patterns are produced by some ordering of the intrinsically anisotropic crystallites in the spherulites. The spherulites are usually observed in a medium of other spherulites or of amorphous material of the same substance. Spherulites can also be formed from a mixture of two polymers as when poly(ecaprolactone) is isothermally crystallized from a mixture with poly(viny1 chloride)." The spherulites are space-filling for up to 50% PVC. Their size goes through a maximum and the intensity of the light scattered decreases with increasing PVC content, indicating increasing disorder of the crystallites in the spherulites. In segmented poly(ether-b-urethanes)a the urethane segments crystallize. The radii ( R ) of the spherulites can be calculated from the angle of maximum light intensity (0,) of the H, pattern according to133 (16.4.1)

where A is the wavelength of the light. Similar measurements on poly(ester-b-ether)spherulites yield dimensions of 1-5 pm in agreement with microscopic observations.134 Spherulite patterns have also been observed from poly(styrene-bbutadiene-b-~tyrene).~~~ The superstructure revealed by these scattering patterns is composed of the more-or-less spherical grains, measuring a few microns in diameter, in which the polystyrene rods are coherently aligned.136 The required anisotropy is not due to molecular orientation in R . S. Stein and M. B. Rhodes, J . Appl. Phys. 31, 1873 (1960). A. Lilaonitkul, J. C. West, and S. L. Cooper, J . Mucrumol. Sri.. Phys. 12, 563 (1976). lSs

R. S. Stein and G . L. Wilkes, J . Pulym. Sci.. Part A-2 7, 1695 (1969). R. S. Stein, Polym. Lett. 9, 747 (1971).

lSB

3 00

16. POLYMERIC

ALLOYS

the block copolymer,137but to the geometry of isotropic rods parallel oriented in an isotropic matrix.138 Additional molecular anisotropy is introduced on stretching such sample^.^^^,^^^ Low-angle light scattering in stretched samples of poly(styrene-bbutadiene-b-styrene), in which polystyrene spheres are originally isotropically dispersed, may have its origin in a periodic deformation in the concentration profile and also in void f o r m a t i ~ n . ~ ~ ~ J ~ ~ 16.4.2. Small-Angle X-Ray Scattering (SAXS)l

Absolute SAXS intensity measurements as a function of the scattering vector h = (4n/A) sin 8 over the whole range in which the scattering intensities Zh are above the constant background scattering allow the study of any two-phase system. A is the wavelength, most often 1.54 A, from the Copper K a radiation and 8 is one-half the angle between incident and scattered beam. When measured with a Kratky camera with infinite slits, the invariant Qh is given by140 Qh

=

1-(Ih 0

(2) 2

-

Wh dh

=

A3RDAsio+A+B(Ap)2,(16.4.2)

where U is the constant background scattering, e and m the charge and mass of the electron, c the speed of light in vacuum, R the distance from the sample to the detector, D the sample thickness A& the intensity of the attenuated beam, and ( A P ) ~the experimentally observed square of the electron density difference between the two phases present in the sample. When the latter is identical with (Ap)',,,, for the case of the pure, completely phase-separated components one can conclude that this is the state of the mixture. When (A# < (Ap);,,, there is either partial mixing of the components or there is a diffuse boundary between the phases, or both occur simultaneously. Which case holds for a particular mixture cannot be resolved from SAXS data Electron density differences in mixtures of poly(viny1 chloride) and poly(ecapro1actone) obtained from the invariant are lower than calculated for completely phase-separated mixtures, indicating residual mixing and/or a large i n t e r p h a ~ e . ~ ~ In the case of poly(styrene-b-butadiene-b-styrene)the experimentally Is'

M. J. Folkes, A. Keller, and F. P. Scalisi, Polymer 12, 793 (1971). M. J. Folkes and A. Keller, Polymer 12, 222 (1971). T. Inoue, M. Moritani, T. Hashimoto, and H. Kawai, Macromolecules 4, 500 (1971). G. Porod, Kolloid-Z. & Z . Polym. 124, 83 (1951); 125, 51 (1952). R. Bonart and E. H. Miiller, J . Macromol. Sci., Phys. 10, 177 and 345 (1974). See also Chapter 6.2 (this volume, Part B).

16.4. SCATTERING TECHNIQUES

30 1

derived ( A P ) ~through Eq. (16.4.2) agreed well with the calculated value for a completely phase-separated Other measurements, however, indicate ( A P ) ~< ( A P ) ; ~ , ~for similar block polymers.143 Absolute intensity measurements on segmented poly(ester-b-urethanes) indicated considerable intermixing of the two components and an interphase thickness of about 20 8, is derived. Such interphase is appreciable when the domain size is of the order of 100 Intensity maxima obtained from relative intensity measurements of SAXS arise from the presence of periodic variations in the electron densities in the sample. The angles of maximum intensity are related to the spacing d of the periodicity by Bragg's law,

2 sin 8 --

A

n

-

z

(16.4.3)

in which n is the scattering order. When the periodicity is randomly oriented on the scale of the x-ray beam, Debye-Scherrer rings are observed. From samples with highly oriented order, arch or pointscattering patterns may be obtained.138 Block copolymers with their regular alternating domains are highly suited for study by SAXS. From Eq. (16.4.3), for example, it follows that an 88 8, periodicity has a first-order ( n = 1) scattering maximum at a 1" scattering angle. This periodicity represents the center-to-center distance between two dispersed domains. Figure 6 shows the diffraction pattern of a poly(styrene-b-isoprene). Information on the spatial geometry of the domains is extracted from the ratios of the Bragg spacings in the scattering pattern. The ratios of the Bragg spacings for spherical, cylindrical, and lamellar geometries are given in Table II.144.145 From the periodicity d and knowing the morphology of the domains R A ( T A ) , the characteristic dimension of the dispersed phase can be calculated. The equations are given in Table 11. The volume fraction of the block components should be obtained from analytical procedures. The results of x-ray studies are complementary to those of electron microscopy on block copolymers. The x-ray scattering pattern establishes the center-to-center distance and the morphology. Electron microscopy yields the domain size primarily. Successful comparisons of both methods on samples of the poly(styrene-b-diene) type have been re142

14s

lH

H. Kim, Macromolecules 5, 594 (1972). D. G. LeGrand, Polym. Lert. 8, 195 (1970). V. Luzzati, H. Mustacchi, A. Skoulios, and F. Husson, Acta Crystallugr. 13, 660

(1960). 141i

P. Grosius, Y. Gallot, and A. Skoulios, Makromol. Chem. 132, 35 (1970).

302

16.

POLYMERIC ALLOYS

2 B lmin I

FIG. 6. Relative SAXS intensity as a function of scattering angle (28) for a poly(styrene-b-isoprene)film. MW, 105,000: QPS = 0.55. The correction shown is for the tota! reflection. The maxima appear at angular ratios 1 :2 : 3 :4 typical for the lamellar morphology (with the kind permission of the publisher and the authors, taken from Hashimoto et a / .9.

p ~ ~ e d . ~ Results ~ ~ ,from ~ ~SAXS ~ - have ~ ~been ~ .included ~ ~ ~in Fig. 5. From SAXS on poly(ester-b-urethane) and poly(ether-6-urethane) segmented elastomers periodicities of 100-300 A have been ~ b t a i n e d ,in~ ~ ~ . ~ ~ ~ agreement with crystalline polyurethane domains of 30- 100 A observed in the electron In some poly(ester-6-urethane) samples SAXS maxima are absent, indicating intensive mixing or lack of regular periodicity.148J4e In these samples urethane-ester hydrogen bonding competes favorably with urethane-urethane hydrogen bonding. Phase separation is also more pronounced when the urethane blocks are 10nger.l~~ The SAXS technique allows the study of gels made of block polymers swollen with solvents. The spacing, morphology, and dimensions of the domains are found to be a function of the block lengths and the solvent fraction in each block. All morphologies described for bulk block copolymers have also been found for these gels. For example, poly(styrene-b-vinyl-2 pyridine) (40/60) forms lamellae with 0-40% toluene, which preferentially solvates the polystyrene blocks, but forms poly(viny1-2pyridine) cylinders when more toluene is present.lW Similar 140 A. Keller, E. Pedemonte, and F. (1970).

M. Willmouth. Kolloid-Z. & Z. Po/.vm. 238, 385

R. Bonart, J. Mucromol. Sci.. Phys. 2, 1 I5 (1968). S . B. Clough, N. S. Schneider, and A. 0. King,J. Macromol. Sci., Phys. 2,641 (1968). 149 R. Bonart, L. Morbitzer, and G. Hentze. J . MAcromol. Sci.. Phys. 3, 337 (1969). lM P. Grosius, Y.Gallot, and A. Skoulios, Mokrornol. Chem. 127,94 (1%9). 14'

16.4

SCATTERING TECHNIQUES

303

studies were made on poly(styrene-6-vinyl-4 p ~ r i d i n e )poly(styrene,~~~ b-ethylene oxide),151 poly(ethy1ene oxide-b-propylene oxide),lS2 poly (styrene-b-isoprene),ll2poly(butadiene-b-styrene-b-butadiene)153 and poly (butadiene-b-a-methyl styrene).l" For poly(methy1 methacrylate-b-nhexyl methacrylate) a complete phase diagram, in which spheres, cylinders, lamellae, and inverted cylinders are found, was constructed as a function of block copolymer composition and volume fraction of acetonitrile.15s Within a single morphology, the domain size of a block decreases smoothly with decreasing amount of solvent in that block. The domain size of the nonsolvated block increases with decreasing amount of solvent. In some cases the domain size of the solvent-free block polymer can be found by a small extrapolation to zero solvent concentration.112,150.151.1~~1s7 Since the solvent-containing systems are likely to be in thermodynamic equilibrium, such extrapolation to values for bulk samples suggest strongly that the latter are also at equilibrium. When the solvent is a polymerizable monomer, SAXS before and after polymerization have shown that the domain sizes of the block polymer change little during the p o l y m e r i z a t i ~ n . ~ ~The ~ * viscosity ~ ~ ~ ~ ~ -of~ the ~ polymerizing medium and grafting possibly prevent the formation of a third polymer phase. The polymerized samples can also be studied by electron microscopy. This allows a comparison of SAXS and electron microscopy data obtained from the same sample.lw-lse SAXS studies have shown that block copolymers can incorporate the parent homopolymers in their domains, provided that the molecular weight of the homopolymer is not larger than that of the block.1soB161 When a block takes up its own weight of homopolymer, the characteristic domain size almost doubles. Rejected homopolymer will aggregate to form large particles several microns in diameter.61 The relative intensities of the consecutive orders of scattering can be used to calculate the individual domain size and the thickness of the interphase between the domains. For example, the intensity maximum A. Skoulios and G . Finaz, J . Chim. Phys. 59, 473 (1%2). G. Tsouladze and A. Skoulios, J . Chim. Phys. 60, 626 (1%3). lPA. Douy and B. Gallot, Makrornol. Chem. 156, 81 (1972). A. Douy, G. Jouan, and B. Gallot, C . R . Hebd. Seances Acad. Sci. 281, 355 (1975). lsa H. Ailhaud, Y. Gallot, and A. Skoulios, Kolloid-Z. & Z . Polym. 248, 889 (1971). Is A. Douy and B. Gallot, Mol. Cryst. Liy. Cryst. 14, 191 (1971). 'sI A. Douy, G. Jouan, and B. Gallot, Makromol. Chem. 177, 2945 (1976). 'lYl G. Finaz, A. Skoulios, and C. Sadron, C. R . Hebd. Seances Acad. Sci. 253,265 (l%l). le A. Douy and B . Gallot, C . R . Hebd. Seances Acad. Sci. 274, 498 (1972). la, A. Skoulios, P. Helffer, Y. Gallot, and J. Selb, Makromol. Chem. 148, 305 (1971). lel B. F'taszynski, J. Temsse, and A. Skoulios, Makromol. Chem. 176, 3483 (1975). lal

3 04

16.

POLYMERIC ALLOYS

found in the scattering from a stack of lamellae varies as1sz,1s3 =

sin(n?r) T A / d sin(n?r) X / d n?rTA/d n d / d

[

1*

'

(16.4.4)

where TA is the thickness of one lamella and X the thickness of the interphase. The scattering intensities are very sensitive to nTA/d, especially at high n. Intensity maxima will be absent whenever nTA/d = n ~ is$ an ~ integer. The first term in Eq. (16.4.4) is for the case of a sharp interface between the layers. If an interphase with thickness X is present, the scattering intensity will be modified by the second term in Eq. (16.4.4). Estimates of the interphase thickness in poly(styrene-&-isoprene)have Such measurements establish an order been given as 12 8,lSZ and 30 of magnitude. According to Eq. (16.2.14), and taking X from SAXS equal to a I , these interphase thicknesses correspond to x12 values for this polymer pair of 0.20 and 0.03, respectively. The electron micrographs of block copolymers with one crystalline block indicated that lamellae of crystalline polymer sandwiched between layers of the amorphous polymer are encountered most often. From the volume fraction of the crystalline block, its lamellar thickness can be calculated, and from the degree of crystallization, the thickness of the crystalline and amorphous layer is obtained. In the case of poly(ethy1ene oxide-b-ethyl methacrylate) SAXS has revealed that the spacing increases gradually with the temperature of crystallization. The thickness of the crystalline layer itself increases gradually from 80 to 250 8, with the temperature of crystalli~ation.~~ This is different from the behavior of narrow-molecular-weight-distribution poly(ethy1ene oxide) in which the crystalline lamellae tend to form discrete thicknesses that are exact subdivisions of the chain length.184 Therefore, the presence of the attached amorphous polymer prevents a high degree of order to be realized in the crystal layers. Similar reduction in crystalline ordering was found in a poly(ethy1ene oxide) segmented with urethane groups,1s5 in poly(n-propyl methacrylate-g-ethylene oxide),lBs and in poly(styrene-&-ethylene ~ x i d e ) . l ~This ~ J ~lower ~ crystalline order is paralleled by lower degrees of crystallization. A. Skoulios, in "Block and Graft Copolymers" (J. J. Burke and V. Weiss, eds.), p. 121. Syracuse Univ. Press, Syracuse, New York, 1973. T. Hashimoto, K. Nagatoshi, A. Todo, H. Hasegawa, and H. Kawai, Macromolecules 7, 364 (1974). Is( J. P. Arlie, P. Spegt, and A . Skoulios, Makromol. Chem. 104, 212 (1967). J . C. Galin, P. Spegt, S. Suzuki, and A . Skoulios, Makromol. Chem. 175,991 (1974). A . Thierry and A. Skoulios, Makromol. Chem. 177, 567 (1976). '81 Y. Shimura and T. Hatakeyama, J . Polym. Sci., Polym. Phys. Ed. 13, 653 (1975).

16.4 SCATTERING TECHNIQUES

305

To a lesser extent low-angle electron diffraction in the electron microscope has been used in lieu of SAXS.188 Applications to block copolymers give information on their domain spacing, in excellent agreement with SAXS results.115-117J6s 16.4.3. Small-Angle Neutron Scattering (SANS)2

The large difference in the neutron coherent scattering length for deuterium and hydrogen provides contrast by which a normally “hydrogen-rich’’ polymer can be studied in a perdeuterated medium or vice versa. Moreover, the mixture can always be so composed that, with regard to its scattering behavior, the scattering polymer is in dilute state, while the effective polymer concentration is higher. This is realized by, for example, mixing a small amount of hydrogenated polymer A with a large amount of perdeuterated polymer A in perdeuterated polymer B. Intermolecular interference can thereby be avoided. Under these conditions the weight-average molecular weight M w is obtained d t e r extrapolation to zero scattering angle 8 and zero concentration c: (16.4.5)

The radius of gyration is obtained from the angular scattering dependence at zero concentration: Ph = 1 -*he(r2>

+

*

*

a,

(16.4.6)

where Ph is the scattering function, in which h = (47r/A) sin (8/2). In contrast to x-ray terminology, 8 is the angle between the incident and scattered beam. In this way, it was established that polystyrene is mono-molecularly dispersed in perdeutero poly(a-methyl styrene), and that its radius of gyration ( r 2 ) ,is slightly smaller than the unperturbed at least up to 10% by weight. Poly(styrene-co-acrylonitrile)with 10-28.7 wt% acrylonitrile yields normal Zimm plots and the correct molecular weight when mixed (up to 1.5%) with perdeutero poly(methy1 metha~rylate).’~~ At 19 G. A. Bassett and A. Keller, Philos. M a g . [8] 9, 817 (1964). G. Karnpf, H. Kromer, and M. Hoffmann, J . Macromol. Sci., Phys. 6 , 167 (1972). I7O D. G. H. Ballard, M. G . Rayner, and J . Schelten, Polymer 17, 640 (1976). 171 W. A. Kruse, R. G.Kirste, J. Haas, B. J. Schrnitt, and D. J. Stein, Makromol. Chem.

lee

177, 1145 (1976).

See also Chapter 5.4 (this volume, Part A).

306

16.

POLYMERIC ALLOYS

wt% acrylonitrile, poly(styrene-co-acrylonitrile) exhibits a radius of gyration and second virial coefficient characteristic for a polymer in a good solvent. Temperature variation of the radius of gyration and the second virial coefficient indicate that the polymer-polymer interaction is exotherm for this pair. Copolymers with 10 and 28.7 wt% acrylonitrile have negative virial coefficients, indicating that these compositions form the limits of solubility with poly(methy1 methacrylate). Optical clarity, glass transition measurements, and electron microscopy on mixtures of poly(methyl methacrylate) and poly(styreneco-acrylonitrile) yield the same copolymer compositions for the limits of mutual solubility.172 When a polymer is not monomolecularly dispersed in another polymer, abnormal Zimm plots are observed in SANS.'" The molecular weight is then a high multiple of the real molecular weight and the radius of gyration is that of a collapsed coil.173

16.5. Glass Transition Temperature Measurements The glass transition temperature Tg is the main transition temperature found in amorphous polymers. The transition is characterized by a change in thermal expansion coefficient and in the heat capacity. The underlying molecular process is that at the glass transition temperature frozen backbone sequences begin to move. Therefore, Tg is determined not only by the main-chain architecture but also by its immediate surroundings. The concept of an isofree volume state at Tggoes a long way in explaining many of the factors that influence Tg.174 Chain ends and low-molecular-weight plasticizers lower the Tg of a polymer. A sufficiently large number of cross links will increase Tg. The Gordon-Taylor-Wood equation is the most general one describing the Tgllof a homogeneous mixture of two compounds:17s (16.5.1)

where w1 and w2 are weight fractions and TgI and T , the glass transition temperatures of the pure compounds. When both compounds are 17*

D. J . Stein, R. H. Jung, K.-H. Illers, and H . Hendus, Angew. Makromol. Chem. 36,89

(1974). 173

R. G. Kirste, W. A. Kruse, and K. Ibel, Polymer 16, 120 (1975). For example, see F. Bueche, "Physical Properties of Polymers." Wiley, New York,

1%2. lT5

L. A. Wood, J . Polym. Sci. 28, 319 (1958).

16.5.

GLASS TRANSITION TEMPERATURE MEASUREMENTS

307

polymers Eq. (16.5.1) can be used to assess the level to which two bolymers mix. Indeed, Eq. (16.5.1) will be applicable to two polymers on the condition that the mixing is homogeneous to the level "seen" by the T , measurement. On the other hand, when two polymers in a blend are separated into their individual large zones, each polymer will exhibit its own Tg. If it is assumed that the constant k in Eq. (16.5.1) for a pair of miscible polymers is the same as that for their random copolymer, as has been found e~perimentally,~~.'761'7~ then the composition of homogeneous phases of two polymers can be calculated from their Tgs. It should be mentioned that the T, of a polymer has a kinetic aspect to it, whereby the glass transition occurs over a time and temperature range. The value of T, depends also on the experimental method used. There are semistatic methods, for example, dilatometry and calorimetry, in which the external parameters can be changed very slowly and the sample allowed to respond fully, and dynamic methods (dynamic mechanical, dielectric, NMR,ESR-spin probe) in which the sample response to an externally imposed frequency is observed. Tgmeasurements can only be used to assess the compatibility of two polymers if their Tgs are sufficiently far apart from each other. The minimum temperature difference required to distinguish two T,s depends on the width of the two transitions and on the sensitivity of the experimental method. In mixtures of two polymers, glass transition regions are often broader than in the pure polymers under identical experimental conditions. They are an indication for the presence of a large amount of variable intermixing of the polymers in the sample. A large interphase will also widen the glass transition region. This review focuses on borderline cases in which changes in the T, of a polymer are observed in the presence of a second polymer and tries to assess their significance in terms of mixing of the two polymers. 16.5.1. T, of Mixtures of Polymers

The use of homogeneous random copolymers allows the study of the different T, phenomena encountered in mixtures of polymers. For example, it is found that poly(methy1 methacrylateco-butyl acrylates) (for the homopolymers ATg = 155°C and a1 - = 0.2) form optically clear mixtures when their compositions do not differ more than 10- 15% depending on the molecular weight of the copolymer^.^'^ The electron microscope 17'

L. H . Sperling, D. W. Taylor, M. L. Kirkpatrick. H. F. George, and D. R. Bardman.J.

Appl. Polym. Sci. 14, 13 (1970). '" S. Krause and N. Roman, J . Pulym. Sci., Part A 3, 1631 (1965).

308

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POLYMERIC ALLOYS

shows the presence of 30 8, heterogeneities. A single T,, intermediate between that of the pure copolymers but slightly broader, is found for such mixtures.178 Two resolved Tgsabout 20°C apart are found when the composition differs by 20%; the zone size is then more than 1 pm and the mixture 0 p a q ~ e . lA ~ ~3.4 wt% difference in the copolymer composition of two poly(styreneco-acrylonitriles) causes opacity of their mixtures (6, - = 3.7; ATg = 12°Cfor the homopolymers). The small difference in the Tgsof the homopolymers does not allow its use in phase ~tudies."~ In mixtures of two different poly(styrene-co-butadienes)(6, - = 1.0, ATg = 180°C for the homopolymers) two Tgs are found when the copolymer compositions differ by more than 20% and their Tgs differ by about 35°C. Otherwise, a single Tgis found intermediate between that of the individual copolymer^.^^ Within the polyacrylates and polymethacrylates, poly(methy1 methacrylate) and poly(ethy1 acrylate) (ATg = 120°C) show partial compatibility among various polymer pairs tested.180 In the presence of some graft copolymer and in interpenetrating networks, a single rather broad intermediate Tg is found.17BJ80JB1The average Tg corresponds to that of the random copolymer of the same composition, but the broadness of the transition suggests wide composition fluctuations. Fine structure of the order of 100 A is revealed in the electron microscope.18* Poly(isopropy1 methacrylate) and poly(isopropy1 acrylate) blends (ATg = 85°C) show only a single Tg,always corresponding to that of the random copolymer of identical composition. 177 The Tgs of blends of polystyrene ( T g = 100°C) and poly(2,6-dimethyl1,Cphenylene oxide) (PPO, T g = 220°C) have been studied by different experimental techniques. Single intermediate TG have been found by an optical technique, by differential scanning calorimetry (DSC), and by dynamic-mechanical methods.183-183bIn other instances, two intermediate Tgs are found in blends of polystyrene with PPO by a dynamicmechanical method, indicating two partially mixed phases coexist.lM In these samples DSC detects only one T g , similar to the Tg of the polystyrene-rich phase seen by the mechanical method.lM Dielectric F. Kollinsky and G . Markert, Makromol. Chem. 121, 117 (1969). G. E. Molau, Polym. Lett. 3, 1007 (1965). L. J. Hughes and G. L. Brown, J . Appl. Polym. Sci. 5 , 580 (1961). 181 V. Huelck, D. A. Thomas, and L. H. Sperling, Macromolecules 5, 348 1972). V. Huelck, D. A. Thomas, and L. H. Sperling, Macromolecules 5, 340 1972). la W. M. Rest and R. S. Porter, J . Polym. Sci., Part A-2 10, 1639 (1972). laa A. R. Shultz and B. M. Beach, Macromolecules 7, 902 (1974). lab R. A, Fava and C. E. Chaney, J . Appl. Polym. Sci. 21,791 (1977). J. Stoelting, F. E. Karasz, and W. J. MacKnight, Polym. Eng. Sci. 10, 33 (1970). In

Ire

16.5.

GLASS TRANSITION TEMPERATURE MEASUREMENTS

309

relaxation measurements reveal one broad T,, that of the major phase.2 Two explanations can be advanced. Either there are two partially mixed phases with not every experimental method being sufficiently sensitive to detect both (e.g., the higher Tgmay be broad and obscured by the larger and lower T, and is therefore not detected in the DSC measurement) or each technique measures average compositions on a different scale. The sample may therefore appear homogeneous to a technique measuring over a large volume element but heterogeneous to the method measuring at a smaller level. The Tg measured over a large volume element is expected to approach that of the overall composition of the blend. Note that if two phases are present in polystyrene-PPO blends their size is relatively small, as indicated by the optical clarity of the blends.lM PPO is incompatible with poly(p-chlorostyrene) as shown by two unchanged Tg~.183a*18s Two T,s are observed in blends of PPO with poly(styreneco-p-chlorostyrene) containing more than 66 mol% p-chlorostyrene.laa Partial mixing occurs between PPO and poly(a-methyl styrene).lB6 The T, of blends of polystyrene and poly(viny1 methylether) show dependence on the casting solvent. Cloudy films exhibiting the Tgs of the two homopolymers are produced from chloroform, dichloroethane, and trichloroethylene. Clear films with one intermediate T, are formed on casting from benzene and t01uene.l~~This clearly indicates that the polystyrene-poly(viny1 methylether) compatibility is based on specific interactions between the two polymers, which are destroyed in some polar, weakly hydrogen bonding solvents. This is in agreement with a negative x~~~~and a lower critical solution temperature for this polymer paiF-" (Fig. 2b). The DSC and dielectric relaxation measurements on the blends show broad Tg regions, indicating some heterogeneity in the blends.lB7 Pulse NMR data suggest two mixed microphases coexisLq A single T, has been observed for blends of poly(viny1 chloride) with poly(butadieneco-acrylonitrile) containing about 40% acrylonitrile.70J88-101This single Tgprogressively varies with the blend composition between those of the pure corn pound^.^^ The zone size is about 30 A in these blends.07 A single Tgwas also found for blends with a copolymer containing 32% acrylonitriIelg2and 29% a c r y l ~ n i t r i l e . ~ Blends ~ ~ of PVC F. E. Karasz, W. J. MacKnight, and J. J. Tkacik, Polym. Prepr. 15, 415 (1974). L. M . Robeson, M. Matzner, L. J. Fetters, and J. E. McGrath, Polym. Sci. Techno/. 4, 281 (1974).

M. Bank, J . Leffingwell, and C. Thies, Macromolecules 4, 43 (1971). L. E. Nie1sen.J. Am. Chem. Soc. 75, 1435 (1953). M. Takayanagi, M e m . Fac. Eng., Kyoto Univ. 23, 41 (1963). 1 8 0 P. Zitek and J . Zelinger, J . Polym. Sci.. Part A - / 6, 467 (1%8). Y. J. Shur and B. RAnby, J . Appl. P o / y m . Sci. 19, 2143 (1975). In G. A. Zakrzewski, Polymer 14, 347 (1973). IBB

310

16. POLYMERIC

ALLOYS

with copolymers containing less acrylonitrile form larger zonese7 and exhibit two Tgs.180~1e2*189 Even then the Tgs are not those of the pure components but indicate partial mixing. Poly(butadiene-co-acrylonitriles) containing less than 30% acrylonitrile often present two Tgs as a result of compositional heterogeneity introduced by copolymerization to high conversion.'Bz When such copolymer samples are mixed with poly(viny1 chloride) the latter is compatible only with the copolymer of highest acrylonitrile content.lg2 Blends of poly(viny1 chloride) with poly(ethy1eneco-vinyl acetate) resemble those with poly(butadiene-co-acrylonitrile). Copolymers with between 60 and 75 wt% vinyl acetate form blends with a single Tg intermediate between the Tgs of the pure homopolymers (92 and - 3 T ) and varying according to the composition of the blends.'" The Tg of the blends is only slightly broader than that of poly(viny1 chloride), indicating nearly random mixing. This high degree of mixing is obtained by milling the polymers together at 190°C;1" lower milling temperatures give only partial mixing.1g5 Poly(ethylene-co-vinylacetate)with 45% vinyl acetate and poly(viny1 chloride) form blends with two very broad Tgs, moved inward from those of the pure corn pound^.^^^^^^ A single Tg has been observed by dielectric loss measurements.1g8 Obviously, partial mixing occurs. The state of the blends is very sensitive to their history. It was shown that heating the samples promotes phase separation. 1g7*1w The electron microscope revealed 1 pm poly(viny1 chloride) zones surrounded by copolymer,gsbut comparison with Tgdata is difficult in view of the different treatment given to the samples. A copolymer containing 40 wt% vinyl acetate and poly(viny1 chloride) form blends with two unchanged T&'" Poly(viny1 chloride) can be compatibilized with other copolymers when they contain the right amount of polar groups. Examples of other copolymers designed for their compatibility with poly( vinyl chloride) are and poly poly(a-methyl styrene-co-methacrylonitrile-co-ethylacrylate)200 (ethyleneco-vinyl acetateco-sulfur dioxide).201 A. H. Jorgensen, L. A. Chandler, and E. A. Collins, Rubber Chem. Techno/. 46, 1087 (1973). lD( C. F. Hammer, Mucrrimoleculcs 4, 69 (1971). lo5 Y. J . Shur and 8 . Rhnby, J . Appl. f o l y m . Sci. 19, 1337 (1975). K . Marcincin, A. Romanov, and V . Pollak, J . Appl. f o l y m . Sci. 16, 2239 (1972). le7 C. Elmqvist and S. E. Svanson. Eur. f o l y m . J . 12, 559 (1976). lm D. Feldman and M. Rusu, Eur. Polym. J . 10, 41 (1974). C. Elmqvist and S. E. Svanson, Eur. folym. J . 11, 789 (1975). J. F. Kenney, J . Polym. Sci., folym. Chem. Ed. 14, 123 (1976). *01 J. J . Hickman and R. M. Ikeda, J . Polym. Sci.. Polym. f h y s . Ed. 11, 1713 (1973).

16.5.

GLASS T R A N S I T I O N TEMPERATURE MEASUREMENTS

311

The quenched blends of poly(viny1 chloride) ( T , = 92°C) and poly(ecaprolactone) ( T , = - 61°C) have sharp glass transitions, which follow Eq. (16.5.1), indicating that this polymer pair forms a compatible mixture.,02 From blends with more than 30% poly(c-caprolactone) the latter tends to ~ r y s t a l l i z e . ~ ~ " ~ ~ Similar behavior is found in blends of poly(methy1 methacrylate) ( T , = 100°C) and poly(viny1idene fluoride) (Tg= -50°C). Quenched samples have an intermediate T,.44*203*204 In samples containing more than 35% PVF, the latter polymer crystallizes on annealing.ea The amorphous phase left has a Tg between 40 and 45"C, indicating some residual mixing of the polymers. Solvent-cast films exhibit a crystalline endotherm down to 10% PVF,. The T, of the residual amorphous phase lies between 70 and 80"C.44 Blends of poly(ethy1 methacrylate) (Tg= 65-70°C) with poly(viny1idene fluoride) show a single T, dependent on the amount of poly(viny1idene fluoride) in the quenched sample.4s2ad05 In samples that are more slowly cooled (e.g., 40"C/min) and that contain more than 40% PVF, two constant T,s are observed, one at - 50°C identical to that of pure poly(viny1idene fluoride) and one at about 27°C belonging to a mixed amorphous phase containing about 45 wt% poly(viny1idene f l ~ o r i d e ) .A~ ~single T, at about 30°C was observed in the composition range 40-80% poly(vinylidene fluoride) but differently interpreted.40s 16.5.2 T, of Block Copolymers

In samples of block copolymers in which domain formation has occurred, the measured domain sizes are of the order of 100 to a few hundred angstroms. When the T,S of the two blocks are sufficiently different, two T,S will be observed.2w This establishes clearly that intermediate Tgs are the result of mixing to below the 100 A level. In some solvent-cast films of poly(styrene-b-butadiene-b-styrene)a weak third transition, variably found between 0 and 6o"C, has sometimes been obmeasure~ e r ~ e d It. is~usually ~ ~ observed ~ ~ ~ in- dynamic-mechanical ~ ~ ~ J. V. Koleske and R. D. Lundberg, J . Polym. Sci., Part A-2 7,795 (1969). zm J. S. Noland, N . N.-C. Hsu, R. Saxon, and J. M. Schmitt, Adv. Chem. Ser. 99, 15

(1971).

D. R. Paul and J. 0. Altamirano, Polym. Prepr. 15, 409 (1974). R . L. Imken, D. R. Paul, and J . W. Barlow, Polym. Eng. Sci. 16,593 (1976). R . J . Angelo, R. M. Ikeda, and M. L. Wallach, Polymer 6, 141 (1965). z07 T. Miyamoto, K. Kodama, and K. Shibayama, J . Polym. Sci., Par? A-2 8,2095 (1970). zo(l V. N. Os'kin, Yu. G. Yanovskii, A . Ya. Malkin, V. N . Kuleznev, V. S. Al'tzitser, and I. A. Tutorskii, Polym. Sci. USSR (Engl. Trans/.) 14, 2489 (1972). me A. Beamish, R. A. Goldberg, and D. J. Hourston, Polymer 18,49 (1977). *lo G. Kraus, C. W. Childers, and J. T. Gruver,J. Appl. Polym. Sci. 11, 1581 (1%7). z05

312

16.

POLYMERIC ALLOYS

ments, especially at low frequency.211 It is suggested that it originates from a mixed interphase between the polystyrene and polybutadiene domains. It may, however, be a weak transition of polystyrene.20e The Tg of polystyrene in block copolymers is often lower (up to 10°C) than that of the homopolystyrene of the same molecular weight.3sJo5*110-212 A correlation of the Tg lowering with the surface-to-volume ratio of the polystyrene domains strongly suggests that it is due to partial mixing of the two blocks and that an average Tgfor the whole sample is The interphase would be too small to produce its own T g . The study of or the polyisolow-molecular-weight poly( styrene-b-butadiene-b-styrene) prene variant (molecular weight < 2 x lo4) in which the interphase becomes important shows two broad Tgs intermediate between those of both homopolymers suggesting considerable mutual mixing in the two domains and the formation of wide composition variations.214315Some block copolymers are tapered, i.e., the composition of the chain near the block junction is a copolymer in which the monomer sequences gradually change from pure A to pure B. Tg measurements indicate that a larger amount of mixing occurs in tapered than in pure block A few cases of block copolymers in which both blocks are highly compatible are known. When the molecular weight of poly(styrene-6isoprene) is very low, a region of thermodynamic compatibility must be reached [Eq. (16.2.71. Single Tgs have been observed for poly(styreneb-isoprene) with block molecular weights of 5000 and 1000, r e s p e c t i ~ e l y . ~ ~ The Tg region is narrow, indicating high homogeneity. The Tg is independent of the block sequence, and is equal to that of the random copolymer of the same composition. In poly(styrene-b-a-methyl styrene) two highly compatible chains are linked together. Single Tes, intermediate between those of the homopolymers, have been observed for molecular weights up to 4 x 105.186316-218 For higher molecular weights two Tgs are observed.35 At 50/50 block composition the Tg region of poly(styrene-b-a-methyl styrene) becomes larger with increasing molecular weight This observation indicates that even when a single Tgis observed, the concentration R. E. Cohen and N . W. Tschoegl, Trans. SOC.Rheol. 20, 153 (1976). R. A. Robinson and E. F. T. White, in "Block Polymers" ( S . L. Aggarwal, ed.), p. 123. Plenum, New York, 1970. llS J . Bares, Macromolecules 8, 244 (1975). 114 N . S. Surkova, G . T. Tkachenko, Ye. A. Sidorovich, G . M. Tolstopyatov, A. I . Marci, and Ye. V . Kuvshinskii, Polym. Sci. USSR (Engl. Transl.) 16, 1434 (1974). *I5 G . Kraus and K. W. Rollmann, J . Polym. Sci., Polym. Phys. Ed. 14, 1133 (1976). *lo M. Baer, J . Polym. Sci., Parr A 2, 417 (1964). P. Black and D. J. Worsfold, J . Appl. Polym. Sci. 18, 2307 (1974). )Is D. R. Hansen and M. Shen, Macromolecules 8, 903 (1975). 211

*Iz

16.5. GLASS TRANSiTlON TEMPERATURE MEASUREMENTS

3 13

fluctuations in the sample are sufficiently large to broaden the glass transition range. Blends of polystyrene and poly(a-methyl styrene) differ from their block polymers in that they exhibit two Tgs when their molecular This is in agreement with theory35 weights are higher than 105.186*21s*217 and is illustrated in Fig. 1. On the one hand, small-angle neutron scattering shows that low-molecular-weight polystyrene is mono-molecularly dispersed in poly(a-methyl styrene); on the other hand the single Tgregion of the same blends are broader than those of the parent h o m ~ p o l y m e r . ~ ~ ~ This suggests that Tg measurements can register composition gradients occurring in the volume pervaded by a single polymer chain with a 63 A radius of g ~ r a t i 0 n . l ~ ~ Blends of poly(styrene-6-a-methyl styrene) with their homopolymers ( S O / S O ) mostly exhibit a single Tg intermediate between that of the block and the added polymer.186*218At higher molecular weights and in ternary blends, the polystyrene Tg appears beside that of the block polymer, suggesting that poly(a-methyl styrene) is more compatible with the block polymer than is polystyrene.186 In segmented block copolymers, block lengths are usually small and have a wide molecular weight distribution. This increases the surface to volume ratio and the likelihood of polymer mixing. Segmented pol y( ether4-uret hane) and pol y(ester-&-urethane) exhibit the low-temperature Tg of the polyether or p o l y e ~ t e r . ~ This ~ ~ - Tg ~ ~in-~ creases with increasing urethane content of the sample, suggesting that some domain mixing occurs.221 Two novel endotherms occur in these polymers. The first between 50 and 80°C is assigned to the dissolution of low-order regions in which many ester-urethane interactions occur, and the second between 130 and 150°C where higher, semicrystalline order disappears. The latter is also found in dynamic-mechanical tests.220 The relative importance of these two endotherms depends on the chemical structure of the elastomer and the urethane, qn their relative abundance, and on the sample history.221 The first endotherm is strongest in poly(ester-6-urethanes) in agreement with SAXS, which also indicates more mixing and larger interphases than in poly(ether-b-~rethanes).'~~*~~~ Infrared measurements prove ester-urethane hydrogen bonding and allow an estimate to be made of the amount of m i ~ i n g . ~The ~ ~ melting n~~~ of the urethane crystallites occurs above 180°C. S. B . Clough and N . S. Schneider, J . Macromol. Sci., Purr B 2, 553 (l%8). S . L. Cooper and A. V . Tobolski, J . Appl. Polym. Sci. 10, 1837 (1965). 221 N.S. Schneider, C. S. Paik Sung, R. W. Matton, and J. L. Illinger, Macromolecules 8, 62 (1975). 122 R . W. Seymour, G . M . Ester, and S. L. Cooper, Macromolecules 3, 579 (1970). 223 C. S. Paik Sung and N . S. Schneider, Macromolecules 8, 68 (1975). 218

220

314

16. POLYMERIC

ALLOYS

The Tgof amorphous blocks seems little affected by the presence of a crystalline block unless mixing occurs. The polystyrene Tg in poly(styrene-b-ethylene oxide) agrees well with that of the homopolymer with the same molecular eight.'^^.'^'

16.6. Conclusion The study of the thermodynamic and morphological behavior of polymeric alloys is still very qualitative. Most progress has been made in the study of the simplest block polymers. The quantitative determination of polymer concentrations in two mixed-phase systems and improvements in the measurement of the interphase between two polymers promise to yield the most significant results. On the detailed knowledge of the morphology of polymer blends will be built an understanding of their many interesting practical properties.

17. PERMEATION, DIFFUSION, AND SORPTION OF GASES AND VAPORS By R. M. Felder and G. S. Huvard

17.1. Introduction The earliest studies of the transport of fluids in polymers consisted of measuring permeation rates of various gases in natural rubber membranes and correlating the measured rates with the partial pressure difference across the membrane, the membrane thickness, and the temperature. The objectives of later studies have been to define and model the sorption and transport processes that underlie permeation; to devise and implement experimental methods for determining model parameters: to seek correlations that allow the estimation of transport properties of unstudied penetrant-polymer pairs, or of studied pairs under previously unstudied conditions: to apply physical and thermodynamic theories of solution, diffusion, and stress relaxation in polymers to predict the dependence of transport and solubility coefficients on measurable system variables: to modify polymer compositions and structures in order to achieve or enhance desired permeability characteristics, and use measured transport rates and their dependences on experimental conditions to elucidate structures of polymers. The principal objectives of this part are to review the experimental methods used in the studies described above, and to outline methods of data analysis and currently accepted theories to an extent sufficient to make the material on experiments meaningful. More detailed expositions of the theory may be found in the comprehensive volume of Crank and Park,' and in more recent review^^-^ and We confine our discussion to the transport of gases and vapors in J. Crank and G . S . Park, eds.. "Diffusion in Polymers." Academic Press, New York, 1968.

* V . Stannett, H. B. Hopfenberg, and J . H. Petropoulos, in "Macromolecular Science" (C. E. H. Bawn, ed.). p. 329. Butteworth, London, 1972. H . B. Hopfenberg and V. Stannett, in "The Physics of Glassy Polymers" (R. N . Haward, ed.), p. 504. Appl. Sci. Publ., London, 1973. 315 METHODS OF EXPERIMENTAL PHYSICS, VOL. 16C

Copyri#ht @ 1980 by Academic Ress, Inc. All rights of reproduction in any form reserved. ISBN 0-12475958-0

316

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polymers. Detailed discussions of liquid-transport processes such as dialysis, pervaporation, ultrafiltration, reverse osmosis, and electrodialytic and electro-osmotic processes may be found in the text of Hwang and Kammermeyer' and in recent review^.^.^*^ Swelling and crazing phenomena are discussed in Part 15 of this volume.

17.2. Historical Perspective 17.2.1. Theory

The first recorded observation of the permeation of a gas through a membrane appears to be that of Thomas Graham,'O who in 1829 observed that a wet bladder became inflated to the bursting point when inserted in an atmosphere of carbon dioxide. Graham attributed this phenomenon to the CO, dissolving in the absorbed water, permeating through the "capillary canals" of the membrane, and evaporating into the bladder interi0r.t A systematic study of rates of permeation of gases in polymers appeared in 1831, when Dr. J. K. Mitchell, the inventor of the toy rubber balloon, discovered that his balloons collapsed at different rates when they were filled with different gases.', Mitchell then measured the relative rates of permeation of ten gases through india rubber (caoutchouc) membranes, finding among other things that carbon dioxide permeated at roughly six times the rate of hydrogen. He also observed that a solid

* H. B. Hopfenberg, ed., "Permeability of Plastic Films and Coatings to Gases, Vapors, and Liquids." Plenum, New York, 1974. S. Hwang, C. K.Choi, and K. Kammermeyer, Sep. Sci. 9,461 (1974). R. McGregor, "Diffusion and Sorption in Fibers and Films," Vol. I . Academic Press, 1974. I S . Hwang and K. Kammermeyer, eds., "Membranes in Separations." Wiley, New York, 1975. * S. Sourirajan, ed., "Reverse Osmosis and Synthetic Membranes." Nat. Res. Counc., Canada, Ottawa, 1977. N. N. Li, ed., "Recent Developments in Separation Science," Vols. 1 and 2. CRC Press, Cleveland, 1972. lo T. Graham, R. Insr. J. (1829). I I K. Sollner, in "Membranes in Separations" (S.Hwang and K. Kammermeyer, eds.). pp. vii-xvi. Wiley, New York, 1975. lg J. K. Mitchell, PhiladelphiaJ. Med. Sci. 13, p. 36; reprinted in R. Inst. J. 2, 101 and 307 ( I83 1). t This description is provided by Sollner,ll in an interesting set of notes on the history of membrane separation processes.

17.2.

HISTORICAL PERSPECTIVE

317

piece of rubber could absorb nearly its own volume of carbon dioxide when left long enough in pure CO,, and he correctly connected this result with the relatively high permeation rate observed for this gas. Although a number of reports of gas permeation through rubber appeared in the years following Mitchell’s work, the next major step in understanding the process came with the publication of a paper by Graham13in 1866, in which it was postulated that the permeation process entailed dissolution of the penetrant, followed by transmission of the dissolved species through the membrane as though through a liquid-a process Graham called “colloidal diffusion.” t In this extraordinary study, Graham laid the foundation for the bulk of modern membrane transport science and technology. Among other things, he (a) devised and tested an apparatus in which a gas permeated through a membrane into a vacuum, displacing a mercury column, (b) established that permeation rates are independent of whether the receiving chamber contains a vacuum or a gas other than the penetrant, (c) observed that temperature increases have two competing effects on permeation rates-decreasing the penetrant solubility, but also making the rubber more permeable (“softer, more liquid in nature”) to the dissolved penetrant, (d) noted that extended exposure at elevated temperatures alters the gas retention capacity of a rubber, (e) suggested that the differences in the permeabilities of rubber to different gases could be exploited to achieve a dialytic separation of gases, and (f) demonstrated the feasibility of this method by producing a gas containing roughly 41% oxygen and the balance nitrogen from atmospheric air, and showed that varying the thickness of a membrane used for this purpose alters the rate of permeation but not the selectivity. In 1855, Fick proposed his law of mass diffusion by analogy with Fourier’s law of heat conduction, and in the mid-1870s ExnerI4 and StefanI5 demonstrated that the rate at which a gas permeates through a soap film is proportional to the product of the solubility of the gas in water l3 I‘

T. Graham, Philos. M a g . [4] 32,401 (1866). Exner, Sitzungsber. Wien. Akad. 70, 465 (1875). Stefan, Sitzungsber. Wien. Akad. 77, 371 (1878).

t In the years between his first observation of gas permeation in 1829 and his formulation of the still-accepted solution-diffusion model for gas permeation, Graham invented the processes of pervaporation and liquid dialysis.

318

17.

GASES A N D

VAPORS

and the Fick's law diffusion coefficient of the dissolved species. Wroblewski'* in 1879 adapted these results to the permeation of gases in polymers, and thereby established the quantitative foundation for Graham's solution-diffusion model. He postulated that the sorption of a gas at the surface of a rubber exposed to a gas obeys Henry's law

c = sp,

(17.2.1)

where C is the dissolved species concentration in equilibrium with a gas whose partial pressure is p , and that the absorbed gas diffuses through the membrane in accordance with Fick's law, which states that the flux of penetrant through a membrane is proportional to the local concentration gradient J

=

DVC.

( 17.2.2)

The solubility and diffusion coefficients S and D were assumed independent of concentration. Wroblewski showed that if these assumptions are valid, the steady-state permeation rate per unit area through a membrane of thickness h is (17.2.3)

where Ap is the partial pressure difference across the membrane. The product P = DS is the permeability of the membrane to the penetrant. (The customary definition of P is J h l A p . ) Kayser" in 1891 provided further support for the solution-diffusion theory by demonstrating the validity of Henry's law for the absorption of carbon dioxide in rubber over a range of pressures from 125 to 1125 mm Hg. He also derived quadratic formulas to fit observed temperature dependences of hydrogen and carbon dioxide solubilities over the range 0-70°C. A series of studies performed between 1917 and 1920 by Shakespear'*

supported Graham's observation that the permeability of a gas is independent of the presence of other permeating gases. Shakespear also found that the temperature dependence of the gas permeability was independent of the partial pressure difference across the membrane. Edwards'O in 1918 found a highly nonlinear effect of temperature and a slight effect of total pressure on measured permeabilities of hydrogen in rubber. He also measured or estimated errors due to sorption of water in lT

S. von Wroblewski, W i d . Ann. 8, 29 (1879). H . Kayser, Wied. Ann. 43, 544 (1891). G. A. Shakespear. Adv. Commun. Aeronuut. Rep. T.1164 (1918). J. D. Edwards, U.S.Bur. Sfund., Tech. Pap. No. 113 (1918).

17.2.

HISTORICAL PERSPECTIVE

319

the membrane, evolution of volatile substances in the membrane, and effects of gas flow rate across the membrane surface. In the same year, Dewaf1° measured permeabilities of several gases.in rubber over a range of temperatures, and found that a plot of log P vs. T for a given gas consisted of two straight lines that intersected at 0°C. He attributed the discontinuity to the presence of water in the membranes. Dayneszl in 1920 developed the mathematical analysis of Fickian diffusion in a flat membrane, and used it to derive the familiar time lag method for determining the diffusivity and solubility from the transient response to a step change in penetrant partial pressure on one side of a membrane. Barrerz2in 1939 further developed the technique, and established that D, S, and P all exhibit Arrhenius law dependences on temperature. In 1944, Matthesz3observed a systematic deviation from Henry's law in the sorption of water by hydrated cellulose membranes. He postulated that two competitive phenomena were occurring: dissolution, which obeyed Henry's law, and adsorption, which followed a Langmuir isotherm, and he wrote the total isotherm as the sum of the isotherms corresponding to each phenomenon: C=*+kp.

(17.2.4)

A similar hypothesis forms the basis of the modern theory of the transport of gases in glassy polymers, as discussed in Chapter 17.3. Beginning in the 1940s with the work of King, Crank, and Park, and continuing in the 1950s and thereafter with the work of Stannett, Michaels, Barrer, and their colleagues, investigators have found and, to a large degree, explained many systematic deviations from the ideal case of Henry's law sorption and Fickian diffusion with concentrationindependent coefficients. The nature of these deviations and of the phenomena that lead to them are discussed in Chapter 17.3. 17.2.2. Experimental Methods

Unlike the theory of gas transport in polymers, experimental methods for permeation and sorption rate measurements have undergone relatively few innovations in the past five decades, and a survey of the early literature contains references to a surprising number of recently rediscovered techniques. An interesting review of methods and devices used before J. Dewar, R . Insr. Proc. 21, 813 (1918). H. A. Daynes, Proc. R. SOC., London, Scr. A 97, 286 (1920). zz R. M. Barrer, Trans. Faraday SOC. 35, 628 (1939). 99 A. Matthes, Kolloid-2. 108, 79 (1944).

zo *I

3 20

17. GASES AND VAPORS

1920 is provided by B a ~ rwhose , ~ ~ work is the source of much of the material that follows. MitchelP in 1831 determined permeation rates by measuring rates of shrinkage of balloons. Grahamls in 1866 stretched a rubber film over a tube partially filled with mercury open to the atmosphere at the bottom, and inserted the closed end into the desired gas atmosphere. The rate'of fall of the mercury in the tube provided a measure of the rate of permeation of gas through the film. Wroblewskils in 1879 performed both permeation and equilibrium sorption experiments on samples of vulcanized rubber. He used a gravimetric technique to determine solubilities of Hz, C02, and NzO, and then stretched the samples until their thicknesses were as low as hundredths of a millimeter to measure permeabilities. Lord Rayleighz5in 1900 allowed air to permeate through a rubber membrane, measured the percentage of oxygen in the permeate, and then liquefied the oxygen and measured the percentage of argon in the remaining gas. From these results and the known composition of atmospheric air, Rayleigh was able to calculate the relative permeabilities of oxygen, nitrogen, and argon in the rubber. DewarZ0in 1918 used a variation of Rayleigh's device to measure permeation rates of eight gases in a 0.01 mm thick stretched and supported rubber membrane. He allowed the gases to penetrate into an evacuated chamber, and determined the rate of permeation by monitoring the pressure rise in the receiving chamber. Four general approaches to permeation rate measurements were adopted prior to 1923, and are still used singly or in combination. 17.2.2.1. Volume Loss Method. A container closed entirely or in part by a membrane is filled with the gas penetrant, and the rate of decrease of the container volume at constant pressure is monitored. This technique was used for all studies prior to 1909, going back to the original work of MitchellI2in 1831. B a r P describes the Renard-Surcouf balance, a device in which a portion of the chamber seal was provided by a column of water that could be maintained at an adjustable but constant head to achieve constant pressure during the course of the run. Corrections of the raw data for back-diffusion of air and variations in temperature and barometric pressure were performed. 17.2.2.2 Continuous-Flow Method. A penetrant permeates through a membrane into a flowing gas stream. The effluent from the permeation O1 G. Barr, in "Dictionary of Applied Physics" (R. Glazebrook, ed.), Vol. V. Macmillan, New York, 1923. z5 Lord Rayleigh, Philos. M a g . [5] 49, 220 (1900).

17.2.

HISTORICAL PERSPECTIVE

32 1

cell is analyzed to determine the penetrant concentration; the product of the concentration and the stream flow rate equals the permeation rate. Rosenhain and Barr2sin 1909 designed a cell that worked on this principle. Hydrogen was purified by passage through absorption and drying cells, and passed through the chamber on one side of a rubber membrane. Air was also cleaned and dried, and passed through the chamber on the opposite side of the membrane. The air and any hydrogen that permeated through the membrane passed through sulfuric acid drying tubes, and entered an electrically heated silica tube in which the hydrogen was burned. The water formed was absorbed in columns packed with calcium chloride and pumice-sulfuric acid. The rate of accumulation of water was determined by periodic weighing, and the rate of permeation of hydrogen was calculated from the result. Modifications of this apparatus were effected by several investigators; of these modifications, the most important was the use of a refractometer (interferometer) in place of the combustion tube and water collectors to measure the quantity of hydrogen in the chamber effluent. This idea was proposed and implemented by Frenze2’ in 1914, and was improved by Edw a r d ~ in ’ ~ 1918. 17.2.2.3. Constant-Volume Methods. A gas permeates into a closed constant-volume chamber. If the chamber is initially evacuated, the pressure rise is monitored, while if the chamber contains air or another diluent gas, an analytical technique is used to measure the change in the penetrant concentration. In 1917, Shakespear2*designed a cell of this type that consisted of two hollow copper blocks with matched cavities, joined on opposite sides of a flat membrane supported by concentric rings. Shakespear and Daynes20 in 1920 incorporated a thermal conductivity detector called a “Katharometer” into this cell. Fine platinum wires, which formed the arms of a Wheatstone bridge, were inserted in two cavities-one containing pure air, and the other within the receiving chamber of the permeation cell. The change in thermal conductivity of the gas in the receiving chamber caused by the presence of the hydrogen penetrant was monitored by balancing the bridge galvanometer. indicates that this device was also used in flow cell measurements. 17.2.2.4. Gravimetric Method. A vapor permeates through a membrane into a vessel that contains an absorbing material (e.g., a desiccant for water vapor), and the weight gain of the vessel is recorded as a func2eRosenhain and Barr, A.C.A. Tech. Rep. (1909-1910). 27 Frenzel, Z . Flugtech. Mororluftschijfahrr, Bed, p. 264 (1914). l e G . A. Shakespear, A.C.A. Rep. Mem., Nos. 317 and 516 (1917). 28 G. A. Shakespear and H. A. Daynes, Proc. R . Soc. London 97, 273 (1920).

3 22

17.

GASES AND VAPORS

tion of time. Alternatively, a liquid is placed in a container, its vapor permeates (or the liquid itself pervaporates) through a membrane, and the weight loss of the container is recorded. The first of these techniques was used in 1920 by Edwards and Pickering30 in their determination of the rate of permeation of water vapor through rubber. They placed phosphorus pentoxide in a shallow dish, closed the top with a rubber membrane, exposed the membrane to saturated air or liquid water, and periodically weighed the container. Harvey3I in 1924 may have been the first to use the weight loss method, in his measurements of the water permeability of an asphalt-filled board. Wilson and Lines32 in 1925 used a gravimetric method to determine water vapor permeabilities and swelling properties of shoe leather. They concluded (among other things) that the sensitivity of those who predict weather changes by the degree of pain in their corns is increased by wearing shoes made of chrome-tanned leather. A variation of the gravimetric method was reported in 1936 by Wosnessensky and D u b i n k o ~who , ~ ~absorbed permeating water vapor in a desiccant suspended from a calibrated quartz spring, and measured the extension of the spring as a function of time. This appears to be the first published application to polymer transport measurements of the spring extension technique devised by McBain and Bakl-3' in 1924 for measurements of gas adsorption on charcoal. The principal alternative to permeation rate measurements for transport coefficient determinations is the direct measurement of the rate of uptake or loss of a penetrant by a polymer sample. The advantages of the sorption approach are that it eliminates the need for elaborate sealing and the extreme leak sensitivity associated with permeation methods, it reduces uncertainties in the sample surface area caused by membrane supports, and it is well-suited to lengthy experiments on materials in which diffusion coefficients are low. Direct gravimetric measurements of equilibrium solubilities have been commonplace since the pioneering work of Mitchell.I2 Although many early studies report equilibrium sorption measurements obtained by techniques that could easily have been used to measure rates of sorption (e.g., a 1926 paper by Hedges3% sorption rate data first appear in a 1940 paper by King and C a s ~ i ewho , ~ ~used the spring extension method to measure J . D. Edwards and S. F. Pickering, Nufl. Bur. Sfund. (U.S.), Sci. Pup. No. 387 (1920). A. R. Harvey, Pup. Trude J . 78, TS256 (1974). 32 J . A . Wilson and G . 0. Lines, Ind. Eng. Chem. 17, 570 (1925). S . Wossnessensky and L. M. Dubinkow, Kolloid-Z. 74, 183 (1936). 3 ( J . W. McBain and A. M. Bakr. J . Am. Chern. Soc. 48,690 (1926). s5 J . J . Hedges, Truns. Furuduy Soc. 22, 178 (1926). G . King and A. B. D. Cassie, Truns. Furuduy Soc. 36, 445 (1940). 30

17.2. HISTORICAL PERSPECTIVE

323

rates of absorption of water vapor by wool fibers. Thomas and Gent3' in 1945 derived the solution of the diffusion equation for the fractional uptake as a function of time, and used sorption data taken with a quartz spring to determine diffusion coefficients. BaughanS8in 1948 was the first to obtain sorption data by placing a sample in the chamber of an electromagnetic balance. In yet another approach to transport rate measurement, the dissolved concentration of a penetrant at a particular time may be measured as a function of position in the membrane. A curve-fitting technique may then be used in conjunction with known analytical solutions of the diffusion equation to estimate the diffusion coefficient. In some instances, the determination of the extent of penetration may be performed by visual inspection, as when the penetrant advances in a well-defined front and swells or crazes the sample. HermansSOin 1946 observed such a phenomenon when measuring the diffusion of water into cellulose model filaments, and HartleyJoin 1949 used the same principle to study the diffusion of organic penetrants in cellulose acetate. In a variation of this method, Newitt and Weale4' in 1948 sorbed gases in polystyrene at pressures up to 300 atm, warmed the samples at ztmospheric pressure, and observed the position of the bubbles that subsequently formed within the polymer. When penetration is more gradual, a technique may be used in which a polymer film is clamped between two transparent plates and immersed in the penetrant, which enters through the edge of the film. The concentration-distance curve may be followed by measuring the refractive indexJ2or the visible, UV, or x-ray absorbanceJ3of the film as a function of time and position. Between 1831 and the present, almost every conceivable combination of closed and open permeation and sorption chambers has been used. NewnsQ4cites measurements made before 1950 that involve permeation of vapors from (a) closed chambers containing gases, pure penetrant liquids, and saturated solutions of the penetrant, and (b) flowing streams of the pure penetrant or gas mixtures containing the penetrant, into (c) closed A. M. Thomas and W. L. Gent, Truns. Faruduy Soc. 57, 324 (1945). E. C. Baughan, Trans. Furaday Soc. 44, 495 (1948). 30 P. H. Hermans, "Contributions to the Physics of Cellulose Fibres," p. 23. Elsevier, Amsterdam, 1946. 40 G. S. Hartley, Truns. Furaduy Soc. 45, 820 (1949). 41 D. M . Newitt and K . E. Weale, J . Chem. Soc. p. 1541 (1948). C. Robinson, Proc. R . SOC. London. Ser. A 204, 339 (1950). F. Grun, Experienriu 3, 490 (1947). * A. C. Newns, Shirley Inst., Mem. 24, 27 (1950). ~3'

38

32 4

17.

GASES A N D VAPORS

evacuated chambers, (d) closed chambers containing absorbing or reactive materials, and (e) flowing streams. Particular methods have often been lost and then rediscovered, their origins forgotten. The isostatic method of permeation rate measurement (i.e., any method in which the total pressures on both sides of the membrane are kept approximately equal) has been reinvented any number of times, most recently in 1973, since Mitchell used it in 1831. Cells with built-in resistance thermometers for continuous thermal conductivity measurement have been invented in each of the past four decades; Shakespear may not live on in the memories of polymer physicists, but his 1920 Katharometer seems destined for perpetual reincarnation.

17.3 Phenomenology According to the simplest form of the classical solution-diffusion model, sorption follows Henry's law (Eq. (17.2. l)], diffusion follows Fick's law [Eq. (17.2.2)], and the steady-state permeation rate through a flat membrane is given by Eq. (17.2.3). The solubility and diffusion coefficients are independent of applied pressure and dissolved gas concentration, and depend on temperature according to the Arrhenius law: D = Do exp( -Ed/RT),

(17.3.1)

S = So exp( - AH,/RT),

(17.3.2)

P = Po exp( -E,/R7') = (DOSO) exp[-(Ed

(17.3.3)

+ M,)/RTl,

where Edis the activation energy for diffusion, AH, the heat of solution of the gas in the polymer, and E , = Ed + AH, is the apparent activation energy for permeation. Most research on the transport of gases in polymers carried out in the past two decades has been devoted to detecting and analyzing violations of this model. It appears now that the model provides a consistently accurate representation of the permeation of gases above their critical temperatures, at atmospheric and subatmospheric pressures, through nonswollen rubbery polymers that do not contain crystallites or fillers. For other systems and conditions, the model must be applied with considerable caution. Permeability coefficients of gases and vapors in polymers are usually reported with the units cm3(STP)/cm - sec cm Hg, as required for dimensional homogeneity when the flux is expressed in units of cm3(STP)/cm2 sec and the partial pressure gradient is in cm Hg/cm.

-

17.3.

PHENOMENOLOGY

325

Other units are occasionally used, however. Y a ~ u d ahas ~ ~ recently suggested adoption of the unit cm2/sec, a unit commonly encountered in liquid-phase studies when fluxes are expressed in moles/cm2 sec and the concentration gradient has the unit (moles/cms)/cm. A table of conversion factors for common permeability units is given by Bixler and S~eeting.~~ A number of tabulations of permeabilities, diffusivities, and solubilities can be found in the literature. Stannett4’ has compiled diffusivities and energies of activation for diffusion for simple gases in 32 polymers. A rather complete listing of transport parameters, which includes a separation of the materials cited into crystallinity classes, has been given by Bixler and S ~ e e t i n g .Rogers, ~~ Fels, and Li40 give permeability and diffusivity data and separation factors for several gases, organic vapors and water vapor. Felder, Spence, and FerrelP tabulate transport coefficients for sulfur dioxide in various polymers. Finally, the Polymer Handbook5’ is a good source of physical property data of almost every kind, including transport properties. 17.3.1. Correlation and Estimation of Transport and Solubility Coefficients 17.3.1.1. Free-Volume Theory. Diffusion of a dissolved gas in a polymer is viewed as a series of activated jumps from one vaguely defined cavity within the polymer matrix to another. Qualitatively, any agent that increases the number or size of cavities in a polymer or renders chain segments more mobile increases the rate of diffusion; such agents include plasticizers and penetrant molecules present in amounts sufficient to induce swelling. Structural entities such as crosslinks or crystallites decrease the size or number of cavities or immobilize chain segments, and thereby decrease the diffusion rate. Most analyses of the solution and diffusion of penetrants in polymers are based on one of several free-volume theories. The theories differ in their definitions of free volume, and in the assumptions made that permit

H. Yasuda, J . Appl. Polym. Sci. 19, 2529 (1975). H. J. Bixler and 0. J . Sweeting. Sci. Technol. Polym. Films 2, 7 (1971). “ V. Stannett, in “Diffusion in Polymers” (J. Crank and G. S. Park, eds.), p. 41. Academic Press, New York, 1968. ’* H. J. Bixler and 0. J. Sweeting, Sci. Technol. Polym. Films 2, 41 (1971). I°C. E. Rogers, M. Fels, and N . N. Li, Recenr Dev. Sep. Sci. 2, 107 (1972). R . M. Felder, R. D. Spence. and J . K . Ferrell, J . Chem. Eng. Dara 20, 235 (1975). s1 J . Brandrup and E. H. Immergut, eds., “Polymer Handbook,” 2nd ed. Wiley (Interscience), New York, 1975. Is

326

17.

GASES AND VAPORS

the calculation of transport parameters such as Do and Ed from various physical properties of the penetrant and polymer. A review of the elements of free-volume theory is given by Kumins and K ~ e i .Recent ~ ~ studies apply the theory to predict effects of pressure on gas permeabilities,= and to establish relationships between free volume and the glass transitionJ4 and tacticityS5of polymers. Other studies extend the free-volume model, incorporating elements of Flory Huggins solution theory and a structural theory of Bueche to obtain estimates of mutual diffusion coefficients in amorphous penetrant-polymer systems over a wide range of condition^.^^ The free-volume theory provides a good qualitative representation of observed variations in D with changes in temperature, penetrant concentration, and penetrant molecular weight, but it is cumbersome to apply in its most general form, and requires physical property data that are not generally available. Recent comparisons of measured and predicted diffusion coefficients for ethylbenzene in styrene indicate that the model provides estimates accurate only within a factor of two to five,% suggesting that the simpler empirical and semiempirical correlations to be described in the following sections still retain their usefulness. 17.3.1.2. Permeation Rates and Permeabilities. The permeabilities of two gases at a given temperature may vary by orders of magnitude from one polymer to another, but their ratio is often nearly invariant; similarly, the ratio of the permeabilities of a specific gas in two different polymers tends to be relatively constant from one gas to another. The ratios of activation energies for permeation and diffusion exhibit a similar constancy for a given pair of gases in different polymers, and for different gases in a given pair of polymers. These invariances can be used to estimate to within a factor of three unknown permeabilities and diffusivities of a given penetrant-polymer combination from data for other gases and polymers.47 A relationship of the form (17.3.4)

where a , 6 , and c are constants, may be used to correlate the permeation rate 4 of a vapor penetrant with the penetrant vapor pressure p o and pars C. A. Kumins and T. K. Kwei, in "Diffusion in Polymers" (J. Crank and G . S. Park, eds.), Academic Press, p. 107. 53 S . A. Stern, S. M. Fang, and H. L. Frisch, J . Po/ym. Sci.. Prrrr A-2 10, 201 (1972). K . D. Ziegel and F. R. Eirich, J . Polyrn. Sci.. Polym. Phys. Ed. 12, 16 (1974). ss W. R. Brown and G . S . Park, in "Permeability of Plastic Films and Coatings to Gases, Vapors, and Liquids" (H. B. Hopfenberg, ed.), p. 207. Plenum, New York, 1974. J. L. Vrentas and J . S . Duda, J . Polym. Sci.. Polyrn. Phys. Ed. 15, 403 & 417 (1977).

17.3.

PHENOMENOLOGY

327

tial pressure p . This correlation has been applied to the permeation through polyethylene of propane and pr~pylene.~' 17.3.1.3. Diffusion Coefficients. Measurements of diffusion coefficients of a homologous series of gases in a specific polymer have shown that both the pre-exponential factor Do and the activation energy Ed increase with increasing penetrant molecular diameter, with the net effect being a decrease in D.47 The activation energy generally varies as a power of the diameter: the power is close to 1 for rubbery polymers, close to 2 for polar and other relatively stiff polymers,47and in some instances. may be as high as 3.58 Improved correlations are obtained by applying empirical correction factors to the diameter for anisotropic penetrant molecule^^^ and inert gases.60 A plot of log Do vs. Ed for several gases permeating through a rubbery polymer often appears although the linearity may be deceptive, and should be verified by one of several alternative plotting methods." This correlation implies a linear relationship between the enthalpy and entropy of activation, a relationship that has been shown to have a theoretical basis.a-BB The values of Do and Ed frequently change with temperature in the regions of the glass transition temperature Tg and the polymer melting point T,. Correlations derived by Lundstrom and Bearmanso may be used to estimate the effects of changes in temperature on gas permeation rates in polymers with TBand T,,, known, or alternatively to estimate the values of Tg and T,,, from measured permeation rates. 17.3.1.4. Solubility Coefficients. Solubility coefficients of various gases in amorphous, rubbery polymers have been found to fall on straight lines in plots of log S vs. the boiling point, critical temperature, and Lennard-Jones force constant of the penetrant specie^.^^-^^ The law of corresponding states provides a good basis for correlating D. M. Lei150 and M. L. dos Santos, J . Polym. Sci., Part A-2 10, 1 (1972). ss R . Ash, R. M. Barrer. and D. G. Palmer, Polymer 11,421 (1970). A. S. Michaels and H. S. Bixler, J . Polym. Sci. 50,413 (1%3). O7

J. E. Lundstrom and R. J. Bearman. J . Polym. Sci.,Polym. Phys. Ed. 12,91(1974). R. M . Barrer, Trans. Faraday Soc. 38, 322 (1942). G. J. van Amerongen, Rubber Chem. Techno/. 37, 1065 (1964). R. McGregor, in "Permeability of Plastic Films and Coatings to Gases, Vapors, and Liquids" (H. B. Hopfenberg. ed.), p. 87. Plenum, New York, 1974. A. W. Lawson, J . Phys. Chem. Solids 3,250 (1957). 05 A. W. Laws0n.J. Chem. Phys. 32, 131 (1%0). R. W. Keyes, J . Chem. Phys. 29,467 (1958). G . J. van Amerongen, J . Polym. Sci. 5 , 307 (1950). an R. M. Barrer and G . Skirrow, J. Polym. Sci. 3, 564 (1948). A. S. Michaels and H. J. Bixler, J . Polym. Sci. 50, 393 (1950). M,

328

17.

GASES A N D VAPORS

the sorption behavior of both gases and vapors. A reduced temperature is defined as TIT,, and a reduced pressure as p J p , , where p s is the pressure above which the application of Henry’s law yields an error of 5% or greater in the calculated solubility of a penetrant, and T, and p c are the critical constants of the penetrant. The solubilities of a number of gases and vapors in polyethylene correlate well with the square of the reduced temperature, and the reduced pressure varies linearly with the reduced temperature. 70-7z 17.3.2. Effects of Polymer Composition and Morphology on Transport Rates

The composition and morphology of a polymer play a significant role in determining sorption and transport properties. The factors having the greatest influence fall into two categories: those relating to the chemical composition and structure of the polymer, such as degree of saturation, presence of side chains, and crosslinking, and those that involve heterogeneities in the polymer, such as orientation, crystallinity, and the presence of plasticizers and fillers. 17.3.2.1. Structure and Crosslinking. The diffusion coefficient of a given penetrant in some instances has been found to vary inversely with polymer density, but in many other cases the two properties are apparently unrelated. The value of D tends to decrease with an increasing extent of saturation of the polymer, and the presence of side chains, particularly polar chains, reduces segment mobility and hence decreases D.47 The effects of polymer crosslinking on gas transport properties have been studied by several investigator^.^^-^^ Diffusivities and permeabilities invariably decrease with an increasing degree of crosslinking owing to lowered chain segment mobility. Chemical crosslinking can alter the composition of a polymer, and can consequently increase or decrease the solubility. If crosslinking converts a rubbery polymer to a glassy polymer, S increases, and if the crosslinking decreases the degree of crystallinity the same effect results (albeit for different reasons). 17.3.2.2. Crystallinity. A number of polymers undergo partial crystallization upon being cooled from the melt. (See Parts 9 and 10 of this volume, Part B.) The volume fraction of the crystalline portion of the S. A. Stem, J. J. Mullhaupt, and P. J. Garris, AIChEJ 15, 64 (1969). S. A. Stem, S . M. Fang, and R. M. Jobbins, J . Mucromol. Sci., Phys. 5, 41 (1971). 7* L. I. Stiel and D. F. Hamish, AIChEJ. 22, 117 (1976). 7s R. M. Barrer, J . A. Barrie, and P. S. L. Wong, Polymer 9, 609 (1968). 74 R. Y. M. Huang and P. J. F. Kanitz, J . Mucromol. Sci.. Phys. 5, 71 (1971). 75 P. J . F. Kanitz and R. Y. M. Huang, J . Appl. Polym. Sci. 14, 2739 (1970). 70

71

17.3.

329

PHENOMENOLOGY

polymer is generally in the range 0.4-0.6, although it has been known to exceed 0.8.4s In an unstrained homogeneous polymer, the effective solubility coefficient of a penetrant varies linearly with the amorphous f r a ~ t i o n , and ~~-~~ the effective diffusion coefficient depends on the amorphous fraction in a manner that varies from one polymer to another.le The magnitudes of these variations increase sharply with increasing molecular size of the penetrant; the variations may be negligible for small molecules such as Hz and He, and pronounced for large penetrants such as dyes.‘# The effects are even more dramatic if asymmetry is present naturally or if it is induced by drawing or otherwise straining the polymer.B0 Orientation effects influence the observed transport; in addition, if plastic or elastic deformation takes place, the effective free volume changes, with concomitant changes occurring in the values of the effective solubility and diffusion coefficients. If the volume fraction of the amorphous component is a, then the quantity of penetrant absorbed per unit volume of dry sample is

w

= aw,,

(17.3.5)

where wa is the solubility in the amorphous polymer. The effective diffusion coefficient is D

=

JiDaIB,

( 17.3.6)

where Da is the diffusion coefficient in the amorphous polymer, Ji a “detour ratio” that accounts for the reduction in penetrant mobility caused by the tortuosity of the flow channels between the crystal lamellae, and B a “blocking factor” (also called an immobilization factor) that accounts for the physical crosslinking between lamellae. The inverse of Ji, called the “tortuosity factor,” is sometimes used in Eq. ( 17.3.6).7e The value of Ji has been found to be proportional to a power of the amorphous fraction in several investigations:

Ji

=

(1

- a)”.

(17.3.7)

The power n is close to 1 for helium, oxygen, argon, carbon dioxide, nitrogen, and methane in crystalline poly(ethy1ene terephthalate),sl and for water vapor in unoriented samples of the same material and in nylon 6-10 A . W. Myers, C. E. Rogers, V. Stannett, and M. Szwarc, Tuppi 41, 716 (1958). C . H . Klute, J . Appl. P d y m . Sci. 1, 340 (1959). 78 A. Michaels and R. Parker, J . Pulyrn. Sci. 41, 54 (1959). 70 A. Peterlin, J . Mucrumol. Sci., f h y s . 11, 57 (1975). H. Yasuda and A. Peterlin, J . Appl. Polym. Sci. 18, 531 (1974). A . Michaels, W. Vieth, and J. Barrie, J . Appl. Phys. 34, 1 (1963). 77

330

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GASES A N D VAPORS

films.82 By contrast, n has been found to vary from 1.25 to 1.88 for polyethylene prepared in a variety of ways.SB Expressions for JI in an unoriented polymer may be obtained by analogy with expressions for the electrical conductivity of a semicrystalline c o n d ~ c t o r . ~ ~ The value of the blocking factor B depends on the size of the penetrant molecule: it is approximately I for helium, and has been found to be as high as 11.5 for SFBin a 77% crystalline polyethylene. The values of the factors t+b and B and their variations with the amorphous volume fraction depend strongly on the micromorphology of the polymer, and therefore provide a sensitive means of evaluating morphological models derived from microscopic, x-ray, or mechanical investigation^.^^ 17.3.2.3. Plasticizers and Fillers. The introduction of a plasticizer into a polymer-either during fabrication or subsequently by permeation-increases the mobility of chain segments and consequently increases the effective diffusion coefficient, primarily by lowering the activation energy for diff~sion.~' If a plasticizer converts a polymer from a glassy to a rubbery state, the resulting elimination of microvoids should as a rule decrease the solubility of gases in the polymer.46 On the other hand, a plasticizer that is a solvent for a specific gas may increase the solubility and hence the permeability of this gas, thereby providing a means for selective separation. For example, the addition of 3-methyl sulfolene to poly(viny1idene fluoride) films increases the film permeability to hydrocarbons,Mand the permeability of the same polymer to sulfur dioxide is increased by plasticizing films with sulfolane (tetrahydrothiophene 1,1-dioxide), sulfolene (2,5-dihydrothiophene-1,1-dioxide),and N,N,N' ,N'-tetraphenyl-p-phenyldiamine."s5 Solid fillers are generally impermeable to gases and thus reduce effective solubilities and diffusivities of penetrants. However, adsorptive fillers can increase gas solubilities, as was observed for carbon black in rubbeP and for molecular sieve spherules in silicone rubber.87 A good discussion of the analysis of transport in filled polymers is given by Barrer 17.3.2.4. Copolymers and Laminates. Depending on its chemical S. W. Lasoski and W. H. Cobbs, J . Polym. Sci. 36, 21 (1959). as F. P. McCandless, Ind. Eng. Chem., Process Des. Dev. 12, 354 (1973).

D. R. Seibel and F. P. McCandless, Ind. Eng. Chem., Process Des. Dev. 13,76 (1974). L. C. Treece, M.S. Dissertation, North Carolina State University, Raleigh (1975). BB G. J. van Amerongen, Rubber Chem. Technol. 28, 821 (1955). D. R. Paul and D. R. Kemp, J . Polym. Sci.. Polym. Symp. 41, 79 (1973). 8(1 R. M. Barrer, in "Diffusion in Polymers" (J. Crank and G. S. Park, eds.), Academic Press, New York, 1968. p. 165.

17.3.

PHENOMENOLOGY

33 1

composition and how it was fabricated, a membrane may consist of a single homogeneous or microheterogeneous phase, a series of laminae, a discrete phase dispersed in a continuous phase, or a complex of interwoven regions of different compositions. (See Part 16 of this volume.) For homogeneous copolymers, the transport properties may often be obtained by averaging those of the homopolymer constituents, and laminates may be treated in a straightforward manner as resistances to mass transfer in series.88 The behavior of two-phase copolymers is more complex in that the copolymer properties may or may not exhibit predictable variations with composition changes. For example, in the sorption of n-hexane in compatible blends of poly(pheny1ene oxide) and polystyrene, the craze front velocity varies smoothly with the weight fraction of PPO at low penetrant activities and temperatures, but decreases to a value lower than that for either polymer at high activities and temperature^.^^^^^ Anionic copolymers of nylon 6 and polyethylene are two-phase systems whose permeability characteristics are by and large those of nylon, but cationic copolymers exhibit partial miscibility and averaging of transport properties.O1 Graft copolymers can be prepared specifically for the purpose of achieving permselectivity to a specific component of a gas An extensive review of permeation in heterogeneous polymers has recently been published by Hopfenberg and P a d w

17.3.3.Transport of Water Vapor The high cohesive energy and hydrogen-bonding capacity of water give rise to several phenomena that, when they occur, cloud the interpretation of sorption and permeation data. The solubility of water in polymers under some circumstances follows Henry's law, and under others follows Flory-Huggins theory or exhibits features of multilayer (BET) adsorption. The diffusion coefficient of water in hydrophilic polymers generally increases with increasing concenC. H. M. Jacques, H. B. Hopfenberg, and V. Stannett, Polym. Eng. Sci. 13,81 (1973). C. H. M. Jacques and H. B. Hopfenberg, Polym. Eng. Sci. 14,449 (1974). M. Matzner, D. L. Schober, R. N. Johnson, L. M. Robeson, and J. E. McGrath. in "Permeability of Plastic Films and Coatings to Gases, Vapors, and Liquids" (H. B. Hopfenberg, ed.), p. 125. Plenum, New York, 1974. C. E. Rogers, S. Yamada, and M. I. Ostler. in "Permability of Plastic Films and Coatings to Gases, Vapors, and Liquids'' (H. B. Hopfenberg. ed.), p. 155. Plenum, New York, 1974. 93 C. E. Rogers and S. Sternberg, J . Macrornol. Sci., Phys. 5 , 189 (1971). H. B. Hopfenberg and D. R. Paul, in "Polymer Blends" (D. R. Paul and S . Newman, eds.), Chapter 10. Academic Press, New York, 1978.

332

17.

GASES AND VAPORS

tration, while in hydrophobic polymers D may vary inversely with C, behavior generally attributed to the clustering of dissolved water molecules in the polymer. This effect manifests itself in a difference between solubilities determined from permeation rate measurements, which are influenced by changes in D, and equilibrium sorption measurements, which are n ~ t . ~ ~ ~ ~ ~ ~ Other difficulties associated with water include its tendency to bind to the polymer, and to adsorb on the walls of the receiving volume; its low vapor pressure (and hence low driving force for permeation) at ambient temperatures; and its high heat of condensation, which makes isothermality difficult to achieve in measurements of transport in hydrophilic polymers. Yasuda and Stannetto5present a detailed discussion of these and other difficulties in water permeation and sorption measurements, and offer various means of overcoming them. Their principal suggestions are summarized in Chapter 17.9. The degree to which the presence of water affects permeation rates of other gases in a polymer depends on the solubility of water in the polymer: an increase in humidity usually increases gas permeabilities in highly sorbing membranes, and leaves them unchanged in nonsorbing membranes. Examples of this effect have been cited for the permeation of sulfur dioxideJo and carbon dioxidee7 in several polymers. 17.3.4. Concentration-Dependent Fickian Diffusion in Rubbery Polymers

At temperatures well above the glass transition, the diffusion of organic vapors generally follows Fick’s law; however, the presence of the penetrant weakens the molecular interaction between adjoining chains, augmenting the magnitude of micro-Brownian motion within the polymer and thereby increasing the rate of penetrant diffusion. This effect increases with the amount of penetrant present, so that the diffusion coefficient of organic vapors in rubbers generally exhibits a concentration dependence. Even if an appreciable amount of swelling takes place, the mobility of the chains is sufficiently great for D to reach its limiting value at a given concentration almost i n s t a n t a n e o ~ s l y . ~ ~ As long as the sorbed penetrant does not exceed roughly 10 wt% of the H. Yasuda and V. Stannett, J . Macrornol. Sci. Phys. 3, 589 (1%9). J . A. Barrie, A. Nimis, and A. Sheer, in “Permeability of Plastic Films and Coatings to Gases, Vapors, and Liquids” (H. B. Hopfenberg, ed.), p. 167. Plenum, New York, 1974. Y. Ito, Chem. High Polym. ( J p n . ) 18, IS8 (I%]). H.Fijita, in “Diffusion in Polymers” (J. Crank and G . S. Park, eds.), p. 75. Academic OB

Press, New York, 1968.

17.3.

PHENOMENOLOGY

333

polymer, the diffusion coefficient typically varies exponentially with concentration, D(C) = D(0) exp(AC),

(17.3.8)

where A is a “plasticizing parameter” that can be related to the HoryHuggins interaction parameter.4e~eeFor higher sorbed concentrations, replacement of the concentration by the penetrant activity yields better correlations.loo The most obvious manifestation of a concentration-dependent diffusion coefficient is a dependence of the permeability on applied pressure, although a pressure-dependent solubility coefficient may also be responsible for this effect. Another manifestation is a hysteresis effect in a conjugate sorption-desorption measurement. Details of the data analysis required to determine D(C) from permeation and sorption rate measurements are given in Chapter 17.4. 17.3.5. Dual-Mode Sorption and Diffusion in Glassy Polymers

Concentration vs. pressure sorption isotherms for gases in glassy polymers are frequently concave to the pressure axis, approaching linearity at high pressures. This behavior has been modeled by Barrer er a/.loland Michaels and Vieth et a/.81J02-104 in terms of a dual sorption mechanism. The polymer is presumed to consist of a continuous amorphous matrix, and microvoids (“holes”) that may be discrete or continuous. Sorption in the amorphous region is assumed to follow Henry’s law, and that in the microvoids obeys a Langmuir isotherm, so that the total isotherm is of the form of Eq. (17.2.4). A transport model has been formulated based on the dual mode sorption model, according to which the species absorbed in the holes are in equilibrium with the species in the matrix, and are immobilized-that is, they do not contribute to the diffusive flux in the polymer. This model can in principle be used to predict permeation rates in any heterogeneous polymer containing an absorbing or adsorbing component in an amorphous matrix. It has been applied to diffusion in crystalline D. Machin and C. E. Rogers, Crir. Rev. Macromol. Sci. 1, 245 (1960). C. E. Rogers, V. Stannett, and M. Szwarc, J . Polym. Sci. 45, 61 (1960). lol R. M. Barrer, J. A . Barrie, and J. Slater, J . Polym. Sci. 27, 177 (1958). I m W. R . Vieth, Sc.D. Thesis, Massachusetts Institute of Technology, Cambridge (I%]). lO9 W. R. Vieth and K . Sladek, J . Colloid Sci. 20, 1014 (1965). IM W. R. Vieth, J . M. Howell, and J. H. Hsieh, J . Membr. Sci. 5 , 177 (1976). D. R. Kernp and D. R. Paul, J . Polym. Sci. 12, 485 (1974). loo

334

17.

GASES AND VAPORS

polymers,103catalysts and adsorptive filler^,*^*^^^^^^ textile fibers,IOBand enzyme mernbranes.Io7 According to the dual-mode transport model, at low penetrant pressures the apparent diffusivity (that which would be measured in a time lag experiment) is less than the true diffusivity in the amorphous polymer, since some of the penetrant molecules are absorbed in the dispersed phase. At higher pressures, the holes become saturated rapidly, and measured transport involves only the matrix; the measured solubility and diffusion coefficients are then those of the amorphous region only. Recent studies show that improved data correlations are obtained by using a partial immobilization model, assigning diffusion coefficients to both absorbed components.108.10Q This point is discussed in greater detail in Section 17.7.2.3. 17.3.6. Anomalous Transport of Vapors in Glassy Polymers

Non-Fickian diffusion is frequently observed for the permeation of organic vapors in polymers below the glass transition, and sometimes above but within roughly 10°C of Tg. The values of D calculated by different techniques-from transient absorption and desorption measurements, steady-state permeabilities, and time lags, for example-and for films of different thicknesses, may differ by more than an order of magnitudelIO;moreover, the rearrangement of polymer molecules in the presence of the penetrant may proceed relatively slowly, so that the apparent diffusion coefficient is a function of both penetrant concentration and time. A frequently reprinted graphical summary of observed transport phenomena derived by Hopfenberg and Frisch112is shown in Fig. 1 as a plot of temperature vs. penetrant activity. For activities from 0 (infinitely dilute vapor) to 1 (pure liquid or saturated vapor) at temperatures well below TB, and for all temperatures at activities close to 0, concentration-independent Fickian diffusion is generally observed. At higher temperatures and activities, the diffusion coefficient exhibits a J. Komiyama and T. lijima. J . Polym. Sci.. f u r t A-2 12, 1465 (1974). W. R . Vieth, S. S.Wang. and S. G . Gilbert, Eiotechnol. Eioeng. Symp. 3, 285 (1972). D. R . Paul and W. J. Koros, J . Polym. Sci.. Polym. Phys. Ed. 14, 675 (1976). W. J. Koros. D. R. Paul, and A. A. Rocha, J . Polym. Sci.. Pol.vm. Phys. Ed. 14,687 (1976). G . S. Park, in Diffusion in Polymers" ( J . Crank and G. S. Park, eds.), p. 141. Academic Press, New York, 1968. 11' J . Crank, J . folvm. Sci. 11, 151 (1953). 112 H. B. Hopfenberg and H. L. Frisch, J . Polym. Sci., Purt E 7,405 (1%9). loo

lo'

17.3. PHENOMENOLOGY

335

CONCENTRATION DEPENDENT DIFFUSION TI

w

a 3 I-

a a

w

Q

z I-

0

1.0 PENETRANT ACTIVITY

FIG.I . Hopfenberg-Frisch chart of anomalous transport phenomena. Reprinted from Hopfenberg and Frisch1Izby permission of John Wiley & Sons, Inc.

dependence on concentration. Still closer to T,, D begins to depend on time explicitly as well as on concentration. The effective value of T, itself decreases with increasing penetrant activity. At moderate penetrant activities when swelling is appreciable and the temperature is less than but within about 10°C of T,, the mechanism of penetration may change from Fickian diffusion to a stress relaxation-controlled process in which the penetrant advances in a sharply defined front at a nearly uniform velocity. This mechanism has been designated case I1 transport.ll* At sufficiently high penetrant activities, stress cracking and solvent crazing may occur. The term “anomalous diffusion” is used to designate transport under conditions where a combination of case I (Fickian) and case I1 diffusion takes place. The two given modes of transport are easily distinguished using the results of sorption measurements. If Mt denotes the cumulative mass ab-

336

17. GASES

A N D VAPORS

sorbed in a sorption run, then in both modes at small times t , Mt

=

kt”.

(17.3.9)

If case I transport occurs, n = 3, while for case I1 transport n = l.l13 A mode of diffusion designated “super-case I1 transport” has been observed in the sorption of n-hexane vapor by a 1.5 mil polystyrene film.l14 Its distinguishing characteristic is a sorption curve convex to the time axis at large times on a plot of penetrant uptake vs. time, where a similar plot would be linear for case I1 transport and concave for case I transport. The phenomenon is attributed to the interaction of the Fickian tails that precede the case I1 fronts advancing toward the film midplane from both membrane boundaries. A quantitative basis for the correlations of Hopfenberg and Frisch112 has been derived by Vrentas ef al.,115*116 who define a dimensionless group called a diffusional Deborah number: characteristic stress relaxation time of the polymer-penetrant system (Deb)D = characteristic time for diffusion =

(17.3.10)

7m/(xz/D*),

where T~ is the mean relaxation time of the polymer-penetrant system at the conditions of interest, D* the molar average of the self-diffusion coefficients of the polymer and penetrant, and x a characteristic dimension of the polymer (e.g., the thickness of a membrane or the diameter of a sphere). Fickian diffusion occurs for (Deb), << 1 and (Deb), >> 1, while anomalous behavior occurs when (Deb), is on the order of 1. An interesting implication of this correlation is that for a given temperature -activity combination, the occurrence and nature of anomalous behavior- may depend on the size of the polymer sample. Experimental evidence of this effect is provided by Enscore,l17 who observed case I behavior for the diffusion of n-hexane in submicron polystyrene microspheres, and case I1 behavior in polystyrene films of 100 pm thickness and greater. Existing mathematical models for anomalous transport that include the T. Alfrey, E. F. Gurnee, and W. 0. Lloyd, J . Polym. Sci., furt C 12, 249 (1966). C. H. M. Jacques, H. B. Hopfenberg, and V. Stannett, in “Permeability of Plastic Films and Coatings to Gases, Vapors, and Liquids” (H. B. Hopfenberg, ed.), p. 73. Plenum, New York, 1974. l I 5 J. S. Vrentas, C. M. Jarzebski, and J . L. Duda, AIChE J . 21, 894 (1975). 11(1 J. S. Vrentas and J . L. Duda, J . Polym. Sci., Polym. Phys. Ed. 15, 441 (1977). D. J. Enscore, Ph.D. Dissertation, North Carolina State University, Raleigh (1977). 113

17.3. PHENOMENOLOGY

337

effects of Fickian- diffusion and stress relaxation are outlined and discussed by Petropoulos and R o u s s ~ s . ~ ~ ~ ~ ~ 17.3.7. Two-Stage Sorption of Swelling Penetrants in Glassy Polymers

Experiments in which swelling penetrants are sorbed by glassy polymers are frequently characterized by a rapid approach to an apparent equilibrium state, followed by a gradual shift to the true equilibrium state. This phenomenon has been attributed to a gradual relaxation of the elastic cohesive force in the polymer,11eand to a time-varying surface concentration of penetrant.lz0 Berens and Hopfenberglzl have been partially successful in fitting two-stage sorption data by assuming Fickian diffusion with a constant boundary condition for the first stage, and a first-order relaxation process for the second stage. Sorption data interpretation is complicated by the fact that the equilibrium solubility of a swelling penetrant in a polymer is a function of the history of the polymer-in particular, the extent to which the polymer may previously have been swelled. Enscore et a/.l z 2 sorbed n-hexane vapor at an activity of 0.1 at 30°C into polystyrene spheres as received, and also into spheres that had previously been exposed to vapor at an activity of 0.9. Sorption into the as-received sample was a two-stage process, with the second stage being exceptionally slow; in contrast, sorption into the preswollen sample was rapid and Fickian. Moreover, the apparent equilibrium sorption in the preswelled sample was significantly higher than that in the as-received sample, suggesting that the free volume increase caused by the preswelling maintained itself upon rapid desorption. However, the equilibrium sorption in the preswelled sample at a fixed temperature was observed to decrease if the sample were held under vacuum for an extended period of time prior to the sorption run, with the extent of the decrease being a function only of the time under vacuum. The picture that emerges from these observations is as follows. In general, rates of transport of vapors in glasses are governed by the simultaneous occurrence of Fickian diffusion and stress relaxation following sorption. The effect of the relaxation is to open the channels through 118 J. H. Petropoulos and P. P. Roussis, in “Permeability of Plastic Films and Coatings to Gases, Vapors, and Liquids” (H. B. Hopfenberg, ed.), p. 219. Plenum, New York, 1974. lie E. Bagley and F. A . Long, J . Am. Chem. Soc. 77, 2172 (1955). 120 F. A . Long and D. Richman, J . Am. Chem. Soc. 82, 509 (1960). lZ1 A. R. Berens and H. B. Hopfenberg, Polymer 19,489 (1978). 122 D. J . Enscore, H. B. Hopfenberg, and V. T. Stannett, Polymer 18, 793 (1977).

338

17.

GASES A N D VAPORS

which diffusion takes place, whatever these channels may be, and hence to increase the diffusion rate. At very low penetrant activities, for which swelling stresses are negligible and relaxation is accordingly immeasurable, diffusion is Fickian and the diffusion coefficient is independent of concentration. As the activity increases, relaxation begins to exert an influence. The relaxations that occur are small, and take place almost instantaneously relative to the rate of diffusion. Fick’s law is still adequate to describe the transport, but the relaxation causes the effective diffusion coefficient to increase in value. The result is concentration-dependent Fickian diffusion. At higher activities, the rates of diffusion and stress relaxation become comparable, and the measured penetration rate is a complex function of time and activity. This is the so-called anomalous diffusion regime. At sufficiently high penetrant activities, the intrinsic diffusion rate far exceeds the relaxation rate, and relaxation accordingly becomes the rate-determining step for the transport process. Relaxation-controlled transport manifests itself in two ways, depending on the size of the polymer sample. In a relatively large sample, the stress builds up to the critical level close to the boundary, and the relaxation occurs in a moving front that travels at a uniform velocity. This is case I1 transport. On the other hand, if the sample is small enough, the rate of diffusion may be rapid enough to saturate the sample in its unswelled condition. The relaxation then occurs uniformly throughout the sample, and two-stage sorption is the observed result. The picture is confused by the dependence of the apparent equilibrium solubility on the history of the polymer sample. At a fixed temperature and penetrant activity, there exists a unique equilibrium solubility and concomitant degree of swelling. In a glassy polymer as normally received, the relaxation time may be extremely long, leading to an underestimation of the equilibrium solubility. On the other hand, if the polymer is preswelled beyond its true equilibrium state for the given temperature and activity, a residual swelling remains, which slowly decays under vacuum. Penetration in this case proceeds by a Fickian mechanism, and the final mass sorbed is higher than that which would correspond to the true equilibrium solubility.

17.4. Categories of Experimental Methods Most experimental techniques for studying the diffusion of gases in polymers involve one of three modes of transport: (1) sorption into or out of a polymer, (2) permeation through a membrane into a closed chamber, and (3) permeation through a membrane into a flowing stream. In principle, solubility and diffusion coefficients for a given

17.4. CATEGORIES OF EXPERIMENTAL METHODS

339

penetrant-polymer pair may be determined from either sorption or permeation rate data. The paragraphs that follow contain general guidelines for selection of a method; if at all possible, however, both sorption and permeation techniques should be used, as inconsistencies between results obtained using these two approaches often reveal the occurrence of anomalous transport phenomena. A. Sorption. Recommended for 1. Measurements of equilibrium solubilities. Sorption methods yield solubilities directly from equilibrium data, while permeation methods require that the solubilities be calculated indirectly from transient data, with an increased probability of experimental error. 2. Two-stage sorption processes (Section 17.3.7). Only a direct sorption measurement is sensitive enough to determine quantitatively the minute changes in equilibrium solubility that take place as a consequence of gradual swelling or stress relaxation in a polymer. 3. Slow processes. Microspheres may be used that yield equilibrium times on the order of minutes, where permeation methods involving thin membranes might require weeks or months to achieve equilibrium. 4. Studies of anomalous diffusion (Section 17.3.6). If non-Fickian transport occurs, the anomalies are more likely to be masked in a permeation experiment than in a sorption measurement. 5 . High-pressure measurements. Permeation measurements are characterized by an extreme sensitivity to membrane rupture, leaks, and changes in surface area due to membrane distention, particularly at high pressures. These problems are minimal for sorption measurements. 6. Studies of solvent cracking and crazing, and direct observation of the position of a penetrant in a polymer. Sorption methods are not recommended for measurement of steadystate permeabilities, or for studies of exceptionally rapid processes. B. Permeation into a closed chamber. Recommended for 1. Measurements of low permeation rates. A cumulative quantity is measured, so that long-time experiments are feasible. (If the rates are too low, however, a sorption technique using microspheres or microfibers might be preferable.) 2. Penetrants for which continuous chemical analysis is impractical, or for which calibration standards with known penetrant concentrations are difficult to obtain.

17. GASES A N D VAPORS

340 3.

Measurement of transport rates of easily condensed vapors. A weighing-cup technique often provides the easiest means of determining the permeability of a vapor at or near its dew point.

C. Permeation into a flowing stream.

Recommended for

1. Measurement of moderate to high permeation rates. The penetrant can be diluted to any desired extent to bring its concentration within the range of an available analyzer. 2. Membranes susceptible to tearing or distention. Equal pressures can be maintained on both sides of the membrane, minimizing sealing and support requirements. 3. Penetrant mixtures. Closed-volume techniques cannot readily be adapted to gas mixture analysis, whereas the efftuent from a continuous-flow permeation cell may easily be analyzed by chromatography or mass spectroscopy.

17.5. Pressure Measurement and Temperature Control All of the permeation and sorption methods to be described have in common the need for precise pressure measurement and accurate temperature control. The pressure-measuring devices most often used are manometers, McLeod gauges, and electromechanical pressure transducers. Manometers are used primarily for measuring system pressures when charging penetrant reservoirs; they are less useful for transient measurements, since movement of the manometer fluid creates a change in the system volume. Table I lists several commonly used gauges and transducers. Lapelle,lZ3Giles,'" and Diels and J a e ~ k e ldiscuss ' ~ ~ the operating principles and working ranges of the vacuum gauges listed. A comprehensive listing of electromechanical pressure gauges is provided by the Instrument Society of America in the ISA Transducer Compendiurnl2"; the listing includes information on operating principles, pressure ranges, and manufacturers' specifications regarding accuracy and linearity. The McLeod gauge has been used both as a reference standard for calibrating electromechanical tra!isducers and for working measurements in R. R. LaPelle, "Practical Vacuum Systems." McGraw-Hill, New York, 1972. A. F. Giles, "Electronic Sensing Devices." William Clowes and Sons, Ltd., London, 1966.

K . Diels and R. Jaeckel, "Leybold Vacuum Handbook." Permagon, Oxford, 1966. G. F. Harvey, ed., "ISA Transducer Compendium," 2nd ed., Part 1 . Plenum, New York. 1969. 126

17.5.

PRESSURE MEASUREMENT A N D TEMPERATURE CONTROL

341

TABLEI . Common Pressure Gauges Gauge type

Range (tom)

1. Inverted magnetron 2. Bayer -Alpert 3. Penning 4. McLeod 5 . Diaphragrn/capacitance 6. Diaphragm/strain gauge 7. Bourdon

the range Torr. A vacuum “watchdog” is usually incorporated into systems in which measurements are made by gravimetric means. A McLeod gauge is suitable for this purpose; the authors have also had good results with a Penning transducer. Diaphragm/capacitance transducers give excellent service for measurements on the downstream side of closed volume permeation systems. High pressures (2-30 atm) in both barometric sorption and permeation systems can be measured with accurate Bourdon gauges. Some investigators have used diaphragm strain gauge transducers for high-pressure measurements.127 Transducers offer marginally greater accuracies than good Bourdon gauges, at a considerably higher cost; their principal advantage is that they may be linked to recorders or minicomputers for subsequent analysis of transient responses. Temperature control may be achieved by wrapping system components with thermostated heating tape and insulation, or by submerging sorption and permeation cells in constant-temperature baths. Glass sorption chambers used for gravimetric measurements are usually provided with jackets through which bath fluids may be circulated. Convective currents may arise in sorption cells due to uneven distributions of heating or cooling sources; high-sensitivity electrobalances are especially vulnerable to this phenomenon. A method of overcoming the problem is to enclose the entire measurement system in a thermally insulated environment equipped with heating or cooling devices and small circulating fans to eliminate local temperature gradients. Simple control systems demand the least maintenance. The authors use a 300-Wlight bulb in series with an accurate controller to regulate this temperature in a 1.2 m3 sorption chamber to well within 0.1”C. A small fan provides sufficient circulation, and temperatures in the range 25 -65°C can be attained. The only required maintenance has been replacement of the bulb every 6 to 8 weeks. I*’

W. J . Koros and D. R. Paul, J . Polym. Sci. 14, 1903 (1976).

342

17.

GASES AND VAPORS

17.6. Sorption Methods 17.6.1. Experiments and Data

Sorption experiments fall into four categories: a. Integral sorption: A sample is abruptly exposed to an atmosphere containing the penetrant. If sorption rates are to be measured, the external penetrant pressure is usually maintained constant throughout the run; if only equilibrium solubilities are desired, only the initial and final pressures need be known. b. Integral desorption: A sample that has been equilibrated at some penetrant activity is abruptly exposed to a penetrant-free atmosphere (normally a vacuum), until an immeasurable amount of penetrant remains in the sample. c. Interval sorption; d. Interval desorption: As above, except that the initial and final penetrant concentrations are both greater than zero. In all cases, the mass of penetrant in the polymer is measured as a function of time, either directly by a gravimetric technique or indirectly by measuring a change in pressure or volume of the atmosphere surrounding the polymer. The data might appear as shown in one of the curves of Fig. 2.. The variable plotted on the ordinate of this figure, M,,is the mass of penetrant sorbed or desorbed from the beginning of the run to the time shown on the abscissa, and M , is the value of MI as t ---* m. In an integral measurement, M, is the final (sorption) or initial (desorption) mass of penetrant in the polymer, while in an interval measurement it is the difference between the initial and final penetrant masses. Figure 2 shows data that might be obtained in a sorption run followed by a desorption run to the original penetrant level. The sorption and desorption curves coincide for Fickian diffusion with D constant (Fig. 2a), and initially vary linearly with t1/2. If D increases with the penetrant concentration C , as it does for the diffusion of organic vapors in rubbers and at low activities in glasses, the sorption curve lies above the desorption curve, as in Fig. 2b. If D varies inversely with C , as, for instance, in the diffusion of water in hydrophobic polymers, the positions of the curves are reversed, as in fig. 2c. If swelling is significant and stress relaxation controls the penetration rate, as in case I1 or anomalous transport (Section 17.3.6), the sorption curve is sigmoidal, but desorption from the swollen polymer is Fickian and initially relatively rapid (Fig. 2d). If relaxation occurs slowly, twostage sorption occurs (Section 17.3.7), and the cycle appears as shown in

17.6.

SORPTION METHODS

TIME

343

’h

FIG.2. Sorption cycles for several transport mechanisms. (a) Fickian diffusion, constant D: (b) Fickian diffusion, D increases with increasing C; (c) Fickian diffusion, D decreases with decreasing C; (d) anomalous or case I1 transport; (e) two-stage sorption.

Fig. 2e. (As a rule, this phenomenon is only observed in interval sorption experiments.) The sections that follow outline the analysis required to extract Henry’s law solubility and Fickian diffusion coefficients from data of the type shown in Fig. 2a-c. The mathematical formulations that underlie most of the given calculative procedures may be found in Crank.lZ8 17.6.2. Calculations 17.6.2.1. Concentration-Independent Fickian Diffusion. If the sorption-desorption cycle curves coincide as in Fig. 2a, the case I transport model probably describes the system: sorption follows Henry’s law, diffusionfollows Fick’s law, and the solubility and diffusion coefficients S and D are independent of the penetrant concentration. The Henry’s law solubility coefficient is obtained from the penetrant mass sorbed at equilibrium Me,, and the final penetrant pressure peq,as

S = Meq/Vpeq,

(17.6.1)

*x8 J. Crank, “The Mathematics of Diffusion,” 2nd ed. Oxford Univ. Ress (Clarendon), London and New York, 1975.

3 44

17.

GASES A N D VAPORS

where V is the volume of the polymer in an unswelled state.t The best estimation method is to measure Me, for several values of peq at a fixed temperature, plot M,,/V vs. peq,and determine S as the slope of the resulting line. A systematic departure of the plot from linearity indicates a deviation from Henry's law sorption. CranklZ8presents solutions of the diffusion equation for different geometries. Several diffusion coefficient estimation methods based on these solutions have been derived; applying at least two of them and comparing the results provides a good consistency check on the data and calculations. 17.6.2.1.1. HALF-TIMEMETHOD.The simplest technique is to note the time tllz at which M t / M , equals 3. Then,

D=- 0.04919 (t1,2/h2)

plane sheet'20 0dx d h

solid cylinder130 (t112/R2) 0 d r d R Dz- 0.03055 (f112/R2) 0 6 r d R

Dz- 0.06306

(17.6.2) (17.6.3) (17.6.4)

17.6.2.1.2. INITIALSLOPEMETHODS. The sorption curve has the following asymptotic forms at small timeslZ8:

MI = M m

surface 2 volume

(-)

t)'" t112

(plane sheet),

(17.6.5)

(cylinder),

( I 7.6.6)

(sphere),

(17.6.7)

(arbitrary geometry). (17.6.8)

In Eq. (17.6.8), the surface is that into which the penetrant sorbs; it is presumed to be both sides of the plane sheet, and the lateral surface of the 1*0 J. Crank and G . S. Park, in "Diffusion in Polymers" (J. Crank and G . S. Park, eds.), p. 1. Academic Press, New York, 1968. R. M. Felder,J. Membr. Sci. 3, 15(1978).

t To clarify the terminology, M e , is the mass of penetrant in the polymer at equilibrium, while M, is the mass sorbed or desorbed from r = 0 to r -+ m. If Mo is the mass initially sorbed, then M , = Mo f M,.

17.6.

345

SORPTION METHODS

cylinder. A plot of MJM, vs. F2should thus be linear at small times for all geometries, although the range over which the linearity holds is relatively small for the cylinder and sphere. This functionality provides a useful test of the assumption of Fickian diffusion: if the M J M , curve initially varies linearly with (say) t rather than tin, anomalous diffusion is indicated. If the initial slope on the plot vs. t1I2is linear, the diffusion coefficient may be determined from Eq. ( 1 7 . 6 3 , (17.6.6), or (17.6.7). An alternative estimation procedure due to Carpenter131is useful for analyzing sorption data when the approach to equilibrium is exceptionally slow, and measurements of equilibrium uptakes are difficult to achieve. For both planes and spheres, at small times

--dM, dt

kDM, exp XZ

(+),

(17.6.9)

where k = 8 and x = h for planes, and k = 6 and x = R for spheres. Taking the logarithm of both sides of Eq. (17.6.9) yields In

(9) =

X2

(17.6.10) + In ( kDM, 7 ) .

M f vs. time data may be differentiated numerically, and D calculated from the initial slope of a plot of In(dM,/dr) vs. time. The amount sorbed (or desorbed) from t = 0 to t + x, M,, may in principle be obtained from the calculated diffusivity and the intercept of the line, although the accuracy of this procedure is likely to be extremely poor. 17.6.2.1.3. MOMENT METHOD. A variation of a method derived by Felder et al. 132*133 for estimating diffusion coefficients from permeation rate data may also be applied to sorption measurements. The quantity (17.6.11) is first calculated by numerical integration. Then130 D=-

h2

127s

D = - R2 8%

131

h,

(17.6.12)

(solid cylinder), 0 s r s R ,

(17.6.13)

OSrSR.

(17.6.14)

(plane sheet),

0 sx

S

A. S. Carpenter, Truns. Foruduy SOC. 43, 529 (1947). R. M. Felder, R. D. Spence, and J . K. Ferrell, J . Appl. Polym. Sci. 19, 3193 (1975).

R. M. Felder, C-C. Ma, and J. K. Ferrell, AIChEJ. 22, 724 (1976).

346

17.

GASES A N D VAPORS

The moment calculation has the advantage of utilizing the entire sorption curve, rather than a single point as in the half-time method, or the portion within a time interval of uncertain duration as in the initial slope method. For slow processes, however, the required numerical integration is susceptible to truncation and round-off errors. 17.6.2.1.4. OTHERMETHODS.Evnochides and Henleyl" outline a frequency response technique for obtaining D and S from a single sorption experiment. The external penetrant pressure is varied sinusoidally, and the values of D and S are determined from the amplitude and phase angle of the resulting weight change as a function of the cycle frequency. Crank12*describes various other curve-fitting and curve-matching procedures for estimating the diffusion coefficient from data of the type shown in Fig. 2a, and outlines modifications in the procedures to account for changes in the boundary condition due to a measurable percentage decrease in the external penetrant pressure duricg a run. 17.6.2.2. Concentration-Dependent Fickian Diffusion. When the diffusion coefficient is independent of penetrant concentration, the value of r/x2 ( x = membrane thickness or, for cylinders and spheres, radius) for which M J M , has a given value is independent of M , , and sorption and desorption curves coincide. As was noted in Section 17.3.4, this behavior is not observed for organic vapors in rubbers. Swelling or penetrant clustering frequently occurs in such systems, and the diffusion coefficient is consequently dependent on penetrant concentration. The standard diffusion coefficient estimation methods may be modified to allow for a concentration dependence; however, if a significant volume change due to swelling occurs, a mass-fixed frame of reference must be adopted in defining the diffusion coefficient. The required procedures are outlined by Crank.12" If a sorption or desorption measurement is carried out in which the dissolved penetrant concentration varies between C, and c,, the application of any of the methods given in the preceding section yields an average diffusion coefficient D( where is some concentration between C,,and C m* A direct estimate of D ( 0 may be obtained by performing a series of interval sorption measurements, and applying the half-time method or the method of moments to estimate D( in each interval. Taking data over a large number of small concentration increments provides a good approximation to the exact D(C)curve, at the expense of time and experimental precision.135 An even better approximation may be obtained by

c>,

c

c>

S. K. Evnochides and E. J. Henley, J . P d y m . Sci.. Part A 2 8, 1987 (1970). Is5

R. J. Kokes, F. A. Long. and L. J. Hoard, J . Chem. Phys. 20, 1711 (1952).

17.6.

SORPTION METHODS

347

performing a sorption-desorption cycle between each pair of concentrations, and taking D ( c ) to be the average of the diffusion coefficients measured in the two branches of each A related approach is to use a long-time curve-fitting technique, and to presume that over the time interval in which the fitting is performed the concentration does not vary significantly from the final concentration Cm, so that what is calculated is D(C,). The D(C)functionality is determined by performing such measurements for a series of concentration steps. Integral sorption data may be corrected for concentration dependence of the diffusion coefficient if the functionality of D(C) is known. Corrections for linear and exponential dependences are summarized by Crank.128 Duda and V r e n t a ~propose l~~ a method of moments to deduce the concentration dependence of the diffusion coefficient from a single sorption curve. (See also Section 7.4 of Crank.12a) The method takes into account effects of phase volume changes and volume changes due to mixing, and requires no assumptions about the nature of the concentration dependence. The drawbacks are that several successive numerical differentiations and integrations are required, and the form of the trial function is circumscribed by the practical upper limit of two moment evaluations, all of which restrict the accuracy of the method. 17.6.2.3. Dual-Mode Sorption. Concentration vs. pressure sorption isotherms for glassy polymers are concave to the pressure axis, as shown in Fig. 3. According to the dual-mode sorption model (Section 17.3.5) the isotherm has the form

c = CD + CH = s p + 1ChbP + bp’

(17.6.15)

where C is the total solubility at equilibrium [cm3(STP)/cm3polymer], S the Henry’s law constant [cm3(STP)/cm3polymer * atm], b the hole affinity constant (atm-I), Ch the hole saturation constant [cm3(STP)/cm3 polymer], and p the equilibrium penetrant pressure (atm). A parameter estimation procedure developed by ViethIogconsists of first determining the value of S as the slope of the linear high-pressure (20-30 atm) portion of the isotherm, then plotting p / ( C - Sp) vs. p , and determining CL and b from the slope (1 /Ch) and intercept (1 /Chb) of the resulting straight line. Koros et ~ 1found . this ~ graphical ~ ~ method to be inaccurate due to the ISo lS7

S . Prager and F. A. Long, J . Am. Chem. Soc. 73, 4072 (1951). M. J . Hayes and G . S . Park, Truns. Farcrdiig SOC. 52, 949 (1956). J . L. Duda and 1. S. Vrentas, AfChEJ. 17, 464 (1971).

17.

GASES AND VAPORS

PRESSURE FIG.3. Dual-mode sorption isotherm.

invalid assumption of hole saturation at the higher pressures; that is, the high-pressure portion of the isotherm has usually not yet reached its true asymptotic slope when the experiment is terminated. They recommend that a nonlinear regression be performed to determine simultaneously the proper values of S, Ck, and 6 . The nonlinearity of the equilibrium sorption isotherm complicates attempts to model the kinetics of sorption of gases in glassy polymers. Vieth and Sladek" and F e n e 1 0 n ~have ~ ~ developed curve-matching methods to estimate the diffusion coefficient from dual-mode sorption rate data. Both methods are based on an assumption of total immobilization of the molecules that sorb according to the Langmuir isotherm. Koros, Paul, and RochalOe have shown that this assumption may be erroneous, and have developed permeation rate analysis methods to account for mobility of the Langmuir-sorbed species (see Section 17.7.2.3). 17.6.2.4. Anomalous Transport of Vapors. A number of mathematical models have been formulated to account for the sorption and nonFickian diffusion phenomena often observed for vapors in glassy polymers (Sections 17.3.6 and 17.3.7). The analysis of case I1 transport, in which a penetrant front advances at P. J. Fenelon, Am. Chem. Soc.. Div. Org. Coat. Plast. Chem.. Pap. 34, 522 (1974).

17.6. SORPTION METHODS

349

a constant velocity, has been outlined by Enscore et al. 140 Peterlin141proposes a model for anomalous transport, i.e., transport that is neither Fickian nor case 11, whereby a Fickian diffusion front precedes a case I1 boundary: stress relaxation is presumed to take place instantaneously when the penetrant concentration in the first front reaches a critical value. Kwei et al. 142-144 develop an alternative model whereby diffusion and convection occur simultaneously; asymptotic solutions of an augmented diffusion equation are used to derive model parameter estimation formulas. Discussions and comparisons of these and other models are presented by Petropoulos and Roussis.118 Stress relaxation models have been applied to the sorption of penetrants that are good swelling agents. The earliest methods are based on the classical diffusion equation: an approach due to Crankll' is to allow the diffusion coefficient to vary with time, and an alternative approach formulated by Long and Richman120holds D constant but allows the surface concentration to vary with time. Numerical solutions are required in both cases. Qualitative features of these two models have been reviewed by Petropoulos and Roussis,118who observe that both models account for the sigmoidal sorption curves characteristic of anomalous transport (Fig. 2d), but only the model of Long and Richman can account for two-stage sorption (Fig. 2e) when relaxation is slow relative to diffusion. Crank"' and Wang et a/.142 propose models that view anomalous transport as the consequence of swelling stresses caused by an uneven distribution of penetrant in a polymer matrix. Though limited from a practical standpoint, these models have a basis in analogies with thermally induced stress phenomena, and provide a link between transport properties and mechanical properties of polymer-penetrant systems.l18 The theory of irreversible thermodynamics accounts for many observed anomalies in the transport of gases in glassy polymers. Discussions of this subject are presented by F r i s ~ h l 146 ~ ~and . Crank.lZ8 17.6.3. Experimental Methods

Most sorption measurements are performed using either a gravimetric or a barometric technique. Gravimetric methods, in which a sorption or D . J . Enscore, H. B. Hopfenberg, A. R . Berens, and V. Stannett, f o / y m e r 18, 1105 (1977). A. Peterlin, J . fol.vm. Sci., Purr B 3, 1083 (1965). 14* T. T. Wang, T. K . Kwei, and H. L. Frisch, J . Polym. Sri., Part A-2 7, 2019 (1969). T. K . Kwei and H. M. Zupko, J . Polym. Sci., Purr A-2 7,867 (1969). '41 T. K . Kwei, T. T. Wang. and H. M. Zupko. Murromolecules 5, 645 (1972). 14s H. L. Frisch, J . f h y s . Chem. 41, 3679 (1964). IW H. L. Frisch, in "Non-Equilibrium Thermodynamics, Variational Techniques, and Stability" (R.J. Donnelly, R. Herman, and I. Prigogine, eds.), p. 277. Univ. of Chicago Press, Chicago, Illnois, 1966.

350

17.

GASES A N D VAPORS

desorption rate is determined directly by following the change in weight of a polymer sample, are generally used for measurements at subatmospheric pressures, while barometric techniques, in which the rate is determined from the rate of change of pressure in a sorption cell, are preferred for higher pressures and penetrant uptakes. Other methods of determining sorption rates include measurement of the change in volume of an isobaric sorption cell, the change in resonant vibrational frequency of a piezoelectric crystal or a fiber, the penetration depth of a swelling front, and the width of a chromatographic peak. The sections that follow outline devices and operating features associated with each of these approaches. 17.6.3.1. Gravimetric Techniques. Periodic removal and weighing of a polymer sample undergoing sorption has been used to study the sorption and desorption kinetics of water in various rnaterial~,'~'and the sorption of organic vapors in polystyrene and other polymer^.^^*-^^^ The repeated interruptions of the sorption run and handling of the sample required by this method can lead to serious errors in rate data, and lesser errors in equilibrium measurements. The McBain balanceg4(Fig. 4) is commonly used for penetrant pressures less than one a t m ~ ~ p h e r e The . ~ polymer ~ ~ ~ is~suspended ~ ~ ~ ~ ~ ~ ~ ~ A. J. Kovacs, J . C h i m . Phys. 45,258 (1948). J . Crank and G. S. Park, Truns. Furuduy Soc. 45, 240 (1949). 14@ G. S. Park, Trans. Furudoy Soc. 46, 684 (1950). lW G. S.Park, Trans. Furuduy Soc. 47, 1007 (1951). lS1 S. W. Benson, D. A. Ellis, and R. W. Zwanzig, J. Am. Chem. Soc. 72, 2102 (1950). le H. B. Bull, J. Am. Chem. Soc. 66, 1499 (1944). lM A. C. Newns, J. Text. Inst. 41, T269 (1950). IM A. J . Stamm, J. Phys. Chem. 60, 76 (1956). D. W. McCall, J . Polym. Sci. 26, 151 (1957). lo D. W. McCall, J. F. Ambrose, and V. C. Tanza, J. Polym. Sci. 26, 165 (1957). D. W. McCall and W. P. Slichter, J . Am. Chem. Soc. 80, 1861 (1958). A. F. Mellon, A. H. Horn, and S. R. Hoover, J . Am. Chem. Soc. 69, 827 (1948). 150 A. F. Mellon, A. H. Horn, and S . R. Hoover, J. A m . Chem. Soc. 70, 114 (1948). A. R. Urquart and A. M. Williams, J . Text. Inst. 15, TI38 (1924). 161 P. M. Hauser and A. D. McCloren, Ind. Eng. Chem. 40, 112 (1948). ld2 S. L. Madorsky. Rev. Sci. Instrum. 21, 393 (1950). P. E. Rouse, J . Am. Chem. Soc. 69, 1068 (1947). T. A. Garrett and G. S. Park, J. Polym. Sci.. Purr C 16, 601 (1956). J . G. Downes and B. H . McKay, Proc. Int. Wool Text. Res. Conf., Is:, 1955 PaperD. p. 203 (1956). G. King, Trons. Furoduy Soc. 41, 325 (1945). ld7 G. S. Park, Truns. Furuduy Soc. 48, 11 (1952). A. W. Myers, J. A. Meyer, C. E. Rogers, V. Stannett, and M. Szwarc, Toppi 44,58 '41

14*

(l%l). J. L. Duda, G. K. Kimmerly, W. J. Sigelko, and J. S. Vrentas, Ind. Eng. Chern., Fundum. 12, 133 (1973).

17.6.

SORPTION METHODS

35 1

SOURCE

SUPPLY

MANOMETER

SORPTION CHAMBER

FIG.4. McBain balance apparatus.

from a calibrated spring within a water-jacketed evacuated glass chamber, into which penetrant is introduced. The mass of penetrant sorbed by the polymer is determined by measuring the spring extension with a cathetometer and correcting for hydrostatic buoyancy forces. Quartz springs, which obey Hooke's law over wide load ranges, are most often used; tungsten springs, which are more durable, have also been used. The relationship between spring extension and load is determined with calibration weights. Weight changes on the order of 2 pg can be observed using this method. McBain systems should be set up in areas free of vibrations, as bouncing of the sensitive springs can cause serious interference with the measurements. The use of a reference fiber leads to considerable smoothing of sorption data by eliminating effects of abnormal shifts in the spring position."' Such shifts can be caused by settling of the water jacket cap to which the spring is attached, inadvertent jolting of the cell, twisting of the H. B. Hopfenberg, V. Stannett, and C. H. M. Jacques, J . Appl. Polym. Sci. 19,2439 (1975). 17'

In In

'I4

H. B. Hopfenberg, V. Stannett, and G . M. Folk, Polym. Eng. Sci. 15, 261 (1975). H. Ochiai, K . Gekko, and H. Yamamura, J . Polym. Sci., Par: A-2 9, 1629 (1971). H . Nakasima, H. Yamakawa, and I . Sakurada, Kobunshi Kagaku 14,333 (1957). J. L. Williams and A. Peterlin, J . Polym. Sci. 9, 1483 (1971). W. R. Brown, R. B. Jenkins, and G. S. Park, J . Polym. Sci.. Polym. Symp. 41, 45

( 1973).

352

17.

GASES A N D VAPORS

spring around its support loop, and small changes in the cathetometer position. Weight gains may also be monitored continuously by mounting the polymer sample at the base of a torsional pendulum. The added mass due to sorption is reflected by a change in the period of oscillation of the pendu1~m.I~~ Many investigators have carried out sorption runs in the sample chambers of Cahn electrobalances or similar i n s t r u m e n t ~ . ~ ~ ~ * ' ~ ~ - ' ~ ' Systems have been devised to supply water vapor at a constant humidity to a balance charnber,I8Oand to follow the uptake or loss of water vapor from multiple samples.181 17.6.3.2. Barometric Techniques. The high sensitivity of gravimetric methods has led to their general adoption in recent years for operation at subatmospheric pressures, although low-pressure barometric systems are still used ~ c c a ~ i o n a l l y .On ~ the ~ ~ other ~ ~ ~hand, ~ ~ ~the~ baromet~ ~ ~ - ~ ~ ric approach dominates the literature on high-pressure measurements. 103~10g~127~13g~185~188 A schematic representation of a typical barometric sorption rate measurement device is shown in Fig. 5 . For pressures not exceeding 2 or 3 atm the system is generally made of glass; at higher pressures stainless steel is used. Calculating a sorption rate from a pressure change requires an accurate knowledge of the system volume. A common method used to determine the volume is first to determine the volume of a reservoir (R of Fig. 5 ) by filling it with a liquid. The reservoir is then emptied, connected to the sorption system, and filled with a gas to a measured pressure with the remainder of the system evacuated. Valves are successively opened to connect each system component to the reservoir, and the component volumes are determined from the resulting pressure changes. The best combination of operational convenience and accuracy is achieved if the reservoir and sorption chamber volumes are approximately A. S. Carpenter and D. E. Twiss, Ind. Eng. Chem., Anal. Ed. 12, 99 (1940). A. R. Berens, Polym. Prepr. 15(2) (1974). lT8T. K. Kwei and T. T. Wang, Macromolecules 5, 128 (1972). I. Cabasso, J. Jagur-Grodzinski, and D. Vofsi, J . Appl. Polym. Sci. 18, 2095 (1974). M. J. Palin, G. J. Gittens, and G . B. Porter, J . Appl. Polym. Sci. 19, 1135 (1975). 181 R. Best and E. Spingler, Chem.-/ng.-Tech. 44, 1222 (1972). G . V. Schultz and H. Gerens, Z . Phys. Chem. 7 , 182 (1956). ISs A. S. Michaels, W. R. Vieth, and H. J . Bixler, J . Appl. Polym. Sci. 8, 2735 (1964). l1 J. L. Williams and A. Peterlin, Makrornol. Chem. 22, 215 (1968). Ins J. L. Lundberg, M. B. Wilk, and M. J . Huyett, J . Polym. Sci. 57, 275 (1962). J. L. Lundberg, M. B. Wilk, and M. I. Huyett, Ind. Eng. Chern.. Fundam. 2,37 (1%3). ln7 W. R. Vieth, P. M. Tam, and A. S.Michaels, J . Colloid Interface Sci. 27,360 ( 1966). W. R. Vieth and J . A. Eilenberg, J . Appl. Polym. Sci. 16, 945 (1972). 18g P. R. Dumll and R. G . Griskey, AIChELI. 12, 1147 (1966). '71

17.6.

I

41

I I

I I

I

353

SORPTION METHODS

++.

lo

I

VACUUM

I I

SOURCE

I

SUPPLY AMPULE

I I

I I I

1RANSDUCER AND RECORDER

I I

I I

I I

MANOMETER

I I

SOUP1ION CELL

I I 1HERMAL ISOLATION

I

I

I

I

FIG.5. Barometric sorption apparatus.

The volume calibration can also be achieved without resorting to liquid filling if two such experiments are performed-the first as above, and the second with a spacer block of known volume inserted in the system. The volumes of both the reservoir and the sorption chamber may be determined from the results of these measurements, using a suitable equation of state. If possible, a single source of thermodynamic data should be used for all PVT calculations in gas sorption measurements1Ov;several ,such sources are a ~ a i l a b l e . ~ @ " J ~ ~ In most analyses of sorption rate data, the penetrant concentration at the upstream membrane surface is assumed to achieve its limiting value from the moment the experiment commences. If the time required for the concentration to build up is a significant fraction of that required for the sorption, data analysis becomes extremely difficult. The chamberreservoir system described above permits the rapid attainment of a desired penetrant pressure. The reservoir is charged with the sorption chamber evacuated, and the valve between the two chambers is then opened. The initial condition for the sorption run is calculated from the initial pressure in the reservoir and the known volumes of the system components. The dual-volume system is also particularly convenient for interval sorption and desorption measurements. All that is required is to close the valve between the reservoir and the sorption chamber after each run, adjust the reservoir pressure, and reopen the valve. F. Din, ed., "Thermodynamic Functions of Gases." Butterworth, London, 1956. U. S. National Bureau of Standards, Nut/. Bur. Stand. (U.S.), Circ. 564 (1955).

lw

17. GASES AND VAPORS

354

An interesting approach to the problem of rapidly achieving a desired penetrant pressure in a sorption run is that of ToilB2 A film sample is mounted at one end of a wire and floats on a mercury bath in a closed sorption chamber. An iron <eight at the bottom of the wire is immersed in the mercury. A magnet outside the sorption cell is placed adjacent to the weight and moved down, submerging the film. The cell is then evacuated, the penetrant is introduced at its desired pressure, and the magnet is abruptly withdrawn; the film quickly rises above the mercury, and sorption commences. Lundberg et al. 1*5~186studied sorption of gases in molten polystyrene in a system interfaced to a digital data-logging facility. From 4000 to 6OOO measurements were obtained, recorded, and analyzed automatically during a three-day series of experiments. Durrill and GriskeylBOalso studied sorption in molten polymers, at pressures up to 20 atm. They achieved temperature control by mounting their cell in a heated fluidized bed. 17.6.3.3. Volumetric Techniques. Devices based on measurements of volume changes accompanying sorption or desorption are relatively uncommon, perhaps due to the complexity of their design relative to the designs of common gravimetric and barometric devices. The technique offers several potential advantages. Rather large and easily measured gas phase volume changes may accompany relatively small amounts of sorption; moreover, the volumetric method permits operation at constant penetrant pressures, thus alleviating the variable boundary-value problem associated with large degrees of sorption in barometric systems. An ingeniously designed volumetric sorption apparatus is that of Rosen'" (Fig. 6). A manometer measures the difference between a preset pressure and the pressure in the sorption chamber. As sorption proceeds, the rising mercury in the chamber arm of the manometer contacts a platinum lead, thereby generating a signal that causes a solenoid pinch valve to open. Mercury flows through the valve into a tube connected to the sorption chamber, compressing the chamber gas to an extent that the manometer returns to its initial position and the solenoid valve closes. A platinum wire in the mercury tube acts as the sensing arm of a Wheatstone bridge, so that a continuous signal proportional to the volume change can be transmitted to a recorder. The volume vs. time signals obtained using this device are not completely smooth, owing to the on-off nature of the feedback control system. However, Rosen states that proper adjustment of the volume control tube diameter and the rate of flow through the solenoid valve can produce acceptably smooth recorder traces. K. Toi, J .

Ira

Polym. Sci., Polym. Phys. Ed. 11, 1829 (1973).

17.6. SORPTION METHODS

355

FIG.6. Volumetric sorption apparatus. Modified and reprinted from R o s e P by permission of John Wiley & Sons, Inc.

17.6.3.4. Miscellaneous Techniques. A piezoelectric sorption device developed by King1@’.1e5 has been used by Bonner and Cheng1B8*1e7 to study the sorption of ethylene by polyethylene. A crystal is coated with a known mass of molten polymer, and the resonant crystal frequency is recorded; subsequent penetrant absorption is followed by observing further changes in the resonant frequency. The method can be used over a wide range of temperatures and pressures. A method due to Gray and Guilletleehas been used to study diffusion in polymer coatings on glass beads.188-200 Penetrant samples are eluted through a column packed with the beads. The shape of the chromatographic peak depends on both the diffusion coefficient of the penetrant in B. Rosen, J . Polym. Sci. 35, 225 (1959). W. H. King, ReslDev. 20,28 (1%9). W. H. King, Ham Radio Mag. 50 (1975). ‘06 D. C. Bonner and Y. L. Cheng, J . Polym. Sci.. Part B 13, 259 (1975). lo’ Y. L. Chengand D. C. Bonner, J . Polym. Sci.. Polvm. Phys. Ed. 15, 593 (1977). D. G . Gray and J. E. Guillet, Macromolecules 6, 223 (1973). J. M. Braun, S. Poos, and J . E. Guillet, J . Polym. Sei.. Polymer Lett. Ed. 14, 257 ( 1976). J . M. Kong and S. J. Hawkes. Macrornolecitles 8,685 (1975). Irn

IDI

356

17.

GASES

A N D VAPORS

the polymer and the carrier gas flow rate; the coefficient is determined from peak widths measured in runs carried out at several flow rates. AssinkZo1has developed a pulsed nuclear magnetic resonance spectrometric method (see Part 4 of this volume, Part A) for the study of gas sorption in polymers, and has used it to study the sorption of ammonia in polystyrene. The results substantiate the fundamental assumption of the dual sorption model (Section 17.3.5) that the species sorbed according to the Henry’s law and Langmuir isotherms are in equilibrium with each other. When heats of solution are large, temperature increases accompanying sorption can cloud the interpretation of rate data. The problem is particularly acute for the sorption of water in various ~ u b s t a n c e s ~ although ~; corrections for nonisothermality have been formulated, the uncertainties in estimated transport parameter values caused by measurable temperature changes can never be completely eliminated. A method developed by Downes and McKayIB5 is useful for penetrant-polymer pairs characterized by large heats of solution. A polymer filament is mounted as the frequency control element of an oscillating circuit, and the penetrant gas is blown over the filament at a rate high enough to assure isothermality during sorption. The sorption curve is determined from measured values of the resonant frequency of the fiber, which vanes as the square root of the total mass of the fiber and sorbed penetrant. Sorption measurements have been performed (usually for liquid penetrants) in which the distribution of the penetrant in a polymer sample is measured directly; the measurement may be by visual observation if crazing occurs, otherwise by a radiation attenuation method. Experimental techniques for this approach are described by Crank and Park,lzB and curve-matching techniques for estimating Fickian and anomalous transport coefficients are reviewed by Crank. lZ8

17.7 Integral Permeation (Closed Receiving Volume) Methods 17.7.1. Experiments and Data

The cumulative mass of penetrant that permeates through a membrane into a closed chamber is determined as a function of time. The penetrant pressure p1at the upstream face of the polymer is generally held constant, and the receiving volume is usually large enough so that the downstream R. A. Assink, J . Polym. Sci., Polym. Phys. Ed. 13, 1665 (1975).

17.7.

INTEGRAL PERMEATION METHODS

357

pressure p z , while measurable, is negligible relative to the upstream pressure. The data appear as shown in Fig. 7. In most cases, the permeation curve asymptotically approaches a straight line at large times. Most transport parameter estimation methods are based on derived relationships among the parameters, the slope of the asymptotic line (i.e., the steady-state permeation rate), and the intercept e of the line on the time axis, or the time lag. 17.7.2. Calculations 17.7.2.1. Concentration-Independent Fickian Diffusion. For Henry's law sorption and Fickian diffusion with both S and D independent of concentration, Eq. (17.2.3) is applicable. The permeability P ( = DS)is obtained from the steady-state permeation rate C#J (the slope of the asymptotic line of Fig. 7), as

(17.7.1)

I I I

8

TIME FIG.7. Data from a closed-volume permeation experiment

358

17.

GASES A N D VAPORS

The diffusion coefficient is obtained from the time lag D = h2/68.

asz1 (17.7.2)

The solubility coefficient S is calculated as P / D . Corresponding formulas for cylindrical and spherical membranes, and the series solutions of the diffusion equation from which the formulas are derived, are given by Crank. lZ8 An assumption underlying Eq. (17.7.2) is that the downstream surface penetrant concentration is negligible compared to the upstream concentration. If the receiving volume is small, this assumption may lead to errors in estimated transport coefficients. Paul and DiBenedetto2O2 solved the diffusion equation assuming a finite receiving volume, and derived formulas to correct values of P , D , and S estimated neglecting the downstream penetrant concentration. They define a parameter r) as 7 = 0.278 STAh/VceI1,

(17.7.3)

where S is the true Henry’s law solubility coefficient, T (“K), temperature, A (cm2)and h (cm) the membrane area and thickness, and Vcell (cm3) the downstream chamber volume. They plot curves of k / P , D / D , and S / S

vs. r), where the caret signifies an estimated value, and give an iterative graphical procedure for determining P , D , and S from k, b, and 3. Since the plots given by Paul and DiBenedetto are close to linear, a direct rather than an iterative solution is possible, using the following equations: 0.278TAh/Vce11,

(17.7.4)

P/b

1 + 0.38ak/b’

(17.7.5)

P = P(1

(17.7.6)

D

+ 0.69aS), = b ( l + 0.29aS).

(17.7.7)

=

S=

This procedure is applicable as long as aS ( = q ) is less than roughly 0.35. For laminates consisting of layers exhibiting Henry’s law sorption and concentration-independent Fickian diffusion, expressions for the steady-state permeation flux, concentration distribution, and time lag have been developed by Barrer and C O - W O ~ ~ ~ ~ Extensions S . ~ ~ ~ J of~ the analysis to media in which S and D vary across the membrane have been D. R. Paul and A. T. DiBenedetto, J . folym. Sci., Part C 10, 17 (1%5). R. M . Barrer. in “Permeability of Plastic Films and Coatings to Gases, Vapors, and Liquids” (H. B. Hopfenberg, ed.), p. 113. Plenum, New York, 1974. 4M R. Ash. R . M. Barrer, and D. G . Palmer, Er. J . Appl. Phys. 16, 873 (1%5). zo2

*09

17.7

INTEGRAL PERMEATION METHODS

359

formulated by Ash et u / . , Petropoulos ~ ~ and R o u s s ~ s ,and ~ ~ ~Frisch and Prager,206and discussed by Barrer,203Stannett, Hopfenberg, and Petrop o u l o ~and , ~ Crank.128 17.7.2.2. Concentration-Dependent Fickian Diffusion. If the diffusion coefficient depends on the penetrant concentration (Section 17.3.4), then a mean value of D in the range 0 s C c C1may be defined as

-

1"'

D = -!- D(C) dC.

c1

( 17.7.8)

0

Pollack and FrischZo7 have shown that if C1is the penetrant concentration at the upstream membrane surface, and the variation of D with C is such that

In

[D(C') dC'

is a convex function of C , then h2 -sDs-. 6

h3 2

(17.7.9)

This relation allows the estimation of 5 to within a factor of 3 from a measured time lag in the absence of information about the functional form of D(0. Rogers et a/.208show that an approximate solution of the diffusion equation valid at small times yields the relation In(t1/2F)= l n [ 2 C , ( D / ~ ) ~-~h2/4Dt, ]

(17.7.10)

where (17.7.11) If the plot of Fig. 7 is differentiated numerically to determine F ( t ) , and I n ( P F) is plotted vs. l/t, at small times the plot should be linear. The values of D and C1may be determined from the slope ( - h2/4D) and intercept ln[2C,(D/~)~/~] of the asymptotic line; moreover, the calculated value of D should be close to the true value of D(C) as C approaches 0, since at small times the concentration is low throughout most of the membrane. If sorption follows Henry's law, the solubility coefficient may be calculated as S = Cl/p,. 205

*07

J. H. Petropoulos and P. P. Roussis, J . Chem. Phys. 51, 1337 (1969). H. L. Frisch and S. Rager,J. Phys. Chem. 54, 1451 (1971). H. 0. Pollack and H. L. Frisch, J . Phys. Chem. 63, 1022 (1959). W. A. Rogers, R. S. Buritz, and D. Alpert, J . Appl. Phys. 25,868 (1954).

360

17. GASES

AND VAPORS

A variety of methods for estimating D ( 0 are reviewed in Chapter 10 of Crank.lZ8 The most widely used is that of F r i s ~ h ,who ~ ~ showed ~ * ~ ~that ~

\ xr,(x) dx // Jo Jo h

B=

\cl

D(C) dC.

(17.7.12)

~

The steady-state concentration profile C,(x) may be obtained by solving the equation

Ifi

D ( 0 dC =

D(C) dC.

(17.7.13)

To use Frisch’s method, a functional form must be specified for D ( 0 . Measurements of 8 for several values of Cl may then be used in conjunction with Eqs. (17.7.12) and (17.7.13) to solve for the coefficients of the assumed function. If D depends exponentially on C,

D ( 0 = D(0) exp(pC),

(17.7.14)

the n21 40(0)8 -hZ

4 exp(pC,) - 1 + exp(2pC1)[2pC1 - 31 , [exp(pC,> - 113

(17.7.15)

and the steady-state permeation rate is

11.

(17.7.16)

The value of D(0) may be determined from the small-time approximation given above [Eq. (17.7. lo)] or by performing a run at a very low value of C,. Thereafter, measurements of 8 and #IJ for several penetrant pressures permit the calculation of p and C , from Eqs. (17.7.15) and (17.7.16). If a system exhibits Henry’s law solubility in addition to concentration-dependent diffusion, the solubility coefficient can be determined by combining two permeation experiments. One is the usual time lag experiment with a membrane initially free of penetrant; in the other, proposed by Fatt,212penetrant is allowed to desorb from a polymer sample equilibrated with a penetrant at partial pressure p l . uses Frisch’s technique to derive a general expression for the steady-state asymptote of the time lag plot, and shows that the intercepts differ for the H. L. Frisch, J . Phys. Chem. 61, 93 (1957). H. L. Frisch, J . Phys. Chem. 62, 401 (1958). P. Meares, J . Appl. Polym. Sci. 9, 917 (196s). 212 1. Fatt, J . Phys. Chem. 66, 760 (1%2). 21s D.R. Paul, J . Polym. Sci., Purt A-2 7, 2031 (1%9). zlo zll

17.7

36 1

INTEGRAL PERMEATION METHODS

two experiments by AQa (0) = S p , h / 2 , from which S may be calculated. A similar analysis has been applied to the time axis intercepts of such plots by Petropoulos and R o u s s ~ s . ~ ’ ~ 17.7.2.3. Dual Mode Sorption. The nonlinearity of the sorption isotherm [Eq. (17.6.IS)] complicates efforts to model the unsteady-state permeation behavior of gas/glassy polymer systems that exhibit dualmode sorption. Most a t t e m p t ~ ~ have ’ ~ * ~been l ~ based upon an assumption that species that sorb according to the Langmuir mechanism are immobilized, and hence incapable of contributing to the diffusive flux. A more recent m ~ d e l assigns ~ ~ ~ diffusion J ~ ~ coefficients to the species in both the amorphous matrix region and the “holes,” and predicts that the permeability varies with the upstream penetrant pressure p1 as P = S D (1

+ A) 1 + bp, ’

(17.7.17)

where K = C’,b/S is determined from sorption data (see Section 17.6.2.3). The parameter F equals the ratio of the diffusion coefficient of the Langmuir species to the diffusion coefficient (D) of the species sorbed according to Henry’s law. The values of D and F may be determined by measuring steady-state permeabilities over a range of values of p 1 and, having previously determined b from equilibrium sorption measurements, plotting vs. ( 1 + bp)-l. Expressions for the time lag derived by Paul and Koroslo8may be used to check the consistency of the calculations. Experimental results for the transport of carbon dioxide in polycarbonate are consistent with a value of F = 0.1. 17.7.2.4. Anomalous Transport of Vapors. As was observed in Section 17.3.6, the analysis of the transport of vapors in glassy polymers is complicated by time and/or concentration-dependent diffusion phenomena. Anomalies may arise due to swelling, stress relaxation, clustering of penetrant molecules, diffusion accompanied by reaction, and temperature increases that arise from the evolution of heats of sorption. Extensions of the time lag method to systems in which the effective diffusion coefficient depends explicitly on time and/or position in the membrane have been formulated by F r i ~ c h , ~ and ~ ’ - reviewed ~~~ by Stannett, Hopfenberg, and Petropoulos,2 Petropoulos and Roussis,lla Barrer,203and J . H. Petropoulos and P. P. Roussis, J . Chem. Phys. 47, 1491 (1%7). D. R . Paul, J . Po/ym. Sci.. Purr A-2 7, 1811 (1969). 216 J . A. Tshudy and C. von Frankenburg, J . P d y m . Sci.. Purr A-2 11, 2027 (1973). 217 H. L. Frisch, J . Phys. Chem. 63, 1249 (1959). 118 H. L. Frisch, J . Chem. Phys. 36, 510 (1%2). 218 H. L. Frisch, J . Chem. Phys. 37, 2408 (1962). 214

*Is

362

17.

GASES A N D VAPORS

Crank.IP8 The proposed methods require the measurement of time lags at both the upstream and downstream faces of the membrane, for both sorption and desorption permeation experiments. The measured lags are then used in conjunction with theoretical expressions given in the references cited above to deduce and quantify the nature of the occurring phenomena.P20-223 17.7.3. Experimental Methods

Many closed-volume measurements are performed by allowing a penetrant to permeate into an initially evacuated chamber, and measuring the resulting rise in pressure at constant volume or the rise in volume at constant pressure. In variations of this method, the chamber initially contains a gas other than the penetrant, and a quantity proportional to the penetrant concentration, such as thermal conductivity or refractive index, is measured. A gravimetric approach is frequently used to measure permeation rates of condensable vapors. The penetrant permeates into a closed cell containing an absorbing medium, and the cell is intermittently or continuously weighed. The rate at which the weight increases is equated with the rate of permeation of the vapor through the membrane. In most systems, penetrant is supplied to the upstream surface of the membrane from a closed reservoir; in some cases, such as when the penetrant is a condensable vapor at a high activity, a gas stream containing the penetrant is passed over the upstream surface. Hubbell, Brandt, and MunirP2‘bubbled air through a water column to obtain a feed stream containing water vapor at a constant humidity. Felder, Ferrell, and SpiveyZP5 produced feed streams containing as much as 21 mol% water vapor for high-temperature permeation measurements by metering liquid water through a packed, heated tee into an air stream. Heating tape was used to vaporize the water in the tee and to keep it from condensing in the feed line to the permeation chamber. 17.7.3.1. Barometric Techniques. In most vacuum systems the pressure in the receiving chamber is measured as a function of time, using one of the gauges listed in Table I. Barometric devices have been used to measure permeation rates of pure gases,s.46~226-232 condensable R. Ash, R. W. Baker, and R. M. Barrer, Proc. R . Soc. London, Ser. A 304,407 (1%8). R. M. Barrier, in “Surface Area Determination,” p. 227. Butterworth, London (1970). OZZ J. H . Petropoulos and P. P. Roussis, J . Chem. Phys. 47, 1491 and 14% (1967). zL1 J. H. Petropoulos and P. P. Roussis, J . Chem. Phys. 48, 4619 (1%8). pu W. H. Hubbell, Jr., H . Brandt, and Z. A. Munir, J . Polym. Sci. 14, 493 (1975). R. M. Felder, J. K . Ferrell, and J. J. Spivey, A n d . Instrum. 12, 35 (1974). 2M R. M. Barrer and G. Skirrow, J . Polyrn. Sci. 3, 549 (1948). 2zo

17.7

INTEGRAL PERMEATION METHODS

363

vapors,228~32.233 and gas -vapor mixtures.2a-u6 Provisions have been made for continuous-pressure measurement and r e c ~ r d i n g ~ 'and - ~ ~for operation with upstream gas pressures as high as 30 atm.loe The ASTM standard method for measuring gas transmission rates and permeabilities of flat membranes is designated D1434-75.240 The barometric technique specified for the ASTM test utilizes the Dow gas transmission cell (Fig. 8) developed by Brown and Sauber2" and discussed by Rogers242and Hwang and Kammermeyer.' The membrane is supported with a filter paper, and sealed with an 0 ring. An open-end mercury manometer is used to measure the pressure in the receiving chamber; the height of mercury in the cell manometer leg is determined visually with a cathetometer, or automatically with a platinum lead wire connected to a resistance recording instrument. Detailed descriptions of the calibration and testing procedures, including formulas to correct for volume changes due to the manometer fluid displacement, are given in the ASTM reference .240 Since the receiving volume in the Dow cell is quite small, assuming that the downstream penetrant concentration is negligible may lead to errors in estimated transport coefficients. The procedure of Paul and DiBenedettoZo2[Eqs. (17.7.3)-( 17.7.7)] should be used to correct estimated values of P , D,and S for this effect. 17.7.3.2. Volumetric Techniques. Variable-volume permeation cells are often used for rapid measurement of relatively high steady-state permeation The volumetric method is generally simpler to imG. J. van Amerongen, J . Appl. Phys. 5,307 (1950). C. E. Rogers, J. A. Meyer, V. Stannett. and M. Szwarc, Tappi 3!3,737 (1956). 229 S. A. Stem, S . K. Sen, and A. K. Rao, J . Marromol. Sci., Phys. 10, 507 (1974). mo J. E. Curry and M. D. McKinley, J. Polym. Sci.. Polym. Phys. Ed. 11, 2209 (1973). S . A. Stem and G. W. Britton, J. Polym. Sci.. Part A-2 10, 295 (1972). m2 J. A. Barrie, A. Quig, and H. G. Spencer, J . Appl. Polym. Sci. 19, 3369 (1975). P. M. Doty, W. H. Aiken, and H. Mark, Ind. Eng. Chem., Anal. Ed. 16,686 (1944). J. A. Meyer, C. E. Rogers, V. Stannett. and M. Szwarc, Tappi 40, 142 (1957). pJs R. M. Barrer, J. A. Barrie, and J. Slater, J . Polym. Sci. 23, 315 (1957). 298 P. E. Rouse, J . A m . Chem. Soc. 69, 1068 (1947). p7 P. Meares, J. Am. Chem. Soc. 26, 3415 (1954). Ian F. H. Muller and E. Hellmuth, Kolluid-Z. 144, 125 (1955). B. Rosen and J. H. Singleton, J. Polym. Sci. 25, 225 (1957). 240 American Society for Testing and Materials, "1976 Annual Book of ASTM Standards," Part 35, p. 463. . Am. SOC.Test. Mater., Philadelphia, Pennsylvania, 1976. W. E. Brown and W. J. Sauber, Mod. Plusr. 36, 107 (1959). ~4~C. E. Rogers, in "Engineering Design for Plastics" (E.Baer, ed.). Chapter 9. Van Nostrand-Reinhold, Princeton, New Jersey, 1964. R. M. Barrer, Trrms. Faraday Soc. 43, 3 (1947). L. W. Elder, Mod. Packag. 16, 69 (1943). 227

2za

3 64

17.

GASES A N D VAPORS J

FIG.8. Dow manometric permeability cell (ASTM D 1434-58). A, Upper plate; B, lower plate; C, rubber gasket; D, porous filter paper supporting sample film; E, swivel bolts, F, mercury storage reservoir; G, calibrated portion of instrument; H, Kovar seals; I, Demi-G valve; J, gas supply tube; K, to vacuum; L, wires to recorder; M, glass supporting clip. Reprinted from Bixler and S ~ e e t i n by g ~ permission ~ of John Wiley & Sons, Inc.

plement but less sensitive than the barometric approach, and so is rarely used for high-precision time lag measurements. Brubaker and K a m m e r m e y e ~ describe -~~~ a simple volumetric device that is reminiscent of the apparatus used by GrahamI3 in his pioneering 1866 study. The penetrant permeates through a film into a precision-bore glass capillary tube containing a small slug of mercury, and the permeation rate is determined from the velocity of the slug. The tube is vibrated slightly to keep the mercury from sticking. MajorZS1describes a portable system of this type that weighs less than one pound, and requires us B. G. Harper, J. Appl. folym. Sci. 1, 50 (1959). T. W. Sarge, Anal. Chern. 19, 396 (1947). A. C. Shuman, Ind. Eng. Chem., Anal. Ed. 16, 58 (1944). g48 V. L. Simril and A. Hershberger, Mod. f l u s f . 27, 97 (1950). H. R. Todd, Mod. f a c k a g . 18, 124 (1944). D. W. Brubaker and K. Kammermeyer, Ind. Eng. Chem. 44, 1465 (1952). C. S. Major, Mod. Puckag. 36, 119 (1963). 28 S . A. Stern, P. J. Gareis, T. F. Sinclair, and P. H. Mohr, J. Appl. f o l y m . Sci. 7, 2035 (1963).

17.7

INTEGRAL PERMEATION METHODS

365

no electrical connections, vacuum sources, or membrane-changing equipment. The ASTM standard volumetric method, which like its constantpressure counterpart is designated D1434-75,240 uses the Linde cell (Fig. 9) originally developed by Stern and co-workers.252 The procedure calls for operation with positive pressures on both sides of a membrane supported on a filter paper. The permeating gas displaces a slug of fluid in a calibrated capillary tube; 4-methyl-2-pentanone is recommended as the fluid in preference to mercury, owing to the relatively high vapor pressure and contact angle hysteresis characteristic of the latter substance. After steady state is achieved, the slug velocity is measured using a cathetometer and a stopwatch, and the permeation rate is calculated as the product of the velocity and the cross-sectional area of the tube. Different tube diameters can be used to accommodate a widely varying range of permeation rates. The cell should be contained within a liquid bath that

FIG. 9. Linde cell. Reprinted from Bixler and Sweeting'@by permission of John Wiley & Sons, Inc.

366

17.

GASES A N D VAPORS

*

can control the temperature to 0.1"C. Detailed operating and calculative procedures are given in the ASTM reference.240 17.7.3.3. Analytical Techniques. Many analytical techniques and devices have been used to determine penetrant concentrations in the receiving volumes of closed permeation cells. They include labeling of penetrant molecules and subsequent concentration determination by radiometric method^,^^-^^^ infrared absorption spectros~opy,~~' electrolytic decomposition of water vapor,258measurement of refractive index using an inte~ferometer,~~~ maSs spectroscopy,260*361 Orsat analysis,262and thermal conductivity measurement using a platinum resistance thermometerZeor a thermistor.2s3 Kaess2wallowed odorous substances to permeate into a closed volume, and after ten days had a panel of impartial observers rate the intensity of the odor on a scale of 0 to 4. Hoffman and co-workers2%also studied the permeation of odorous substances, using a less democratic but more precise radiometric method to analyze for penetrant concentrations. 17.7.3.4. Gravimetric Techniques. In a method used primarily for water vapor transmission measurements, a desiccant is sealed in a container partially or entirely enclosed by the test material. The container is exposed to the penetrant atmosphere, and its weight is monitored with time. Many variations of this method are described by Newns,44and dozens of examples of its use are cited in bibliographies published by the Institute of Paper Chemistry.265One version is the ASTM standard method, designated E96-66, for water vapor transmission of materials in sheet form.266 W. Hoffman, H. Kramer, and V. Linowitski, Chem.-1ng.-Tech. 37, 34 (1%5). V. Linowitski and W. Hoffman, Kunsrstofle 55, 765 (I%% *= A. D. Kirshenbaum, A. G. Streng. and W. B. Dunlap. Jr.. Rubber Age 74,903 (1954). A. I. Abrams, Y.Y.Linde, L. L. Pelekis. and I. Y. Taure. Izv. Akad. Larv. SSR, Ser. Fiz. Tekh. Nauk. No. 3, p. 41 (1965). *57 R. M. Husband and P. J . Petler, Tappi 49, 565 (1966). zw H. J. Lelie, T N O Nieuws 22, 345 (1967). J. Hanousek and J. Zak, Obaly 6, 183 (1960). C. D. Bailey, W. D. Holland, and J. Hulsebos, SAE Prepr. 746D (1%3). H. Eustache and P. Jacknet, Mod. Plast. 45, 163 (1968). pd) A. H. Landrock and B. E. Proctor, Tappi 35,241 (1952). ma H. Yasuda and K. J. Rosengren, J. Appl. Polym. Sci. 14, 2839 (1970). G . Kaess, Z. Lebensm.-Unrers. -Fortsch. 90, 107 (1950). *aC. J. West, W. B. Kunz, and G. R. Sears, eds., "Permeability of Organic Materials to Gases," Bibliogr. Ser. No. 169, Vols. I and 11. Inst. Paper Chem., 1948; S . Wilkinson, C. L. Brown, and J: Weiner, eds., ibid.. Suppl. I, 1956; J. Weiner and L. Roth, eds., ibid.. Suppl. I, 1%4: ibid., Suppl. 111, 1970. ma American Society for Testing and Materials, "1976 Annual Book of ASTM Standards," Part 35, p. 823. Am. SOC.Test. Mater., Philadelphia, Pennsylvania, 1976. *I

IM

17.8.

DIFFERENTIAL PERMEATION

3 67

The ASTM procedure consists of placing a desiccant in a wide-mouth dish, and sealing the opening over the dish with the membrane. Dried anhydrous calcium chloride that passes a no. 8 (2.36 mm) sieve but not a no. 30 (600 pm) sieve is recommended, with the moisture gain during the test being limited to 10% of the starting weight. If CaCl, reacts chemically with the membrane material, activated silica gel or a similar absorbing desiccant may be used, but the moisture gain should be limited to 4%. During the test, the dish is placed in a chamber in an atmosphere of controlled temperature and humidity. The dish is normally inverted so that the desiccant is in direct contact with the membrane, but an upright position may be used if there is a danger of breaking the seal or damaging the membrane. The dish should be shaken before each weighing to assure absorption of all of the transmitted vapor, and if possible, the weighing should be performed in the controlled-atmosphere chamber. The gravimetric procedure is suitable for measurements of relatively high steady-state permeation rates. For low permeation rates and time lag measurements, one of the techniques described in the previous sections or in Chapter 17.8 should be used.

17.8. Differential Permeation and Weighing Cup (Open Receiving Volume) Methods 17.8.1. Experiments and Data 17.8.1.l.Continuous-Flow Cells. A penetrant is introduced into a chamber on one side of a flat membrane or on the inside or outside of a hollow tube, and permeates through the polymer into a gas stream. The upstream chamber may be open or closed; if closed, it may contain a pure gas, a gas mixture, or a liquid. The permeation rate is obtained by analyzing the downstream effluentgas to determine the penetrant concentration, and multiplying the concentration by the gas flow rate. A typical continuous-flowpermeation system is shown in Fig. 10, and data obtained from this system are shown in Fig. 11. The differential continuous-flow approach has several advantages over integral closed-volume methods.lSo Measuring an instantaneous rate of permeation rather than a total penetrant mass permits the attainment of a true steady state, and yields data relatively free of cumulative errors. Equal total pressures can be maintained on both sides of the membrane, minimizing support and sealing requirements, and the motion of the gas on one or both sides of the membrane diminishes the effects of gas phase mass transfer resistance. Upstream penetrant concentrations may be ad-

3 68

17.

GASES AND VAPORS

- - - - - - -THERMOSlAlED - - - - - - - - - - - OVEN - - - - - - --- -OllUTlON AIR

FIG.10. Continuous-flow permeation apparatus. Modified and reprinted from Rodes er by permission of the American Chemical Society.

a/.*"

TIME (min)

FIG.1 1 . Data from continuous-flow permeation experiments-SO,

in fluorosilicone rubber. Reprinted from Felder er u / .ls3 by permission of the American Institute of Chemical Engineers.

17.8.

DIFFERENTIAL PERMEATION

369

justed over wide ranges by varying pressures and stream dilution ratios, and any convenient separation and analysis technique can be used to determine concentrations of single penetrants or penetrant mixtures in the downstream effluent. On the other hand, the closed-volume approach is advantageous when the downstream concentration is near the lower detection limit of an available instrument, or if no convenient analytical technique is available. Several comparisons of closed- and open-volume techniques have been published. Lyssy et ~ 1analyzed . ~ test ~results ~ from seven different laboratories, and found that isostatic measurements consistently yielded less scatter than closed-volume vacuum techniques. Yasuda and Rosengren2Mfound the open-volume approach better for high permeation rates, but obtained better accuracy for low permeation rates using a closed receiving volume, a result probably attributable to the detection limits of the thermal conductivity device used to measure penetrant concentrations. 17.8.1.2. Weighing-Cup Technique. A liquid is placed in a container partially or entirely enclosed by a membrane, and the rate at which the liquid species permeates is determined by intermittently or continuously weighing the container. This technique is useful for measuring steady-state permeation rates of saturated vapors in relatively permeable materials. When it is applicable, it is by far the simplest of all permeation rate measurement methods; it is also intrinsically the least accurate. 17.8.2. Calculations

At time t = 0 a penetrant with partial pressure p1 is introduced on one side of a flat membrane of thickness h and area A, or on the outside of a hollow cylindrical tube of length L and inner and outer radii a and b. The penetrant emerges on the downstream side of the polymer into a flowing stream, and the permeation rate 4 cm3(STP)/sec is measured as a function of time. If diffusion is Fickian, sorption follows Henry’s law, and both D and S are independent of concentration, then the steady state permeation rate is

(flat membrane), (PI - p z )

*07

(hollow tube).

G . P. Lyssy, P. Hieke, and H. Mohler, Tura 17, 465 (1%5).

(17.8.1) ( 1 7.8.2)

370

17. GASES A N D VAPORS

The permeability P ( = SD)may be determined from the measured value of 4mand the measured penetrant partial pressures (usually neglecting pz). Three methods have been proposed for the estimation of the diffusivity D from a curve of the form shown in Fig. 1 1 . In the procedures to be described, 4 need not be the permeation rate itself but may be any measured quantity proportional to it, such as the penetrant concentration in the gas emerging from the permeation cell. The simplest estimation method is to note the time t1,2 at which 4 reaches half of its asymptotic value For a flat membranezm (17.8.3)

and for a hollow tube with inner and outer radii a and

D - ( b - ajz [1 7. 199t1/2

+

o . o ~ i ( b- a ) z ] ab

(17.8.4)

Equation (17.8.4) is accurate to within 0.2% for b / a S 5 . A second method is to utilize an asymptotic solution of the diffusion equation valid at small times. As shown by Rogers, Buritz, and Alpert,zo8 a plot of ln(4/7> vs. l/r at small times is a straight line with slope - h2/4D. (For a tube, h may be replaced by b - a and the same formula used.) The advantage of this method is that it does not require knowing the value of &: however, it requires that the initial time t = 0 be known with precision. Moreover, as shown by Yasuda and Rosengren,zs3the values of D estimated in this manner appear to be more sensitive to the gas flow rate than are the values obtained by the half-time method. Pasternak, Schimscheimer, and HellerZeO outline several other smalltime estimation techniques for flat membranes. Their techniques require a knowledge of &, but appear to be valid over a wider range of times than is the method of Rogers et A third estimation technique is a moment method proposed by Felder who showed that a quantity T ~ defined , as and 7p =

lom y] [l -

dr,

(17.8.5)

equals the time lag of the closed volume experiment. Once T~ is calculated by numerical integration, the diffusivity may be estimated as

MB

K . D. Ziegel, H. K. Frensdorff, and D. E. Blair, J. Polym. Sci., Purr A-2 7,809 (1969). R. A. Pasternak, J . F. Schirnscheimer, and J. Heller, J. Polym. Sci.. Purr A-2 8, 467

(1970).

17.8. DIFFERENTIAL PERMEATION

D

(flat membrane), 0 c x G h, (17.8.6)

= hz/6rp

D =

u2

-

*'

371

(a2 + b2) ln(b/a) (hollow tube), a c r c b. 4rP ln(b/a) +

(17.8.7)

The data shown in Fig. 11 were analyzed by the moment method to determine the diffusion coefficients of SOz in a fluorosilicone rubber tube at three temperatures, and the estimated values were substituted into the solution of the diffusion equation to generate the calculated responses shown on the same figure.133 The moment method has several attractive features. It requires simple numerical integration rather than curve-fitting; it utilizes the complete response, rather than a single point (as in the half-time method) or a portion of the response that falls within the region of validity of a small-time asymptotic solution of the diffusion equation; and it is readily applicable to both planar and cylindrical membranes. Perhaps its greatest advantage, however, is the degree to which it enables the dynamics of system components other than the membrane to be factored out of the measured response,133a point discussed further in Chapter 17.9. 17.8.3. Experimental Methods 17.8.3.1. Permeation Cells. The geometry of a cell of the continuous dialyzer (as opposed to the weighing cup) type may be that of any conventional heat exchanger or membrane separation device: two cavities separated by a flat membrane, membrane-spacer stacks, concentric tubes with the inner tube being the membrane, or hollow-fiber bundles. The latter geometry provides a large surface-to-volumeratio, and should be particularly useful for the measurement of low permeation rates. The design of a flat membrane cell is described by Yasuda and Rosengren,ZsSand the design of a cell for a single hollow tube is given by Rodes, Felder, and Ferre11.270 Weighing-cup methods have been used almost exclusively to measure steady-state permeation rates of water vapor through relatively permeable membranes. A version of the weighing-cup method is part of the ASTM standard technique for water vapor transmission measurements, designated E96-66.2sB A layer of water 3-5 mm deep is placed at the bottom of a shallow wide-mouth dish, and the top of the dish is sealed with the membrane; air at a controlled temperature and humidity is circulated over the membrane, and the dish is weighed periodically to determine the water permeation rate. Another version of the weighing-cup C. E. Rodes, R. M. Felder, and J. K. Ferrell, Environ. Sci. Techno/. 7, 545 (1973).

'"M. Kondrup and J. Mogens, Konserves 5, 33 (1947).

3 72

17.

GASES AND

VAPORS

method was used by Kondrup and M ~ g e n s , who ~ ~ ' wrapped frozen cod fillets in plastic films and measured the subsequent weight loss with time. 17.8.3.2. Measurement of Penetrant Concentrations. In a cell of the continuous-flow type, the permeation rate is determined by measuring the penetrant concentration in the downstream chamber effluent, and multiplying by the known volumetric flow rate of this stream. The same method may be used for a cell of the weighing-cup design when the permeation rate is too low to be detected gravimetrically. In principle, any convenient wet chemical or instrumental method may be used to measure the penetrant concentration. FrenzelZ7in 1914 used an interferometer for this purpose, and Shakespear and DaynesZ9in 1920 used a platinum wire thermal conductivity detector. The latter technique served as the basis of the Cambridge fabric permeator, a commercial flow cell described in a 1941 publication.272 Among the analytical instruments used recently are continuous hyg r o m e t e r and ~ ~ a~ P205 ~ ~ ~electrolytic ~ ce1P4 for water vapor detection, thermal conductivity d e t e c t i ~ n ,a~gas ~ .chromatograph ~~~ with a thermal conductivity detector for mixtures of penetrants,276a helium detector gas chromatograph for detecting oxygen and other penetrants at part-perbillion levels,277and a flame photometric detectorZ7Oand an ultraviolet photometer278for measurements of SOz permeation rates.

17.9. Sources and Minimization of Errors 17.9.1. Operating Procedures

The most common source of error in a permeation rate measurement is leakage between the upstream and downstream compartments of the cell. Pressure or vacuum leak tests should be performed periodically, and if a leak is detected, all data obtained since the last successful test should be regarded with suspicion. When the permeation rate through a membrane is measured, tests should be performed at two or more upstream pressures. If the calculated permeability changes by an abnormally high amount when the pressure is increased, a leak in the membrane or the membrane seal is indicated. If the permeation rate varies in proportion to the total pressure Anon, India Rubber World 104, 55 (1941). M. Karel, Y. Arkawa, and B. E. Proctor, Mod. Packag. 29, 213 (1955). P. E. Toren, Anal. Chem. 37, 923 (1965). 275 R. A. Pasternak and J. A. McNulty, Mod. Packag. 43, 89 (1970). D. G. Pye, H. H. Hoehn, and M. Panar, J . Appl. Polym. Sci. 20, 287 (1976). T. L. Caskey, Mod. Plast. 45,477 (1967). E. G. Davis, M. L. Rooney, and P. L. Larkins, J . Appl. Polym. Sci. 19, 1829 (1975). *7s

17.9.

SOURCES A N D MINIMIZATION OF ERRORS

373

difference across the membrane rather than the partial pressure or penetrant activity difference, the transport mechanism is probably Knudsen flow through micropores rather than activated diffusion of dissolved penetrant, and the data should be analyzed a~cordingly.~ Complete and accurate characterization of the test membrane is important. If a flat membrane is used, its thickness should be measured at no fewer than five locations away from edges. If the barrier is a hollow tube, the outer diameter should be measured at several positions away from edges; a section should then be cut axially, and the wall thickness measured at several points away from the cut. The membrane should be sealed in such a manner that the effective area for permeation is well defined. As many physical characterization tests as possible should be performed on samples of the membrane to define precisely the material for which transport properties are to be determined. If a membrane support is used, such as a filter paper or wire screen for a plane sheet, or a porous, sintered stainless steel or ceramic tube for a plane sheet or thin-walled tubing, its porosity should be as high as possible consistent with its required mechanical properties. If possible, permeation measurements should be carried out using two supports with different porosities; if no difference in the permeation rate is observed, it may be concluded that the supports offer negligible resistance to mass transfer, and measured transport properties are therefore those of the membrane alone. If a flowing stream is involved in a permeation experiment, either in the upstream or downstream compartment, runs should be performed at several flow rates with all other conditions held constant. If the permeation rate is unchanged, gas-phase mass transfer resistance effects may be neglected; if the permeation rate changes, the flow rate should be increased until no further change is observed. A dependence of permeability on membrane thickness may also indicate significant boundary layer resistances, or it may reflect nonuniformities in the membrane struct ~ r e . *Mass ~ ~ transfer across a fluid boundary layer may of course be taken into account in the analysis of permeation data (it must be taken into account in liquid permeation systems) but having to do so introduces uncertainties in the analysis that are better avoided. 17.9.2. Data Analysis

Most permeation or sorption measurements are analyzed assuming Henry’s law sorption and Fickian diffusion with constant coefficients. *‘O S . T. Hwang and K. Kammermeyer, in “Permeability of Plastic Films and Coatings to Gases. Vapors, and Liquids” (H.B. Hopfenberg, ed.), p. 197. Plenum, New York. 1974.

374

17.

GASES A N D VAPORS

As has been observed in the preceding sections, deviations from this model are almost the rule rather than the exception. Experiments should be performed to detect such deviations before calculated values of P, D, or S are accepted. Deviations from Henry's law are best detected by observing nonlinearities in measured equilibrium sorption isotherms. Concentrationdependent Fickian diffusion is indicated by a pressure-dependent permeability, or a hysteresis effect in a sorption-desorption cycle. The occurrence of anomalous (non-Fickian) diffusion is revealed by a sorption rate measurement in which a plot of penetrant uptake vs. is not initially linear and concave to the time axis, but ratherappears as shown in Fig. 2d or 2e. The common assumption that the downstream penetrant concentration in a closed-volume experiment is negligible may break down if the receiving volume is relatively small, as it is in the Dow and Linde cells (Section 17.7.3), or when the penetrant is a substance with a low vapor pressure at the temperature of the experiment. The correction procedures derived from the results of Paul and DiBenedetto202summarized in Section 17.7.2.1 should be applied routinely in such cases to correct for nonzero downstream concentrations. Alternatively, if only membrane permeabilities (as opposed to diffusion coefficients) are to be measured, the downstream chamber may be pumped out for a time long enough to establish a steady-state permeation rate, and the receiving volume may then be closed to commence the rnea~urement.~~ If these measures fail to reduce the potential estimation error to a suitable level, the experiment should instead be performed using a device of open receiving volume design (Chapter 17.8). Incorrect fitting of the asymptotic line in a time lag plot (Fig. 7) leads to errors in estimated values of P and D. The necessary truncation of the curve at a finite time leads to underestimation of the asymptotic slope and hence of the permeability, and overestimation of the diffusion coefficient. Random scatter in the data also leads to estimation errors. Petropoulos and MyratZa0discuss these errors in detail, using statistical analysis to correlate them with the values of adjustable or measurable parameters, including the applied penetrant concentration, the receiving chamber volume, the number of data points and time interval between each point, the type of pressure gauge used, and the gauge sensitivity. The results are used to derive experimental conditions that minimize the expected values ' of estimation errors of the types considered. Adsorption of penetrant on the walls of the receiving volume of a permeation cell may cause a significant increase in the value of a measured ma J.

H. Petropoulos and C. Myrat, J . Mombr.

Sci. 2, 3 (1977).

17.9. SOURCES

375

AND MlNlMlZATlON OF ERRORS

time lag, and hence an error in the estimated diffusion coefficient. The magnitude of this effect is indicated by the degree to which the estimated value of D depends on the size (more precisely, on the exposed surface area) of the downstream chamber. Yasuda and Stannett,e5in a study of special problems associated with water vapor transport measurements, discuss methods of minimizing errors due to vapor adsorption. They observe that such errors can be nearly eliminated by pumping the downstream chamber contents continuously into a prewetted receiving volume. The pumping keeps the water used to presaturate the chamber walls from sorbing into the membrane, and precludes a back-pressure buildup in the downstream chamber, with an attendant decrease in the driving force for permeation. An alternative approach is to coat the receiving volume walls with a nonadsorbing film. Another problem associated with water vapor transport is the temperature increase that accompanies sorption-an increase attributable both to the high heat of condensation of water vapor and the heat associated with hydrogen bonding. Stannett and ~ o - w o r k e r have s ~ ~investigated ~~~ this effect for the sorption of water in fibers and films. If D, is the value of the diffusion coefficient estimated assuming isothermality, then a correction factor ( D I D 3 may be expressed in terms of a dimensionless number X,

X

=

Ha/LWpD,

(17.9.1)

where H is the coefficient for heat transfer from the polymer surface to the surroundings, a the fiber radius or film half-thickness, L the heat of sorption, W the temperature coefficient of the equilibrium sorption regain, p the fiber density, and D the true value of the diffusion coefficient. Yasuda and Stannettes state that a value of H 2 1.5 x lo-* cal/cm2 sec "C was obtained for sorption of water in both wool fibers and ethyl cellulose films; they also outline methods for determining the parameters L and W from sorption isotherm data. The percentage error in D resulting from the temperature rise vanes approximately inversely with X,and is on the order of 2% when X 100. The error can be reduced by increasing the fiber or film thickness (so that the time during which most of the temperature rise takes place is a small fraction of the time required for diffusion), and by decreasing the change in the upstream partial pressure during each run.

-

-

2

17.9.3. System Dynamics

In closed-volume barometric or volumetric permeation experiments, the measured variable provides a direct indication of the amount of penetrant in the receiving volume. The data (Fig. 2) therefore approximate A. A. Armstrong and V. Stannett. Makromol. Chem. 90, 145 (1966).

376

17.

GASES A N D VAPORS

the transient response of the membrane alone to a step change in penetrant concentration at the upstream boundary. On the other hand, if an instrument such as a thermal conductivity detector is used to measure the penetrant concentration, an extraneous lag is introduced into the measured signal. The lag is attributable in part to the time required for the penetrant to diffuse and mix with the gas in the region of the detector probe, and in part to the dynamics of the detection instrument. The problem is compounded if the penetrant is swept out of the receiving chamber to the detector, in which case the residence time in the gas line adds another lag to the measured signal. Moreover, since the sample gas must be fed into the upstream chamber from an external source, lags are imposed due to the residence time of the gas in the feed line and the mixing time in the chamber. In order to determine the correct value of the diffusion coefficient, the dynamic response of the membrane alone must somehow be factored out of the total system response. Many investigators assume, explicitly or tacitly, that the extraneous lags are negligible; others treat them as pure time delays, and simply shift the measured response signal horizontally before calculating the time lag, half-time, or any other quantity used to estimate the diffusion coefficient. DayneP in 1920 appears to have been the first to perform an analysis of the dynamics of a time lag experiment. He showed that the response of his thermal conductivity probe followed a first-order (exponential) decay formula, independently measured the time constant, and subtracted it from the measured time lag before calculating the diffusion coefficient. When several lags occur in series, or when components are used whose responses cannot be modeled as pure delays or first-order lags, the analysis required to isolate the membrane response from the total system response is generally complex, requiring either numerical Fourier transformation of measured signals or iterative evaluation of convolution integrals. Fortunately, it is possible to calculate the correct time lag without performing the deconvolution, by following the procedure outlined beloW130.132.133: 1. Assign a lag time 7i to each gas line, chamber volume, measurement device, and recording instrument between the penetrant flow source and the &vice from which the measured signal is obtained. 2. Calculate the values Of Tifor all components but the membrane. For a flow-through component, such as a feed or effluent line or the upstream or downstream chamber in a permeation cell, ri is simply the mean residence time (volume/volumetric throughput rate). For a component such as a gas analyzer or recorder, Tfmay be determined from the results of a

17.9.

377

SOURCES A N D M I N I M I Z A T I O N OF ERRORS

calibration experiment in which the response to an impulse or step input is measured. If an impulse is applied at the device inlet (i-e.,a pulse of negligible duration on the time scale of the response) and RAt) is the response, then

(17.9.2)

If a step response Rs(t) is measured and R , is its asymptotic value, then (17.9.3) 3. If the permeation run is carried out with an open receiving volume and the response +(t) appears as shown in Fig. 1 1 , the total response moment 7; may be calculated as 7;

=

lom $1 [I

-

dr.

(17.9.4)

The moment due to the membrane alone is then calculated as133 7)) = 7;

-

(17.9.5) all components but membrane

and substituted into Eq. (17.8.5) or (17.8.6) to determine the diffusion coefficient. 4. If a closed-volume permeation measurement is performed and 8' is the time lag obtained from a plot of the form of Fig. 7, then the time lag for the membrane alone is 8=8'-

2

71.

(17.9.6)

all components but membrane

The diffusion coefficient is then calculated from 8 using Eq. (17.7.2). Acknowledgments The authors acknowledge with gratitude the contributions of Professors Vivian Stannett, Donald Paul, and William Koros, and Dr. David Enscore, who reviewed the manuscript and offered many corrections and suggestions for its improvement; Professor Stannett also provided many of the historical references cited in Chapter 17.2. Thanks are also extended to Ms.Penny Horkan for expert assistance with the manuscript preparation.

This Page Intentionally Left Blank

18. ELECTRICAL METHODS 18.1 Dielectric Constant and Loss

By Richard H. Boyd 18.1.1. Introduction

In general the chemical bonds between unlike atoms in polymer molecules possess permanent electric dipole moments. Many polymers have chemical structures such that the bond moments can vectorially accumulate into molecular or segmental moments in many molecular configurations or conformations. These polymers can therefore by virtue of these moments be polarized by an electric field. They are said to be “dielectrically active,” that is, they show polarization due to orientation of permanent dipoles. In some chemical structures the bond moments vectorially add to zero in all important conformations and the polymer is said to be dielectrically inactive since no polarization due to permanent dipole orientation will be induced by a field. The measurement of the polarization induced in dielectrically active polymers has proven to be an extremely useful method for probing polymer structure. The equilibrium polarization induced by a static field can give information concerning the equilibrium structure of the system including averaged conformational properties, which in turn depend on the accessible conformational states and their relative energies. Information can also be deduced concerning the strength and nature of interactions between chains. The time-dependent polarization in response to a changing field is especially interesting because the rate of dipole orientation is controlled by internal motions of the polymeric chain. Thus the response to time-dependent fields becomes a method for studying chain dynamics. Some advantages of the dielectric method are the following. Reasonably straightforward formulas have been derived from Statistical mechanics and electrostatics that connect the magnitude of the induced equilibrium polarization with microscopic quantities such as the number and magnitude of dipole moments and the energetics of interaction between 319 METHODS OF EXPERIMENTAL PHYSICS, VOL. 16C

Copyright @ 1980 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-475958-0

3 80

18.

ELECTRICAL METHODS

FIG.1 . Dielectric constant (c') and loss ( Q " ) of nylon 6.10 as a function of frequency and temperature. The data were taken using two bridges (1-100 kHz and 100 kHz-3 MHz), a resonant circuit (1-80 MHz), and a slotted line (0.3-8 GHz) (from Boyd and Porter').

moments. Thus it is often possible to deduce quantitative molecular information from dielectric measurements. Perhaps the most important advantage, however, is an experimental one. It is, relative to most other techniques, possible to vary conveniently the frequency of the perturbing field over very wide ranges. Thus it is possible to detect or follow molecular motions over a very wide range of time scales from slow to extremely rapid by the same basic technique. It may require a number of experimental configurations to accomplish this. However, in some parts of the time scale a single apparatus may sometimes serve for measurements over a number of decades of frequency. Since the molecular motions in a given polymer tend to have a very broad range of relaxation times at constant temperature and are thermally activated, often with large activation energies, the capability of broad frequency range of measurement is a very important consideration (see Fig. l).l 1

R. H. Boyd and C. H. Porter, J . Po/yrn. Sci.. Purt A-2 10. 647 (1972).

18.1.

DIELECTRIC CONSTANT AND LOSS

38 1

In what follows below we first give a brief account of the phenomenology of dielectrics in order to establish the connection between the material properties sought after and the experimental parameters measured. Then the methods available for experimental investigation are introduced. 18.1-2. Phenomenology of Dielectrics

In order to provide definitions of measured quantities and to establish the connection with measuring techniques, the macroscopic phenomenological theory of dielectrics2” is discussed here. 18.1.2.1. Static Fields. The electric field E in electrostatics may be defined as the force per unit charge acting on an indefinitely small test charge placed in the field. A closely related quantity is the displacement field D, which may be defined through Gauss’ law (in integral form) as @6*dds=q,

(18.1.1)

where q is the charge within a volume enclosed by the surface S. If Eq. (18.1.1) is applied to a single point charge in vacuum by integrating over a sphere of radius R, and if we keep in mind Coulomb’s law, then in vacuum evidently E and D are related as D = L = K ~ E (free space). 4rR2

(1 8.1.2)

Since the units of charge are defined independently of E (the force per unit charge), a constant K~ having the units of (charge)2/work/length and called the permittivity uffree space is required to relate the force between the unit charge and the charge q . In the units coul/V/m (rationalized MKS or SI), it has the value 8.854 x F/m. Application of Eq. (18.1.1) to the case of two infinite plates in vacuum containing surface charges q of density u = q / A is straightforward when it is realized that 5 must, from the symmetry of the problem, be directed normal to the plates (Fig. 2). Integration of Eq. (18.1.1) gives D @ dS = uA

or D = u,

(18.1.3)

* H. Frohlich, “Theory of Dielectrics,’’ 2nd ed. Oxford Univ. Press, London and New York, 1958. C. P. Smyth, “Dielectric Behavior and Structure.” McGraw-Hill, New York, 1955.

18.

3 82

ELECTRICAL METHODS

D =qlA = d

D A = q

FIG.2. Displacement field between charged parallel plates,

and from Eq. (18.1.2) K&

=

(+

(free space).

(18.1.4)

If a dielectric material is inserted between the plates, which already contain surface charges u,the effect of the electric field is to inducepolarization in the material in such a way that the material remains neutral but induced charges uf appear on the plates in addition to the original ones (+ (see Fig. 3). Thus the electric field (still defined as the force/charge on a small test charge) between the plates is now

E

= ((+ -

(+’)/KO.

(18.1.5)

However it is the essence of the definition of the displacement field D that q in Eq. (18.1.1) means only ‘‘real” charges, deliberately placed as sources of external electric fields, and q by definition does not include charges induced in continuous dielectric media by external fields. Therefore, D retains its value in Eq. (18.1.3). Thus comparing E and D we now

have K~= E (a- u’),

D = u.

The induced surface charge density u’is often called the polarization and given the symbol P, so that D =

K&

+ P.

( 18.1.6)

Since (Au’)d/u = P (where u is the volume), the polarization is the induced dipole moment per unit volume and is uniform through the dielectric. For the usual field strengths, P is proportional to E and P = XK~E,

( 18.1.7)

18.1.

383

DIELECTRIC CONSTANT A N D LOSS * d - 6'

(a 1 + 6'

- 6

0 @ @ 8 0 8 @

-6

FIG.3. Induced surface charges (a') appearing from polarization of a dielectric in an electric field (arising from placed surface charges u):(a) induced electronic or atomic polarization, (b) polarization by spatial orientation of permanent dipoles.

where

x is called the susceptibility D

and therefore

+ x)K~E.

= (1

We now define the dielectric constant

D

as

P(uOE)

E

(18.1.8)

(18.1.9)

or c=l+x.

Notice that for free space x = 0 and defined as

= 1.

Since the capacitance C is

C = q / V = -uA V

and V = Ed = du/uoc,where V is the voltage across the plates, then E = -

Cd KOA'

(18.1.1Oa)

or =

c/c,,

(18.1. lob)

where Co is the capacitance of he condenser l..!iththe dielectri removed. Either of the above two equations can serve as an experimental basis for measurement of P. It is characteristic of dielectric measurements that the desired material property, the dielectric constant, is determined by establishing a relationship between it and the parameters (capacitance) of circuit elements constructed from the dielectric.

3 84

18.

ELECTRICAL METHODS

Gauss’ law also serves to establish the connection between capacitance and dielectric constant for various other geometries. Another commonly used one is the coaxial capacitor (Fig. 4). The displacement field for an infinite capacitor has cylindrical symmetry and constant values of D for a given radius r , or dS = 9 ,

D

= EKOE= 9 / 2 rl, ~

and

from which it follows, since C = q/lVl, that (18.1.11)

18.1.2.2. Time-Dependent Fields.2 For the case of a time-dependent field, the quantities D, E, and P in Eq. (18.1.6) will be time dependent.

Due to resistance to motion of charges there will be a delay in P in responding to a change in E . An arbitrary time-dependent field E(t) at the present time t can be regarded as built-up from incremental changes in E , dE, applied over past times u , or (18.1.12)

It is assumed that the polarization follows the applied field through a time-dependent relaxation (or more properly retardation) function equal to the susceptibility in the static case [see Eq. (18.1.7)], or dP(t) =

KoX(f

- U ) dE(u),

(18.1.13)

where the left-hand side is the increment of polarization resulting at time t

-1-

INNER CONDUCTOR

(+GI

DlELEC TRlC OUTER CONDUCTOR (-<)

FIG.4. Displacement field in a coaxial capacitor.

18.1. DIELECTRIC CONSTANT A N D

385

LOSS

from application of a field dE(u) at past time u. Therefore, from Eq. (18.1.6)

The assumption that the increments of P in Eq. (18.1.13) may be summed to achieve Eq. (18.1.14) is a very general assumption but nevertheless is an assumption known as the (Boltzmann) superposition principle. Thus we see that there is a close analogy between the relationship of E to D (or CT) and stress and strain in mechanical experiments (see Part 11). An important case is that of a periodic field of angular frequency w. Since Eq. (18.1.14) is linear in E, we may write E in complex notation and equate the real part of Eq. (18.1.14) to D(t),after the integration is carried out. Substituting E* = Eoefouin Eq. (18.1.14) results in

1

KOEOefot + KoEoeior ioX(u‘)e-fou’du’ , (18.1.15)

where u’ = t - u and Re means “real part of.” It is convenient to define a complex dielectric constant such that ( 18.1 .16)

D(t) = Re[c*~oE&“‘]

or c* = 1

fOM

+

iwx(u’)e-iou’du’.

The real and imaginary parts of €*

written as = E’ - ie’’

(18.1.17)

E*

(18.1.18)

are commonly referred to as E’, dielectric constant, and E”, dielectric loss factor. They are given as functions of ~ ( u ’as ) c’ = 1 E” =

-

+

o fom

xosin

out

du‘,

(18.1.19)

KO

1 x“

cos ou‘ du.

o

(18.1.20)

KO

Both the real and imaginary parts c’ and E” have physical significance. Notice that Q* is a constant for fixed o and does not contain time as a variable. Since this is true, it is also convenient to write D*(f) =

€*KOE*(f),

(18.1.21)

where D* = c*KOEOeiot.If we write E*

= pe-f8

(18.1.22)

3 86

18.

ELECTRICAL METHODS

where p =

tan6 =

(€12

E”/E’

+

€’’2)112

(18.1.23)

= dissipation factor,

(18.1.24) (18.1.25)

Thus the displacement field has the same frequency as E but is phaseshifted through the angle 6 given by Eq. (18.1.24). We now consider how in principle E’ and E” are related to experiment. In the static case, Eq. (18.1.10), we could write for the parallel-plate capacitor, V = qd/K&.

For complex periodic K ~ E * E= * D* = 4*lA9

V* = VOeior , we

voltage

E*V* = q*d/KoA.

( 1 8.1.26)

have

from

(18.1.27)

Differentiating both sides with respect to time to obtain the currentj gives

“ d -.1 E*V* = L K ~ iAw

(18.1.28)

Therefore, the complex impedance Z* defined by v*/z* = J”

(18.1.29)

K A _1 -- iw 0 E*,

(18.1.30a)

is given by

Z*

d

or in terms of the empty-cell capacitance Cot 1/Z* = iwCoe*.

(1 8.1.30b)

Thus the problem of determining E* = c’ - id‘ becomes that of determining the impedance Z* of parallel plates filled with the dielectric (of course, other geometries may be used). Detailed methods for accomplishing this are deferred to Section 18.1.3. 18.1.2.3. Representations of the Relaxation Function and Complex Dielectric Con~tant.S*~ Up to now our treatment has been general and no assumptions other than the linear superposition of polarizations induced from past times embodied in Eqs. (18.1.13) and (18.1.14) have been made.

’

N. G. McCrum, B. E. Read, and G. Williams, “Anelastic and Dielectric Effects in Polymeric Solids.” Wiley, New York, 1%7.

18.1.

DIELECTRIC CONSTANT A N D LOSS

387

In principle, sufficiently detailed measurements of E* at various frequencies would serve to determine X ( t ) , since according to Eq. (18.1.17) the complex dielectric constant is essentially the Fourier transform of the time derivative of x(r). However, in the presence of finite data points, it has become the custom to use various empirical formulations that connect E* and x in terms of a few parameters that are determined by fitting the data (E’ and d’ at a number of frequencies), 18.1.2.3.1. THE SIMPLERELAXATION TIMEMODEL. In specifying the nature of ~(t), the simplest assumption to make is that the rate of change of dipolar polarization is proportional to its displacement from equilibrium (and that the electronic and atomic polarizations respond instantaneously in the time scale of interest). Thus, a simple exponential retardation results,

&/du

=

-Ux

- xo),

or

x(t

- u)

= xI +

xo(l

-

e-(t-u)’r),

(18.1.31)

where xIis the instantaneous susceptibility, xo the equilibrium dipolar susceptibility, and T a relaxation time. Substitution in Eq. (18.1.17)results in €*

=

(I

1 + XI) + 1 + iwrxo.

(18.1:32)

The first term in Eq. (18.1.32)is the portion of the dielectric constant due to polarization induced fast compared to the time scale of the measurement and would be attained even if the frequency were so high as to exclude contribution due to time-dependent relaxation (i.e., even if w >> l/d. This term then is usually called the “unrelaxed” dielectric constant, cU. The eventual value at long times or low frequency (i.e., w << 1/7) is usually called the “relaxed” dielectric constant. Therefore, Eq. (18.1.32)may be written E*

=

€0

+ (QR

- bU)/(1

+ iw7).

The real and imaginary parts of z* written as are

(18.1.33)

388

18. ELECTRICAL METHODS

f -2

-I

a

I

2

LOG wr,, FIG.5. Dielectric constant (c’), loss factor (c”), and distribution of relaxation times F(ln T ) for various values of p in the Cole-Cole distribution. For p = 1 , the curves are for the single relaxation time or Debye equations.

These are plotted in Fig. 5 . The above equations are known as the Debye equations since they were first derived and used by him in his molecular theory of “anomalous dispersion” of polar liquid^.^ However, they are now taken to mean any single relaxation time process of this form. Two parameters (in addition to eu) are required to characterize the dipolar relaxation, a relaxation strength (defined as eR - eU) and a relaxation time T. While dipolar relaxation in many simple liquids obeys Eq. (18.1.33) following the early observations of FUOSS~ and Baker and Yager,7it has nearly always been found in polymers that the dipolar relaxation is considerably broader in the frequency domain than Eq. (18.1.33) implies (see Fig. 1 for example). 18.1.2.3.2. SPECTRUM OF RELAXATION TIMES.The most straightforward method of writing a more general relaxation function than the P. Debye, “Polar Molecules.” Dover, New York, 1947. R. M. Fuoss, J . Am. Chem. Sor. 63,2410 (1941). W. 0. Baker and W. A. Yager,J. Am. Chem. Sor. 64, 2171 (1942).

18.1.

389

DIELECTRIC CONSTANT A N D LOSS

exponential retardation above is to assume that parts of the polarization relax through different paths but that in each path the rate of relaxation is still proportional to the displacement from equilibrium. Thus Eq. (18.1.31) is generalized to (18.1.37)

and E* = EU

x

+

Acn/(1 + iwTn),

(1 8.1.38)

n

E'

=

EU

+ 2 AEn/(l + w ' T , ~ ) ,

(18.1.39)

n

E"

=

z

AE,,w n / ( l

+ w272).

( 18.1.40)

n

where Aen = xn0. The relaxation strength is given by ER

-

Eu =

2 AEn.

(18.1.41)

n

A useful relation concerning relaxation strength is found by computing,

=lr (ER - E U ) .

(18.1.42)

2

The area under the loss curve (plotted against In o or log v) is equal to a/2 times the relaxation strength. This is useful experimentally as a consistency check and also because the entire absorption region often cannot be measured and the loss vs. frequency curves are often more easily extrapolated than the dielectric constant curve. The finite spectrum is often replaced by a continuous one, usually based on a logarithmic time scale. For example, x(t - u ) =

xI + (eR - E

where E* = EU

r

~ )

+ (ER - E U ) J

0

F(ln ~ ) ( 1 - e-(t-u)'r)d In

F(ln ~ ) / ( 1 +

iwT)

d In

T,

7,

(18.1.43)

(18.1.44a)

18.

390

ELECTRICAL METHODS

where F(ln 7 ) is the normalized distribution of relaxation times. Again the problem of numerical inversion from a finite set of data points exists for determining F(ln 7 ) from E*(w). Approximate numerical methods used in viscoelasticity may be used (see Part 11). It has been more common in dielectric work to use parametized empirical functions for F(ln 7 ) . 18.1.2.3.3. COLE-COLEFUNCTION:^ This function is based on an empirical modification of the equation for the complex dielectric constant for the single relaxation time [Eq. (18.1.3311, E*

=

EU

+

(ER

-

EU)/~

+ (im0)@.

(18.1.45)

The effect of the parameter p is to broaden the loss curve compared to the single relaxation time ( p = 1) for values of 0 < p < 1 (see Fig. 5 ) . The dielectric constant and loss resulting from Eq. (18.1.45) are

The distribution function, Eq. (18.1.44a), which leads to Eq. (18.1.45) is

F(ln

7)

=

1 sin p.n 2.n cosh(p In 7 / 7 0 ] + cos pm‘

(18.1.48)

The parameters T ~p,, and eR - eu are determined by the following procedure. The equations for E’ and E” may be combined by eliminating W T ~ between them to give

(18.1.49) If

E”

8

is regarded as plotted against E‘ , this equation is a circle with the

K. S. Cole and R. H.Cole, J . Chem. Phys. 9, 341 (1941).

18.1. DIELECTRIC CONSTANT A N D LOSS

39 1

- eu) cot pw/2 and radius 3(eR - eU)csc @/2. center at (eR + 4 / 2 , Referring to Fig. 6, it may be seen that

sin

e = 1 (eR - eu)

1:

- (eR - eU) csc pw/2

or

e

( 18.1.50)

= pw/2.

Thus eR - eUis determined from the intercepts of the circular arc with the E’ axis and p from the angle 8. The parameter T~ is usually determined from the maximum in the loss curve vs. w ( o ~ = ~ l~) , T but~ can be back calculated from individual points (at w) on the circular-arc plot (Fig. 6)* using the relation V / u = ( o T ~ ) ~ . The plot of the imaginary vs. real parts of a complex function is known in general as an Argand diagram. When applied to the complex dielectric constant, the plot itself has become known in physical chemistry as a Cole-Cole plot regardless of the applicability of the circular-arc function for fitting the data. Such plots are extremely useful in checking the mutual consistency of dielectric constant ( e l ) and loss ( E ” ) measurements at a number of frequencies and are to be recommended whenever possible. 18.1.2.3.4. FUOSS-KIRKWOOD FUNCTION.^ This representation is based on an empirical modification of the loss equation for the single relaxation time [Eq. (18.1.36)] €”

=

-

2 € & ~ ( 0 7 0 ) ~ /+ 1 I t

emax

sech m In

(WTO)’~

(wmax = 1 / ~ ~ ) . (18.1.51)

W/wmax

The introduction of the parameter m has the effect of broadening the loss peak (compared to the single relaxation time, m = 1) for 0 < m < 1 (see Fig. 5). The function is usually fitted by determining E&. visually from a plot of E” vs. log o and then m and wmaxfrom the slope and intercept of a plot of cosh-lE~ax/ettvs. In w. Application of Eq. (18.1.42) to Eq. (18.1.51) results in ER

-

EU

=

(18.1.52)

2c&x/m

for the relaxation strength. The distribution function Eq. (18.1.44) that leads to Eq. (18.1.51) is F ( S ) dS = F(ln 8

T)

d In

T,

R. M. Fuoss and J. G.Kirkwood, J . Am. Chem. Soc. 63, 385 (1941).

392

18.

ELECTRICAL METHODS

2.49 (a)

2.43

1

I

I

0.000 -

:

z'' 0.006 0.004

0.002

(b)

: I

I

I

FIG.6. A Cole-Cole circular arc plot constructed from the L', L" data shown. The data are for the a relaxation process in slightly oxidized linear polyethylene. This process arises in the crystalline fraction of this semicrystalline polymer (from Sayreg').

where S = In

T ~ / T

and F(S) =

(m/7r) cos(rn7r/2) cosh mS cos2(m7r/2) + sinh2 mS '

(18.1.53)

18.1.2.3.5. DAVIDSON-COLE FUNCTION.'^ The Fuoss-Kirkwood and Cole-Cole functions are symmetrical about In T ~ .Davison and Cole introduced the function E* =

lo

eU

+(

E ~

eU)/(l + iw1Iy,

( 1 8.1 .54)

J . A. Sayre, Ph.D. Dissertation, University of Utah, Salt Lake City (1977). D. W. Davidson and R. H. Cole, J . Chem. Phys. 18, 1417 (1950).

18.1. DIELECTRIC CONSTANT A N D

LOSS

393

which leads to (18.1.55) ( 18.1 .56)

where tan 4 = W T ~ ,

urnax T~ =

:1

tan ( y

3

and to (18.1.57)

The low-frequencyend of the plot is circular, but the plot is skewed at the high-frequency end (see Fig. 7). Many measurements on solutions of polymers in low-molecular-weightsolvents often give results11*12 that are fit by the Davidson-Cole function. 18.1.2.4. Temperature as a Variable. It is implicit in the discussions of representations of the relaxation function (Section 18.1.2.3) that scans of dielectric constant and loss vs. frequency are at constant temperature. However, in general, the relaxation times are temperature dependent. Thus a complete study of the dielectric properties might be represented by contour maps of E' or E" vs. frequency and temperature. More often in practice, however, isothermal scans vs. log frequency are plotted as families on a single figure. The frequency at which E" is a maximum is

El-

€6

€;--EL

FIG. 7. Plot of C" vs. C' for which the data are skewed at high frequency (low d) and are fit by the Davidson-Cole function. The data are for solutions of poly(p-chlorostyrene) in various solvents (from Mashimol2). l1 19

M. E. Baur and W. H. Stockmayer, J . Chem. Phys. 43, 4319 (1%5). S. Mashimo, Macromolecules 9, 91 (1976).

394

18.

ELECTRICAL METHODS

determined &), and treated as analogous to a rate constant. Thus a convenient summary of the effect of temperature on the relaxation process is a logf,,, vs. l/Tor "relaxation map" (see Figs. 8 and 9).11J3J4 One defines an activation energy from the slope of such a plot, AHt = -2.303R slope. The convenience of such plots is even more apparent when it is realized that most polymers show several relaxation regions. Since the absorption region in the frequency domain in polymers is often extremely wide, a particular single experimental method often does not cover the region very well. As a convenience especially when characterization is the principal goal, often the plot vs. log frequency is abandoned and the loss plotted vs. temperature (i.e., an isochronal scan; see Fig. 10).15 If a family of such curves at a few frequencies is generated, a log fmax vs. 1/T plot can still be made and an activation energy determined. Plots of logf,,, vs. l/Tfrom isothermal and isochronal scans are usually very similar; however, in general, they need not be the same. When the relaxation strength (cR - cU) is strongly temperature dependent or when the width of the loss process (Cole-Cole p parameter or

LOG

CJ

FIG.8. The central relaxation time ( T ~= 21rfma,) may be considered as subject to a thermally activated rate process and Inf,,, plotted v s . 1/T. Is

G.Williams, Truns. Furrrduy SOC. 61,

1564 (1965).

'' S. Yano, R. R. Rahalkar, S. P. Hunter, C. H. Wang, and R. H. Boyd, J . Polym. Sci.,

Polym. Phys. Ed. 14, 1877 (1976).

18.1. DIELECTRIC CONSTANT AND LOSS

3

395

5

4

10'

FIG. vs. 1/T 'loss map for polypropyleneoxide. The data cover an unusually broad frequency range and involve low-frequency DC transient measurements, bridge measurements, and slotted-line measurements (data from Baur and Stockmayer,lI Williams,1s and Yano er a1.I').

Fuoss- Kirkwood m parameter) is strongly temperature dependent, the two types of scans can lead to significantly different relaxation maps and activation energies for the same material.15*16 An average activation energy can often be defined from the area under a plot of loss factor vs. l/Tfor a single isochronal scan (i.e., one frequency only). McCrum, Read, and Williams discuss these definition^.^ 18.1.3. Experimental Procedurest 18.1.3.1. Lumped Circuits (10-4-10e Hz). As seen earlier [Eq. (18.1.30)], the complex dielectric constant may be determined by measuring the complex impedance of the dielectric as a circuit element. This may be accomplished by employing calibrated purely capacitive and l5

C. R. Ashcraft and R. H . Boyd, J . Polym. Sci.. Polym. Phys. Ed. 14, 2153 (1976). B. E. Read and G . Williams, Polymer 2, 239 (1961).

t See also Volume 6B (Solid State Physics) of this Series, Chapter 7.1.

18. ELECTRICAL

396

E"

t 1

1

.

1

.

1

METHODS

.

1

2

1

3

0

4

b

log f 100Hz

-200

-100

-150

-9

TEMP " C FIG.10. Isothermal and isochronal scan comparison for the loss E" associated with the y relaxation process in slightly oxidized linear polyethylene (data from Ashcraft and Boyd'O).

purely resistive circuit elements that are adjusted to the same impedance as the circuit element containing the dielectric sample. The comparison capacitor and resistor may be effectively connected either in series or parallel. If effectively connected in parallel, comparison with Eq. (18.1.30) yields 1 1 = i d , + - = iWCOE*,

Z*

(18.1.58)

R P

or e' = Cp/CO1

(18.1.59)

l/wR,Co,

(18. I .60)

tan 6 = l/wR,C,.

(18.1.61)

Err

=

If the comparison circuit is effectively connected in series,

z* = -+ R , = l/ioCor*, iwC,

(18.1.62)

18.1.

DIELECTRIC CONSTANT A N D LOSS

397

or (18.1.63) (18.1.64) tan 6 = wRsC,.

(18.1.65)

18.1.3.1.1. BRIDGEMETHODS (Intermediate Frequencies: 10’- lo7 Hz). The most convenient method for measuring the sample impedance is to connect the comparison circuit element and the sample in an ac Wheatstone bridge, i.e., an impedance bridge (see Fig. 1I). Using relatively conventional resistive and capacitive circuit elements in the bridge, a frequency range of approximately 10 Hz to 100 KHz may be covered with high sensitivity and precision. Special-purpose bridges of considerably less accuracy are available in the 1- 10 MHz region. In any real circuit element containing the sample dielectric, there will be additional distributed capacitances due to the leads, etc. The effect of these may be more or less satisfactorily handled by either of two methods:

(a

FIG. 1 1 . A typical impedance bridge. Some bridges regard the equivalent circuit of sample as (a) a series capacitor and resistor and (b) a parallel capacitor and resistor.

398

18. ELECTRICAL

METHODS

(a) two-terminal measurements subtracting out lead capacitance or (b) three-terminal guarded-circuit measurements. The lead capacitance is regarded as an additional capacitance in parallel with the sample. In the parallel equivalent-circuit notation

c, = c, Rp

=

( 18.1.66)

+ Cpl,

Rx,

(18.1.67)

where x refers to the sample and I to the distributed lead capacitance, The latter can be determined in one of two ways. If the cell configuration is such that Co may be calculated confidently [see Eq. (18.1.10)or (18.1.111, for example], then a measurement on the empty cell will serve to determine

c,o = CIJ + CpI,

(18.1.68)

and CPlis determined by difference. CPlis then subtracted from measurements of C, to arrive at Cpx.Alternatively, Cocan be measured by calibrating the cell using a sample of known dielectric constant (and zero loss) in addition to the empty cell measurement, that is,

C, = e’CO+ CPl C,O = Co + CPl

(known dielectric),

( 18.1.69)

(18.I .70)

(empty cell),

or

co = (C,- CP”)/<€’- l), C,I = c,o - co.

(18.1.71) (18.1.72)

Since some bridges read out directly in dissipation factor tan 6, one must be careful to realize that this means tan S for the sample circuit including leads. Therefore, in parallel notation, tan S (bridge) = l/oR,(C,

+ Cd),

tan S (sample) = tan 6 (bridge) -

cpz

(18.1.73) +

CPX

“l.

(18.1.74)

The above equations also point to a method for extending the dissipation factor range of some bridges to allow measurements on very lossy samples. The sample can be loaded in parallel with a known capacitance to lower the measured tan 6. Two-terminal measurements suffer the disadvantage of the fact that the lead capacitance CPl is often not highly stable and can vary somewhat from run to run. This is especially important since it is not unusual in

18.1.

DIELECTRIC CONSTANT A N D LOSS

399

many configurations for the lead capacitance to be comparable to or even considerably greater than the sample capacitance. In three-terminal guarded-circuit measurements, it is possible to largely eliminate the lead capacitance and confine the measurement to a fairly well-defined portion of the sample. For example, in a paralle-plate configuration, one of the plates can be divided into two electrodes: an inner circular one, and a surrounding annulus (see Fig. 12). If the sample leads are used for making the capacitance measurement and the guard lead is kept isolated from but at the same potential as sample lead 1, then the capacitance measured will be well-defined geometrically by the area of the lower inner circle. Further, if lead 1 has a shield connected to the guard this lead will see no stray capacitance since the shield will be at the same potential. Some older bridges have been equipped with separate bridge circuits for the guard so that the sample and guard could be separately and iteratively balanced to the point where the sample is in balance in the bridge and the guard potential is balanced to that of lead 1. However, this can be much more conveniently accomplished by means of bridges with transformer ratio arms. The resistive-capacitive ratio arms Z1,Z , in Fig. 11 can be replaced by the inductances and the voltage applied by means of transformers as in Fig. 13. This arrangement has the important property that the voltage ratio and therefore the impedance ratio Z,/Zl at balance is determined almost entirely by the ratio of turns in the transformers N,/N,. This means that stray capacitances from the point H (high voltage, low impedance) to the guard or ground point G will not affect the balance point. If the guard ring is connected to G, the guard ring to H electrode capacitance will also appear here. If the shield to the L electrode (low voltage, high impedance) connection is connected to G, the stray capacitance of the L lead will appear across the detector. It will therefore shunt the detector and reduce its sensitivity somewhat but not affect the balance point. ThereSAMPLE L E A D 2

ELECTRODE

SAMPLE

LEAD

1

FIG. 12. A dielectric sample in a capacitor equipped with a guard ring.

18.

400

ELECTRICAL METHODS

FIG.13. A transformer arm ratio bridge.

fore, a completely guarded circuit not requiring special balancing provisions can be a~hieved.~'*~* Figure 14 shows a typical three-terminal parallel-plate cell constructed for use with polymeric samples. METHODS(10-4-102 Hz). Both bridge and 18.1.3.1.2. LOW-FREQUENCY transient methods have been used for low-frequency measurements. The ADJUSTING NUT ADJUSTING DIAL

IMPEDANCE ELECTRODE

GUARD ELECTRODE

.

I NSUL AT1 N G WASHER

HI Gtl

IMPEDANCE ELECTRODE

FIG.14. A micrometer adjustable parallel-plate dielectric cell equipped with a guard ring.

Is

R. H. Cole and P. M. Gross, Rev. Sci. Insrrum. 20, 252 (1949). General Rodio Experimenter, 36, 3 (1962).

18.1.

40 1

DIELECTRIC CONSTANT A N D LOSS

low-frequency limit of Schering or transformer arm ratio bridges is approximately 10 Hz, due to the problems in coupling the generator to the bridge using transformers. Direct-coupled bridges have been designed 102 Hz. Such a bridge has been described for operation in the range by Scheiber.’O Transient methods are particularly useful at very low frequencies. Up to now we have discussed primarily response to periodic fields. Indeed the literature places great emphasis on measuring and displaying dielectric data in the form of E’ and E” as functions of frequency. However, much the same information concerning the relaxation function x(t) [Eq. (18.1.14)] can be obtained from other programmed applied fields. Particularly appropriate is the transient response to a suddenly applied static field (step function voltage)(see Figs. 15 and 16).lea Since E = q/C,V [see Eq. (18.1.lo)], we define a charging function proportional to the charging current in response to a suddenly applied constant voltage V , as J ( t ) = dE/dt = j / C o V o .

(18.1.75)

Application of a constant field Eo at u = 0 in Eq. (18.1.14) results in E = D/K&o

(1 8.1.76)

or J(t) = dx(t)/dt.

(18.1.77)

Thus the charging experiment directly measures dx/dr. However, in view of the great interest in expressing data in terms of €’(a), ~”(a it )is appropriate (in order that transient experiments can be compared with periodic ones) to express E ’ , E” in terms of J(r). If Eq. (18.1.17) is integrated by parts we find e* = 1

+ [dX/du’e-bu’ du’

or E* =

E,,

+

1

J(u’)e+”’ du’

( 1 8.1.78)

where in Eq. (18.1.78), we have also recognized that the step voltage and measurement of J ( t ) will not be rapid enough to determine the rapid electronic and atomic polarizations and hence they are included as eU. Equation (18.1.78) may be rationalized in the manner of Eqs. (18.1.19) and D. J. Scheiber, J . Res. Natl. Bur. Stand., Secr. C 65, 23 (l%l). G . Williams, Polymer 4, 27 (1963).

lea

18. ELECTRICAL METHODS

402

j

current

\

charge (V-V,)

I

1

t=O

+

I

/--

V

discharge ( V . 0 )

FIG.IS. Transient current response (schematic) to a step voltage.

FIG.16. A circuit for measuring transient current response to a step voltage. A voltage in the range 60-240 V is applied by closing S, to V. The current is determined by measuring the voltage drop across R 1(lO1o-lO1* fl) by means of a dc amplifier. In order to keep the guard at potential A and to ensure that V appears only across the sample, a feedback circuit in the amplifier maintains the potential across R, equal and opposite to the potential across R, (after Willi'amslsa).

18.1.

DIELECTRIC CONSTANT A N D LOSS

403

(18.1.20) to yield z’ =

EU

+

I

J(u’) cos wu’ du‘,

(18.1.79)

0

E”

=

J(u’) sin wu’ du’.

(18.1.80)

Thus E’(w), €”(a)may be determined from the charging current data by numerical integration of Eqs. (18.1.79) and (18.1.80), a relatively straightforward matter using a computer. There are approximate relations connecting J(r) and E’(o), ~ ’ ’ ( 0 that ) are somewhat useful. If we pass to the exponential form of the relaxation function with a distribution of relaxation times [Eq. (18.1.43)], we find from Eq. (18.1.76) for the step voltage experiment,

in comparison with Eqs. (18.1.44a-c). It is characteristic of polymeric materials that the distribution of relaxation times is very broad. Under these circumstances the approximation (the viscoelastic “first” approximation)20that the exponential build-up of one relaxation behaves as a step function relative to F(T), or F(T)(~ - e-”3

=

1, 0,

T

7

< t, > 1,

(1 8.1.82a) (18.1.82b)

from which follows

or J ( t ) = de/dt

E (QR

- EU)F(T= t ) / t .

(18.I .83)

From Eq. (18.1.44~)we are led to a similar approximation that regards OT = 1, or

W T / I + 02f as sharply peaked at

( 18.1.84)

Therefore, z”(0)

IT =tJ(t), 2

1 t =-

0’

(18.1.85)

A. V. Tobolsky, “Properties and Structure of Polymers.” Wiley, New York, 1960.

404

18.

ELECTRICAL METHODS

For Eq. (18. I .44b) the approximation is made that ( 18.1.86a)

(18.1.86b) Thus

or e'(d

= 4 t ) = eU +

w = l/t.

I,'J(r) dr,

(18.1.87)

Other approximations connecting J(r) and L' (w), e " ( o ) have been made.21 The main interest of the present treatment is on dipolar relaxation. However, polymers in general also transport charge under the influence of an electric field. That is, they have a bulk resistivity. This resistivity will also contribute to the measured dielectric properties. This resistivity will tend to behave as a resistance in parallel with the capacitance of the sample. Unlike the resistive component due to dipolar relaxation, the resistance due to this source will tend to be independent of frequency. That is, if the current is due to bulk charge migration in a quasi-static field uninfluenced by electrode effects, this resistance is characterized by bulk resistivity, a material constant. Looking again at the parallel equivalent circuit formulation, Eqs. (18.1.59) and (18.1.60), we see that the dielectric constant L' is uninfluenced by this resistive contribution. However, we see that it does contribute to the loss

(where we have emphasized the frequency dependence of the dipolar resistance R,(co)). From Eq. (18.1.88) we see that although the dipolar parallel rtsistance becomes infinite at low frequency, the loss due to dc conductaace will continue at rise at low frequency. This increasing loss at low irequency due to dc conductance is frequently observed and can mask or obscure dipolar loss processes. In fact, only in exceptionally low-dc-conductance polymers (i.e., perhaps exceptionally clean, lowionic-impurity-containing, low-dielectric-constant polymers) can dipolar loss processes be studied at frequencies below 1- 10 Hz (see Fig. 17).21a 18.1.3.1.3. HIGHFREQUENCIES-RESONANT CIRCUITS (108-188 Hz).

-

*I N . G . McCrum, B. E. Read, and G . Williams, "Anelastic and Dielectric Effects in Polymeric Solids," p. 214. Wiley, New York 1967. *la C. H. Porter and R. H. Boyd, Mitcrornolecrtles 4, 589 (1971).

18.1.

-

DIELECTRIC CONSTANT A N D LOSS

3

405

10’ 10’ 10‘ FREQUENCY, Hi! 100

I..

FREQUENCY,Hz

FIG. 17. Loss in polyethyleneoxide at low frequency. The very high loss is due to dc conductance (data from Porter and

There is a frequency region (- 1 - 100 MHz) where, although it is difficult to design and assemble the many variable capacitors, resistors, switches, etc. into a bridge because of mutual inductances and stray capacitances, the sample-filled capacitor itself may still be considered a lumped circuit. Resonant methods provide a means of taking advantage of this because the capacitance and loss of the specimen are not determined by comparison with another circuit but can be determined by difference using the resonance condition as indication of constant impedance of the empty and filled cell. One form of the basic circuit of the resonance method is shown in Fig. 18. A generator is loosely coupled (often capacitively) to an inductor ( L ) connected in parallel with variable capacitor CD (which also has a finetuning vernier capacitor Cv), a high-impedance high-frequency voltmeter (V), and the sample (&, C,,). The leads to the sample cell are represented by Clead. The admittance (a, 6) is given by (18.1.89)

18.

406

ELECTRICAL METHODS

Since the generator is loosely coupled (l/joC, >> Zab)to the circuit, it acts as a constant-current source. At resonance the admittance is a minimum (impedance is a maximum) and therefore the voltage V is a maximum. At resonance when tuning with CD+ Cv, dF/d(CD

+ CV) = 0,

CD

+ CV + Clead + c p x = L . (18.1.90)

A sample is placed in the cell and the circuit excited by frequency o/27r. The circuit is tuned to resonance by adjusting capacitor CDto value c D 1 . The sample is then removed and the circuit is returned to resonance with CD,to value C m . The difference between the readings of C m and c D 1 is the difference in the capacitance of the sample and empty cell, CDp - C D 1 = ACD = Cpz - Co.

(18.1.91)

Substitution of Eq. (18.1.10) for Cpz results in ACD/Co =

(E'

- 1).

( 18.1.92)

The value of R, is found by finding the width of the resonant peak. The circuit is tuned to resonance and the maximum voltage is noted. The circuit is then detuned on both sides of the resonant peak to voltage V using the vernier capacitor Cv. The difference in capacitance between V through the maximum and back to V on the other side of the peak is then a I

r

1

I

I

I

1

I

I

1

b COAXIAL CELL

GR-1690 A C E L L

LEADS FIG.18. Schematic diagram for (a) resonant circuit apparatus; (b) resonant circuit where sample is connected to adjustable and vernier capacitances by means of a transmission line (a piece of coaxial line).

18.1. AC,.

DIELECTRIC CONSTANT AND LOSS

407

Since the voltage is in general

where C

=

C,

+ C y + Clead+ C , m = V,,,/V

=

and V,,, = jR,

at resonance, then

( d R 2 , ACv2/4 + l).1/2

The loss of the sample is then (18.1.94) An often practiced variation of the method is for CD,Cv, and C,(R,) to all be in the same cell. That is, the sample is placed between the plates that are adjustable by means of a micrometer. Resonance is achieved near an approximate desired frequency by selecting the inductor L and then fine adjusting the frequency. The width of resonance (AC,) is measured by detuning with the vernier capacitor. The sample is removed and resonance again achieved by adjusting the micrometer electrodes (to Co). A calibration curve of micrometer reading versus capacitance may be established by bridge measurements at lower frequency. Variations in inductance and stray capacitances are kept at a minimum using this method. Measurements of the resonance width on the empty cell generally show the presence of some residual resistance in the circuit. If this is considered to be in parallel with the other circuit components, the loss equation (18.1.94) may be modified to Q”

=

ACv - ACv(empty) 2Co(m2- 1)lI2 ’

(18.1.95)

An apparent dissipation factor of -0.005 for the empty cell is not uncommon. It is often not convenient to use this method over temperature extremes, however, since removal of the sample is not convenient and construction of the micrometer cell may not permit it. A variation that conveniently circumvents these problems is shown in Figs. 18b and 19. The sample is not actually placed in the cell but is placed in a coaxial line connected in parallel with the tuning cell. The coaxial line may be in a thermostat and the tuning cell at room temperature. The resonant method then is considered to measure the impedance at the entrance to the coaxial line. Transmission line equations (see Section 18.1.3.2.2) may then be used to relate the impedance at the entrance to the line to the dielectric properties of the sample. A derivation is given by Porter and

18.

408

ELECTRICAL METHODS

FIG. 19. A pictorial representation of the circuit in Fig. 18b, showing adjustable capacitors (micrometer cell) connected to sample by means of a coaxial line. The sample may be maintained at a desired temperature and the micrometer cell may remain at room temperature.

Boyd” and the final results for a “thin sample” are in place of Eqs. (18.1.92) and (18.1.95), ( 18.1.96)

ACd = CO

E’ -

AE”X + x / A DEN

E’X’ -

AE”’x

-

CRF,

(18. I .97)

where DEN = ( 1 - AXE’)+ (AXE”)’, CRF = ( 1 - AX + x/A - ~ ‘ ) / ( 1 x = tan(2?rfs/c),

-

AX)’,

s is the length of line, c the velocity of light, f the frequency, A = 27zfZ0CO,Zothe characteristic impedance of coaxial line, and Co the empty cell capacitance of sample-filled portion of line (see Fig. 18b). The magnitude of the coaxial line correction depends on the values of E‘ and E”, but typically the corrected values of loss and dielectric constant are 10-15% lower than those predicted by Eqs. (18.1.92) and (18.1.94) at frequercies around 50 MHz, E’ = 5 and E” = 1.2, and S = 12 cm. 18.1.3.2. Distributed Circuits. 18.1.3.2.1. STANDING WAVES I N TRANSMISSION LINES.'^ When the frequency becomes high enough that C. H. Porter and R. H. Boyd, in “Dielectric Properties of Polymers” (F.E. Karasz, ed.), p. 147. Plenum, New York, 1972. A. R. von Hippel, “Dielectric Materials and Applications.” MIT Ress, Cambridge, Massachusetts, 1954.

18.1. DIELECTRIC CONSTANT A N D LOSS

409

#wavelength approaches the dimensions of the electrical paths, mutual inductances and distributed capacitances become important and it is no longer possible to use the lumped-circuit approach. This occurs roughly at a frequency of lo* Hz. Above this frequency it becomes appropriate to study the propagation of waves through the dielectric of interest. An electromagnetic wave propagating in one direction ( x ) has electric and magnetic components E,,, H , that can be written as% E, = E,,Oe-fot-Y’X ( 18.1.98) 9

(18.1.99)

where

is a complex propagation constant. In vacuum, o/iy* = co = velocity of light in vacuum, or y* = i(2w/Ao), ho = wavelength in vacuum, but in a dielectric, S*

(1 8.1.100)

Further, to satisfy Maxwell’s equation EUoand H,O are related as (18.1.101)

where we assume a nonmagnetic medium of permeability p,,. The loss component of E* gives rise to a real part of y*, which in turn leads to attenuation of the wave. If the wavelength and attenuation could be measured in the dielectric, the propagation constant would be known and the complex dielectric constant would thus be determined. However, rather than measure them directly, it is often more convenient to measure the properties of the standing wave setup when the traveling wave propagates from vacuum into the dielectric and is partially reflected. A plane polarized traveling wave such as Eqs. (18.1.98) and (18.1.99) is conveniently set up in a rectangular waveguide. One set of parallel plates serves to conduct the electric field, the set normal to these is the magnetic field direction, and the wave propagates along the length of the tube. Another equivalent configuration is the coaxial line, where the electric field direction is in the radial direction and the magnetic field is circumferential (see Fig. 20). If such a line is empty and is terminated by a short circuit, the traveling wave will be totally reflected and a standing wave set up with nodes of zero voltage occurring every Ao/2. If the line is partially filled by a dielectric sample inserted in the line (say at the shorted end) a standing wave will still be set up. However, due to reflection at the air-dielectric sample interface, the standing-wave pattern will be altered. For example, partial reflection at the interface and total reflection from the short-circuited end (leading to multiple reflections in the sample) J . C. Slater and N. H. Frank, “Electromagnatism.” McGraw-Hill, New York, 1947.

410

18. ELECTRICAL

METHODS

(C)

FIG.20. Transmission lines: rectangular and coaxial waveguides and a coaxial slotted line.

will cause a phase shift of the reflected wave. Then the reflected wave in the air-filled portion will be unable to completely cancel the incident wave at the nodes. Voltage minima rather than zeros will occur. The ratio of the maximum to minimum voltage along the empty line (voltage standing-wave ratio, VSWR) will depend on the dielectric constant (amount of reflection) and length of the sample. Similarly, absorption (c”, y’ 7 0) in the dielectric will cause a phase shift at the interface and attenuation in the sample. Thus loss in the sample lowers the standing wave ratio (see Fig. 21). The standing wave pattern in the empty portion of the line can be measured by cutting a narrow slot along the length of the line and inserting a small crystal detector into the line to monitor the field.25 Precision low-loss slotted lines are available commercially in both the rectangular waveguide and coaxial configurations. A rectangular waveguide has a low-frequency cut-off, depending on its dimensions, below which it will not propagate. There is also a high-frequency limit for propagating only the fundamental mode, and one wishes to avoid mixed mode propagation. Coaxial lines have no low-frequency cutoff but also have an effective high-frequency limit due to mixed mode propagation.28 The dielectric properties of the sample are quantitatively determined from the VSWR and the node position as follows. The traveling-wave *6

S. Roberts and A. R. von Hippel, J . Appl. Phys. 17, 610 (1946). S. Ramo, J. R. Whinnery, and T. Van Duzer, “Fields and Waves in Communication

Electronics.” Wiley, New York, 1%5.

18.1. DIELECTRIC CONSTANT A N D LOSS

41 1

FIG.21. The standing-wave pattern in a transmission line containing adielectric sample at the short-circuited end.

equations (18.1.98) and (18.1.99) are generalized in the presence of a reflected wave to ,qX)= ,r+oeior-ypr + E-o e1(ot+8)+y*r (18.1.102) 9

~

( = H+O ~ 1eior-i*z

+ H -o

e((~t+8n~*r 9

(18.1.103)

where + , - refer to the incident and reflected waves and the y and L subscripts on F and HO are implied. In order to have the direction of the reflected wave reversed, the phases of the reflected electric and magnetic vectors must be reversed with respect to each other in the incident wave. That is, for a wave traveling in the negative direction (~*)ll2E-O=

-( p O ) ' W - O ,

( 18.1.104)

and from Eq. (18.1.101) E-O/E+O = - H -O /H+O.

(18.1.105)

The phase angle 6 of the reflected wave with respect to the incident depends on several factors. If the origin ( x = 0) is taken at an interface between two nonlossy dielectrics E; < €4, then reflection of the electric vector will be out of phase and the magnetic vector in phase or y = 7. If either of the dielectrics is lossy (Q" # 0), there will be a phase shift away from 6 = 7 (see Section 18.1.3.2.2). If the origin is arbitrary with respect to the interface, 6 will depend on the distance from the interface to the origin. If the reflected wave is made up of waves from multiple reflections (air- sample interface, sample-line termination interface, etc.), 6 will ob-

412

18. ELECTRICAL METHODS

viously depend on all of the reflections. Defining the quantities R* = E_Oe‘*/E+O= -H-oei6/H+o = Reis = reflection coefficient Z , = E+O/H+O = “characteristic impedance of the transmission (slotted) line.” (This depends on the dimensions of the rectangle in the rectangular wave guide and the radius ratio in the coaxial line.) Z(x) = E ( x ) / H ( x ) = impedance at point x along the line, if we find Z(x) from Eqs. (18.1.102) and (18.1.103) there results ( 18.1.106)

The impedance at point x2 is thus related [from Eq. (18.1.106) by eliminating R * ] to the impedance at point x1 as ax,) =

Z(x,) - Z , tanh

zo [Z,

-

Z(xJ tanh

( 18.1.107)

where x = x2 - xl. The above equation is known as the impedance transformation equation for transmission lines since it relates the impedance at a point to the impedance at a distance X further along the line. The strategy now is to relate the measured impedance at a voltage minimum in the empty portion of the slotted line to that at the dielectric-air interface using the impedance transformation equation. Similarly, the impedance is known at the short-circuited termination of the dielectricfilled (sample) end of the line. That impedance can be transformed to the air-sample interface also. The laws of refraction require that the tangential components of both E and H be continuous across the interface. Therefore, the impedance is continuous across the interface. Thus equating the impedance transformed to the interface from the air side to that transformed to the interface from the sample side, an expression relating the complex propagation constant of the dielectric sample to the known propagation constant of the air-filled line is obtained. The impedance at the voltage minima may be determined as follows. Let us refer to the air-filled line as region 1, and the dielectric filled portion as region 2. We further assume that the air-filled line is lossless, 7: = iyi’, The rms voltage at any point in region 1 (7: = i$’) according to Eq. (18.1.102) can be written as ERMs= [(E+O + E-o)2 cos2yi’x’+ (E+O - E-o)2 ~ i n ~ ~ ; ’ x ’ )(18.1.108) ]~/~,

18.1.

where x’ = x

413

DIELECTRIC CONSTANT AND LOSS

+ 6/2y;‘.

This equation has maxima and minima,

Emax = (E+O + E-O),

at

Emin= (E+O - E-O),

at xLin = nw/2.

=

nw,

(18.1.109) (18.1.110)

Similarly, the magnetic field has a maximum at the voltage minimum [see Eq. (18.1.105)] and vice versa, HOm n i =

(H+O + H-O),

at xmaX,

(18.1.111)

=

(H+O - H-O),

at ~

(18.1.112)

&ax

1- R= (1

E+O-

+ R ) H+O

~ 1 , .

1-R (1 + R )

(1 8.1.113)

(where R = IR*I) but since Emin - - E+O - E-O Emax E+O.+ E-O

1 =-1

-R 1 + R - 3’

( 1 8.1.1 14)

where S is the voltage standing wave ratio, = Emax/Emin, then Zbmin) = &/S*

(1 8.1.1 15)

We may now write the impedance at the interface Z13 in terms of the impedance at xminusing Eq. (18.1.107):

where xo is the distance from the voltage minimum to the interface. Using Eq. (18.1.115) and y;’xo = (2n/Ao)xo[Eq. (18.1.100)], we have z 1 2

=

5

1

1 - iS tan(2n/Ao)xo S - i tan(2.rr/Ao)xo

[

(18.1.1 16)

The impedance at the short-circuited termination of region 2 (samplefilled end) is zero since the voltage is constrained to be zero by the short circuit. Thus, according to Eq. (18.1.107) using x = - d , Z(xl) = 0, Z2, = Zoztanh $2.

(18.1.117)

Equating the impedances at the interface ZIz = Z,, and relating the characteristic impedances through Eqs. (18.1.100) and (18.1.101) as Zol/Zoz= y2*/y:

=

- i y f ho/2w,

( 1 8.1.1 1 8)

414

18.1.

DIELECTRIC CONSTANT A N D LOSS

we arrive at the slotted line equation:

[

tanhy,*d Y:d

]

ih0 1 - tan(2n/ho)xo 2 r d S - i tan(2r/Ao)xo

*

(18.1.119)

The left-hand side of Eq. ( I 8.1-119) contains the desired propagation constant y t = y; + iy;’, which is related to the dielectric constant through Eq. (18.1.100) or E’ E”

=

(y;” - ~~z)(h0/27r)2,

= 2$’ & ( h o / 2 ~ ) ~ .

(18.1.120) (18.1.121)

The right-hand side of Eq. (18.1.119) contains the experimentally measured quantities ho, wavelength in the air-filled line; S, voltage standing-wave ratio, = Emas/E,,,,”;and xo, distance from a voltage minimum in the air-filled line to the sample interface. Thus it is required that Eq. (18.1.119) be solved for ygd. A convenient method for solving the equation that is often successfulzzis to use an iterative Newton-Raphson procedure. If we let u* = y t d , we may write the Newton approximation as a

un* = 4 - 1 - f(Un*-I)/f’(Un*-l),

(18.1.122)

where

f(u*) = tanh u* = u*C*,

f ’ ( u * ) = 1 - tanh2u* - C*,

and C* is the right-hand side of Eq. (18.1.119). Since computer languages such as Fortran may be programmed directly using expressions of complex variables, it is not necessary to explicitly rationalize Eq. (18.1.122). The value y t d (initial) can be found by using “low-loss” approximations. If tan y2 is sufficiently low, the real and imaginary parts of Eq. (18.1.119) can be separated and approximate values for y;’ and y; can be found, which are then substituted in Eq. (18.1.119) to find y; (initial). This is a satisfactory method, except for some cases where high loss is encountered. The solution for ygd actually has a number of solutions due to the periodicfunctions in Eq.(18.1.119). Equation (18.1.119) wasderived by comparison of impedances at the sample interface, but there is no way of telling without additional experimentation how many half-wavelengths are contained in the sample. This gives rise to an infinite number of possible answers for E‘ and E”, each dependent on the number of half-wavelengths between the interface and the short circuit, of which only one pair is cor-

18. ELECTRICAL M E T H O D S

415

rect. There are a number of methods that can be used to determine the correct set of values. In most cases one has a good idea of what E’ should be at the lowest frequencies, and by using that value as a starting point, and taking all the other values in context, the dielectric constant at the next highest frequency can be deduced. This method is satisfactory where the dielectric constant does not change dramatically as a function of frequency. The principal objection to this method is that at high frequencies the possible values of E’ and e’’ become closely spaced and it is really a question of judgment as to which is the correct solution. A method that ties down the correct solution set is that of making two separate runs with samples of different lengths. The two samples for each frequency then have two families of solution, the solution in common being the correct one. The “low-loss” approximation for the complex part of Eq. (18.1.119) is (18.1.123)

The value of yL‘d can be found by a Newton method starting this solution off by letting (18.1.124)

which is equivalent to starting A2 at +d for the first solution and increasing it by h2/2 for each successive solution. This procedure should take care of the fact that there could be any number of half-wavelengths in the sample. In high-loss samples, however, the low-loss approximation for wavelength becomes poor. and there is no assurance that the procedure employed in Eq. (18.1.124) would increase the true wavelength by halfwavelength intervals. As a result, tile solutions can be skipped. In fact, it has been found that there can be additional solutions for high-loss materials that do not appear to be related to the periodicity of the low-loss equation (18.1.123). In high-loss materials the only reliable method of solution of Eq. (18.1.119) is a pattern search. Computer programs have been presented for this purpose.22 In relatively high-loss materials the standing-wave ratio S can be measured directly by measuring the probe output at Xmlnand X,,,. However, in very low-loss materials S is too high to be measured conveniently by this method. However, it can be determined by investigating the variation of E with distance in the vicinity of the minimum. Typical crystal detectors give an output that is proportional to EL (i.e., “square-law” detector). The voltage, as a function of distance, in the line as measured by

416

18.

ELECTRICAL METHODS

a square-law detector is from Eq. (18.1.108) Eh2 = E2ax sin2 8

+ E&,

cos2 8,

(18.1.125)

where f3 = 7r Ax/ho and Ax12 is the distance from x to the minimum. If EL2 is chosen as 2Egin, which is equivalent to EL2 being 3 dB greater than Egi,,, then Eq. (18.1.125) can be rearranged into (18.1.126)

where 8 = 7r A x ~ ~ B / A , ,and AxBdBis the distance separating El2 = X:,,, through the minima. At high values of S, Eq. (18.1.126) is well approximated by simply

s = ho/W

b3dB.

(18.1.127)

Further details of the slotted-line method are discussed by von Hippel= and Porter and Boyd.22 18.1.3.2.2. TRANSIENT MEASUREMENTS I N TRANSMISSION LINESTIMEDOMAINREFLECTOMETRY (TDR).27We have seen that in lumped circuits one has the choice between point-by-point frequency methods and transient traces that may be converted to the frequency domain. In Section 18.1.3.2.1 we developed the frequency domain principles for distributed circuits (transmission lines). In this section we discuss transient methods for transmission lines. The basic principle of time domain reflectrometry (TDR) is to transmit a voltage pulse (usually a step voltage) along the line, where it is partially reflected from a sample-filled section.28 A sampler inserted in the l h e (with as little disturbance of the pulse as possible) between the generator and sample allows measurement of the incident pulse v k t ) and the reflection uR(t)from the air-dielectric interface (see Fig. 22). The sample must either be long enough that the time for reflection from the opposite end of the specimen leaves a sufficient “window” for the measurement of the first reflection transient or the multiple reflections must be taken into account in the analysis. Advantages of the method include the fact that, in common with all transient methods, all of the frequency information is obtained in a single time domain scan. Thus it is convenient and rapid. Furthermore, the equipment is not highly expensive. The analysis makes use of the principles already developed for transmission lines. That treatment was carried out in the frequency domain in terms of traveling waves of frequency o. However, we recall that tran-

*’ M.J. C. van Gemert, Phirips Res. Rep. 28, 530 (1973). ZB H.

Fellner-Feldegg,J . Phys. Chem. 73,616 (1969).

18.1. DIELECTRIC CONSTANT A N D LOSS

417

STEP GENERATOR

h

OSCILLOSCOPE

FIG. 22. The basic apparatus configuration for the time domain reflectometry method

(TDR).

sient and frequency domain results are interrelated by Fourier transforms (see Section 18.1.3.1.2). We start by relating the voltage pulse to the reflected pulse by a response function (reflection coefficient) in linear superposition [see by analogy Eq. (18.1.14)]

where U' = t - u, t is real time, and u is past time. If the incident and reflected pulses have Fourier components Vt"(iw)=

Vt(io)=

/-: /-:

uI(t)e-*o'dt,

(18.1.129)

uR(t)e-jWtdt,

(18.1.130)

then the Fourier components of the response function (reflection coefficient) (18.1.131)

will be related to V,* and Vg as Vg(iw) = V;F(io)R*(iw).

( 18.1.1 32)

Thus R*(iw) may be determined from the Fourier transforms of the experimentally measured incident and reflected pulses. The problem now is to use transmission line theory to relate R*(iw)to the complex dielectric constant E* (iw). Following Section 18.1.3.2.1 and calling the incident and reflected electric fields in the air-filled line E+(x), E-(x), respectively, and calling Ez(x) the total electric field in the dielectric, continuity at the interface [x = 0,

418

18.

ELECTRICAL METHODS

Eq. (18.1.102)l requires E+(O) + EJO)

+ E-Oei6 = E2(0)

=

E+O

=

H+O -

(18.1.133)

and [see Eq. (18.1.105)l H+(O) - H-(O)

H-Oe'6 =

H2(0).

( 18.1.134)

It then follows, by dividing Eq. (18.1.133) by Eq. (18.1.134), that the impedance in region 2 at the ,interface Z2(0) = E2(0)/H2(0)is given by (18.1.135)

where Zol = E+O/H,O (the characteristic impedance of the air-filled line) and R* is the reflection coefficient E-(O)/E+(O) = Pef6/E+O. This equation, then, forms the basis for TDR measurements. 18.1.3.2.3. SINGLEREFLECTION.^^ If the sample is very long, the electric field in the sample is a traveling wave to the right and therefore Z2(0)= ZzO,the characteristic impedance of the dielectric-filled line. From Eqs. (18.1.100) and (18.1.101), it follows then that ZO2/Z0, = ( E ; ) " ~ / ( E ~ ) ~= ' ~ 1 / ( ~ $ ) ~ or ' ~ for the dielectric constant of the sample E*

=

1-R* Vf - v* (1 + R*) = ( V f + V ; )

'

( 1 8.1.136)

Thus the dielectric constant is determined using this equation from the Fourier transforms of the incident and reflected voltage pulses. Several methods are available27 for calculating the transforms. Loeb and his co-workers2e use S a m u l o n ' ~ adaptation ~~ of Shannon's3' sampling theorem. Cole32advocates the analytical transform for a ramp function of rise time T for the incident step voltage and finds simple rectangular summation with constant interval Ar satisfactory for the response. Brehm and S t ~ c k m a y e fit r ~an ~ empirical function to the reflection coefficient. Figure 23 shows schematically the appearance of the response to a step voltage. The single-reflection method, of course, suffers the disadvantage of requiring a very long specimen (to observe only the first reflection) for low-frequency (long-time) response. 18.1.3.2.4. FINITESPECIMENS.27d2 TO extend Eq. (18.1.135) to finite specimens merely requires using the transmission line impedance transzD H. W. Loeb, G. M. Young. P. A. Quickenden, and A. Suggett, Ber. Bunsenges. Phys. Chem. 75, I155 (1971). J' H. A. Samulon, Proc. IRE 39, 175 (1951). C. Shannon, Proc. IRE 37, 10 (1949). s* R. H. Cole, J . Phys. Chem. 79, 93 (1975). G.Brehm and W. H. Stockmayer, J . Phys. Chem. 11, 1348 (1973).

18.1. DIELECTRIC

CONSTANT A N D LOSS

419

t-

OPEN T P T I O N

(b

-t

FIG.23. Responses in typical time domain reflectometry (TDR) experiments (schematic): (a) single-reflection method, (b) finite sample (multiple reflections) with open-circuit termination.

formation equation [Eq. (18.1.107)lto express Z,(O) in terms of a boundary condition at the other end of the sample. There are a variety of terminations possible (open circuit, short circuit at end, etc.). The sample placed at the end of an open coaxial line is a convenient arrangement. In that case the impedance is infinite at the end of the sample (of length d) and Eq. (18.1.107)leads to Z,(O) = Z,/tanh ygd.

(18.1.137)

Therefore Eq. (18.1.135)becomes =

(Q*)'"

1 - R* (m) coth ygd.

( 18.1.138)

Cole32rearranges this equation slightly. Using Eq. (18.1.100)and if we let p ( t ) = u,(t) - uR(t) and P*(io) = V,*(io)- V$(io),then €*

=

cg

2iod 1

P* ( i d / V,*(iw) yg d coth ygd. P(iw)/ZVf(io)

-

(18.1.139)

Cole3, advocates approximating ygd coth ygd as 1 - 4(KWd/C,J2e* and thus Eq. (18.1.139)gives an explicit solution for E*. For very thin specimens Eq. (18.1.139)reduces to e* =

_cO P*(io)/vf(iw). 21wd

( 1 8.1.140)

18.

420

0

ELECTRICAL METHODS

2 0 3 0 40 t (nseclFIG.24. Experimental TDR results for short-circuit termination method (data are for nheptanol; from de Loor et d . 9 . 10

This latter approximation has been used” but Cole32points out it can lead to appreciable error for typical “thin” samples. The short-circuit termination has also been which from Eqs. (18.1.135), (18.1.107), and (18.1.100) leads to

(’1 + R*

jwd - R*> tanh ygd CO

=

rid.

(18.1.141)

This may be solved by a Newton-Raphson procedure35similar to that discussed in conjunction with the standing-wave method.22 Figure 24 shows results obtained using this method. TDR methods are limited on the high-frequency side by the rise time of the step voltage and recording of the signal. Tunnel diode generators and sampling oscilloscopes permit measurements up to slightly greater than 10 GHz in the frequency domain. With the finite-sample methods, results down into the megahertz region have been obtained at the low-frequency end or even below. It has been suggested by Cole36that there are advantages to remaining in the time domain and determining the relaxation function directly, rather than Fourier-transforming the incident and reflected voltages u,(f), uR(t) and working in the frequency domain. Then if desired, the relaxation function can be Fourier-transformed to the frequency domain to determine E*. In this form the method is more akin to the low-frequency transient method as exemplified by Eq. (18.1.78). Cole36has presented a4 55

H. Fellner-Feldegg. J . Phys. Chem. 76, 2116 (1972). G. P. deloor, M. J. C. van Gemert. and H. Gravesteyn, Chem. P h y . Lerr 18, 295

(1973). s8

R. H . Cole, J . Phys. Chem. 79, 1459 (1975).

18.1.

DIELECTRIC CONSTANT A N D LOSS

42 1

methods for determining the relaxation function from time domain data on finite specimens. These have been extended by Chahine and B O S ~ . ~ ' Clarkson et ~ 1 have . discussed ~ ~ sources of error in TDR measurements in some detail. Acknowledgment The author's research in dielectric properties of polymers has been supported by the National Science Foundation.

R. Chahine and T. K . Bose, J . Chem. Phys. 65, 2211 (1976). T. S. Clarkson, L. Glasser, R. W. Tuxworth, and G . Williams, Adv. Mol. Relax. Inr. Proc, 10, 173 (1977). 58

18.2 Static Electricity By D. Keith Davies 18.2.1 Introduction

Static electricity may be defined as the phenomenon of charge generation and retention on low-conductivity dielectrics. The classical concept of a dielectric is that of an aggregate of bound charges-dipoles-which may be capable of local movement in response to excitation, but in which large-scale carrier injection or migration is precluded. It has been shown that ideal insulators do not exist and that space charges may readily be injected from metallic electrodes under applied electric fields. The process of charge injection across a metal insulator interface without applied fields-contact charging-is the commonest electrification phenomenon. Historically,1 the milestones in the development of modern electrostatics are extremely erratic. The Greeks were certainly familiar with the attractive forces caused by charged surfaces through observing the adhesion of particles to their polished amber jewelry. Systematic investigations did not start, however, until some 2000 years later, when Gilbert (1600) established that the amber effect could be induced by rubbing a large class of bodies. He called the phenomenon electric after the Greek term for amber. Over a century elapsed before Du Fay (1733) discovered that there are, in fact, two kinds of electricity, but the terms positive and negative stem from Franklin in the 1740s. Based on this polarity concept, Wilke (1757) proposed the idea of arranging materials into a triboelectric series (a misnomer from the Greek tribos, to rub) whereby contact between any two numbers of the series left the upper positive and the lower negative. Formulation of such series-largely inconsistentlyremains a current pastime, a recent example being shown in Table I.* The second half of the eighteenth century was particularly fruitful. The laws of charge conservation and interaction were both discovered at Chambers’s Encyclopaedia, Vol. V, p. 52. International Learning Systems, C o p . , 1970.

* J. Henniker, Nature (London) 1%. 472 (1962). 422

M E T H O D S OF EXPERIMENTAL PHYSICS, VOL.

1 6 ~ All

Copyright 0 1980 by Academic Ress. Inc. rights of reproduction in any form reserved. ISBN 0- 12-475958-0

18.2

STATIC ELECTRICITY

423

TABLEI. Triboelectric Series (Positive End)' Silicone elastomer with silica filler Borosilicate glass, fire-polished Window glass Aniline-formol resin (acid catalyzed) Pol yformaldehyde Poly(methy1 methacrylate) Ethylcellulose Polyamide I I Polyamide 6-6 Rock salt, NaCl Melamine formol Wool. knitted Silica, fire-polished Silk, woven Poly(ethy1eneglycol succinate) Cellulose acetate Poly(ethy1eneglycol adipate) Poly(dially1 phthalate) Cellulose (regenerated) sponge Cotton, woven Polyurethane elastomer Styrene-acrylonitrile copolymer

Styrene-butadiene copolymer Polystyrene Polyisobutylene Polyurethane flexible sponge Borosilicate glass, ground surface Poly(ethy1ene glycol terephthalate) Poly(viny1 butyral) Formo-phenolique, hardened Epoxide resin Pol ychlorobutadiene Butadiene-acrylonitrile copolymer Natural rubber Polyacrylonitrile Sulfur Polyethylene Poly(diphenylo1propane carbonate) Chlorinated polyether Poly(viny1 chloride) with 25% D.O.P. Poly(viny1 chloride) without plasticizer Polytrifluorochlorethylene Polytetrafluoroethylene

After Henniker.'

this time. The former independently by Watson (1746) and Franklin (1747) and the latter by Coulomb (1785). The period culminated in the theoretical ideas of Volta (1789), which remain the basis of modern theories of contact charging. Helmholtz, almost a century later, elaborated Volta's ideas in proposing that charge redistribution occurred at a solid-solid interface forming a double layer. The only other novel proposition was that of Coehn (1898), who attempted to describe established triboelectric series in terms of the dielectric constants of the materials. This idea was based on ion transfer, the image attraction of an ion being smaller to materials of larger permittivity. This, however, was soon disproved. Investigation of electrification was somewhat desultory in the first half of the present century, but the advent and rapid exploitation of plastics with their concomitant electrification problems has resulted in progressively more intensive investigation over the past 25 years. Quantitative models describing both contact and rubbing electrification have been proposed. Charge retention-the largely neglected aspect of electrification-has also been investigated partially stimulated by the

424

18.

ELECTRICAL METHODS

commercial viability of the electret as a transducer. Charge carriersthe ionized entities whereby electric current is transported-may be retained indefinitely, located at special structural sites or chemical groups (traps). The preeminently successful electrostatic process-xerog raphy -clearly continues to foster large investigational effort into charge imaging, development, and neutralization mechanisms. This specialized field has been thoroughly reviewed3 and only those aspects describing unique properties of materials are described here. Essentially an account of modern experimental techniques and the way those techniques are elucidating the basic properties underlying the electrostatic characteristics of insulators-particularly polymers-is given. 18.2.2 Methods of Measuring Charge

Solid materials may be obtained in discrete particulate form or as continuous sheets. The total charge on the former may be measured directly, while the charges on or in the latter may only be detected either by their accompanying electric fields or by the attractive forces they exert. A uniform charge density v generates a field E given by Gauss’ equation E = U / E K ~ , where E and K~ are the relative permittivity of the medium and the permittivity of free space, respectively. An isolated plane electrode of area A acquires an induced charge density u and experiences an attractive force K o A E ’ / ~ .The field intensity and hence the charge density may be determined either by measuring the electrode potential (knowing its capacity to ground) or by measuring the force. 18.2.2.1 The Faraday Cup Technique. The apparatus, shown schematically in Fig. 1, comprises two metal containers, one being totally enclosed by but well insulated from the other. The outer case is usually earthed. An object with total charge q induces the same charge on the screened container and hence generates the potential V given by V = q / C , where C is the combined capacitance of the inner container and the potential measuring system to ground. Clearly, predetermination of C permits the direct measurement of the charge q. The ideal system, where all field lines from the charged object end on the inner container, can clearly be approximated by using a long narrow shape, thus obviating the necessity for a lid. This form of device may be used in a dynamic mode, a continuous input of charged material producing a measurable current rather than a fixed potential. Such systems J . H. Dessauer and H. E. Clark, “Xerography and Related Processes.” Focal Press, London, 1965.

18.2

STATIC ELECTRICITY

425

FIG.1 . The Faraday pail apparatus providing net charge determination by measuring potential V on pail capacitance C.

have been employed to examine the charging of powdet' or extruded polymers5 for example. 18.2.2.2 The Ponderomotive Effect. Devices based on the attracted disk principle are rarely employed to determine charge.' Essentially the attracted disk forms an independent central section of a large uniform field electrode. On exposure to a second electrode at a high potential (or a charged surface) the disk is deflected. An equal restoring force is applied electromechanically, the energy required being measured. 18.2.2.3 Field Mills. Field mills are by far the commonest devices employed to determine electrostatic potential gradients, both axiala*' and radial design^^^^ having been described. The principle of operation of a planar field mill is shown in Fig. 2. Segmented sensor plates are periodically exposed to the field being sampled by a rotating, earthed, segmented shutter. The alternating potential induced on the plates is amplified and, after rectification, its amplitude measured. This simple device would be insensitive to the field direction (polarity) but recent versions, having reference signals generated in synchronism with the rotating shutter and using phase-sensitive detection, are thereby polarity sensitive. It has been shown'O that if the input impedance is predominantly capacitive so that the input time constant CR and shutter rotation speed W sat-

' J. van Turnhout, Adv. SfaficElecrr. I, 56 (1971). D. M. Taylor and T. J . Lewis, DECHEMA-Monogr. 72, 125 (1974). P. E. Secker, Insf. Phys. Conf. Ser. 27, 173 (1975); J . Electrostatics 1, 27 (1975). ' 1. E. Pollard and J . N. Chubb, Inst. Phys. Con$ Ser. 27, 182 (1975). R. T. Waters, J . Phys. E. 5, 475 (1972). J . M. van de Weerd, Insf. Phys. Con$ Ser. 11, 158 (1971). lo W. W. Mapleson and W. Whitlock, J . A m o s . Terr. Phys. 7, 61 (1955).

426

18.

ELECTRICAL METHODS

Roior Chopper

FIG.2. Schematic diagram of field mill with polarity-sensing circuit. After SeckeP and Pollard and Chubb.' isfy the condition ( W C R ) 2>> 1, then the amplitude V of the signal generated on sensor plates of area A is given by

v

=

K, ,AE/ ~C ,

which is independent of the shutter speed. The signal is linearly dependent on the field and sensitivities in the range lo2-los V m-l have been quoted.6 Caution must be exercised in interpreting field mill readings, however, since the geometry of the field mill in relation to other local metal objects will certainly influence the data. Ideally, the complete capacitive environment of the detector should be defined for accurate and unambiguous data. These conditions usually result in the use of a probe system. 18.2.2.4 The Electrometer Probe. Investigations of polymer films are almost exclusively based on probe techniques. A probe comprises an open-ended coaxial capacitor, the outer sheath forming an earthed screen. There are generally two modes of operation. The charged film may be passed rapidly beneath a fixed probe11.12(or vice versa) or, as with field mills, a shutter is placed between the probe and the The D. K. Davies, J . Sci. Instrum. 44, 521 (1967). T. R. Foord, J . Sci. Instrum. 46, 411 (1%9). l3 D. K . Davies, Br. J . Appl. Phys. [2]2, 1533 (1969). l1

lP

18.2

427

STATIC ELECTRICITY

-lol oscILu)xopE

r-----------I-----------

A MPLlFl E R

CP

I

I

-FIG.3. Principle of the electrometer probe together with the equivalent circuit. From Davies."

output pulse may be displayed and measured on an oscilloscope or its amplitude determined by a meter. The charged dielectric is usually mounted on an earthed plate to facilitate absolute measurements. A schematic diagram of a probe system together with the equivalent circuit is presented in Fig. 3.11 A uniform surface charge density a generates a surface potential V, in the presence of the probe. The probe, of effective area A and input capacity C, itself acts as a capacitive divider and, for small probe-to-surface separation d, where no field divergence may be assumed, the displayed pulse amplitude V is given by

v= ,

C

(1

+ d/d*)-',

(18.2.1)

where, g is the amplifier gain and e and d, are the specimen permittivity and thickness, respectively. More elaborate analyses to include the contribution of volume-distributed charges have been published." However, since generally the precise charge distribution is not known, the assumptions necessary to obtain explicit expressions return to determining an equivalent surface charge. For a geometrically precise system, absolute charge determination is evidently possible. The unknown quantity in Eq. (18.2.1), having measured the probe input capacitance, is the probe effective area. The capacitance of overlapping parallel electrodes of differing area is defined by that of the smaller. Thus for a uniformly charged surface of area greater than the geometric area of the probe, the latter defines the probe I4

H. J. Wintle, J . Phys. E. 3, 334 (1970).

428

18.

ELECTRICAL METHODS

effective area. However, if the area of charge is smaller than that of the probe, then the effective area is that of the charged Usually the charged area is much greater than that of the probe and there is no ambiguity. However, care must be exercised in interpreting the measurements for rapidly varying surface charge distributions. A modified expression for the pulse amplitude has been obtained assuming a sinusoidal bipolar charge d i s t r i b u t i ~ n , ~but ~ * the ~ ' interpretation of the problem is not eased, particularly as the observed charge distributions rarely conform to the assumed ideal. Calibration experiments involving examination of the variation of the probe sensitivity with probesurface distance are preferable. Probe systems have proved extremely versatile and have been used to examine both line and area12Jadistributions of surface charge, and novel designs continue to be described.lg 18.2.3 Contact Electrification

It is generally agreed that, contrary to common belief, rubbing is not essenrial to cause charge transfer across an interface. However, debate continues as to the relative importance of electrons and ions or gross material transfer, the significance of the ambient environment, and the effect of rubbing on the transfer mechanism. Even for a specific model such as electron transfer, whether the charge penetrates the bulk or is localized in surface states is also debated as, in fact, is the whole applicability of the concepts of solid-state physics to crystalline/amorphous polymers. Notwithstanding the arguments about fundamental concepts, charge transfer experiments yielding novel and definitive data for synthetic polymers have been performed over the past decade. 18.2.3.1 Metal-Dielectric Contact. A major improvement in the consistency of contact charge experimental data has resulted from taking practical precautions to counter the influence of environmental factors. Ambient gases can affect the surface properties of materials and also the equilibrium charge densities retained under normal atmospheric conditions can be drastically reduced by surface electrical discharges. An electrical discharge (or spark under normal atmospheric conditions) is the disruptive passage of current through the atmosphere, resulting in the D. K. Davies, J . Phys. D. 6, 1017 (1973). K . Murasaki, M. Kono, M. Matsui, and M. Ohno, Electron. Eng. ( T o k ~ o90, ) 187 (1970).

R. Elsdon and F. R . G . Mitchell, J . Phys. D. 9, 1445 (1976). K. A. Hughes and P. E. Secker, J . Phys. E . 4,362 (1971). ID F. Nordhage and G . Backstrom, J . Elertrcistritirs 2, 91 (1976). IT

In

18.2

429

STATIC ELECTRICITY

MULTIMETAL CONTACT PROBE- SHUTTER SYSTEM

OSCILLOSCOPE

FIG.4. Diagram of rolling-contact experiment. From Davies.IS

collapse of the initiating electric field. ExperimentsPo-z*have shown that the charge injected by metals into both inorganic insulators and various commercial plastics was dependent, both in magnitude and polarity, on the metal work function. The work function, definable as the minimum energy necessary to emit an electron from a solid, may be considered a means of defining the electron population of a solid material. An apparatu~'~ designed specifically to investigate the charge-metal work function relationship is shown schematically in Fig. 4. Polymer film specimens, nominally 50-200 pm thick, are mounted on the periphery of an earthed metal drum. A small contact wheel, comprising five metals of differing work function, namely, aluminum, gold, cadmium, platinum, and zirconium, is maintained, under slight pressure, in rolling contact with the polymers as the drum rotates. A probe shutter system continuously monitors the charges produced on each polymer film successively and the work functions of the metals are concurrently determined with respect to a gold reference electrode. The experiment, therefore, is wholly self-contained and, being conducted under vacuum, variable surface contamination is minimized and electrical discharges precluded. The charge densities measured for several samples of film are plotted against the respective metal work functions in Fig. 5. The correlation is evident. These charge levels should be noted. To fix ideas, a charge density of 3 x 10-BC cm-2 is equivalent to a field of 3 x 1V V cm-', the accepted electrical breakdown field between parallel electrodes in air at STP. These charge densities would therefore inevitably result in *O 21

R . G. C. Arridge, Br. J . Appl. Phys. 18, 131 1 (1%7). D. K . Davies, Insr. Phys. Phys. SOC. Conf. Ser. 4, 29 (1967). I . I. Inculet and E. P. Wituschek, Insf. Phys. Phys. SOC. Con$ Ser. 4, 37 (1967).

430

18.

ELECTRICAL METHODS

sparking between the separating surfaces after contact in the normal atmosphere. Based on equilibration of the Fermi levels in the dissimilar materials in contact, expressions may readily be derived linearly relating the charge density u to the contact potential difference across the interface, i.e., (18.2.2)

-7Nylon 66

2

FIG.5. Charge densities observed on nylon 66 (in the rolling-contact experiment) plotted against the respective metal contact potential difference (cpd) with respect to the gold reference. From Davies.ls

18.2

STATIC ELECTRICITY

43 I

TABLE11. The Work Functions of Dielectric Materials’ Material PVC Pol yimide Pol ycarbonate PTFE PET Polystyrene Nylon 6.6 a

Work function (eV) 4.85 4.36 4.26 4.26 4.25 4.22 4.08

* 0.2 0.06 0.13 2 0.05 2 0.10 0.07 2 0.06 2 2

*

From Davies.I3

where I#JM and I#JD are the metal and dielectric work functions, respectively, and A is a constant. Hence, values of work function have been determined for a number of synthetic polymers, having ascribed a nominal value for that of the gold reference electrode. These data are presented in Table 11. Charge transfer models based on penetration of the charge to a depthz1 A or a trapping at surface states of density D,2334 give A = Q K ~ / ~ C A or,

A = eD,/(I

+ ezD,d/4,

respectively. These are, of course, experimentally indistinguishable. The predominant uncertainty in any interfacial experiment is the area of real contact. This is reflected in attempts to examine the temporal growth of charge. It has long been established that the injected charge increases steadily to saturation in successive short duration contacts. This could well be caused by each successive contact making fresh microscopic areas of r e d contact, yielding a progressive increase in the total area contacted and hence an apparent temporal increase in charge. Even single contacts of varying duration are not unambiguous since the viscoelastic characteristics of polymers result in a time-dependent contact area.25 Recent experimentsz6have confirmed the earlier observations, but introduced a third dimension with the proposition that mechanical deformation of the surface transports filled carrier traps into the bulk of the material. Another investigati~n,’~ on the other hand, showed that for small contact forces, at least, the macroscopic contact area was well H . Bauser, W. Klopffer, and H. Rabenhorst, Adv. Sruric Elecrr. 1, 2 (1971). H . Krupp, Insr. fhys. ConJ Ser. 11, 1 (1971). *5 F. P. Bowden and D. Tabor, “The Friction and Lubrication of Solids.” Vol. 11, p. 233. Oxford Univ. Press (Clarendon), London and New York, 1W. 2a J . Lowell, J . fhys. D. 9, 1571 (1976). 24

432

18.

ELECTRICAL METHODS

described by Hertz2' elastic theory and that the injected charge density was apparently independent of the contact pressure. Resolution of temporal effects must await more definitive experiments, although it is generally agreed that the equilibrium charges densities are defined by the material work function even with violent rubbing. There has been little investigation of contact charge transfer between polymers, probably owing to the difficulty of repeat experiments. An investigation of rolling contact2ebetween polymers showed that the polarity of transferred charged was always consistent with the work function values of the materials. The scatter in the data was large, but the maximum charge densities were well described by the values of work function and charge penetration depths obtained in metal -dielectric contact experiments.21 18.2.3.2 Rubbing Contact. Rubbing was long thought to be a prerequisite for charge transfer; however, it is now considered that it merely modifies the contact phenomena. One undoubted effect is the enhancement of the area of actual contact, but the charging observed29on rubbing apparently identical surfaces indicates additional effects. This classical experiment, in which a rod was "bowed" at a single point by a second rod of the same material, showed the significance of the rubbing asymmetry. The effect was attributed to the temperature rise at the rubbed spot since reversal of the roles of the two rods reversed the polarity of the transferred charge. A model based on ion transfer caused by differential ionic mobility in the heated layer was proposed. Also, disruption of a charged (contaminating) layer compensating an intrinsic charge in the material has been suggested.30 However, in the former the motive for preferred migration out from the heated layer rather than into the bulk material was not given, while in the latter some asymmetry-either in thickness or charge density-is necessary to account for net charge transfer on mixing the layers. Given any form of asymmetry electron migration is equally possible. of the charges inSeveral investigations have been jected into polymers on scribing the surfaces by metal styli. A typical 27 F. P. Bowden and D. Tabor. "The Friction and Lubrication of Solids," Vol. 1, p. 10. Oxford Univ. Press (Clarendon), London and New York, 1950. D. K. Davies, Adv. Static Electr. 1, 10 (1971). P. S. H. Henry, Br. J . Appl. fhys. 4, Suppl. 2, 531 (1953). M. 1. Kornfeld, Sov. fhys. -Solid Srare (Engl. T r a n s / . )11, 1306 (1969). s1 A. Wahlin and G . Backstrom, fnsr. fhys. Conf. Ser. 11, 52 (1971); J . Appl. f h y s . 45, 205 (1974). F. Nordhage and G . Backstrom, f t i s t . Phys. Conf. Ser. 1 1 , 84 (1971). xi B. C. O'Neill and T. R. Foord, fnsr. fhys. Conf. Ser. 11, 104 (1971).

18.2

STATIC ELECTRICITY

43 3

-FIG.6. Schematic diagram of surface-scribing experiment displaying both charge and current measurement.

experiment is shown schematically in Fig. 6. The polymer film is mounted on an earthed plane, which can either be rotated or moved linearly, the motion effectively drawing the stylus, held under a known load, across the surface. The stylus is grounded via an electrometer so that the injected charge is determined from the stylus current. A probe system may also be employed to scan the charged track. Data may be presented as charge per unit track length or, if the track width has been determined independently using a microscope, as charge density. A linear dependence of charge on stylus work function has been observed for €TFE3land polyethylene, for e ~ a m p l e ,charge ~ ~ * ~being largely independent of the scribing velocity. Contrarily, both dependence on velocity and independence of metal work function have also been observed.” In all these investigations the observed dependence of charge on the square root of contact force is consistent with charge increasing linearly with contact area. Several analyses%of the real area of contact at a spherical electrode/polymer interface have predicted a force (F)dependence by combining both elastic deformation (PI3)and plastic flow (PI3). Material transfer u n d o ~ b t e d l yoccurs ~ ~ at high contact pressure and this has been advocated as a charging mechanism. It is not clear, however, why the shearing of a surface film should, of itself, continuously produce a nett charge of one polarity. 18.2.3.3 Interfacial Applied Fields. The charge injected on contact is augmented or diminished by applying an electric field across the interface A. Wahlin, Ph.D. Thesis, p. 28. University of Umea, 1973. B. J. Briscoe, C. M. Pooley, and D. Tabor, in “Advances in Polymer Friction and Wear” (L-H. Lee, ed.), p. 191. Plenum, New York, 1975. Bl

434

18.

ELECTRICAL METHODS

in accordance with the field direction. This has been confirmed recently for polythene both contacted by a mercury electrodeS6 and rubbed by known work function styli.32 No temporal effects were observed and instantaneous charging was inferred. This, together with the observed sensitivity to surface modification (particularly by ozone)3s, suggested trapping at surface states. The latter effect must be considered carefully. Contact experiments using mercury have been thoroughly reviewed3?and its behavior, particularly when contamination or surface oxidation is possible, is far from simple. The differences in behavior of saturated and unsaturated polyethylene is unambiguous and terminal vinyl groups clearly play an important role in contact charging. As will be described later they are also significant in bulk-carrier trapping processes. It is further interesting to note that branching or bulky side groups appear to be significant in the frictional properties of p ~ l y e t h y l e n e . ~ ~ 18.2.3.4 Charges and Adhesion. Materials in contact almost invariably exhibit some degree of adhesion. Apart from circumstances involving chemical reaction or interdiffusion of the materials, the adhesive ~~ effects arise from van der Waals forces or electrostatic a t t r a c t i ~ n .The observed adhesion between rubber and glass surfaces has also been described recently in terms of changes in surface energy at the interface,39 although it has been pointedI5 out that the "energy" could equally arise from the double-layer attraction. An electrostatic contribution to the adhesion of polymers to metals has long been suggested,4O these forces being advocated particularly to account for the data obtained in peel tests of adhesion. These tests have, however, been criticized, the suggestion being that most of the observed energy is absorbed in mechanical deformation of the polymer. In recent investigationsq1 of both the charges produced, and adhesive forces observed, in mutual rolling contacts between glass (both clean and aluminized), steel, natural rubber (lo1? cm), and electrically conducting, carbon-loaded natural rubber (lo2 cm), it was concluded that D. A. Hays,J. Cham. Phys. 61, 1455 (1974). W. R. Harper, "Contact and Frictional Electrification," Chapter XVI, p. 318. Oxford Univ. Press, London and New York, 1967. sB D. Tabor, in "Surface Physics of Materials" (J. M. Blakely, ed.), Vol. 2, p. 475. Academic Press, New York, 1975. 3B K. L. Johnson, K. Kendall, and A. D. Roberts, Proc. R. SOC.London, Ser. A 324,301 ( 197 1 ). B. V. Derjaguin, in "Recent Advances in Adhesion" (L.-H. Lee, ed.), p. 513. Gordon, Breach, New York, 1973. A. D. Roberts, J . f h y s . D. 10, 1801 (1977). 37

18.2

435

STATIC ELECTRlClTY

not more than 10% of the observed adhesion could be attributed to static charges. 18.2.4 Radiation Charging

Charges may be produced on insulating surfaces on bombardment by either energetic particles or ionizing photons. Corona dischargesbursts of ionization produced by the local intensified electric fields adjacent to sharp electrodes-are commonly used to provide controlled charging in electrostatic reprographic machines or static eliminator^'^ and polymer electrets-used in practical transducer d e v i c e d a r e often produced by electron irradiati~n.'~Added stimulus for investigation of radiation charging is currently afforded by spacecraft operational problems associated with charging of insulators on exposure to space radiation.'4 18.2.4.1 Photoelectric Emission. The emission of electrons from metals on illumination is a familiar phenomenon. The emission from semiconductors has also been thoroughly analyzed4sand inve~tigated.'~ In fact, the analysis of photoelectron energy spectra has emerged as a powerful technique for studying surface properties of materials." Rather less is known about insulators, however, largely because of the transient nature of the currents owing to the generation of repressive surface charges Investigations of photoelectric emission from think 100 nm, solution-castfilms of polyethylene have been d e ~ c r i b e d . ' ~The * ~ current, collected at a positively biased electrode located adjacent to the illuminated surface, was measured for several wavelengths of monochromated ultraviolet light. The dark conductivity was, surprisingly, equal to or greater than the emission current and so no charging problems occurred, and hence equilibrium currents could be measured. The experimental data obtained are summarized in Fig. 7, which presents a plot of the square root of the yield (electrons per photon) as a function of the incident

-

A. R. Blythe, Inst. Phys. Conf. Ser. 27, 238 (1975). G. M. Sessler and J. West, in "Electrets, Charge Storage and Transport in Dielectrics"

(M.M. Perlman, ed.), p. 587. Electrochem. Soc., Princeton, New Jersey, 1973. 44 Papers in "Photon and Particle Interaction with Surfaces in Space" (R. J. L. Grard. ed.). Reidel, Dordrecht, 1973. *1 E. 0. Kane, Phys. Rev. 127, 131 (1962). G. W. Gobelli and F. G. Allen, Phys. Rev. 127, 141 (1%2). " M. W. Roberts, in "Surface and Defect Properties of Solids" (M. W. Roberts and J. M. Thomas, eds.), Vol. 1, p. 144. Chem. SOC.,London, 1971. F. M. Lay, J . Vuc. Sci. Techno/. 11, 605, (1974). M. Fujihira and H. Inokuchi. Chem. Phys. Lei(. 17, 554 (1972). K . J. Less and E. G. Wilson, J . Phys. C. 6, 31 10 (1973).

18.

436

ELECTRICAL METHODS

.

401

FIG.7. The square root of the absolute photoemission yield (ratio of photocurrent i, to incident photon intensity N , ) as a function of incident photon energy. After Less and Wilson.m

energy. The data are in reasonable agreement and the photothreshold is readily discernible at about 8.8 eV. The quadratic dependence of photocurrent iphon photoelectron energy E, that is, iph = K(E - ET)',.where ET is the threshold energy, is commonly observed for both metals and semiconductors. In general terms the current is given by

i

=

K(E

-

ET)",

(18.2.3)

where K is a constant that includes the light intensity and n may have half-integer values between 1 and 5/2, depending on the excitation and scattering process.4s Direct investigation of the charge produced on polymers by far UV irradiation has also been des~ribed.~'The film specimens, -50 pm thick, mounted on an earthed, rotatable table, were irradiated while located beneath a fixed counterelectrode. The charge produced was determined using the probe technique described earlier. The surface potential rises to precisely balance the emission energy of the electrons. Positive potentials applied to the counter electrode produce an equivalent increase in the surface charge. The equilibrium charge density is described in terms D. K. Davies, Nature (London) 262, 277 (1976).

18.2

STATIC ELECTRICITY

437

of the specimen capacitance w o l d sby (18.2.4)

where & is the photothreshold, V , the applied extracting potential, and the other quantities are as defined previously. No data on the temporal growth of charge have been published, although it has been noted51that there is a significant difference in charging characteristics with and without applied extracting fields. Information on photoemission by high-energy (nucleonic) radiation is sparse. X-Ray photoelectron spectroscopy, XPS,a familiar and powerful technique for investigating surface properties of conductive materials, has been used successfully for polyethylene.= The electron spectra obtained indicated a band structure in good agreement with theoretical calculation. These photoelectric effects generally tend to confirm the applicability of solid-state physics concepts to synthetic polymers. 18.2.4.2 Electron and Ion Bombardment. Energetic electrons incident on a solid surface lose energy at a rate inversely proportional to their instantaneous energy and dependent on the material characteristics.53 Nonpenetrating monoenergetic electrons have therefore a precise maximum range, losing most of their energy at the end of their track. Electrons with a bimodal energy distribution are emitted from the solid, those with high energy being elastically scattered primaries, the true secondaries, generated by inelastic collisions, having very low energies. It is well established that the total emission from dielectrics can exceed the incident flux for a finite range of primary electron energies. Figure 8 presents a schematic plot of the secondary emission coefficient S vs. incident electron energy. Below the lower threshold El (- 10 eV), the primary energy is too low to ionize the material, while above the upper threshold E2 (- 1 keV), the primary electron penetration exceeds the secondary electron range. Primary electrons with energies within this range can produce a positive charge on a dielectric surface, however, only under an extracting field, since the low energy of the secondaries precludes significant unaided charging. Incident electrons with energies greater than E2 charge the dielectric negatively. The first primary electrons penetrate to their full range, but those following are retarded by the space field of the former." The range a M. H. Wood, M. Barber, I. H. Hillier, and J. M. Thomas, J . Chem. Phys. 56, 1788 (1972). A. J . Dekker, "Solid State Physics," Chapter 17, p. 418. Macmillan, New York, 1958. sI B. Gross, J. Dow, and S. V. Nablo, J . Appl. Phys. 44, 2459 (1973).

18. ELECTRICAL METHODS

438

2.0

I i O

I El

I

l I

I

2 i

3 X 10'

E2 Accelerating voltoge ( V )

FIG.8. Typical variation of secondary emission coefficient, 6 with incident electron energy showing the crossover energies E , and E , (6 = 1).

is thus progressively reduced, equilibrium being achieved when the primary electrons are balanced by equal secondary emission, that is, the incident electron energy is exactiy E t . This has been demonstrated for a number of dielectric^,^^ the equilibrium charge under bombardment being precisely defined by the necessary retarding potential and the geometric capacitance of the specimen. In equilibrium, the negative trapped charge exerts an emissive field and so a positive compensating excursion of the surface is possible. An oscillatory effect may be envisaged where the retardation of the primaries varies as the nett field varies. At very high rates of injection of very energetic primary electrons (-MeV) such oscillations have been observed. In an e ~ p e r i m e n tusing , ~ ~ an elegant te~hnique,~' the potential of the dielectric surface under continuous bombardment was probed by a second electron beam skimming the surface, potential changes being observed directly by deflection of the spot on a fluorescent screen. Random rises D. K. Davies, Nature (London) 262, 279 (1976). A. Watson and J . Dow. J . Appl. Phys. 39, 5935 (1968). C. N. W. Litting, Br. J . Appl. Phys. 5 , 289 (1954).

18.2

STATIC ELECTRKITY

43 9

and collapse in external field were observed, these being attributed to local emission and electrical breakdown in the surface layers. There have been numerous investigationss8of induced conductivity in the metal/polymer/metal sandwich configuration, most of which have been associated with measurement of thermally stimulated currents.s0 Some of the assumptions made, such as a uniform carrier density in the irradiated region and time-invariant fields in the nonpenetrated film, may also be suspect, but despite this activity the detailed physical processes occurring remain obscure. There is also little known about the interaction of ionized or excited molecules with insulating synthetic polymers. It has been shown for an inorganic insulatoeO that at low kinetic energy charge is generated by electron emission, the ion potential energy being transferred directly by Auger neutralization, a process familiar for metals.e1 At higher energies, penetration of the solid by the ion is probably involved. Direct energy transfer between excited molecules and zinc oxide has also been observed.e2 Controlled charging of poly(ethy1ene terephthalate) (PET) in air has been achieved by both positive and negative ions generated by a Nernst filament.s3 The nature of the ions and the interactions with the polymer remain to be established. 18.2.5 Charge Migration

At normal temperatures, in the absence of ambient neutralizing ions or enhanced conduction caused by surface contamination, charge may be retained indefinitely in low-conductivity materials. The electret-the electrical equivalent of the magnet but with the vital difference of a realizable monopole-is a practical reality, charge lifetimes of many years having been observed. The charge storage properties of synthetic polymers have been thoroughly reviewed.& Heating causes increasingly rapid charge migration and this forms the basis of methods for examining the carrier localization and transport processes. Carrier mobility has also been investigated by direct time-of-flight measurement. 18.2.5.1 Thermally Stimulated Currents. The thermally stimulated technique is based on monitoring the current or charge decay produced as sa

B. Gross, G . M. Sessler, and J . E. West, J . Appl. Phys. 45, 2841 (1974). D. M . Taylor, J . Phys. D.9, 2269 (1976). D. W. Vance. J . Appl. Phys. 42, 5430 (1971). H. D. Hagstrum, Phys. Rev. 91, 543 (1953). J . P. Dauchot, J. P. Verhaegen, and J. van Cakenberghe, Nuture (London) 223, 825

(1969).

E. A. Baum and T. J. Lewis, Inst. Phys. Con$ Ser. 27, 130 (1975). J . van Turnhout, “Thermally Stimulated Discharge of Polymer Electrets.” Elsevier, Amsterdam, 1975.

440

18.

ELECTRICAL METHODS

the precharged material experiences a steadily increasing temperature. There are essentially two configuration^,^^^^ one with both electrodes (usually evaporated) in contact with the charged material and the other with an electrode held away from one uncoated surface. Current only may be determined in the fully contacted mode, but remnant charge may also be measured in the latter mode. The heating rate is usually about 1°C min-’, the temperature range extending from cryogenic temperatures, say - 180”C,to approaching the softening range of the polymer. Numerous TSC experiments have been conducted for a large number of materials and for differing experimental conditions and methods of charging. For example, electron-irradiated PTFE and p ~ l y e t h y l e n e , ~ ~ corona-charged poly(ethy1ene terphthalate),66 electron- and ?-irradiated FTFE,67and a number of friction-charged polymers.6* A typical TSC is shown in Fig. 9, displaying the series of peaks commonly observed. In order to decipher the complex spectra “peak cleaning” techniques have been devised.66 These involve partial heating and rapid cooling to remove the shoulders or peaks at temperatures lower than the peak of interest; partial heating to the same temperature several times while observing trends in the initial rise of current; or partial heating to a series of steadily increased temperatures that eventually span the whole range. The spectrum is eventually reduced to two or three activation energies associated with carrier traps. The data obtained are consistent with the activation energies observed in conduction experiments and have indicated the significance of structural features and certain chemical groups in trapping and the role of high-energy radiation in trap filling.65 The latter is consistent with data from the (possibly more familiar) thermoluminescence glow curve technique. Simultaneous investigation of thermoluminescence (TL) and TSC in polyethylene showedse correlations between some of the TL and TSC peaks, which were attributed to the release of electrons from traps. Despite these correlations, however, there is still some caution in interpreting TSC data. 18.2.5.2 Isothermal Charge Decay. Experiment2I has shown that charge injected into a small area in the center of a relatively extensive, but thin polymer film specimen mounted on an eaithed electrode dissipates by transport through the volume of the specimen. In the absence of contamination-induced enhanced surface conduction, spreading is negM. M. Perlman and S. Unger, J . f h y s . D. 5,2115 (1972). aa R . A. Creswell and M. M. Perlman, J . Appl. f h y s . 41,2365 (1970). B. Gross, G . M. Sessler, and J. E. West, J . Appl. Phys. 47, 968 (1976). 05

aa P. H. Ong and J. van Turnhout, DECHEMA-Monogr. J. Randle, J . Phys.

se A. E. Blake, A. Charlesby, and K.

72, 105 (1974).

D. 7, 759 (1974).

18.2

STATIC ELECTRICITY

44 I

x lo-'2 5.4

Temperature P C 1

FIG.9. Typical TSC from electron-irradiatedPTFE for different doses: -, 10' rad; -.-, 6 x 10' rad; ... , 1.8 x lo5 rad; -6 x l(r rad. From Perlman and Unger.05

ligible. A simplified a n a l y ~ i sindicated ~ ~ . ~ ~ that carrier mobility could be determined by observation of the rate of decrease of surface charge or potential. More elaborate analyses have been p u b l i ~ h e d , ~the ' . ~ ~latest of which accounts for crossovers observed in the charge-time curves and an apparent inverse dependence of mobility on the initial injected charge density.73.74 The technique is extremely simple. While maintaining the specimen at the desired temperature, charge is injected into the free surface and the charge density or surface potential subsequently measured at intervals appropriate to the rate of decay. The values of mobility p obtained for synthetic polymers are extremely small, in the range 10-13-10-0 cm2 V-l sec-', say, the temperature dependence being described by a single activation energy W, i.e., p = he-W'kT. Investigations have shown the carrier mobility in polyethylene to be inM. Ieda, G . Sawa, and U. Shinohara, Electron. Eng. (Tokyo) J38,67 (1968). H. J. Wintle, J . Appl. Phys. 43, 2927 (1972). 7 p R . M. Hill, J . Phys. C 8, 2488 (1975). M . M. Perlman and T. J. Sonnonstine, Inst. Phys. Conf. Ser. 27,74 (1975). M. M. Perlman, T. J . Sonnonstine, and J . A. St. Pierre, J . Appl. Phys. 47,5016 (1976). 70

'I

442

18. ELECTRICAL

METHODS

fluenced by the density (crystallinity) of the Also the mobility in low-conductivitydielectrics may be radically increased by introducing electron accepter specie^.'^,^' The latter effect, in polyethylene, is again associated with trapping at terminal unsaturated groups as indicated by the TSC data described earlier. 18.2.5.3 Time-of-Flight Techniques. The velocity of a carrier can be assessed by measuring the interval required to traverse a known distance. Carrier mobility in inorganic materials-both crystalline and amorphous -has been determined by measuring the transit time of injected carriers under known electric stress. A potential difference is applied to the electrodes (usually evaporated) on a thin specimen of material. A pulse of electrons is injected through one electrode and their transit observed either directly by monitoring the specimen current or by integrating the latter on a capacitor and measuring the potential. In the former case, the constant current during carrier transit terminates rapidly on their arrival at the opposite electrode, while in the latter, the steadily increasing potential becomes constant. In either case, ideally, the transit is readily recognized. In practice, however, the sharp transition is rarely observed, owing to spreading of the carrier layer. This technique has been used to examine mobility in synthetic polymer^,^^^^^ but the results have been somewhat inconsistent. It is a critical experimental situation; the injected charge has to be large enough to yield detectable currents, yet not so large as to cause local field distortion. Further, there is an interpretational problem of deciding whether the carrier is moving forward through the neutral material or back through the ionized volume penetrated by the incident electron beam. These ambiguities remain unresolved. '5 76

I7

D. K. Davies, J . Phys. D. 5 , 162 (1972). D. K. Davies and P. J. Lock, J . Electrochem. Soc. 129, 266 (1973). D. K. Davies and R. J. Loveland, IEE. Conf. Publ. 129, 269 (1975). E. H. Martin and J. Hirsch, J . Appl. Phys. 43, 1001 (1972); J. Hirsch and E. H. Martin,

ibid. p. 1008.

K . Hayashi, K . Yoshino, and Y. Inuishi, Annu. Rep., Conf. Electr. Insul. Dielectr. Phenon . 424 ( 1974).

18.3 Electric Breakdown

By 8.R. Varlow 18.3.1 Introduction In any experimental determination of the electric strength of an insulator, whether polymeric or otherwise, it must first be established that all of several possible extraneous breakdown mechanisms have been excluded. These include breakdown due to electrical discharges either on the surface of the dielectric or in cavities within the body of the material. Thermal breakdown and electromechanical breakdown are two further mechanisms of failure, which in plastic materials may individually or ‘in concert contribute to breakdown under the application of an electric field and lead to a value of electric strength that is dependent upon external parameters such as the mode of stressing, electrode material, and specimen geometry. A clear distinction must be made between those measurements of electric strength obtained in short-term conditions in the laboratory and those obtained in service over a period of years. The action of surface or internal discharges, long-term chemical and electrochemical activity, and the growth of electrical and water trees will produce a much lower effective electric strength over a period of several years than that obtained in the laboratory in times ranging from minutes to microseconds. Due to the large variance in the results observed in breakdown experiments, the necessity for large numbers of measurements cannot be stressed too heavily. In particular, when small differences are being sought in the breakdown strength due to changes in an experimental parameter, such as electrode material, crystallinity, or temperature, 100 breakdown measurements under each experimental condition cannot be considered excessive. Small differences can only be justified as a result of a rigorous statistical analysis. No attempt is made here to give a comprehensive account of the theoretical treatment of dielectric breakdown. This is not the principal con443 METHODS OF EXPERIMENTAL PHYSICS. VOL.

16C

Copyright 0 I960 by Academic Press. Inc. All rights of reproduction in any form =served. ISBN 0-12475958-0

444

18. ELECTRICAL METHODS

cern of this volume and is more than adequately dealt with elsewhere.’ It is therefore considered sufficient to outline the various mechanisms that may lead to breakdown, with a very brief theoretical background in order that an understanding may be obtained of the different terms to which reference is made when discussing experimental methods and in order to interpret the philosophy behind the different experimental approaches. It is not possible in the space available to describe in detail all the work carried out in this field, and the object of the author has been to give a general impression together with detailed accounts of selected experimental methods that are either typical of their particular branch of the subject or reflect significant advances in either the experimental method or the physical understanding of breakdown measurements. 18.3.2. Mechanisms of Breakdown 18.3.2.1 Intrinsic Breakdown. There are theoretical reasons for believing that the electronic structure of an insulator will become unstable and the material consequently disrupt when fields in excess of lo6 V cm-I are applied. This is intrinsic breakdown, and the field in which it occurs, is the intrinsic electric strength. Theoretically, the magnitude of the breakdown field strength does not depend on the size and shape of the sample, or on the material and configuration of the electrodes. For this reason the process of breakdown is called “intrinsic” since it is regarded as being characteristic of the dielectric alone at a given temperature. In laboratory experiments to determine this intrinsic parameter of the material, it is first necessary to ensure the absence of the numerous other forms of breakdown that ‘are not so insensitive to experimental conditions. In practice, intrinsic electric strength cannot be verified by direct observation but must be inferred from evidence indicating the absence of these other extraneous breakdown mechanisms. The best way of ensuring intrinsic breakdown will depend on the material and the temperature of the experiment, but there are a few general principles to be followed. It is preferable to use impulses rather than continuous ac or dc voltages because this renders breakdown by cumulative actions of ambient discharges less likely, and it may raise the thermal breakdown voltage above the intrinsic breakdown value. Evidence pointing to the absence of these mechanisms is provided. by showing that the electric strength does not depend on the impulse duration or on the number of impulses applied before breakdown. Failure of the specimen may be induced by thermal, mechanical, or

’ J . J. O’Dwyer, ”The Theory of Dielectric Breakdown of Solids.” Oxford Univ. Press, London and New York, 1964.

18.3

ELECTRIC BREAKDOWN

445

electrical effects arising from breakdown of the ambient. This may occur by flashover from electrode to electrode over the surface of the specimen, or by the passage of a spark from an electrode to the surface and through the specimen. The use of specimens that provide long flashover distances and electrodes without sharp edges will eliminate surface flashoveri but not the second form of ambient breakdown. It can be seen from the position of the puncture whether or not discharges have caused breakdown in a specimen of considerable area. However, the presence of mechanical defects and other inhomogeneities reduce the electric strength, and as their presence is more likely the bigger the specimen, it is advisable to stress as small a volume as possible by using sphere-sphere or sphere-plane electrodes. The puncture produced by the discharge may then be of appreciable size compared with the stressed volume and the absence of ambient discharges cannot be established by inspection. Liquids and compressed gases have been used as ambients to control the effect. In general, pure liquids have a greater permittivity and a greater electric strength than compressed gases, and they are more convenient in use. It is advantageous to increase the specimen thickness as the edges of the electrodes are approached. because this decreases the voltage across the ambient when the permitivity of the ambient is greater than that of the specimen material. Recessed specimens are used whenever the properties of the material permit their manufacture. The classic electrode arrangement is shown in Fig. 1, in which the voltage is conveyed from the main electrodes to the thin part of the specimen by auxiliary electrodes deposited on the surface. The removal of the external electrodes from the thin region of the specimen avoids mechanical effects. The trigatron operates within a fraction of a microsecond and diverts the discharge current from the specimen, thus permitting precise location of the puncture, which should be at the bottom of the cavity. More recently, solid-state current diverters have been employed.2 A uniform electric field is assumed in determining the intrinsic electric strength. This is not strictly true in plane-recessed specimens, and the electric stress at the surface of the cavity exceeds that at the plane by an amount that increases with decrease in the ratio of cavity radius R to specimen thickness L. Formulas are available for estimating the ratio of maximum to minimum field ~ t r e n g t h . ~Provided R / L > 10, Emax/Emln < 1.05. Field distortion may also arise from the flow of prebreakdown current and the resulting trapping of charge carriers in the bulk of the mate~ial.~ R. R. Opoku, E. G . Robles, and J. H. Mason, IEE Con$ Publ. 129, 323 (1975). G . W. Carter and S. C. Loh, IEE Monogr. 1W, No. 325M, 108 (1959). ' A. Bradwell, R. Cooper, and B. Varlow. Proc. Insr. Elecrr. Eng. 118, 247 (1971).

18.

446

Auxiliary

ELECTRICAL METHODS

I

I

Trigatron

FIG. 1. Classical circuit arrangement for intrinsic strength determination, using a planerecessed specimen.

18.3.2.2. Thermal Breakdown. In general, the heat generated by the passage of conduction current through the specimen is partly conducted away to the surroundings, and partly absorbed to cause a rise in the temperature. For energy balance we have, with an applied field F,

C dT/dr

+ div(k grad T ) = UP,

(18.3.1)

where C is the specific heat per unit volume, dT/df the rate of change of temperature (T) with time ( f ) , k the thermal conductivity, u the electrical conductivity, or the equivalent when heating is by dielectric losses, and grad T the space gradient of the temperature. It is assumed thatrhe insulating properties are lost if the temperature at some point in the dielectric exceeds a critical value T , ,and the solution of Eq. (18.3.1) gives the time to reach T , for a given applied field and boundary conditions. This equation cannot be solved in general terms because C, k, and u may all be functions of T, and c may also depend on field strength. The experimental realization of thermal breakdown depends on the period of application of the applied voltage and there are two extremes to consider. In the first case, when the applied field is increased very slowly (i.e., steady-state or dc thermal breakdown) C dT/dt = 0 and the field necessary to raise the temperature of the hottest part of the dielectric to T , is calculated. This gives the lowest field for thermal breakdown and the “minimum thermal breakdown voltage” is obtained, when the field is uniform, in terms of u, k, and the ambient temperature. This form of breakdown develops slowly and is essentially a phenomenon associated with continuously stressed dielectrics. The time required for breakdown to develop is at least milliseconds and in most cases much longer. The breakdown field strength depends on the size and shape of the sample and on the geometry and thermal properties of the electrodes and the ambient medium. The other limiting case occurs when the field is applied rapidly, and it

18.3 ELECTRIC BREAKDOWN

447

can be assumed that no heat is conducted away from the specimen. The heat generated is consumed entirely in raising the temperature of the specimen, in which case

C dT/dt

=

(18.3.2)

uF2.

The electric strength corresponding to this form of breakdown is called the impulse thermal breakdown strength. Assuming a dependence of u on temperature of the form u = uoexp(- W / d T ) ,

(18.3.3)

where uoand W are constants of the dielectric, and assuming a linearly rising voltage wave so that Fb = atb, where a is a constant and f b the time for the field to reach the breakdown value F b , the thermal impulse breakdown strength is, to a close approximation, 2kT0

exp W/2kT0;

(18.3.4)

To being the initial temperature. This breakdown strength varies with time of application of the field, being larger for applied voltage pulses of short duration, but does not depend greatly on the size and shape of the sample. 18.3.2.3.Electromechanical Breakdown. A compressive pressure of several kilograms per square centimeter is experienced by most dielectric materials when subjected to applied fields of about lo6 V cm-1 due to the force of attraction between the surface charges. In materials such as the thermoplastic polymers, the initial thickness of a specimen (Lo) will decrease as the voltage is raised, and for a.plane slab in a uniform field, the equilibrium thickness L when the strain is appreciable is given by Stark and Gartons as ( 18.3.5)

where Q is the relative permittivity of the dielectric, K~ the permittivity of free space, and Y Young’s modulus. These authors point out that a stable state is possible only if L / L o is greater than 0.6. If the intrinsic strength has not been reached at this value, the specimen will collapse. This follows from the fact that L 2 log(L,/L) has a maximum when L / L o = exp(-4) = 0.6. The critical electric stress is F , = ( Y / E K ~ ) ~ ’ * and the highest apparent intrinsic strength that can be observed is Fa a

=

V / L o = exp(-#) F , = ( Y / Q K ~ ) ~x’ ~0.6.

K. H.Stark and C.G . Garton, Nature

(London) 176, 1225 (1955).

(18.3.6)

448

18.

ELECTRICAL METHODS

An objection to the above analysis is that Y describes the material under uniaxial tension or compression. It is experimentally impossible to deform the entire area of the dielectric under the electrode in uniaxial compression. In practice, fields are not perfectly homogeneous and dielectric films are not geometrically smooth. Local regions subjected to higher than average electric fields thus experience a mechanical shear stress tending to form an indentation. As the indentation proceeds, the electric field becomes even more inhomogeneous at the deformation site, leading to instability. The sharp depression thus formed will serve to concentrate the field at this point. The material at the center of the depression will flow outward and form a ridge around the depression. As the field is increased further, the electric strength, either intrinsic or electromechanical, is reached at the center and breakdown occurs. 18.3.2.4 Nonuniform Fields. I t would be of interest to know if the criterion of breakdown in nonuniform fields is simply the achievement of the intrinsic electric strength at some point in the dielectric or a more complicated one, which depends, for example, on the field distribution. Using a concept of breakdown that involves some intrinsic material parameter as described in Section 18.3.2.1, it seems reasonable to expect breakdown when the maximum electric field becomes equal to the intrinsic electric strength. In the event of the nonuniformity in the field arising from the accumulation in trapping centers of injected charge carriers, of whatever polarity, density, or spatial arrangement, a depression of the local field at some point in the material must, for a constant applied voltage, produce field enhancement elsewhere in the sample. Consequently, using this model, space charge must always produce a reduction in breakdown voltage. This is contrary to experimental observation4 since injection of negative charge in the vicinity of the cathode, by depressing the electric field, is able to increase the apparent electric strength of several polymeric insulating materials. In order to interpret such results, a critical field at the cathode has been postulated as the breakdown criterion. This critical cathode field concept has been ably demonstrated experimentally in uniform fields in alkali halide crystalsa and treated theoretically' to explain the breakdown process in terms of a model involving collision ionization. It should prove interesting to apply the high-speed photographic methods of Cooper and Elliotta (see Section 18.3.4.8) and the critical cathode field concept to highly nonuniform field situations in which, due to the geometry of the electrode arrangement, the cathode is in a region of low electric stress. The injection of space charge may then

R. Cooper and C. T. Elliott, Br. J . Appl. Phis. 17, 481 (1966). J. J. O'Dwyer, J . Elecrrochem. SOC. 116, 239 (1969).

18.3

ELECTRIC BREAKDOWN

449

be found to have a bearing on the polarity effect observed in point-plane systems, in which a lower breakdown voltage is required for a positive point than for a negative one. 18.3.2.5. Treeing. It has been established for some considerable time8-” that the breakdown of solid insulation in divergent electric fields may occur in stages by the action of discharges that cause a treelike structure of channels to propagate from the region of high electric stress into the body of the dielectric, until a continuous conducting path is formed between the electrodes. The time taken for this to occur depends on circumstances and may vary from many hours to a few seconds. The average speed of the process is, however, not comparable with that of intrinsic electric breakdown, which is about lo8 cm sec-’ in a uniform field. This phenomenon is called electrical treeing and has been observed between point electrodes using alternating and unidirectional impulse voltages and in both laboratory samples and in insulation withdrawn from service. It is generally accepted that the extension of electrical trees is induced by a gas discharge in the existing tree channel, and that the diameter of a tree channel is of the order of 1 pm. The work of Bolton et aL8 and Cooper and AucklandI2 has, however, cast doubt on the previously held view that the electric potential of a needle electrode, from which an electrical tree is growing, is transferred to the tip of the existing tree channel through the conductive plasma of a discharge or that tree channels can be considered as conducting extensions of the needle electrode. The term “electrochemical treeing” or “water treeing” has been applied to the deterioration process observed in recent years in punctured and healthy cables withdrawn from service in a wet en~ir0nment.l~From vacuum treatment and infrared spectroscopy of the samples containing these trees, it has been shown that they contain water. As with electrical trees, complete bridging of the dielectric between the electrodes does not necessarily ensure breakdown of the sample. However, water trees, unlike electrical trees, are characterized by the absence of any corona discharge activity. These trees tend to originate in areas where foreign substances or defects are present in the insulation.

’ B. Bolton, R. Cooper, and K. G. Gupta, Proc. Inst. Electr. Eng. 112, 1215 (1965).

* J. lo

H. Mason, Proc. Inst. Electr. Eng., Part C 102, 254 (1955). D. W. Kitchinand 0. S. Pratt,Annu. Rep., Conf. Elect4*.Insul. Dielectr. Phenom., 1957

p. 43 (1958). W. P. Baker, Nature (London) 177. 887 (1956). R. Cooper and D. W. Auckland, Annu. Rep., Conf. Electr. Insul. Dielectr. Phenom. 1972, 151 (1973).

’*

G . Bahder, C. Katz, J. Lawson, and W. Vahlstrom, IEEE Trans. Power Appar. Sysr. PAS-93, 977 (1974).

18.

450

ELECTRICAL METHODS

18.3.2.6. Breakdown by Discharges. 18.3.2.6.1 INTERNAL DIS-

Due to the concentration of electric stress in regions of low permitivity (dielectric constant), the stress in a gas-filled cavity is greater than the value in the surrounding solid insulation and discharges will take place in the cavity for a sufficiently high applied voltage. This voltage, however, is low compared with the intrinsic breakdown value of the solid insulation. Internal discharges occur during each half-cycle of alternating voltage. If the voltage is just equal to the inception voltage, the first discharge occurs at the peak of the first half-cycle. At the site of this discharge, the stress across the cavity is then the algebraic sum of the applied field and the reverse field due to surface charge. A reverse discharge can occur at this site soon after the field is reversed, unless the leakage of surface charge is unusually rapid. When direct voltage is applied, initial discharges will probably occur at surface imperfections and if the voltage is maintained constant, further discharges occur only when charge leakage through the material, or over the surface of the cavity, has restored the voltage across the cavity. The interval between successive discharges at one site depends on the leakage time constant. With high-quality insulation this may be several minutes or hours, so that, unless leakage occurs over the surface of the cavity, discharges are only observed when the direct voltage is being raised or lowered. Examination of polyethylene samples, after tests of varying duration shows that internal discharges cause two distinct types of deteri0rati0n.I~ Initially, the surfaces of the cavity are eroded by the discharges. The rate of erosion increases rapidly with increasing stress, and the discharges concentrate, forming deep pits at a few sites. These deeper pits are eroded with increasing rapidity, until they attain a critical length and treeing commences, leading to ultimate failure. 18.3.2.6.2. AMBIENT DISCHARGES. The edges of a rounded electrode in contact with a planar specimen are illustrated in Fig. 2. On application of an impulse of amplitude V, the voltage V1 across the ambient at a small distance x from the point of contact of the rounded electrode and the specimen is approximately CHARGES.

v1

= Vdl/[dl +

(&/&?)41,

( 18.3.7)

where K , and K 2are, respectively, the permittivity of the ambient and the specimen. The empirical relation between the breakdown voltage V, , and the thickness dl for a uniformly stressed gas or liquid is approximately

v, = v, + ad,, "

J. H.Mason, Proc. I n s t . E/ectr. Eng. 98, 44 (1951).

(1 8.3.8)

18.3 ELECTRIC BREAKDOWN

K2db(V.J

45 1

Dielectric

FIG.2. Edge discharge inception.

where Vo and a are constants. If V, exceeds V,for all values of d, , discharges will not occur in the ambient. The above relationships are plotted in the figure, As the applied voltage V increases, curve (a) moves towards curve (b) and a quenched discharge will occur when they touch. The applied voltage Vb above which these discharges will occur i s obtained from the condition that the two roots of the equation ad, + vo = Vbdl/[dl + (Kl/K2)41

(18.3.9)

are equal, and is

Vb =

vo + a(Ki/Kz)d2 + 2[a(K,/K~)d~Vo]"~.

(18.3.10)

The long-held belief that an ambient discharge between the electrode edge and the surface of a dielectric sample constitutes a conducting plasma that behaves as a needlelike extension of the electrode and thus produces breakdown of the dielectric by stress concentration at the discharge site and subsequent local intrinsic breakdown is not consistent with recent photographic work on composite s p e ~ i m e n s . ~ ~ J ~ The behavior of surface discharges, in the long term, is similar to, though usually more complex than, that of internal discharges. There are many sites on the surface, which will discharge at about the same voltage. Also when the voltage is raised, larger discharges can occur to sites more distant from the electrode, whereas the magnitude of internal discharges is limited by the size of the cavity. The rate and nature of the deterioration depend on the ambient medium as well as on the applied stress and the type and thickness of the solid insulation. 18.3.3 Specimen Preparation 18.3.3.1. Molded Specimens. The problem of discharges from the edges of electrodes onto the surface of plane specimens of dielectric matel5 D. W. Auckland, A. B. Borishade, and R. Cooper, Annu. Rep., Con$ Electr. Insul. Dielecrr. Phenom. 1976, 318 (1977).

452

18.

ELECTRICAL METHODS

rial and the resulting errors in the determination of the electric strength of this material were evident to researchers working in the field of dielectric breakdown long before the advent of polymeric insulation. The solution adopted by them in the case of such materials as the alkali halides has been applied with equal success to polymers as they have become available for service as insulation. Spherical recesses, normally in one face only, but occasionally in both, are produced either, in the case of thermoplastic materials, by compression molding or by machining, in more rigid materials. With the accompaniment of metallized surfaces this method allows breakdown to be achieved in the thinnest part of the recess at voltages well below those necessary to produce discharges to the relatively thick material at the electrode edge. The method of compression molding first attributed to Austen and Pelzer16 and Oakes" about 30 years ago has been used and refined by many workers since that time. Although used principally on polyethylene,18-21where a steel ball is pressed into the surface of sheet material at temperatures ranging from 120" to 150°C, this method serves equally well for polystyrene,22,23 poly(viny1 chloride)% and ethylene-propylene In contrast to most other workers Fischer and Nissenzs have used low-density polyethylene in granule or powdered form as their starting point. This material is heated in an evacuated glass tube to 160°C. The polyethylene is compressed under a pressure of 106 N m-2 (140 psi), and the resulting rod is allowed to cool to room temperature over a period of several hours. Two precision ball-bearings are held magnetically against the end faces of short lengths of this rod, and after the temperature has been raised to a maximum of approximately 140"C, during which the electrodes move into the polymer under spring pressure, a cooling phase allows relaxation of the stresses caused by the insertion of the electrodes. This total immersion of the electrodes in the dielectric material eliminates the problem of surface discharges most effectively. Although it should be possible to investigate the effect of electrode material in such an arrangement by coating the ball-bearings with the desired electrode material before immersion in the polymer, it is more common in A. E. W. Austen and H. Pelzer, J. Insr. Elecrr. Eng. 93, 525 (1946). W. G. Oakes, J. Inst. Elecrr. Eng. 95, 36 (1948). Is R. A. Fava, Proc. Insr. Elecrr. Eng. 112, 819 (1965). W. G. Lawson, Nature (London) 206, 1248 (1965). *O D. B. Watson, W. Heyes, K. C. Kao, and J . H. Calderwood, IEEE Trans. Elecrr. Insitl. EI-1, 30 (1965). C. G. Garton and N . Parkman, Proc. Inst. Electr. Eng. 123, 271 (1976). H. G. Riddlestone, Proc. Inst. Elecrr. Eng. 100, 159 (1953). p3 J. Artbauer and J. Griac, IEEE Trans. Electr. Insitl. EI-5, 104 (1970). M. Kosaki, K . Sugiyama, and M . Ieda. J. Appl. Phys. 42, 3388 (1971). m P. H. H. Fischer and K. W. Nissen, IEEE Trans. Electr. Insul. El-11, 37 (1976). l6

18.3

ELECTRIC BREAKDOWN

453

this instance to adopt the less elaborate methods using sheet material from which the molding sphere can be removed after cooling. Compression molding with needle electrodes has also been used for both p ~ l y e t h y l e n eand ~ * ~poly(methy1 ~ methacrylate) (PMMAhZs A principal difficulty in the production of specimens with needle-shaped cavities is the application of the conducting surface layer. For painted suspensions the solvent must possess a sufficiently low surface tension to allow penetration to the tip of the cavity without leaving a gaseous inclusion. Graphite in water has been used by Watsonz7and graphite in kerosene and silver in toluene by Bolton et a/.* As an alternative to painted suspensions Bolton et d . have used vacuum evaporated silver and aluminum but claim that generally this procedure offers no advantages in their particular experiment and the time taken *-oprepare specimens is increased appreciably. If the nature of the electrode material is not a variable parameter in the investigation to be carried out using needle-shaped electrodes, the simplest solution would seem to be to leave the molding needle embedded in the p ~ l y m e r . ~In* this ~ ~ event, however, careful annealing and cooling may be required in order to minimize mechanical stresses arising from differential thermal contraction between the needle and the polymer. Careless preparation of such specimens will either produce fracture of brittle materials or a gaseous inclusion at the needle tip. Because of the difficulty in defining the electrode end in such systems, McMahonZ8has proposed the use of a predefined gaseous inclusion. In order to produce smooth surfaces and specimens free from mechanical stress, whether manufactured by the impression of a sphere or a needle, some workers have found it necessary to anneal their specimens after The annealing process varies somewhat arbitrarily between workers. For example, Blok and LeGrandzgreheated above the melting point for a few seconds followed by slow cooling, while Tsutsumi and Kako30annealed their polyethylene specimens for five hours at 100°C. Where cross-linked polyethylene samples are to be investigated, it may be found convenient to cross-link the material during the specimenforming process. In one case31 the cross-linkable polyethylene compound has been prepared by rolling low-density polyethylene, a chemical la

M . Ieda and M. Nawata, Annu. Rep., Conf Elerrr. Insul. Diclecrr. Phenom. 1972, 143

(1973).

D. B. Watson, J . Phys. D . 2, 1681 (1969). E. J. McMahon and J . R. Perkins, Annu. Rep., Conf. Elecrr. Insul. Dielecrr. Phenom. 1972. 133 (1973). 28 J. Blok and D. G. LeGrand, J . Appl. Phys. 40, 288 (1969). 3o Y. Tsutsumi and Y. Kako, Prcic. Insr. Elertr. Eng. 122, 223 (1975). 31 H . Matsubara and S. Yarnanouchi, Annu. Rep., Conf. Elecrr. Insid. Dielectr. Phenom. 1974, 270 (1975). 27

454

18. ELECTRICAL METHODS

cross-linking agent, and an antioxidant for 10 min before extruding this compound over the conductor. Cross-linking was then accomplished by heating in nitrogen gas at 10 kg mm-2. Eichhorn’s chemically cross-linked specimens32 were cross-linked in the press after molding and subsequently the reaction products of the peroxide decomposition were removed in a vacuum oven. 18.3.3.2.Machined Specimens. The machining of specimens from materials of a more rigid nature is not entirely confined to PMMA although specimens of this material are most commonly produced in this Spherical burrs are used to produce recesses with diameters in the region of 1 mmZ7and pointed cavities with radii of curvature of the order of 25 km may be obtained with the aid of dentists’ flame-tipped drilk8 In addition to PMMA, T ~ u r e i l l ehas ~ ~ machined recesses with a base thickness of about 500 pm in 1.2 mm thick sheets of polyethylene and polystyrene. Thereafter, the test piece is subjected to a fine grinding and polishing until the desired sample thickness is attained. Polyurethane samples that neither melt nor dissolve have been made by Blok and LeGrandZ9by grinding out a spherical depression with a steel burr. All aspects of machining of specimens, whether drilling or grinding, must be performed with great care if thermal and mechanical stresses, which may influence the measured electric strength, are to be avoided. 18.3.3.3.Thin Films. As the specimen thickness is decreased, the production of identical specimens by compression molding becomes increasingly more difficult. Thin film studies in many instances are therefore accomplished using samples of commercially produced polymer films. The practical advantages are patently evident; being principally the mass production of large numbers of specimens of nominally uniform thickness and the freedom from thermal treatment other than that occurring during the vacuum evaporation of electrodes. The former simplifies the interpretation of observations by the exclusion of the contribution to variance ~ - ~latter ~ of the effect of specimen thickness on electric ~ t r e n g t h . ~The avoids involvement in the complications of morphology and its change during and following heat treatment.3s~40 The effect of morphology on the electrical properties of polymers is discussed more fully later. R. M. Eichhorn, Annu. Rep., Conf. Electr. Insul. Dielectr. Phenom. 1973, 289 (1974). A. Toureille, J . Appl. Phys. 47, 2961 (1976). V. M. Morton and A. W. Stannett, Proc. Inst. Electr. Eng. 115, 1857 (1968). 35 R. Cooper, C. H. Rowson, and D. B. Watson, Nature (London) 197, 663 (1963). P. Fischer, Annu. Rep., Conf. Electr. Insul. Dielectr. Phenom. 1974. 661 (1975). 37 S. N . Kolesov and G. I. Geifman, Izv. Vyssh. Uchehn. Zaved., 8, IS5 (1971). 38 H. Luy and F. Oswald, Elektrotech. Z . . Ausg. A 92, 358 (1971). 39 R. Cooper, B. R. Varlow, and J. P. White, Proc. Inst. Electr. Eng. 123, 187 (1976). ‘O R. Cooper, B. R. Varlow, and J. P. White, J . Phys. D 10, 1521 (1977). 31

18.3

ELECTRIC BREAKDOWN

455

Film specimens with conventional vacuum evaporated electrodes have been widely used for low-field conductivity measurements, but at the voltages normally required for breakdown studies the problem of edge discharges arises again. A method of specimen preparation described by Saito41 before the advent of polymeric insulating materials, but that appears to have been used very rarely4* since in the study of dielectric breakdown, has recently been revived by Cooper, Varlow, and White and used predominantly for polyethylene ,39n40 but also for poly(ethy1ene terephthalate) polystyrene, polypropylene and p~lycarbonate.~~ The longer axes of the opposing electrodes are aligned but the electrodes overlap by only about 3 mm, thus producing great electric stress over a very small area. The probability of breakdown at an inclusion or pore is thus rendered small. The mask used to define the electrodes is placed some distance in front of the film so that a diffuse region is produced at the periphery. This avoids stress concentration and the consequential unwanted breakdown between opposite electrode edges. An alternative way of producing specimens from film for electric breakdown experiments and again eliminating surface discharges has been used by M c K e ~ w nwho , ~ ~ encapsulated his specimens together with spherical electrodes in epoxy resin. The surface of the films must be treated to bond them to the epoxy resin, or breakdown may occur along the interface between the resin and the film. Polyolefins have been treated by a wet-chemical oxidation of the surface. For example, polyethylene was immersed by McKeown in a saturated solution of potassium dichromate in concentrated sulfuric acid for between 1 and 5 min. It is important neither to use an organic solvent such as acetone to hasten the drying nor to wipe the surface with paper tissue, as either treatment destroys the bonding ability of the surface. A saturated solution of potassium dichromate in concentrated sulfuric acid quickly degrades polymers containing ester or amide groups. The surface of these films has been activated using an air glow discharge at low pressures. Neither of the above treatments have proved successful for modifying the surface of polytetrafluoroethylene or polychlorotrifluororethylene. One technique that has been found to be more effective is immersion in a solution of sodium dissolved in liquid ammonia. This technique is adequate for polychlorotrifluoroethylene and poly(viny1 fluoride) but not totally effective for polytetrafluoroethylene. After encapsulation in epoxy resin, McKeown's samples were cured at 70" C for 16 hours.

4p

Is I4

Y. Saito, J . Inst. EIectr. Eng., Tokyo 56, 1036 (1936). Y. Inuishi and D. A. Powers, J . Appl. Phys. 28, 1017 (1957). R. Cooper, B. R. Varlow, and J. P. White, IEE Cony. Pub/. 129, 209 (1975). J . J . McKeown, Proc. I n s t . Electr. Eng. 112, 824 (1%5).

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18.

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FIG. 3. Assembly for glow-discharge polymerization. I , Quartz back-up plate; 2, silica microscope slide specimen plate; 3, aluminized “bottom” electrode; 4, tinned copper strip electrical contact; 5, Teflon support pieces with adjustable jaws for clamping slides; 6, vacuum-tight electrical feedthrough; 7, base plate for bell-jar seal; 8, evacuation system and connection to monomer reservoir. (From Bradley and ham me^.^')

The breakdown strengths measured for such specimens of polyethylene were found by McKeown to give higher values than those obtained by other workers. The cause of these high values is not readily understood and may arise from his chemical treatment of the surface or the curing procedure, either singly or in combination. McKeown’s other polymer films did not produce abnormally high breakdown strengths, and in some cases his results were lower than those reported by other authors.23 18.3.3.4. Ultrathin Specimens. For the determination of the electric strength of even thinner specimens(- 1 pm), glow discharge polymerization45-40 and casting from are two methods that have been extensively used for specimen production. The basic equipment for preparing glow discharge films consists of an evacuated chamber with a source of organic “monomer” vapor and electrodes wired to an external source (Fig. 3). Bradley and Hammes4’ used pairs of flat plate electrodes placed approximately 1 cm apart. With a vapor pressure of about 1 mm Hg, the glow discharge was usually initiM. Stuart, Narrire (London) 199, 59 (1%3). J. Goodman, J. Polym. Sci. 44, 551 (I%O). 47 A. Bradley and J. P. Hammes, J . Elecrrochem. Sot. 110, I5 (1963). S. Sapieha, W. Jabionski, and M. Kryszewski, Elecrrocompon. Sci. Techno/. 1, 65 (1974). S . Sapieha, M. Kryszewski, and J. Tomczyk, I n f . Microsymp. Polar. Cond. Insrtl. Poly.. 1972 p. 125 (1972). 5o A. F. Burmester and V. J. Caldecourt, J. Polym. Sci., Part A-2 6 , 1639 (1%8). Is 46

18.3

ELECTRIC BREAKDOWN

45 7

ated at 300-400 V. Good films were obtained when the discharge was An altersustained with a current density between 1 and 3 mA nating voltage in a frequency range of 10-50 kHz was most satisfactory in the production of uniform, adhesive films. In most experiments, the two electrode plates became coated with identical “polymer” layers of thickness dependent only on the discharge current and time (Fig. 4). The film thickness is self-regulating through its own impedance. It is essentially free from pinholes because of the mechanism of its formation, in which the electric field is stronger at a hole or thin spot in the dielectric film, so that increased deposition occurs at that point. The organic vapor is introduced through a needle valve from an external gas cylinder or liquid reservoir. Solid components are placed directly in the reaction chamber and the equilibrium vapor pressure adjusted to meet optimum discharge requirements by heating the entire system. Discharges conducted at 200°C and above appear to give films as good as those produced at room temperature. Films over a range of thickness down to 0.1 pm have been produced in this way from an extensive list of monomers, styrene being the most common, but also including vinylcarbazole, naphthalene,48*4sbenzene, toluene, and tetraflu~roethylene.~~ These films resemble bulk polymers that have been subjected to high-energy irradiation. In the formation of the polymer films in the discharge, there is little doubt that ionized particles, including probably ion radicals and excited species, are carried to the electrodes where they polymerize. The newly formed polymer is bombarded by more ions and irradiated by ultraviolet light from the discharge, so that the polymer may be expected to react as it does under high-energy i r r a d i a t i ~ n .Under ~~ such conditions degradation and crosslinking51as well as the formation of trapped free r a d i c a l ~ all ~ ~occur , ~ in the polymer. Bradley and Hammes4’ have observed that films produced by glow discharge have higher melting points, greater thermal stability, and lower solubility than conventional polymers of similar composition. Films of thickness down to 0.1 pm may also be produced by casting from solution. Burmester and CaldecourtSohave cast thin films of polystyrene from solution in toluene in the following manner. Glass slides, coated with evaporated aluminum, were used as a substrate on which was placed a drop of dilute (0.5-5%) solution of polystyrene in toluene. The drop was spread uniformly along the length of the slide by means of the 51

A. Charlesby, Proc. R. SOC. London 22, 60 (1954).

sz G. K .

Fraenkel, J. Hirshon, and C. Walling, J . Am. Chrm. SOC. 76, 3606 (1954).

* J. C. Bevington and D. E. Evans, Nature

(London) 178, 1112 (1956).

458

18. ELECTRICAL METHODS

STEP1

STEP2

STEP3

STEP4

FIG.4. Preparation of organic polymer film specimens. Step 1: Application of vacuum metallized bottom electrode (Al). Step 2: Organic film from glow-discharge polymerization. Step 3: Top electrode (Al). Step 4: Electrical connections. (From Bradley and Hammes.")

polished edge of a second slide. This spreading slide was supported at a constant distance from the substrate by thin shims and held at a constant angle and pressure by a movable clamp constructed for this purpose. In order to reduce the number of gross imperfections in the fdm, the stock polymer, a commercial polystyrene without additives, was carefully washed before being placed in solution and the mixing and casting operations were carried out in a dust-free glove bag. The resulting films were then devolatized in vacuum for several days to remove solvent. Even with this procedure, it was found by mass spectroscopic analysis that toluene was present in the amount of 0.1-0.5%. By changing the concentration of the casting solution and by changing the thickness of the shims, the thickness could be controlled, over the range 0.1- 10 pm. Films greater than 3 pm in thickness, however, developed crazing after devolatization. Attempts to avoid this by annealing were not successful. Electrical contact to the base electrode was made by pressing platinum foil onto the aluminum film in an area not covered by the polystyrene film. The counterelectrode was contacted by a drop of mercury that was supported on a copper wire. The uniformity of such films may be checked by observing the interference pattern visually under a sodium vapor lamp.50A change in thickness of 0.15 pm produces a complete interference fringe. The better films produced in this way have general thickness variations of less than 10% over the width of the slide and less than 2% over the area of the counterelectrode. The measurement of thickness in this thin film range may be attempted in a variety of ways. Burmester and C a l d e c ~ u r tdetermined ~~ the thickness of their cast fdms by measuring the attenuation of infrared light in the 1495 cm-l absorption band of polystyrene and comparing it with the attenuation caused by a film of known thickness. An alternative method

18.3

ELECTRIC BREAKDOWN

459

is to use a capacitance technique. This has been done by placing an array of small (6.75 mm2) aluminum counterelectrodes on top of the film by vacuum evaporation. These electrodes form small parallel-plate capacitors with the base electrode previously evaporated onto the slide and the polymer as dielectric. Assuming an appropriate value for the dielectric constant, this method gives thicknesses within 10% of those obtained by, the infrared-type measurements. Bradley and Hammes4' have also used a capacitance method for the determination of the thickness of their glow discharge polymerization films. For transparent material, assuming a value of 1.5 for the refractive index of their material, they were also able to determine thickness by interferometry. 18.3.3.5. Single Crystals. Senecal and Ham" have been able to produce single crystals of high-density polyethylene on which to make electrical measurements. These platelike crystals were grown from 0.01-0.03% solutions of polyethylene in p-xylene held at 90°C for 24 hours or longer. A suspension of crystals with maximum lateral dimensions of several hundred microns and thicknesses of about 130 A is deposited in air on a microscope slide that has been coated with an array of aluminum squares 1 mm square, 500 thick, and separated by clean glass aisles 0.2 mm wide (Fig. 5 ) . When the solvent has evaporated, an undeformed crystal situated broadside down and overlapping an edge of one of the aluminum squares is located with the aid of phase contrast optics. The crystal is then masked with thin layers of mica and slivers of cover glass in such a way that the aluminum square serves as the lower electrode. The upper electrode is formed by a new deposition of a 500 A thickness of aluminum evaporated at 5 x Ton- and the crystal itself insulates the lower electrode from the upper one to form a miniature parallel-plate capacitor. Lead wires are attached to either end of the microscope slide and appropriate additional conducting connections are made with silver paint. Again, thickness measurements may be made by either capacitance measurements, assuming a bulk dielectric constant of 2.23, or by the multiple-interference technique of T~lanski.~ In~the latter, randomly selected crystals on a clean glass microscope slide are coated with silver. A cover glass coated with silver is placed on top of a suitable crystal to form an optical wedge, and an interference pattern is produced by using a helium-neon laser with a Leitz epi-illuminating microscope objective. 18.3.3.6. Cast Specimens. In complete contrast to the thermoplastic polymers described in general above, the thermosetting polymers lend 51

G. Senecal and J. S. Ham,J . Appl. Phys. 42,2714 (1971). S. Tolanski, "Surface Microtopography." Wiley (Interscience).New York, 1960.

18.

460

Region of upper electrode

ELECTRICAL METHODS

Clean glass aisle A

0.2 mm

h

Region of lower elect rode ,

/"

/

FIG. 5. Schematic diagram of an electroded polyethylene crystal. The substrate is a clean glass microscope slide, onto which aluminum squares 1.0 mm on a side and 500 A thick have been deposited. After solution containing crystals has evaporated from the slide, a crystal suitable for masking and the deposition of an upper electrode is located. (From Seneca1 and Ham.")

themselves to specimen preparation methods exclusively their own. A whole range of electrode configurations may be used with relative ease by inducing polymerization after immersion of the electrodes in the resin while it is still in the liquid phase. The epoxide resins are dominant among the thermosetting polymers and specimens of this material are commonly produced with and needle e l e c t r ~ d earrange~~*~ ments. To avoid dissolved air, it has usually been found adequate to heat and degas the materials prior to mixing and degas again before pouring into a mold, which may be of polypropylene or cellulose acetate.58 18.3.3.7.Artificial Voids. The study of breakdown by partial discharges in gas-filled cavities, or voids, in insulating polymers has led to the use of samples containing artificial voids. It has been found possible to cast such specimens from epoxy resin.5gTo ensure a smooth void of the desired size by casting, Kind and Konig coated that part of the mold which formed the void with a thin, adherent film of silicone and heated it for about three hours at 220°C before use in the casting operation. After casting, this coated component can then be easily removed. Kind and se J. H. Beard and S. Orman, Proc. I m r . Elrcrr. Eng. 114, 989 (1967). ST 58

S. Zoledziowski and S. Soar, IEEE Trrins. Ekcrr. /mid. EI-7,84 (1972). M . Olyphant, Jr.. IEEE Truns. Power Apprrr. Sysr. PAS-82, I 1 0 6 (1%3).

18.3

ELECTRIC BREAKDOWN

46 1

KonigS9also produced voids by drilling in epoxy resin samples, taking great care in cleaning out the drilled cavity. Despite their care, the surfaces of the drilled voids were not free of scratches, whereas the surfaces of the cast voids were, and this led to much shorter lifetimes for drilled specimens under alternating electric stress. Specimens containing an artificial void have been manufactured from polyethylene,60-62but in a quite different manner. The samples are assembled from sheets of plane material, one of which contains a circular hole. It is common practice to situate the hole adjacent to one of the electrodes. This facilitates periodic inspection of the void surface during life tests without dismantling the sample. 18.3.3.8.Artifical Trees. The phenomenon of electrical “treeing” in solid polymeric insulation has been known for many years but its mechanism remains unclear. It is known that the extension of a tree is connected with the occurrence of electrical discharges in one or more of the gas-filled tubules that constitute the branches of the tree. Actually “tree” tubules are tortuous and the diameter, which is of the order of 1 pm, varies along the length so that detailed quantitative investigations are difficult. However, it has proved possible to produce cylindrical tubules of air confined in PMMA of diameter down to 10 pm, which effectively demonstrate the effects of the wall on the breakdown voltage.a.M In PMMAs3tubules in excess of 25 x mm in diameter may be prepared easily by drilling perpendicular to and at the center of the face of a rectangular block about 1 cm thick. Using a spherical burr and machining along the drilling axis, recesses are cut into the block to the same depth from each of the opposite faces, thus producing the desired length of tubule. The surfaces of the recesses are polished and foil electrodes (e.g., aluminum) are pressed in and sealed at the edges with acrylic cement. Tubules mm may be cast as follows. Two polof diameter less than 25 x ished blocks, measuring 2.5 x l .25 x l cm are clamped together with the edges of two 2.5 x 1 cm faces coincident. Spherical recesses are machined as above at the center of each 2.5 x 2.5 cm face of the clamped block. The clamps are then removed and the opposing 2.5 x 1 cm faces are dipped in chloroform and reclamped with a suitable tungsten wire stretched along the axis of the recesses. Each recess is next washed with chloroform, which leaves a smooth, regular polished surface. The speciD. Kind and D. Konig, IEEE Trans. Electr. Insrtl. EI-3, 40 (1968). L. L. Alston and P. G. Dawson, Proc. Inst. Electr. Eng. 112, 814 (1965). S . Beg and B. Salvage, Electron. Lett. 4, 530 (1968). S. Beg and B. Salvage, Electron. Lett. 5, 388 (1%9). D. W. Auckland, A . B. Borishade, and R. Cooper, Annu. R e p . , Conf. Electr. Insul. Dielectr. Phenom. 1974. 472 (1975). ~4 D. W.Auckland, A. B. Borishade, and R. Cooper, IEE Conf. Publ. 129, I5 (1975). M,

* + I

462

18.

ELECTRICAL M E T H O D S

men remains clamped for two days and the wire is then withdrawn, leaving a clean, cylindrical tubule extending from recess to recess. The surfaces are allowed to harden for a further five days, after which the electrodes are applied as described above. 18.3.3.9 Cable Samples. Considerable attention has been given in recent years to dielectric breakdown in polymeric insulation under service conditions, particularly in power cables. It has been a r g ~ e d that ~~.~~ specimens especially prepared for laboratory tests, usually by molding, while providing some information about the tested polymer, do not necessarily provide any information relevant to a cable produced from this polymer. This argument is based on the suspicion that the dielectric strength of a molded sample may be different from that of the cable product, whose quality depends directly upon extrusion conditions and control. Studies of breakdown by electrical treeing in a needle/plane electrode configuration have therefore been carried out in samples of crosslinked polyethylene derived from extruded-type cables by inserting the needle through the insulation ~ h i e l d . ~ ~ * ~ ~ The difference between molded and extruded specimens has also been raised by McKeanss in a study of microporosity and its effect on electrical breakdown in full-scale cable samples rather than modeled specimens in order to minimize any unknown scaling factors. Model cables have been produced from tapes of a number of dielectrics including polyethylene, polypropylene, polycarbonate, poly(ethy1ene terephthalate), polyamide, and polystyrene, in an investigation of the impulse electric strength of such cables at cryogenic temperatures by Rigby and Weedy.68 These models consisted of six layers of the tape wound helically with a 65/35 registration. A butt gap of 1 mm between adjacent turns will ensure, with this registration, two tape layers between butt gaps in any radial section. The high-voltage electrode, on which the tape was wound, consisted of a polished stainless steel tube of 28 mm outside diameter. Stress cones were built up at the ends of the model and the ground electrode was formed, rather crudely, by winding tin tape over the insulation test section and the stress cones. 18.3.4. Experimental Methods 18.3.4.1. Electrodes. The range of electrode materials used in the determination of the electric strength of polymeric insulation is very exG. Bahder, A. L. McKean, and C. Katz, Annu. R e p . , Conf. Elecir. Insul. Dielecir. Phenom. 1971, 110 (1972). A. L. McKean, IEEE Trans. Power A p p a r . S y s t . PAS-95,253(1976). J. Densley, Annu. R e p . , Conf. Elecir. Insul. Dielecir. Phenom. 1972, 177 (1973). S . J. Rigby and B. M. Weedy, Cryogenics 16, 167 (1976).

18.3

ELECTRIC BREAKDOWN

463

tensive, though not all of these electrode materials are compatible with all polymers. The simplest electrode system uses pressure contact between the polymer surface and bare metallic spheres or plates. Specimens produced by casting electrodes in thermosetting polymers, such as epoxy resin, and those manufactured by the insertion of spheres or needles during a molding or extrusion process may be included in this class of bare metallic electrodes, although the contact between metal and polymer, if produced with care, should be more intimate than that of simple pressure contact. Vail and GausteP claim to have achieved such an intimate contact between 2 in. diam electrodes of highly poliched brass and thick polyethylene sheet into which they were pressed by a heated hydraulic press. Working at cryogenic temperatures, Bob0 et ~ 1 . 'were ~ satisfied with only the minimal pressure to achieve contact between stainless steel spheres and a series of polymeric films. A higher pressure, in addition to possibly damaging their thin fdms [50 pm of poly(ethy1ene terephthalate), polyamide, and polytetrafluoroethylene], would presumably have inhibited their experimental method, which involved drawing the fdm between the electrodes so as to obtain many breakdowns on the same sample. Great care must be taken in interpreting results obtained with spherical electrodes on specimens of film material since the influence of edge discharges will almost certainly be involved. Indeed, partial discharges were observed by Bob0 et a/. and the effect of the ambient medium and the high-voltage source impedance on their magnitude was investigated. By coating their stainless steel spheres with phenolic varnish to limit field emission at the cathode and secondary emission at the anode, the amplitude and energy of partial discharges in their system were reduced, resulting in higher electric strengths. Colloidal suspensions of silver or graphite in a number of solvents have been used as electrode materials on molded, machined, and film specimens as an alternative to metals deposited by vacuum evaporation. Such evaporated metals have included aluminum, indium, copper, gold, silver, platinum, tin, and magnesium, although good adhesion between all these metals and the range of polymers used as insulating material is not universal. Toluene, the solvent commonly used for a colloidal suspension of silver, attacks polystyrene, and this paint is not stable in some liquids in which electric strength is determined. An alternative solvent for cola@C. R. Vail and W. 'O

F. Gauster, IEEE Trans. Power Appar. Sysr. PAS-76, 38 (1957).

J. Bobo, M. Pemer. B. Fallou. and J. Galland, Vacuum 18,397 (1968).

464

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ELECTRICAL METHODS

loidal silver in such circumstances is methylisobutylketone. With colloidal graphite, the adhesion to some common polymers is not always satisfactory. To improve the adherence of silver paint on polyethylene and graphite on polystyrene, surface treatment with chromic acid has been In order to permit viewing of the specimen surface during application of voltage, Blok and LeGrand,29after evaporating gold onto the surface used an aqueous solution of sodium chloride as the bulk electrode. The use of electrolytic electrodes with unmetallized samples was adopted by Swan7’ for reasons peculiar to his experimental investigation in which he studied the effect of iodine on the conduction and breakdown of polyethylene, by varying the concentration of iodine in a solution of sodium iodide. The film was clamped vertically between two glass flanges using silicone grepse for sealing purposes. There is no reason why the use of electrol y t k electrodes mounted in this manner should be restricted to this limited application. If evaporated metal electrodes are applied in extremely small thicknesses (of the order of 0.1 pm or less) then, in addition to the dielectric itself, the electrode material surrounding the breakdown site may be evaporated, isolating the destroyed area.42.72 This has been referred to as “self-healing b r e a k d o ~ n ”and ~ ~ in addition to clearing weak spots permits a number of consecutive experiments on a single sample. The possibility of obtaining “nondestructive breakdown,” as described by Reihl et a/.,73 by restricting the charge transported through the sample during breakdown by the use of high loading resistors cannot be very great since the quantity of energy stored in a typical polymeric sample, at fields approaching the electric strength, is more than sufficient to cause vaporization of the material in the breakdown channel. The principal purpose of such current-limiting resistors is to keep the destruction within tolerable bounds in order that the breakdown site may be accurately ascertained. 18.3.4.2. Mode of Voltage Application. For those concerned with the use of polymeric insulation in practical service, methods of testing using alternating voltages are of most immediate interest. Such tests can be roughly separated into two quite distinct groups: those which achieve breakdown in a very short period and life tests that cover several years. The mechanisms of breakdown are almost certainly different. In shortD. W. Swan, J . Appl. Phys. 37, 464 (1%6). M. Kryszewski, W. Jablonski, and S . Sapieha, Int. Microsymp. Polar. Cond. Insul. Poly., 1972 p. 125 (1972). N . Reihl, H. Baessler, S . Hunklinger, W. Spanning, and G . Vaubel, Z . Angcw. Phys. 71

72

27, 261 (1969).

18.3 ELECTRIC BREAKDOWN

465

time laboratory tests the alternating voltage is either raised at a constant rate to achieve breakdown in minutes or less,23or increased stepwise with intervals ranging from minutes to days.ss Long-term life tests in which breakdown usually results from the development of electrical or electrochemical trees, or by general erosion resulting from the action of partial discharges, may be carried out by observations on cable samples removed from service after many years. Many attempt^^*^^*^^ have been made to accelerate these long-term effects by increasing the frequency of the alternating voltage into the kilohertz range after developing empirical relationships for the frequency dependence of the breakdown process. To a very crude approximation, insulation lifetime is inversely proportional to the frequency of the test voltage. Breakdown under direct voltage application is in the main carried out under laboratory conditions in short-time tests. The direct voltage is raised gradually to breakdown over a period of usually less than a It is of some importance when determining the direct voltage breakdown strength to exclude, where possible, extraneous factors such as electromechanical and thermal breakdown, which give rise to erroneous results. These two breakdown mechanisms are dealt with in more detail in Sections 18.3.4.5 and 18.3.4.6, respectively. Impulse voltages are used in breakdown strength measurements for a variety of reasons. The impulse voltage waveform attempts to simulate that occurring in power systems due to lightning surges. The British standard, which is similar to most others, takes the form of a 1/50 psec wave. This is defined as an impulse wave that rises from 10 to 90% of its peak value in 1 psec and falls from its peak to 50% of its peak value in 50 psec. The short duration of the impulse voltage pulse also allows the experimenter to avoid to a considerable degree the involvement in thermal and possibly electromechanical breakdown. With impulses of sufficiently short duration, the effect of field distortion from the nominally uniform configuration by injected space charge can also possibly be a~oided.~ There has been, and still is, controversy as to the correct mode of breakdown using impulse voltages. On the one hand, breakdown on the wavefront of an impulse with a peak value greatly in excess of the anticipated breakdown voltage, while having the advantage of a single voltage application, can lead to erroneously high values of electric strength due to statistical and formative time lags. Indeed, Vail and GausteP attribute W. K . Hogg and C. A. Walley, Pruc. Inst. Electr. Eng. 117, 261 (1970). , G. Bahder and C. Katz, Annu. Rep., Conf Electr. Insul. Dielectr. Phenom. 1972, 190 (1973). ”

75

466

18.

ELECTRICAL METHODS

tneir scatter in me breakdown values occurring on the rront or impulse voltage waves applied to polyethylene to the scatter in the time to break down. The alternative is to obtain breakdown using a succession of flat-topped impulses (e.g., 1/8000 psec waves4) of progressively increasing amplitude. This latter method avoids the possibility of overvolting but the experimenter must, by the use of auxiliary experiments, ensure that no permanent effect has been obtained by the application of prior impulses. Without sufficient care a lowering of the electric strength may be produced if structural damage has occurred or an increase in apparent electric strength may be observed if any injected space charge is not allowed to discharge between successive impulses. The use of 1/50 psec impulse waves for wavetail breakdown may give misleading results when the impulse voltage is determined from calibration curves, which relate the charging voltage to the peak of the output impulse wave. Direct oscillographic observation of the breakdown impulse is necessary if there are significant time lags to breakdown, since by the time breakdown occurs the voltage may have fallen appreciably below its peak value. Combinations of direct and impulse voltages have been applied to molded specimens of polyethylene4 and to film samples of a number of polymer^,^^*^^ in order to discover the effect of injected space charge on the electric strength of polymers and to explain the often reported difference between the impulse and direct voltage breakdown strengths. Values of electric strength were obtaining corresponding to the following modes of stressing: (a) With direct voltage, increasing steadily from zero to the breakdown voltage in about one minute. (b) With a series of 1/8000 psec impulses, increasing in 3% increments to produce wavetop breakdown within ten successive impulse applications. Between impulses 30 sec were allowed to elapse, during which time the specimen remained short-circuited through the impulse generator wavetail resistor (& in Fig. 6). (c) With impulse voltage, following a period of prestressing with direct voltags, when the impulse voltage either produced a reversal of polarity on thr: specimen, referred to as “fields opposing” or when the polarity remained unchanged, referred to as “fields aiding.” In each case the prestressing voltage was applied initially for four minutes. Subseouent work40 showed that the effect of prestress on the electric strengt3 of polyethylene was established in a period of the order of 100 psec and sufficient time to allow prestressing to be established is available in the time occupied in resetting the magnitude of the succeeding impulse voltage. This result is substantiated by the work of Watson78in

18.3

ELECTRIC BREAKDOWN

467

I

#

,

I -

trigger 2 to oscilloscope

FIG.6. Conditioned breakdown circuit.

which the rise time of impulse voltage waves, applied to polyethylene samples, was varied and an increase in breakdown voltage M’as observed for wavefronts in excess of about 70 psec. The reported results4 in which the impulse breakdown strength could be raised toward that of the direct voltage strength by prestressing in the “fields-aiding’’ sense or dramatically reduced by prestressing in the “fields-opposing” sense, has been attributed to the injection and accumulation of space charge. The decay of this space charge, which must be allowed for between successive impulse applications to obtain an accurate impulse breakdown strength, was measured by interposing a delay between the removal of the direct prestressing voltage and the application of the breakdown impulse in the fields-opposing sense. The abnormally low values of electric strength obtained with no such delay recovered to the unprestressed impulse value with a decay of less than 1 s ~ c . ~ Further evidence of the influence of injected space charge on the electric strength of polyethylene has been given in an interesting variation of the above work by Perret et ~ 1 in which . ~ the ~ prestress was applied at temperatures of 20, 60,and - 196°C (Fig. 7). Following prestressing all breakdowns were obtained at - 196°C by impulses applied in the fieldopposing mode. Prestressing at - 196”C, where the prebreakdown current was negligible and produced no appreciable injected space charge, had no effect on the impulse electric strength at - 196°C. Room temperature prestressing produced reductions in impulse electric strength at - 196°C comparable with those obtained at room temperature by BradD. B. Watson, J . Phys. D 4, L19 (1971). J. Perret, R. Jocteur, and B. Fallou, Rev. Gen. Elecrr. 81,757 (1972).

18.

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ELECTRICAL METHODS

well ef u I . ~despite the low temperature at breakdown. The high prebreakdown current flowing during prestressing at 60°C generated a high injected space charge and produced prestressing effects at - 196°C greatly in excess of those at room temperature. Another effect of trapped space charge has been reported in the measurement of the impulse electric strength of electrets of PMMA.27 Under the influence of an applied direct voltage, the specimens were heated to 112°C in one hour and maintained at this temperature over a further two hour period. The direct voltage was then removed and breakdown of the samples achieved by the application of 1/50 psec impulse voltages at room temperature. Injected space charge accumulated during the above thermal cycle and subsequently “frozen-in’’ on return to room temperature, was able to lower the impulse electric strength compared with that of samples having undergone the same heat treatment without the application of the direct “forming” voltage. The experiments of Bradwell ef al.4 have a bearing not only on the reported differences between the electric strengths measured with impulse and direct voltages but also between the electric strengths of planerecessed specimens of polyethylene when measured with direct voltage and with 50 Hz alternating v ~ l t a g e . ~It~ *appears ~~ that the electric strengths are about equal at temperatures less than about -5o”C, and greater than about 50°C. At temperatures between these limits, the

2 3 1 5 6 7 Polarizing field, MV cm-’ FIG.7. Variation of breakdown field at - 196°C as a function of the prestressing field. Breakdown is by direct voltage after 30 min application of direct prestressing voltage. Polarizing temperature (a) - 196“C, (b) 20°C. and (c) 60°C. (From Perret, PI d.’’)

O’

1

18.3 ELECTRIC BREAKDOWN

469

50 Hz strengths are less than the corresponding direct voltage strengths, and the difference is a maximum between 0 and 20°C. The effect has been attributed to heating caused by dipolar impurities, or by ambient discharge^.^^ Neither Oakes" nor Riddlestonell took precautions to minimize error due to electromechanical deformation, because its possible magnitude was unknown at the time. Above 50°C, with both dc and with ac voltage, breakdown in both investigations was very probably due to electromechanical in~tability.~With space charges present, breakdown with an alternating voltage approximates to the condition of fields opposing, as defined by Bradwell cf u I . ~ At low temperatures, the prebreakdown current and the effect of prestress have been shown to be negligible'' and it is therefore not surprising that the difference between the direct voltage electric strength and the alternating voltage electric strength also becomes negligible. However, at temperatures greater than about -2o"C, with alternating voltage, modulation of the space charge may occur, and a temperature-dependent effect on the apparent electric strength is only to be expected. Most of what has been said above with regard to the mode of application of voltage in electric strength measurements is based on experience obtained on a limited range of polymers, with a strong emphasis on low-density polyethylene. This limitation must be borne in mind when applying these results to polymers in general. The relevance of the conclusions will vary between polymers and will depend on such factors as the rate of formation of trapped space charge and the dependence of electrical, mechanical, and thermal properties on each other and on temperature. With this constraint in mind, a number of important points may be summarized:

(a) Alternating voltage breakdown in the long term is likely to be accomplished in practice by the deleterious effects of electrical or electrochemical erosion in one form or another. (b) Short-term alternating voltage breakdown, unless carried out at low temperatures may be subjected to the influence of injected space charge. (c) Breakdown due to the application of direct voltage will in general always be effected by injected space charge, except at low temperatures, and the possibility of thermal or electromechanical breakdown cannot be excluded, particularly at high temperatures. (d) Impulse breakdown approaches most closely the "intrinsic" concept of electric strength, being uninfluenced by space charge injection if the impulse duration is sufficiently short. (e) Care must be taken in interpreting impulse breakdown measurements using either wavefront or wavetail breakdown to avoid errors due

470

18. ELECTRICAL METHODS

to overvoltage and prestressing respectively. Oscillographic observation of the breakdown impulse is strongly recommended. 18.3.4.3 Time to Breakdown. For many years the standard method of measurement of the time elapsing between the application of voltage and breakdown of the specimen has been by o s c i l l ~ g r a p h y .Breakdown ~~~~~~~~ is obtained by the application of an impulse voltage wave, a small proportion of which is applied to a single-shot oscilloscope by means of a simple capacitance divider. In practice, this capacitance divider can also act as the wavefront capacitor of the impulse generator (see Fig. 6). Breakdown of the specimen under the influence of the applied impulse is observed on the oscilloscope as a collapse of the voltage wave. In their investigation of treeing, Bolton et a/.* devised a method for determining the time interval between the application of an impulse voltage and the appearance of luminescence associated with a discharge in the sample. Synchronism of their system, which involved a photomultiplier, video amplifier, and single-sweep oscilloscope, was effected by triggering the impulse generator, to produce a 5/800 psec wave, with a voltage pulse generated upon initiation of the time base sweep. A marker pulse was derived from the impulse wavefront and applied to the oscilloscope in order to indicate the instant of peak voltage across the specimen, from which the time lag in the occurrence of the discharge was observed by the photomultiplier and transmitted via the video amplifier to the oscilloscope for reference with the marker pulse. In their particular work the time lag was extremely variable and of the order of 10 psec. This method is clearly limited to transparent polymer samples. The formative time lag, as observed in the photographic studies initially in alkali halides by Cooper and Elliotts and subsequently in the same labowould appear to consist of a period of about 30 nsec, ratory in PMMA,12*15 during which the progressive ionization of the eventual breakdown channel takes place, followed by the time required for the collapse of voltage across the specimen. The time for the collapse of voltage has been measured in polyethylene by Bradwell and P~lfrey,~O who used a distributed capacitance potential divider within a coaxial line (Fig. 8). The specimen was placed between two hemispheres at one end of the air-insulated transmission line which was arranged to have a characteristic impedance of 68 a. A length of 68 fl polyethylene cable connected the voltage source to the other end of the coaxial line via a tapered section and the voltage generated by the source was a 400/6000 psec impulse wave of magnitude 35 kV. The degree of overvolting due to statistical time lags of order 10 psec was thought '~3

R. Cooper and D.T. Grossart, Proc. Phys. Soc.. London, Ser. B 69, 1351 (1956). A. Bradwell and D. L. hlfrey, J . Phys. D 1, 1581 (1%8).

18.3

ELECTRIC BREAKDOWN

47 1

FIG.8. Schematic diagram of transmission line system. C, capacitance potential divider. (From Bradwell and F'~lfrey.'~)

unlikely to exceed 3%. When a specimen ruptures, the voltage across the specimen collapses and a voltage removal wave is propagated down the line. The wavefront duration of this impulse is equal to the time of voltage collapse across the specimen and is recorded oscillographically via a distributed capacitance potential divider. The latter consisted of a 20 pm thick Mylar poly(ethy1ene terephthalate) insulating film, pressed against the inside wall of the outer conductor of the air-spaced line, and backed with a 20 p m layer of copper foil. This foil was connected through holes in the mylar film and the outer conductor of the coaxial line to the oscilloscope. With this arrangement, a voltage divider ratio of 1:3420 was achieved and subnanosecond rise time transients could be successfully recorded. The voltage collapse time in their polyethylene samples was of the order of 1 nsec. 18.3.4.4. Morphology. The structural characteristics of polymers have a pronounced influence on their electrical properties, including their electric strength. This dependence of electric strength on morphology may be investigated by a direct comparison between polymer samples of differing densitieseoor on samples of one polymer that are subjected to either mechanicale1or thermal treatment.30~4082 Electrical trees, which grow under a stress of 2 kV mm-l at a frequency of 3 kHz, are eliminated when low-density polyethylene samples having a crystallinity of 50% and spherulite dimensions between 2 and 3 p m are replaced by high-density polyethylene with a much higher crystallinity (%) and larger spherulites (dimensions of 50 pm).80 The importance of the spherulites has also been demonstrated by Kolesov and KherasoP using samples of highly crysM. Prigent, J. Bobo, and I . Eyraud, fEEE Con$ Pub/. 129,32 (1975). K . Yahagi and S. Mita, IEE Con$ Pub/. 129, 187 (1975). P. Fischer, Elekrrorech. 2.. Ausg. A 95, 516 (1974). S. N. Kolesov and L. N. Kherasov, Vysokomol. Sordin., Srr. B 12, 266 (1970).

18.

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ELECTRICAL METHODS

talline polypropylene, in which large spherulites may be obtained with relative ease. They claim a much higher electric strength for the central region of the spherulites than for the interspherulite regions and conclude that the breakdown of macroscopic crystalline polymer films occurs primarily in the less dense portions of the film. By subjecting film specimens of polyethylene to heat treatment over a range of temperature between 20 and lOo"C, followed by quenching to room temperature, Cooper et al. have observed both a transient39and a permanent40 influence of the heat treatment on the electric strength of their specimens. In the first hour following quenching from lOo"C, the electric strength of polyethylene film samples falls to a very low value before recovering to approximately the value obtained on similar specimens without heat treatment (Fig. 9). Breakdown was by impulse voltages following prestressing in the fields-opposing sense. This demonstrates the influence of the transient morphology following heat treatment on the ability of the polymer to accumulate the trapped space charge to which the prestressing phenomenon has been a t t r i b ~ t e dThis . ~ conclusion is emphasized by the absence of any transient or permanent effect on the electric strength of the specimens determined using impulses alone, where

20

10

60

Time after quenching.

80 min

100

120

FIG. 9. Variation of impulse electric strength of prestressed polyethylene at room temperature following quenching from 100°C. 62.5 pm thick sheet. Evaporated copper electrodes. (a) 2.9 MV cm-' prestress, (b) 3.8 MV cm-' prestress, (c) 0 prestress. (From Cooper et a/.s0)

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ELECTRIC BREAKDOWN

473

there is insufficient time for the development of the prestressing space charge. Permanent changes in the prestressed electric strength vary in magnitude depending on the temperature of the heat treatment, producing a maximum effect between 60 and 70°C. They have been correlated with changes in high-field conduction in identical samples undergoing identical heat treatment and with changes in crystallinity evinced by density measurements. Density measurement with a sensitivity better than one part in ten thousand may be obtained in a 1 m long density column using a mixture of appropriate liquids [ethanol and distilled water for polyethylene, hexane and carbon tetrachloride for poly(ethylene terephthalate), etc.]. The principal precaution in the assembly of such a density column is the exclusion of most of the dissolved gases, which may form bubbles on the polymer samples producing erroneously low values of density. Modest degasing of the liquid mixtures before introduction into the column is usually adequate for avoiding bubble formation. Mechanically deforming polyethylene film samples to up to approximately 300% elongation changes the electric strength of this material in a manner that does not depend on the mode of voltage application. Yahagi and Mitasl have found that the electric strength passes through a minimum at about 3 w o elongation and a maximum at about 160% elongation while maintaining an approximately constant differential between the direct voltage strength, the impulse voltage strength, and the peak alternating voltage strength. The decreasing order of magnitude of these three modes of stressing at room temperature is compatible with the space charge prestressing model of Bradwell et al. discussed above. The general picture emerging from these morphological investigations stresses the importance that must be attached to the mechanical and thermal history of the material used in the measurement of electric strength. Differences can be expected between specimens obtained from blow-extruded or calendered film and those obtained by glow discharge polymerization or cast from solution because of their different mechanical history. The thermal history of samples produced by injection molding, extrusion, or melt compression may also have an effect on their electrical properties. 18.3.4.5. Electromechanical Breakdown. Softening of thermoplastic polymers with increase in temperature permits mechanical deformation of the sample to take place under the high electromechanical forces generated by the applied voltages used in electric strength determinations. This mechanism has been invoked to explain the results of direct voltage and impulse voltage electric strength measurements obtained using plastic materials, which show that, above a critical temperature, the electric

474

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ELECTRICAL METHODS

--------* screw adjustment

Faraday cage $ '

\micrometer foot FIG.10. Optical lever system. (From Fava.'*)

strength falls with increase in temperature and that the rate of this fall is less when measured with impulse than with direct v ~ l t a g e . ~ * ~ ~ Using recessed samples of low-density polyethylene having a range of melt flow indices (0.3, 7, 20, and 200), Favale has investigated this phenomenon by applying direct and impulse voltages over a temperature range from - 195 to 80°C. As the temperature was increased above about O"C, divergence in the direct voltage breakdown strength between samples with different melt flow indices occurred, lower strengths being progressively obtained at increasing temperature as the MFI was increased. In order to observe the deformation of his samples due to these electromechanical forces, Favale made use of a system of optical levers (Fig. 10). On the assumption that compressions should occur equally on either side of the specimen at the center of the recess, a system of two optical levers was devised to follow movements on either side of the recess at its center. The fulcrum of each lever was a short, thin strip of spring steel, carrying at its free end a small light mirror and a short probe tipped with a 1 mm diam steel ball that touched the surface of the specimen. The pressure exerted on the specimen by this type of optical lever was insignificant. Vertical movements of the probe flexed the spring and rotated the mirror. Screw adjustments allowed the lever probes to touch either side of the center of the specimen recess so that each spring was slightly flexed. Using a lamp and scale arrangement, the movements of the probe were 8(

J. Artbauer and J. Gnat, Proc. Inst. Electr. Eng. 112, 818 (1965). W. G . Lawson, Br. J . Appl. Phys. 16, 1805 (1965).

18.3 ELECTRIC

BREAKDOWN

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magnified by approximately 1W and calibration of each lever was performed in position using the built-in micrometer gauge. Compressions were recorded directly on photographic recording paper. The paper was wrapped around a drum, which when turned about a vertical axis increased the voltage supplied by a variable autotransformer to the high-voltage circuit. A spot of light from each optical lever mirror was focused onto the recording paper and deflected vertically. The horizontal axis was calibrated directly in kilovolts and the vertical axis in microns of compression. The voltage on the specimens could be raised to a maximum of 15 kV without flashover. To prevent gross movement of the recess, the lower optical lever of the system described above was replaced by a filter plate against which the flat side of the plane-recessed specimen was held firmly by suction. Compressions of less than 0.5 pm were evident in this single-lever system in which the specimen was constrained and higher breakdown strengths were obtained. The dual system revealed that, instead of symmetrical compression, the whole recess bulged out toward one side. At 80°C the movement of material at the base of the recess amounted to more than 30 p m at 15 kV in a 50 pm specimen. Movements of approximately 10 pm were also evident at 20°C. In order to confirm this effect at 80"C, a 49 pm thick specimen, which had withstood 3.1 MV cm-* with no measurable compression in the single-lever system, was retested with the filter plate removed, so that the recess was unconstrained. A movement of 24 pm,was recorded before breakdown occurred at 2.4 MV cm-'. The encapsulation of polymer specimens, in either epoxy r e ~ i n for s ~ ~ ~ ~ ~ measurements at room temperature and above, or in a silicone rubbede for liquid-nitrogen temperatures, where the cured rubber becomes a hard and brittle transparent material, has been used in attempts to reduce or eliminate electromechanical deformation. It is difficult, however, to attribute the reported increases in electric strength exclusively to the restraint produced by encapsulation since other factors, such as the removal of partial discharges or the chemical action occurring during the curing processes, will also influence the magnitude of the breakdown voltage. 18.3.4.6. Thermal Effects. Thermal breakdown, as described theoretically in Section 18.3.2.2, is rarely observed in isolation. In thermoplastic materials, for example, thermal and electromechanical breakdown are inextricably interwoven. In materials of even slight conductivity at low field strengths, significant heating may take place as breakdown is approached due to the strong field dependence of the conduction current. Any softening of the polymer at such high electric field strengths is almost certain to be accompanied by electromechanical deformation. The re-

476

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sulting reduction in specimen thickness enhances the field strength, with constant applied voltage, to produce a further increase in conduction current and a subsequent temperature rise. The process may be cumulative, in which event breakdown will rapidly ensue. Thermal breakdown was probably the first breakdown mechanism to be systematically investigated and the factors involved were established over 50 years ago.86 More recently, other mechanisms have been of interest and experiments were designed so as to exclude thermal breakdown. As a consequence of this, there is a dearth of information in the literature of techniques designed specifically to study this mode of breakdown. It is possible, nevertheless, to conceive the experimental conditions required for such an investigation. For example, such experiments will usually have to be conducted at temperatures above that of the normal ambient. The application of direct voltage is a necessary prerequisite for the determination of the minimum thermal breakdown voltage, as is the use of thick samples and electrodes having a small thermal capacity and which inhibit the conduction of significant quantities of heat from the specimen. The application of impulse voltages with a variety of rise times will enable the impulse thermal voltage to be determined and will illustrate its dependence on the duration of the applied voltage. The more rapid the rise in voltage, the higher will be the breakdown voltage; a property that distinguishes this form of breakdown from that taking place under the influence of injected space charge, which has the opposite characteristic. Other thermal phenomena associated with the breakdown of polymers have, however, been investigated. In particular, Winkelnkemper and KalkneF have used two quite different methods to investigate the temperature rise at the surface of a polymeric sample prior to breakdown. In the first instance, they determine the temperature distribution over the surface of their recessed specimens by placing a layer of liquid crystals in the recess on top of the evaporated aluminum electrode to act as temperature indicators. Their second method involved the use of infrared photography to the same end. Temperature images with great local and time resolution are claimed for the latter method. Measurements on different polymers gave a pronounced temperature rise before breakdown. 18.3.4.7. Cryogenics. Breakdown measurements at cryogenic temperatures present problems additional to those experienced at more moderate temperatures and arise mainly from loss of electrode adhesion or changes in the mechanical properties of the polymer. In order to avoid BB

S. Whitehead, "Breakdown of Solid Dielectrics." Ernest Benn. London, 1932. H. Winkelnkemper and N . Kalkner. ElrXrrori~c4.Z . . Airsg. A 95, 261 (1974).

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the former problem, which arises from the differential thermal contraction between the electrode material and the sample, many workers seem to prefer to work using bare metallic contact with their polymer specime n ~despite ~ the ~ obvious * ~ risk ~ of~partial ~ discharges ~ at electrode edges and the consequential possibility of erroneous results. Despite the adhesion problem, evaporated metallic electrodes have been used successfully.29*90Some polymers show poor mechanical properties in liquid nitrogen or helium. For example, when polyethylene is bent or rolled at these temperatures its resistance to cracking is worse than that of polytetrafluoroethylene or poly(ethy1ene terephthalate) and this leads to inconsistent values of dielectric The technical difficulty of working at low temperatures exacerbates the need to test many samples in the same experimental rig without recourse to dismantling the system or constantly returning it to room temperature in order to change the sample. Kobayashi and Takebe8* have constructed two devices that enable them to test several film samples without dismantling their cell. In the first of these devices, 11 specimen films are sandwiched between 12 copper plane electrodes whose edges are surrounded by rings of PTFE, and from which 12 leads emerge through hermetically sealed terminals located on the vertical stainless steel cylinder in which they are suspended. The dielectric strength of the nth sample from the top of the pile is measured by the application of a common voltage to the n electrodes above the specimen, while earthing the remaining 12 n below the specimen. Thus 11 measurements can be made without opening the cylinder. Many more breakdown measurements can be made using their second device, which has 10 brass plane electrodes and a movable brass hemispherical electrode (Fig. 11). A specimen is placed on each of the plane electrodes and the hemispherical electrode, which is driven from outside the cylinder, can move circularly on a specimen. When the hemispherical electrode comes to an opening in the specimen and its plane electrode, it drops onto the specimen below. As the alignment of each opening is successively shifted by 30°, the hemispherical electrode can go round almost a full circle before descending to the next specimen. By earthing the hemispherical electrode and applying a common voltage to all the plane electrodes, measurements can be made at several points on each specimen and in this way, more than 100 breakdown measurements can be made without disturbing the cryogenic system. en K. Kobayashi and H. Takebe, Cryogenics 12,97 (1972). F. T. Stone and R. McFee, Rev. Sci. I n s t r i m 32, 1400 (1961). K . Arnakawa and Y. Inuishi, Jpn. J . Appl. Phys. 12, 755 (1973).

47 8

18. ELECTRICAL METHODS

b

c:

FIG.1 1 . Cryogenic assembly. (a) Brass hemispherical electrode, 6 mm in radius. (b) Brass plate electrode, 890 mm in diameter. The three cuts at the edge are for the rods that fasten the teflon pile. (c) Specimens (thick solid line), electrodes, and Teflon insulators are piled. (From Kobayashi and Takebe.=)

The direct immersion of polymer samples in liquid nitrogen or helium relies on the dielectric properties of these materials to eliminate partial discharges through the ambient from the edge of the metallic electrodes to the polymer surface. These ambient liquids are not ideally suited to this purpose and breakdown measurements made in this way are more likely to be characteristic of breakdown under the influence of discharges than due to some intrinsic property of the material under test. In order to reduce the effect of these discharges on the apparent electric strength in cryogenic media, Bob0 et studied the behavior of solid insulation under high vacuum. By replacing liquid helium by a vacuum of lo-' Torr, a large increase in dielectric strength was observed for certain polymers, especially polyethylene terephthalate. The reduction of the effect of prebreakdown discharges in this manner led these authors to conclude that high vacuum may afford a solution to many insulation problems occurring at cryogenic temperatures. 18.3.4.8 Phototechniques. The photographic recording of the development of a spark in solid dielectrics prior to disruptive breakdown is made difficult by the very rapid nature of the process. The first successful application of high-speed photography to the study of dielectric breakdown was made using alkali halide sampless and has since been extended to transparent polymers, in particular PMMA.IS The initial method is illustrated schematically in Fig. 12 and is based on the use of a Kerr cell and crossed polaroids acting as an electrooptical shutter in a darkened laboratory. Briefly, the operation is as follows. As the unidirectional voltage applied to the specimen is raised slowly, it charges the

18.3

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Transmission line

FIG.12. Optical and photographic system. Lenses 1-3: anastigmats; focal length 5, 5, 3 in., resp.;fnumber 1.9. Lens 4: field. Lens 5: condenser; focal length 1.5 in.;fnumber 1.0. Lens 6: copying;fnumber 1.0. (From Cooper and Elliott.e)

transmission line and opens the shutter. When the specimen breaks down a voltage removal wave propagates along the line and closes the shutter after a time interval I,/c where, I, is the length of the transmission line and c the velocity of electromagnetic waves on the line. Light from the prebreakdown discharge traverses an optical path of length I,, which can be adjusted. The time taken for the light to travel from the specimen to the shutter is I,/c and when I, exceeds I, light is recorded that was emitted from the specimen up to a time (I, - Ie)/c before the collapse of voltage across the specimen. The shutter does not close instantaneously because of its self-capacitance and a correction must be made for its closing time 1, to give the true time of cut-off. In order to obtain a sharp cut-off and a small t, the transmission line characteristic impedance and the shutter capacitance must be made as low as possible. The closing time also depends on the breakdown voltage of the specimen and was estimated by Cooper and Elliott to be about 2 nsec in their experiments. The luminosity is very weak and it is necessary to use a large-aperture optical system and an image intensifier. The maximum working voltage was limited to about 35 kV by technical problems arising from flashover risks and the requirements of the shutter, and consequently this defined the specimen thickness. Since optical paths up to 8 m in length were required, very careful alignment of the specimen on the optical axis was required. The ambient liquid must be transparent, have an adequate electric strength, and satisfy the optical requirement that its refractive index should be close to that of the specimen material.

480

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The polaroids when crossed did not stop light transmission completely and precautions had to be taken to prevent the recording of light emitted in the later stage of the discharge, after the collapse of voltage. These consisted of connecting in parallel with the specimen a current diverter, which operated well within I psec of breakdown, and of limiting the maximum gain of the image intensifier to 1.5 x 103 when the aperture of lens L , was stopped tof/3.5. In these circumstances, no light emission could be detected when the specimen broke down with the electrical connection between it and the Kerr cell broken. The use of the current diverter had the further advantage of minimizing damage to the specimen so that a comparison could be made of the position of the breakdown channel with the pattern of light emission. A photograph of each specimen was taken before breakdown in order to check the alignment and focussing. The shutter was partially opened by applying voltage to the specimen and the transmission line and illuminating the specimen weakly from the rear. The illumination was then removed and the voltage was raised until breakdown occurred. A qualitative interpretation of the photographs may be made by visual inspection. However, photographs may be scanned with a photomicrodensitometer to confirm the features observed visually and to obtain a quantitative measure of the exposure. Plots of the exposure of the photographs are obtained by scanning them in a direction perpendicular to the discharge track, the scans being made at 1 mm intervals along the length of the discharge. One further factor must be taken into account when interpreting results obtained in this manner. The photographs show an integration of the light that has been emitted up to the closure of the shutter. Each group, at a particular length of the optical path, should show the features of the previous group with a longer light path, plus any new features that have appeared during the intervening period (typically 5 nsec). In order to separate the shutter action from the breakdown event, a camera capable of providing single frame photographs with an exposure of 13 nsec at any instant, specified to within 5 nsec, in a period extending over 300 nsec, has more recently been deve10ped.l~ Emission of light in the earliest stage of spark formation is detected by a photomultiplier located close to the specimen and connected through a variable-length delay line to the trigger input of a pulse generator. When the impulse attains 4 V a negative flat topped impulse of 13 nsec duration and amplitude 3.9 kV is applied to the photocathode of the image intensifier by capacitive coupling from a fine wire grid. The intensifier is held off by a positive bias of 3.9 kV applied to the photocathode. The pulse removes this bias and the intensifier is switched on. The grid, which has an optical trans-

18.3

ELECTRIC BREAKDOWN

48 1

mission of 85%, covers the input window of the intensifier but it does not lie in a focal plane and therefore causes no distortion of the final image. The circuits used in the generation of the intensifier switching pulse produce a measured delay of 50 nsec between the illumination of the photomultiplier photocathode and the appearance of the pulse at the intensifier photocathode, this delay being compensated by a 15 m long optical path. Image converter framing methods (typically up to lo7 frames/sec) which are used in liquid breakdown studiese1 are not sufficiently fast for solid breakdown applications. 18.3.4.9. Discharges. Breakdown of polymeric insulation under the action of discharges can be divided into two two groups: (a) ambient discharges and (b) internal discharges. Ambient discharge studies are commonly conducted using a small air gap of 1 mm or less between a needle or spherically ended rod and a plane sample situated on a plate electrode74,e2*93 but have also been carried out in other media, e.g., insulating oil. Acceleration of breakdown under the action of these ambient discharges is achieved by an increase in frequency into the kilohertz range,74*92.93 an inverse dependence of life on frequency being assumed. Using a temperature-controlled test cell, accommodating 10 specimens, Greenshields and HoggsZwere able to determine the life of their samples sequentially by disconnecting each sample from the high-voltage supply using low-current fuses that operated as each sample failed. The time to failure was recorded on a 10-channel detector. Partial discharge measurements for each sample were made at selected times throughout the experiment by connecting each sample in turn to a high-voltage bridge and recording discharge activity with a pulse counter/height analyzer. This bridge gave a system resolution of 0.8 psec and a discharge detection sensitivity of 0.1 picoCoulombs for a 10 pF sample capacity. The manufacture of specimens for studying the effect of internal discharges on the breakdown of polymers has been described in Section 18.3.3.7. The void is usually formed as a hole punched in one layer of a multilayer sample produced from film materia12*60-s2 but may be cast or machined in the thermosetting resins.5g These latter specimens and those assembled from only two layers60-62necessarily have one metallic face in the void and, though they can be readily dismantled to observe erosion in the cavity before breakdown, they are clearly less versatile than multilayer systems in which the cavity may be totally enclosed in the insulating materiaL2 Such a multilayer system, assembled by Opuko et al., had the P. B. McGrath and J. K . Nelson, IEE Conf. fubl. 129, 315 (1975). J. R. Greenshields and W. K . Hogg, IEE Conf. Pub/. 129, 1 (1975). 93 Y. Toriyama, H.Okarnoto, and M. Kanazashi, IEEE Trans. Electr. Insul. EI-6, 124 (1971). 9l 91

482

18. ELECTRICAL

METHODS

additional facility of a vent from the void to the ambient to ensure that the gas in the cavity would be at the same pressure as in the tank in which the electrodes were mounted. This tank contained nitrogen or sulfur hexafluoride at pressures up to 2 atm. 18.3.4.10. Electrical Trees. Rebreakdown conditions in the form of a treelike pattern have been observed in solid synthetic insulation for many years,l0nmand it has long been established that the simplest way to form such a treelike pattern is to subject the insulation to a very nonuniform electric field, applying either an alternating voltage or impulses. Hence, the so-called needle test, based essentially on a point/plane electrode configuration, has been adopted by many laboratories as a criterion for the evaluation of the quality of polymeric i n ~ u l a t i o n . ~ ~ There are clearly two stages in the development of such trees under the application of an alternating voltage: an inception period of perhaps several hours and a formation period of minute^.*^,^^ These two periods may be observed by monitoring the discharge activity in the specimen.65 The apparent corona discharge remains essentially stable during the inception or induction period, and then at a critical inception time it suddenly starts to increase. Finally after several tens of minutes, the sample breaks down (Fig. 13). Dissection and examination of these samples indicate that an increase in corona discharge apparently occurs when a tree forms at the tip of the needle. Specimens dissected and examined, after several hours without an increase in corona discharge, did not show any evidence of treeing. Bahder et d B who 5 also performed such tests at 5000 Hz suggest that the time of tree propagation is inversely proportional to the frequency. Although confirming such a decrease in the time to breakdown by treeing with increase in frequency (up to 800 Hz),Densleya7 claims that this decrease is nonlinear. I

I

-m

-100

:58

-10

x J

g

I

u I

I

2

3

9

21O;O

minutes

6 TREE

I

I

I

I

60

120

180

210

I NCEPTlON POINT

-

I

300

seconds TIME

a -' $ -O

% u I-

z w

a

d

a

a

FIG. 13. Typical record of apparent corona charge during electrical tree propagatlon. (From Bahder cr

18.3 ELECTRIC BREAKDOWN

483

The introduction of so-called voltage stabilizers into cross-linked cable insulation has been found to elevate the tree-starting voltage without having any effect on the rate of growth of the tree, once started.gq The possibility of prestressing affecting breakdown by treeing has recently been ~ u g g e s t e d and ~ ~ . ~interpreted in terms of injected space charge in a manner not dissimilar from that described by Bradwell et ul.' with regard to uniform field breakdown in the absence of trees. Nakayama et a / .,gs using short wavefronts of nanosecond duration on point-plane samples of epoxy resin, found tree propagation easier than with slower wavefronts of microsecond duration. Space charge effects are postulated to explain this difference and also to explain the easier tree propagation from a positive point than from a negative one. Using polyethylene specimens, Noto et ~ 7 1 combined . ~ ~ direct voltage and impulse voltage application and found that the tree initiation voltage was decreased when the impulse voltage, superposed on the direct voltage, was of opposite polarity to it and increased when the direct and impulse voltages were of the same polarity. Almost all work on this particular aspect of dielectric breakdown, including that described above, is either concerned with observations of complete trees as an end product of an experimental exercise or with the indirect monitoring of the progress of tree growth by measuring corona discharge activity. In contrast, a step-by-step observation of tree growth was carried out in PMMA by Bolton et a/.* Starting with a voltage less than the discharge inception voltage, several 1/30 psec impulses of the same amplitude and polarity were applied to a point plane sample of PMMA. If no discharge occurred, the amplitude was increased by a small amount and a step-by-step procedure continued until a discharge was observed at the point. The final voltage increment was less than 5% of the discharge inception voltage. Having thus determined the discharge inception voltage, impulses of this magnitude were applied to the specimen under test and the following procedure was adopted. After observing a discharge with the naked eye, the specimen was inspected using a 32x microscope. With reference to a graticule in the eye piece on which were marked a series of concentric circles, a scale drawing was made showing the length and orientation of the projection of each channel in a plane through the axis of the point and parallel to the face through which obserBL H. Kato, N . Maekawa, S. Inoue, and H. Fujita, Annu. Rep., Con$ Elecrr. Insul. Dielectr. Phenom. 1974, 229 (1975). Ds F. Noto, N. Yoshimura, and T. Oota, IEEE I n t . Symp. Elecrr. Insul.. 1976 p. 205

( 1976). 86

H. Nakayama S. Nagata, T. Yoshida, and Y. Inuishi, Technol. R e p . Osuku Univ. 25,

367 (1975).

484

18.

ELECTRICAL METHODS

FIG.14. Spatial growth of an electrical tree. On, limit of growth produced by nth discharge. (From Bolton et u / . ~ )

vations were made. Thus, step by step, the growth of a tree was observed and recorded. It was noted that channels appeared to have no well-defined termination but merely tapered rapidly at their ends and seemed to disappear at a point determined by the microscope magnification. However, the channel lengths appeared to increase by only about 10% when the magnification was increased from 32x to 200x. A typical result is shown in Fig. 14, in which the point electrode is shown diagrammatically with its tip at the center of the reference system. The points numbered 1, 2, 3, etc. indicate the extent of the growth as it appeared after the first, second, third, etc. discharge. The length of each new extension can therefore be determined by proceeding backwards along each channel from its tip to the junction with a previously established part of a channel. As a result of this work it was concluded that the channels produced by the discharges were hollow tubes with nonconducting walls and containing the gaseous products of the discharges. Olyphants7 showed, by introducing fluorescent dye, that the channels consist of hollow tubes, and he found no difference in the resistance between the electrodes of "

M.Olyphant, Jr., Int. Conf:

Gus Dischorges. Electr. Supply Ind.. 1962 p. 441 (1962).

18.3 ELECTRIC

BREAKDOWN

485

specimens with trees extending from electrode to electrode and specimens containing no trees. Bolton e? a1.8 measured the resistance of several channels in parallel by drilling a blind hole into the tree and filling the hole with mercury. The resistance between the mercury pool and the original point electrode exceeded 10” R. The electric strength of the gas within the hollow channels is clearly important in determining the discharge inception involved in the propagation of the tree. The observation that discharge channels, other than the original one, emanate from the point electrode means that the electric strength of the channels must be comparable with that of the surrounding PMMA. Since these tubules can be as small as 1 pm in diameter, the growth of avalanches is inhibited by the loss of electrons to the dielectric wall and by the field distortion caused by the resulting surface charge. When avalanches are inhibited, the electric strength of the gas is increased, so that the observation that complete penetration of a specimen by a tree channel does not lead to breakdown is perhaps understandable.w Actual “tree” tubules are tortuous and the diameter varies dong the length so that detailed quantitative investigations are difficult. However, it has proved possible to produce cylindrical tubules of air confined in PMMA of diameter down to 10 pm that effectively demonstrate the effects of the wall” (see Section 18.3.3.8). Measurements of the direct voltage, alternating voltage, and 5/3600 psec impulse voltage electric strengths were made on these manufactured tubules with striking results (Fig. 15). For tubule diameters in excess of 200 pm, the breakdown strength of the confined column of air using direct voltages was identical to that of unbounded air. As the diameter of the tubule was decreased below that of the head of an electran avalanche, equal in length to that of the tubule, calculated according to the formula of Llewllyn-Jones,BBthe electric strength rose precipitously. This increase is clearly due to the stifling of avalanches by the trapping of charge at the tubule walls. When using impulse voltages to break down such specimens it is desirable to irradiate the specimen to avoid erroneously high values. These high values are attributed to long statistical time lags because of the small volume of stressed air. With irradiation the characteristics of the tubules broken down under impulse conditions are similar to those obtained using direct voltage. Breakdown strengths using 50 Hz alternating voltage are considerably lower than both impulse and direct voltage strengths. This difference is ea A . Kelen and L.-E. Larsson, Acta Polytech. Scand., Electr. Eng. Ser. [N.S.] 6, 133 (1967). F. Llewellyn-Jones, ”Ionization and Breakdown in Gases.” Methuen, London, 1966.

18.

486

ELECTRICAL METHODS

75 w x

80.

fm

:

C

b a

i5 LO. .-0 c L

-

u w

10

20 30 40 Diameter x 102mm

50

60

FIG.IS. Electric strength of air-filled tubules under the application of direct voltage. (From Auckland e l

believed to be indicative of the effects of deposited wall charges. Charge deposited on one half-cycle aids the applied field in the next half-cycle, thus counteracting the effects observed with unidirectional test voltages. The magnitude of the effect of these wall charges and their rate of decay can be investigated by prestressing specimens with about 90% of their mean direct breakdown voltage for about 100 sec before allowing a waiting period between the removal of the direct voltage and the application of a breakdown impulse. The impulse electric strength is raised above its unconditioned value using impulse and direct voltages of the same polarity and reduced when the impulse voltage is of opposite polarity to that of the prestressing direct voltage. This increase and decrease, which is approximately 50% with no waiting period, becomes negligible if the interval between the removal of the direct voltage and the application of the breakdown impulse exceeds a few hundred seconds. The rate of charge leakage is independent of tubule length which suggests that the wall charges are neutralized by conduction through the body of the dielectric rather than by leakage along the surface of the tubule. The channels comprising a tree may have a length of a few millimeters and a tapering diameter of a few microns. Extrapolation of the results of the above experiments indicate that the direct voltage and impulse voltage electric strengths of such a channel are very high and may well approach

18.3

ELECTRIC BREAKDOWN

487

that of the surrounding dielectric. The alternating voltage electric strength, however, due to the wall changes and polarity reversal, may be very low. During the process of tree growth, the incomplete channel constitutes a composite specimen, comprising an air column in series with a layer of solid dielectric. It is necessary therefore to understand the breakdown mechanism in such a composite specimen if the process of tree growth is to be fully elucidated. Cooper and his c o - w o r k e r ~ have ~ ~ J ~manufactured composite specimens in PMMA as an extension of their work on tubules. The diameter of the air column in their specimens was 70+ 10 pm. Using the photographic techniques described in Section 18.3.4.8, they have cast doubt on existing conceptions of composite breakdown by observing that the initial luminosity, under the influence of 30/1800 psec impulse voltage waves occurred in the solid rather than the air if the air column was negative with respect to the solid. This phenomenon occurred despite the fact that only 2% of the applied voltage was developed across the solid component of the specimen by capacitive division. It was not known initially whether light generated in the air column would be transmitted as efficiently as light generated within the solid because of internal reflections at the column walls. This was resolved by the use of a specimen formed from two rectangular blocks of PMMA, each containing a cylindrical hole extending from one face to the opposite cm, and the other was 18 x lop3cm in face. One hole was 38 x diameter. The blocks were mounted with the holes coaxial and a tungsten wire, coated with a paste of zinc sulfide in an acrylic cement, was drawn coaxially through the specimen from the small to the larger hole. The former was completely filled but an air gap existed between the coated wire and the hole wall of the latter. As the zinc sulfide had been irradiated previously a photograph of the luminescence enabied the assumption of equal transmission efficiencies to be verified. An investigation of the technically more important breakdown by alternating voltage had to await the development of a camera capable of providing single-frame photographs at specific instants.15 In order to interpret their results Auckland et d . l 5 found it necessary to postulate the penetration of electrons (but not positive ions) from a gas discharge through as much as 50 pm of PMMA. This was experimentally established as follows. A glass plate supporting a layer of microcrystalline silver chloride was overlaid by a film of PMMA deposited by allowing a solution of the polymer in chloroform to evaporate. A corona discharge was struck in the air about a point electrode above the PMMA. With the point negative, any electrons that enter the PMMA and do not become trapped pass into the silver chloride and cause blackening. This can be assessed quan-

488

18.

ELECTRICAL METHODS

titatively by removing the PMMA layer, illuminating the glass plate uniformly from underneath, and scanning across the darkened region with a photomicrodensitometer. With otherwise similar experimental conditions but with the point positive, no blackening of the silver chloride occurred, demonstrating that light from the discharge was ineffective and that electrons were the agent causing blackening in the previous experiment. 18.3.4.11. Water/ElectrochemicaI Trees. Although there is considerable literature on the subject of water trees and electrochemical trees, very little of it contributes anything to a discussion of experimental methods, being simply reports of observed trees in cable samples withdrawn from service, either before or after failure. Deterioration of the alternating voltage electric strength of such samples can be accelerated by an increase in and temperature,'OO and deliberate attempts have been made to induce water to enter polyethylene samples using a surface activator and small amounts of electrolyte.'OO Water trees were ' maintaining samples in a vaccaused to disappear by Fukuda er ~ 1 . ' ~by uum of 5 mm Hg for over a week, and restored by reimmersion in water. The location of treeing sites in a length of cable in the absence of obvious visual surface defects, such as breakdown craters, may be facilitated by immersion of the bared core in a bath of glycerol or similar medium heated to a temperature of 120-170°C.102 At these temperatures both polyethylene and polypropylene lose their crystallinity and become transparent. If the core is illuminated from a transmitted bright source of light any treeing sites will become apparent. Where the extent of the treeing is only slight, a microtome sectioning technique is a p p l i ~ a b l e . ~ ~ A*small ' ~ ~ piece of the insulation is embedded in a block of low-melting-point wax, which enables the sample to be clamped firmly to a microtome stage without damage or distortion. Using a sledge-type microtome, thin sections about 15-30 pm in thickness are cut. The sections are mounted between glass slides in a medium of refractive index close to that of the polymer, e.g., oil of cloves, and with the ability to wet its surface. Unless these two conditions are met, knife marks and trapped air bubbles become visible in the image and obscure the full detail of the treeing. The sections are then examined using either bright or dark-field illumination. The latter method enhances contrast and enables the fine structural detail to be more clearly seen. T. Miyashita. IEEE Trans. Electr. Insul. EI-6, 129 (1971). T. Fukuda, S. Suzuki, H. Goto, and Y. Nitta, Annu. Rep., Conf. Electr. Insul. Dielectr. Phenom. 1972, 211 (1973). E. H. Reynolds, R. M. Hinde, and R. M. Black, Annu. Rep., Conf. Elecrr. Insul. Dielectr. Phenom. 1972, 125 (1973). Io1

18.3

ELECTRIC BREAKDOWN

489

Eichhorn3* investigated the morphology of “wet” trees, obtained by exposure to steam at a gauge pressure of 14 psi (approximately lo5 N m-2) in a sealed tube for 30 min and subsequently containing in excess of 0.12% moisture by weight, by microtoming sections of specimens containing trees prior to complete breakdown. These thin sections were cut normal to the general direction of tree propagation so that their surfaces would reveal cross sections of the tree channels. The surfaces of several sections, of both wet and dry specimens, were vacuum coated with a thin layer-of-gold and examined with a scanning electron microscope. Using the high magnification and great depth of field of this instrument, it is possible to look down into holes and depressions that may lie in the surface and to recognize second phases such as those which develop from mechanical crazing. 18.3.4.12. Industrial Testing. The experimental methods employed in industrial testing of electrical insulation will only be of marginal interest to the polymer physicist and no details will be given here. For the interested reader, a brief statement is included that is mainly intended as a reference source to more detailed information. Though the general principles underlying all the national standards are the same, there are many differences in detail, particularly in the electrodes used. ASTM specification D149 specifies pairs of identical electrodes of a variety of sizes and VDE 0303-2 (which is identical with DIN 5348 1) provides a variety of electrode shapes: disk-plane, with different disk diameters; sphere-plane-; sphere-sphere; needle and Rogowski electrodes. BS2918 and IEC243 define different electrode types for different sample forms (e.g., sheet, tape, tubes) and include pairs of cylinders and rod-plane electrode arrangements. The procedures for raising the voltage are defined for determining the breakdown voltage on a “rapidly applied,” 20 sec step-by-step or “one minute” basis; the American, British, and International standards being generally comparable. The process of breakdown in industrial test methods is likely to be compounded from several of the breakdown mechanisms discussed previously acting together. The results are not relevant to these individual breakdown mechanisms and serve more as a means of quality control and for comparing one material with another.

18.3.5. High-Field Conduction 18.3.5.1. Introduction. The experimenter embarking on a study of high-field conduction in polymeric dielectrics is faced with two major problems. First, the achievement of a high field requires either the pro-

18.

490

ELECTRICAL METHODS

duction of very thin specimens or the use of high voltages, or both. There is a limit to the thinness of specimens below which reproducible manufacture becomes exceedingly difficult or the material no longer resembles that occurring in practical circumstances. High voltages, on the other hand, present technical difficulties regarding the suppression of partial discharges. Low discharge activity, which may be tolerated in some short-term electric strength measurements, cannot be permitted in conduction experiments since the electrical noise generated by it soon renders accurate current measurement impossible. Suppression of these partial discharges by total immersion in a suitable dielectric liquid, while adequate in breakdown investigations, cannot be used in the measurement of conduction currents (see Section 18.3.5.2). The second major difficulty follows from the perennial argument as to the very existence of a steady-state conduction current. This argument is based on the observation of long-term current transients generated by the application of voltage. It may be necessary to set an arbitrary time limit to the current transient beyond which subsequent changes are considered negligible in order to carry out experiments that require the repeated application of voltage in a finite period of time (see Section 18.3.5.4). Available space does not permit an account of the immense amount of published work devoted to the interpretation of the results of high-field conduction measurements and the many attempts to fit observed voltage/current characteristics to theoretical models, much of which is contradictory and inconclusive due to the value of crucial parameters and the boundary conditions that have to be assumed. Speculation exists on even the most basic aspects of the conduction process. For example, the charge carriers in polymers have been holes,4,110ions,24*42~105-11s and suggested variously as protons 18.3.5.2. Specimens. The specimens used in high-field conduction in-

IM IoJ

loo lo'

D. M. Taylor and T. J. Lewis, J . Phys. D 4, 1346 (1971). A. C. Lilly, Jr. and J. R. McDowell, J . Appl. Phys. 39, 141 (1968). H. St-Onge, Annu. Rep., Con$ Electr. Insul. Dielectr. Phenom. 1973, 70 (1974). E. Sacher,J. Phys. D. 5, L17 (1972). M. Ieda, M. Kosaki, and M. Yoda,Annu. Rep., Con$ Electr. Insul. Dielectr. Phenom.

1972, 405 (1973). loo lop

J. F. Fowler and F. T. Farmer, Br. J . Radiol. 24, 118 (1956). T. Tanaka and Y. Inuishi, Electron. Eng. (Tokyo) 89, 1 (1969). M. L. McCubbin and I . D. C. Gurney J . Chem. Phys. 43,983 (1965). L. E. Amborski, J . Pofym. Sci. 62, 331 (1962). A. J. Warner, F. A . Muller, and H. G . Nordlin, J . Appl. Phys. 25, 131 (1954). P. Y. Feng and J. W. Kennedy, J . Am. Chem. SOC. 11, 847 (1955). S. Mayburg and W. L. Lawrence, J . Appl. Phys. 23, 1006 (1952).

18.3

ELECTRIC BREAKDOWN

49 1

vestigations resemble more those used in electric breakdown measurements than those used for the measurement of low-field conductivity. The possibility of ambient discharges again demands the use of diffused-edge evaporated electrodes on film material or the molding or machining of recessed specimens from thicker material (see Section 18.3.3). One vital additional requirement is the application of guard electrodes to intercept leakage conduction over the surface of the specimen. Spherical recesses have an undefined conducting area and accurate estimates of current density are impossible. The use of an appreciable flat region at the base of the recess, sufficiently large to render conduction in the thicker material on the periphery of the recess negligible, is thus common p r a ~ t i c e . ~ 3 l $ ~ There is some doubt as to the applicability of insulating liquids for the suppression of ambient discharges as used in breakdown studies. Care must be taken in the use of such liquid ambients to ensure that they do not provide a parallel conduction path and render the guard electrode ineffective. A cup-shaped specimen with a recess molded in the center of its base will overcome this difficulty by allowing the use of separate ambient discharge suppressing liquids for the high- and low-voltage electrode, and will maintain an effective guard electrode. Such an arrangement, devised by Bradwell ef (Fig. 16), incorporated a pool of mercury within the cup to act as the electrode contact and its thermal capacity reduced the possibility of thermal breakdown. Amalgamation of the mercury and the evaporated metallic electrode may be prevented by covering the latter with a layer of colloidal graphite. This specimen arrangement also permits the use of suction via a sintered brass disk in the plane electrode to restrain the specimen and reduce electromechanical deformation (see Section 18.3.4.5). Restraint by ento electrometer specimen

\

,guard -ring

sintered disc

FIG.16. Specimen and electrode arrangement used for investigating conduction. (From Bradwell et d.9

18.

492

ELECTRICAL METHODS

capsulation within a thermosetting resin may not be suitable for conduction measurements since a component of the measured current arising from conduction in the encapsulating resin cannot be entirely excluded and its magnitude will in general be indeterminate. The use of high vacuum as an ambient medium has been considered ideal by some investigators since current magnitudes have been found to be much lower in specimens in this ambient than in atmospheric air.21Jw*105.11s The efficacy of a vacuum of about loFBTorr as a discharge suppressant cannot be disputed, but the relevance of measurements on such “dry” specimens to the practical applications of polymeric dielectrics is debatable. Specimen restraint, either by suction or encapsulation, is not possible in this situation and electromechanical deformation at high fields and temperatures may in any event limit its suitability. 18.3.5.3. Electrode Material. The range of electrode materials used in conduction measurements on polymeric dielectrics is very much the same as that employed in breakdown studies (Section 18.3.4.1) although the nature of the electrode material is of much greater consequence in the former than in the latter case. In many cases, though not all, electrodes of painted colloidal suspensions tend to give rise to larger conduction current densities than do those of evaporated metal^.^*^^ Bare metallic electrodes are prone to partial discharges at high voltages, resulting in a large noise component in the measured current, although no problem of this sort is mentioned in published work using brass and stainless s t e e P in this way. Not only is there a strong electrode effect on the measured current density, observed using identical anode and cathode material^,^^*^^^-^^' but the nature of the individual anode and cathode influences the magnitude of the conduction current4 (Fig. 17). Double injection of charge carriers is therefore invoked to explain this phenomenon in polyethylene4 and a similar conclusion has been drawn from thermally stimulated current measurements in poly(ethy1ene terephthalate)lls 18.3.5.4. Transient Phenomena. The transient behavior of the observed current flowing in polymeric insulation in response to sudden changes in the applied voltage (Fig. 18) is a major impediment in the determination of the conductivity of such materials and has been a source of contention for many years. There is evidence that the steady state may 115

T. Mizutani and I. B. Jordan, Annu. Rep., Conf. Electr. Insul. Dielectr. Pheonom.

1974, 640 (1975). 118 11’

B. R. Varlow, Int. Microsymp. Polar. Cond. Insul. Poly., 1972 p. 59 (1972). M. Aozasa and K. Yahagi, J . Phys. Soc. Jpn. 30, 584 (1971). A. C. Lilly, Jr., R. M. Henderson, and P. S. Sharp, J . Appl. Phgs. 41, 2001 (1970).

18.3

ELECTRIC BREAKDOWN

rdOJ .

0.5 Electric

493

1.0 1.5 field, M V c d

FIG. 17. Dependence of I , on anode as well as cathode. 75 pm thick polyethylene at 20°C. (a) Silver in toluene cathode;'evap. indium anode. (b) Silver in toluene cathode and anode. (c) Evap. indium cathode; silver in toluene anode. (From Bradwell et a/.')

not be reached for several days, especially at normal room temperatures21*116 and at low field strengths.11B*120This has led some workers to restrict their conduction current measurements to those at elevated temperature^.^^**^ In thermoplastic materials, such as polyethylene, this restriction may compel the researcher to approach temperatures at which high-field conduction measurements may be influenced by electromechanical deformation of the specimen, unless some form of restraint is applied. It has been found possible to distinguish between the major source of the transient in low and high electric fields. In low fields, the ohmic dependence of the current at any specified instant during a transient on the applied voltage that has given rise to it, and the equivalence of absorption and desorption transients, indicate a dipolar origin to the phenomenon. At fields within one or two orders of magnitude of the breakdown strength, however, the transients have been shown to reflect the accumulation in trapping centers of injected space ~ h a r g e . ~ At high field strengths, the steady-state current may in some instances W. Reddish, Sci. Monogr. 5, 138 (1959). V. Adamec, Proc. Insr. Elecrr. Eng. 112,405 (1%5).

18.

494

ELECTRICAL METHODS

t

Time

FIG.18. Absorption and desorption current transients. I , , absorption current; I , , charging current; I , , steady-state current: /,, , desorption or discharging current. (From Bradwell et al.')

be so large as to dominate the measured current and the detection of small transient changes is thus rendered both impossible and irrele~ant.~In general, at normal ambient temperatures, if changes in conduction current in response to changes in some external parameter, such as heat treatment,"O electrode material, or specimen geometry, are to be investigated on any reasonable time scale, it is necessary to set an arbitrary limit to the transient observation beyond which, although absolute steady state may not have been achieved, subsequent changes in the measured current are negligible compared with those taking place in response to changes in the controlling parameter. This is especially true in the observation of aging phenomena arising possibly from a whole range of physical, chemical, or morphological a ~ t i v i t y . ~ ~ * ~ l ~ The practice of conditioning the specimen by the initial application of the highest stress and temperature to be considered, in order to reduce the time scale of the transient p h e n ~ m e n a , ~may ~ ~induce ~ ~ ~ the , ' per~ ~ ~ ~ ~ ~ ~ manent polarizations that others have sought to eliminate prior to conduction current measurement by subjecting specimens to several thermally stimulated discharges. lo5 18.3.5.5. Morphology and Aging. The conduction of charge carriers at high fields in polymers depends on the treatment undergone by the material either in its process of manufacture or in specimen preparation. Many of these changes are of a transient nature and are negligible if sufficient time is allowed to elapse between specimen preparation and use. Other effects of specimen treatment may take a more permanent form.4o lzl

D. K . Das Gupta and K. Joyner, J . Phys. D 9, 2041 (1976).

18.3

495

ELECTRIC BREAKDOWN

The degree of crystallinity has been demonstrated to have a controlling influence on the magnitude of the conduction current at both OW^^^,^^^ and high electric field^,^^*^^^^^^^ the general conclusion being that conduction takes place principally via the amorphous rather than the crystalline phase. Differences can therefore be expected between measurements made on film specimens and those made on molded specimens. The rate of cooling from the mold may also be of some importance. Changes of a more transient nature can result from the heat treatment to which the specimen is subjected during the evaporation of metallic electrodes, though a period of one or two days is normally adequate to permit the disappearance of this effect. An aging phenomenon has been observed in molded and film polyethylene samples in which the electrode materials have included painted colloidal suspensions as well as vacuum evaporated metalP6 (Fig. 19). The daily measured value of steady-state current was found to decrease over a period of days following electrode application. Over a period of a week or more, this steady-state current may fall from its first day value by more than a factor of three, and apparently does not depend on the number or duration of previous voltage applications, if any. The explanation of these various phenomena is still subject to much speculation, though the relevance to the experimenter is obvious and

m c T ui

" 16.

4

75pm molded polyethylene

c ln

Silver in toluene

>;

g

12.

3

0.

c 0

Run immediately a f t e r electrode application x x x Dormant for seven days 0.0

,

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3

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5

6

7

a

g

Period a f t e r electrode application. days

FIG.19. Aging phenomena in the measurement of conduction current in polyethylene. (d) 1.28 MV cm-I. (From V a r l o ~ . ~ ~ ~ )

(a) 0.86. (b) 0.96, (c) 1.07,

496

18.

ELECTRICAL METHODS

FIG. 20. Electrolyte test-cell arrangement and associated circuitry. (a) High-voltage breaker, (b) rectifier, (c) relay. (From Swan.")

must be taken into account when devising experimental methods and interpreting the results of conduction current measurements. 18.3.5.6. Conduction Enhancement. A number of high-field conduction measurements in polyethylene have centered on the ability of iodine to diffuse into this polymer and greatly enhance its conductivity. Because of its high electron affinity, iodine is known to form donoracceptor complexes with several organic materials, though the effects of iodine on conduction observed in polyethylene have not been found in pol y (ethylene terephthalate) or pol ytetrafluoroeth ylene. 'I1 The experimental methods used by different workers are generally very similar though differences do occur between one group and another. The polymer, almost invariably in film form, is clamped between two glass flanges, with a thin layer of silicone grease on each flange for sealing purposes (Fig. 20). Electrical connections to the film are made by means of aqueous solutions of either sodium iodide or potassium iodide. Current transients have been studied following either the sudden application of voltage to a system in which the electrolytic electrodes contain iodine122J23 or the sudden injection of iodine into electrolytic electrodes to which the voltage has already been applied.lZ4In some cases, films were presaturated in iodine solutions of various concentrations, prior to testing, in order to eliminate the time required for diffusion of the iodine from the electrode solution into the polymer. Iodine-saturated solutions D. W. Swan, J . Appl. Phys. 38, 5051 (1967). D. W. Swan, J . Appl. Phys. 38, 5058 (1%7). lir T. J. Lewis and D. M. Taylor, J . Phys. D 5 , 1664 (1972).

lz3

18.3

ELECTRIC BREAKDOWN

497

may be used but only where the temperature is not a variable, since iodine will crystallize from solution at the lower temperatures. Acknowledgment The author wishes to thank Professor R. Cooper for his critical reading of the manuscript. His comments are greatly appreciated.

This Page Intentionally Left Blank

AUTHOR INDEX K I R PART

C

Numbers in parentheses are reference numbers and i d e a & that an author’s work is referred to although the name is not cited in the text. A

A. S,2% 251. 252, 253(48), 255, 26498, s6h 265(4ul Arkawp, Y., 372 A r k , J. P., 364 Anastroaa. A. A.. 375 Axridge. R G.C., 129 Mboutr. 1.. 4S2,456(23), -23). 465(23), A-

Abagyan. G. V., 189, 195, 201(17, 3.4). 202(34) Abrams, A. I., 366 Adams, V., 493 Aggamal, S. L., 295, 300(11Q), 311(110) Aiken, W. H.. 363 Ailhaud, H., 303 Akhmed-Zade, K. A., 1%. 20ltSO) Alexander, L. E., 175 Alfonso, G. C., 295,297(109) Alfrey, T., 17. 336 Alger, R. S., 186 Allen. F. G., 435 Allen, G.. 283, 285 Alpert, D., 359, 370 Alston, L. L., 461, 481(60) Altamirano, J. 0.. 311 Al’tzitser, V. S.. 311 Amakawa, K., 477 Amborski. A. J., 490, 493 11 1) Ambrose, J. F., 350 Amelin. A. V.. 227 American Society for Testing Materials, 363, 363240). 366. 371(266) Anderson, R. L., 36 Anderson, R. S.. 186 Andersson, P., 101 Andrade, E. N. da C. A,, IS, 27 Andreatch, P.. Jr., 61 Andrews, E. H., 186, 214, 216, 253,254, 255, 256, 265(51) Andrews, R. D., 33 Angelo, R. J . , 311 Anon, 372 Aozasa, M., 492 Arakawa, T.. 91. 115(4) Ar’ev, A. M., 229 499

474

Artemov, P. G.. 108 ASPi, H.. 295.298( IOI) Asay. J. R.. 66. 74.75.77(16. 22). 78(22) Ash. R.. 327.358.359.362 Asheraft. C. R., 395.3% Assink. R. A., 356 Ast. D. G., 244, 246, 247(32. 35). 248(32), 253(32), 256(21). 271(32) ASTM Standards. 122 Auckland, D. W.,449,451,461,470(12, IS), 478(15), 48q15). 486, 487 Austen. A. E. W., 452

B Backman, D. K., 190(19), 191, 202, 203(19) Backstfom, G., 428, 432, 433(31, 32). 434(32) Backstrom, J., 101 Baer. E., 91. 93, 94. 100. 244, 245, 249, 250(41), 257, 260. 261(73) Baer, M., 289. 312, 313(216) Baessler, H., 464 Bagley, E. B., 47, 337, 350(119) Bahder. G . , 449, 462, 482, 488(75) Bailey, C. D., 366 Bailey, W. J., 104 Baird, J. C., 186,’ 188(10) Baker, R. W., 362 Baker, W. O., 388 Baker, W. P., 449 Bakr, A. M., 322

5 00

ALITHOR INDEX FOR PART C

Balchan, A. S.,94 Ballard, D. G. H., 305. 313(170) Ballman, R.,L., 50 Ballou, J. W., 148 Bandaret, A., 288 Bank, M.,2K3,3119 BaptizmanSkii, V. V., 195, t86, 197(41), 199(39, 41,&),%0lc.S0),2ISW) Baramboim, N. K., 197 Barber, E. I$., 47 Barber, M . , M Bardman, 5). iR., 307,3QB(J?6) Barentsen, W.. *M.,294 Bares, J., 3 U Barham, f.J , -140 Barlow, J.\w.,311 Barnet, F.Bp,c92, 98, .109(35), 111 Barr, G.,XB0,3El Barrer, I& M.,319,327, 328, 33Q, 333,iBR. 359, 361,362., 363 Barrie, J. A%.,32% 379, 332, 333, IiWXZ32h 363 Bartenev, G. M+289 Bassett, G. A.. 2S+p991131), W131). 305 Bauer, M. E.,393. XMtl I), 395 Bauer, R. G..288,293 Baughan, E. C.. 323 Baughman. R. H.. 1 1 Baum, B.. 288.293(5S) Baum, E. A.. 439 Bauser, H.,431 255 Bauwens, J. C..246. Beach, B. M.. 308, W183a) Beahan, P.. 244 245,270 Beamish, A., 311 Beard, I. H., 460 Beardmore. P., 237. 252 Bearman, R. J., 327 Becht, J., 192, 203, 204(21), 606, 207(21), 223, 229(21), 231(124) Beecher, 3. F., 295, 300(110), 311(110) Beg, S., 461, 481(61, 62) Bekkedane, N., 99, 107 Belcher, H. V., 75 Bell, J. D., 45 Bender, B. W., 292 Bennett, J. E., 197 Benson, S. W.,350 Berens, A. R.,337, 349, 350(140). 352 Berge, J. W., 29, 30(65), 40(65), 41(65)

ma),

Bergen, R. L., 256 Berlin, A. A., 195, 201(26) Berry, G. C., 30, 50. 51(69) Berry, J. B., 271 Bersohn, M., 186, 188(10) Bert, C. W.,109 Bessonov. M. I., 253, 265(55) Best, R., 352 Bettelheim, F. A., 165 Beran, L., 256 Bevington, J. C., 457 Bevis, M.,244, 245(19), 270(19) Bi, L.-K., 298 Bibeau, A. A., 292 Biros, J., 287 Bishop, E. T., 7 Birnboim, M. H., 35, 36.42 Bixler, H. J.. 325, 327,329,330(46,59), 352, 362(46), 364, 365 Black, P., 312, 313(217) Mack, R. M..488 Blair, D. E.. 370 Wak. A. E., 440 Bhsius,J., 229 BLprr, B. J., 217(101), 2 U Bk&o. W. C., 288,293(59)

Blitz. 1.

80

B b k , S., Il@d Bbk. 1.. 453,&4,47X29j, 477(29) Blydse. A. R., 435 Blylmweldd L A. 1%. m!M26) Boag. J. W., 188 B o b . J., 463,Qll.4?7C;rOJ, 878 B o b , L.. 287. tssIss). 2tW53 Bolton. B., 449,453.454,470,483,484,485 Bonart, R., 300,301(141). 302. 313141) Bonner, D. C.. 355 Bonnin, M. J.. 108 Bonting, E. W.,92,93(15). 101(15) Bordelius, N. A.. 89 Borishade, A. B., 451,461.470(15). 478(15), 480(15), 486(64), 487(15) Bose, T. K., 421 Bossler. F. C., 121 Bowden, F. P., 431, 432 Bowden, P. B., 249 Boyd, R. H.,380, 394, 395, 396, 404, 405, 408, 415(22), 416, 420(22) Boyer, R. F., 14, 41 Braden, M.,214

AUTHOR INDEX FOR PART C

Bradford, E. B., 2% Bradford, R. D., 295, 300(110), 311(110) Bradley, A . , 456, 457, 458, 459 Bradwell, A., 445, 448(4), 4634). 467(4), 468, 469, 470, 471, 472(4), 473, 483, 490(4), 491, 492(4), 493, 494 Brandrup, J., 325 Brandt, H., 362 Brandt, J.. 106 Braun. J. M., 355 Brehm, G., 418 Bresler, S. E., 195, 199(24, 25). 201(24. 25) Bridgman, P. W., 91, 94,99 Briscoe, B. J., 433 Britton, G.W., 362(231), 363 Broadbent, J. M., 49 Broadhurst, M. G., 85 Brodryan, T. G., 50 Broeckman, A., 285 Brown, C. L., 366 Brown, G. L., 308 Brown, N., 129, 130, 131(17), 132, 235, 236( I), 237, 240, 241, 243, 249, 250, 251, 253(47), 255, 257, 259(65, 66), 260, 261(79), 262(7a), 263, 264(42), 265(47), 265(72), 266(72). 269, 270, 271 Brown, R., 214, 216(88, 89) Brown, W. E., 363 Brown, W. R., 326, 351, 366 Brubaker, D. W., 363(250), 364 Bubeck, R. A., 250, 253, 254, 260 Bucknall, C. B., 273, 292 Bueche, F., 27, 306 Bull, H. B., 350 Bundy, F. P., 100 Bunn, C. W., 167 Burchard, W.. 286 Burell, H., 286 Buritz, R. S.,359, 370 Burke, J. S., 36 Bunnester, A . F., 456, 457,458 Busse, W. F., I5 Butcher, A. F., 49. 50(132), 51(132) Buthenuth, G., 138 Butyagin, P. Yu.,189. 195, 197(31), 199(38), 201(17, 26, 34). 202(34) C

Cabasso, I., 352 Cahn, J. W., 291

Caldecourt, V. J., 456,451,458 Calderwood, J. H.,452, 465(20) Callan, J. E., 289 CUIIPOS-LO~~Z, E., 283, 284(31),

50 1

295, 296(107), 297(107) Campbell, D., 203,204(66), 206(66), 207(66) Cannon, M.R., 45 Cannon, S. L., 145 Cantow, H. J., 285 Capaccio, C.. 137. 140(4) tapla, M.,195, 199(46) Carpenter, A. S.,345, 352 Carrington, A., 186 Carter, G. W., 445 Casale, A., 197, 202(62), 276 Caskey, T. L., 372 Cassie, A. B. D.,322 Chahine, R., 420 Chalidze, V. N., 212 Chalmers, B., 27 Chambers' Encyclopaedia, 422 Chandler, E. F., 116 Chandler, L. A., 310 Chaney, C. E., 106,308 Charlesby, A., 440, 457 Chartoff. R. P., 36 Chasset, R.,45 Chen, F. C., 137 Cheng, Y. L., 355 Chiang, T. C., 203, 204(74, 73, 206, 207 Childers, C. W., 311 Choi, C. K., 315(5), 316, 362(5) Chow, M. T., 297, 298(115), 302(116), 30x115, 116) Choy, C. L., 137 Chubb, J. N., 425, 426 Clackson. T. S.,421 Clark, E. S.,140, 145 Clark, H. E., 424 Clauser, J. F., 18 Claver, G. C., 289, 292 Clough, S.B., 302, 313 Cobbs, W. H.,330 Coe, J. R.,Jr., 21 Cogswell, F. N., 50, 51, 52(156) Cohen, R. E., 312 Cole, K. S.,390 Cole, R. H.,390,400, 418.419, 420 Coleman, B. D., 47 Collins, E. A., 310

5 02

AUTHOR INDEX FOR PART C

Colwell, R. E., 3, 28(9), 45(9), 46(9), 47(9), 48(9) Conaghan, B. F., 288 Cooper, R., 445, 448, 449, 451, 453(8), 454, 455, 461, 46341, 466(39, 431, 407(4), 468(4), 46!9(4), 470,471(40), 472(4,39, 40), 473(4), 478(6, 151, 479, 480( 1 3 , 483(4, 8), 484(8), 485(8), 486(64), 487,490(4), 491(4), 492(4), 493(4), 494(4, 40), 495(40) Cooper, S. L., 290, 298,299, 302(127), 313 Cooper, W., 298 Corish, P. J., 276, 3120) Corsaro, R. D., 94, 100 Coutts, L. N., 271 Courie, J. M. G., 286 Crank, J., 315, 334, 343,344, 346,347, 349, 350, 352(129), 356, 358, 359, 360, 362 Creswell, R. A., 440 Crist, B., 222, 223(117), 229 Crothers, D. M., 45 Crystal, R. G., 290, 298, 299(132), 314(132) Cuevas, J. E., 92, 98, 109(35), 111(35) Culver, L. E., 253, 254(53), 262(53) Cunningham, A . , 175 Curran, D. R., 220 Curry, J. E., 362(230), 363

D Dallquist, C. A., 27 Dannhauser, W., 6, 42(24), 50(24) Das Gupta, D. K., 494 Daubeny, R. deP., 167 Dauchot, J. P., 439 Davidse, P. D., 86 Davies, D. K . , 426, 427, 428, 429,430, 431, 432, 434(15), 435, 437(51), 438, 440(21), 441(21), 442 Davis, C. M., 94, 100 Davis, E. G., 372 Davis, L. A., 140, 203, 204(75), 206(75), 207(75) Dawson, P. G., 461, 481(60) Daynes, H. A., 319, 321, 358(21), 366(29), 372, 376 Dealy, J., 50 Dean, G. D., 3, 27(15), 38(15), 42(15) Debye, P., 388 De Candia, F., 295. 298 Dekker, A. J., 437 Dekking, P., 38

DeLoor, G. P., 420 Denbigh, K. G . , 167 De Nicola, J. P., 35 Densley, J., 462,482 Denson, C. D., 50 Dejaguin, B. V., 434 Dessauer, J. H., 424 DeVries, K. L., 186, 190(19), 191, 193, 195. 196, 201(48, 49). 202, 203, 204(45, 47). 207(67, 71, 80, 811, 208, 209(76a), 210, 211(4. 801, 212(23, 76a), 213. 214, 215(85, 871, 216, 218(4), 219(4), 220, 221(80, 81, 831, 226, 229 Dewar, J., 319, 320 De Witt, T. W., 41, 42(98), 43(98, 109), 49 DiBenedetto, A. T., 358, 363(202), 374 Diels, K., 340 Dietz, A. G. H., 87 Din, F., 353 Dlugosz, J . , 183(66), 297, 302(117, 118). 305(117) Dobry, A., 288 Doll, W. W., 91 Dolzhenkova, N. G., 229 Dondero, G . , 295, 298 Dosi’ere, hl., 141 Dos Santos, M. L., 327 Doty, P. M., 363 Douy, A., 2% Douy , A., 303 Dow, J., 437, 438 Downes, J. G., 350, 356 Drickamer, H. G., 94 Drinkwater, I. C., 292 Drislane, C. J., 35 Drozdovskii, V. F., 195 Druin, M., 145 Dubinkow, L. M., 322 Dubinskaya, A. M., 195 Duda, J. S., 326, 336, 347, 350 Duden, M. C., 277, 286(8) Dumbleton. J. H., 167 Dunlap, W. B., Jr., 366 Dunn, C. M. R., 26, 108 Dunn, D. J., 283, 312(35), 313(35) Durrill, P. R., 352, 354

E Eby, R. K., 89 Eckert, R. E., 195(51), 196, 203(51)

AUTHOR INDEX FOR PART C

Edwards, J. D., 318. 321, 322 Eichhorn. R. M . , 454, 488(32), 489 Eichinger, B. E., 278 Eilenberg, J. A . , 352 Einaga, Y . , 30 Eirich, F. R., 326 Eisenberg, H., 45 Elder, L. W., 363 Elliott, C. T., 448, 470(8), 478(6), 479 Ellis, D. A., 350 Elmqvist, C., 310 Elsdon, R., 428, 432(17), 433(17) (17) Elyash, L. J . , 49 Enscore, D. J . , 336, 337, 349, 350(140), 351(117) Erhardt, P. F., 290, 298, 2!99(132),314(132) Ester, G. M., 313 Eustache, H . , 366 Evans, D. E., 457 Evans, W. W., 33 Everage, E. A,, 50 Evnochides, S. K., 346, 352(134) Exner, 317 Eyraud, I., 471 Eyring, H., 220

F Fakirov, S., 150 Fallou, B., 463, 467, 468(77), 477(70), 478(70) Fang, S. M.. 326, 328 Farber, R., 50 Farmer, F. T., 490 Fatt, I., 360 Fava, R. A., 106, 308,452,474 Felix, M. P., 74 Felder, R. M., 325, 332(50), 344, 345, 362, 367(130), 368, 369( 130). 371, 372(225), 376(130, 132, 133), 377(133) Feldman, D., 310 Fellers, J. F.,236 Fellner-Feldberg. H., 416, 418(28), 420 Fels, M., 325, 333(49) Fenelon, P. J . , 348, 352(139) Feng, P. Y., 490 Ferguson, J . , 50 Ferrell. J. K., 325, 332(50), 345, 362, 368(133), 371, 372(225), 376(132, 133). 377( 133) Ferry, J. D., 3.6, 12, 13, 14, 133). 17,28(3),

503

33, 35(3), 37, 39, 40, 42, 50(24). 55, 83, 84(31), 88(31) Fetters, L. J., 56, 295, 2%(107), 297, 298, 309, 3 13( 186) Fielding-Russell, G. S., 298 Fikhman. V. D., 50 Fillers, R. W., 52 Finaz, G . , 303 Fischer, E. W., 150 Fischer, H., 192, 203,204(21), 206, 207(21), 229(21) Fischer, P., 452, 454, 471 Fischer, S., 235, 250, 251, 253(47), 255, 265(47) Fitzgerald, E. R., 37, 42 Flory, P. J . , 277, 278 Folk, G.M., 351 Folkes, M. J., 165, 297, 300, 301(138). 302( 1 18) Foord, T. R., 426, 428(12), 432, 433(33) Forster, E. O., 49 Foster, G. N., 94 Foumet, G., 176, 230 Fowler, J. F.,490 Fox, T. G., 45 Fraenkel, G. K., 457 Frank, F. C., 137, 145(3) Frank, N. H., 409 Frankenfeld, K., 42 Franz, W., 108 Franzblau, M. C., 121 Fraser, R. D. B., 149, 174(33) Freeguard, G. F., 292 Frei, E. H., 45 Frensdorff, H. K., 370 Frenzel, H. H., 321, 372 Friedland, K. J., 226 Friese, K., 288 Frisch, H. L., 326, 334, 335, 349, 359, 360, 36 1 Ffohlich. H., 381 Fujii, T.. 42 Fujihira, M., 435 Fujimoto, K., 289 Fujino, K., 33 Fujita, H., 33, 332, 483 Fukitani. K., 207(118), 211(118), 222, 223( 118), 224( 118) Fukuda. T., 488 Fuoss, R. M., 388, 391

504

AUTHOR INDEX FOR PART C

Gradowczyk, M. H., 18 Graessley, W. W., 15, 51(41) Gafurov, U. G., 227 Graham, P. H., 218, 22q108) Galin, J. C.. 304 Graham, T., 316. 317, 320, 364 Galland, J . , 463, 477(70), 478(70) Granato, A., 63 Gallot, B.. 2%, 303. 304(79) Graphispot Recorder, 30 Gallot, Y., 290, 301. 302, 303, 304(79) Gratch, S., 45 Garber, C. A., 145 Gravesteyn, H., 420 Gareis, P. J., 363(252), 364, 365(252) Grayson, M. A., 228 Garris, J., 328 Green, A. E., 117(4), 118 Garrett, T. A., 350 Greenshields, J. R., 481 Gasliins, F. H., 3, 50 Griac, J., 452,456(23), 464(23), 465(23), 474 Carton, C. G., 447, 452, 4690). 490(21), Griffith, A. A., 218 491(21), 492(21). 493(21) Griggs, D. T., 94 Gauster, W. F., 463, 463691, 470(69) Griskey, R. G., 94, 352, 354 Gee, G . , 283, 285 Gross, B., 437, 438 Geifman, G. l., 454 Gross, P. M., 400 Geil, P. H.. 141 Grossart, D. T.,470 Gekko, K., 351 Grosuis, P., 301, 302, 303(145) Gent, A. N., 45, 214 Groves, G. W., 184 Gent, W. L., 323 Grubb, D., 183 George, H. F., 307, 308(176) Grun, F., 323 Gerasimov. V. I., I84 Gruneisen, E., 112 Gerens, H., 352 Gruner, C. L., 256, 257(61), 258(62) Gesner, B. O., 288, 293(57) Gruver, J. T.,3 1 1 Gezalov, M. A., 230 Guenther, A. H., 66, 74, 75, 77(16, 22). Gezovich, D. M., 141 78(22) Giardini, A. A., 98 Guillett, J. E., 355 Gibson, R. E., 107 Guillod, M. S., 293 Gielessen, J., 75 Guinier, A,, 176, 230 Gilbert, S. G., 334 Gupta, K. G., 449, 453(8), 454(8), 470(8), Giles, A. F., 340 483(8), 484(8), 485(8) Gillam, E., 117(7). 118 Gurnee, E. F., 336 Gillham, J. K., 41 Gurney. I. D. C., 490 Gilliot, M., 141 Guth, E., 61 Gilman, J. J . , 264 Gilmore, GD.,94 H Gittens, G. J., 352 Haas, J . , 305, 306(171) Glandt, C. A., 41 Hagstrum, H . D., 439 Glasser, L., 421 Hale, P. T., 298 Glenz, W., 175 Hall, H. T.. 92, 94(14), 100 Gobelli, G. W., 435 Ham, J. S., 459, 460,490(54) Godfrey, T. B., 21 Goldbach, G., 52 Hammer, C. F., 310 Goldberg, R. A.. 31 1 Hammes, J. P., 456, 457, 458, 459 Golub, P. D., 76 Hanoosh, J . G., 250, 251(48), 252(48), 253, 255(48), 264(48), 265(48) Goodman, J . , 456, 457(46) Hanousek, J., 366 Gordon, D. J., 297 Hansen, D. R., 312 Goto, H., 488 G

AUTHOR INDEX FOR PART C Harnish, D. F., 328 Harper, B. G., 363(245). 364 Harper, R. C., 42, 43 Harper, W. R . , 434 Hartley, G. S., 323 Hartmann, B., 60, 72, 73, 74, 81, 84, 86, 87(36,43),88(15), 89(36), 98, 100, 107(32), 108(32), 111 Harvey, A. R., 322 Harvey, G. F., 340 Hasegawa, H., 304 Hashimoto, T., 281. 283(21), 288, 290, 295(21), 2%(21), 299(80), 300, 302, 303(61), 304 Hatakeyama, T., 304, 3 l4( 167) Hauser. P. M . , 350 Hawkes, S . J., 355 Hay, I . L., 141, 178, 181(18) Hayashi. K . , 138, 442 Hayes, M. J . , 347 Hays, D. A., 434 Hedges, J . J . , 322 Heffelfinger, C. J . , 142, 181(21) Heijboer, J., 38 Heikens, D., 294 Helfand, E., 281, 282, 297(25) Helffer, P., 303 Heller, J . , 370 Hellmuth, E., 363 Hellwege, K . H., 51, 94 Henderson, C. B., 218, 22q108) Henderson, R. M., 492 Hendricks, J. 0.. 27 Hendus, H., 295, 306 Hengstenberg. J . , 138 Henley, E. J . , 346, 352(134) Henniker, J . , 422, 423 Hentze, G . , 302 Henry, P. S. H., 432 Hercey, E. E., 21 Hermans, P. H., 148, 149, 323 Hershberger, A,, 363(248), 364 Hess, W . M . , 289 Heyes, W., 452, 465(20) Hickman, J . J., 310 Hieke, P., 369 Hien, N . V., 298, 302(127) Hill, R. M . , 441 Hillier, I. H., 437

505

Hinde, R. M., 488 Hirata, E., 290, 299(80) Hirsch, J., 442 Hirsch, P. B., 184 Hirshon, J., 457 Hoard, L. J., 346 Hoare, J., 263, 264 Hobayashi, K., 477 Hoehn, H. H., 372 Hoffman, W., 366 Hoffmann, M., 283, 294, 295, 2%(29, 101, 106h 298(101), 302(106), 305 Hofman-Bang, N., 33 Hogg, W. K . , 465, 481(74) Holden, G., 7 Holik, A. S., 243 Holland, W. D., 366 Holley, W. H., 288, 293(58) Holliday, L., 116 Holmes, D. R., 167 Holt, W. L., 27 Hoover, S. R., 350 Hopfenberg, H. B., 315, 316, 331, 332(2), 334, 335, 336, 337, 349, 350(140), 351, 359 Horio, M., 42 Horn, A. H., 350 Horowitz, H. H., 49 Hourston, D. J . , 3 1 I Houser, E. A., 87 Howell, J. M., 333, 334(104) Hsiao, C. C., 245, 246, 250,253(46), 265(46) Hsieh, J. H., 333, 334(104) HSU,N. N.-C., 311 Huang, R. Y. M., 328 Hubbell, W. H., 362 Hudson, N. E., 50 Hudson, R. W. A., 292 Huelck, V., 308 Hughes, E. J., 121, 133, 134(21) Hughes, K. A., 428 Hughes, L. J . , 308 Hull, D., 244,245( 19). 263,264,265,270( 19) Hulsebos, J., 366 Hunklinger, S., 464 Hunter, S. P., 394, 395(14) Husband, R. M., 366 Husson, F., 301 Huyett, M. J., 352, 354(185, 186) Hwang, S., 315(5,7),316,362(5), 363,373(7)

506

AUTHOR INDEX FOR PART C

I Iavorsky, P. M.,42, 43 Ibel, K., 306 Ieda, M., 441, 452, 453, 482(26), 490, 493 107) Iijima, K., 38 Iijima, T., 334 Ijitsu, T., 290, 299(80) Ikeda, R. M.,310, 311 Illers, K.-H., 295, 306 Illinger, J. L., 313 Imada, K., 137 Imai, Y.,129, 130(14),235,236(1), 240, 257, 259(65, 66),260, 265(72), 266(72) Imken, R. L., 311 Immergut, E. H.,325 Inculet, 1. I., 429 Ingram, D. J. E., 186 Iniushi, Y.,442, 455, 464(42). 477, 483, 490 Inokuchi, H., 435 Inoue, S., 483 Inoue, T., 281, 283(21), 288, 295(21), 2%, 300, 303(61) Israel, S. J., 244, 245, 270(20) Ito, Y.,332 h e y , D. G., 61

J Jablonski, W., 456, 457(48), 464 Jachym, B., 214, 216(92a, 92b) Jacknet, P.,366 Jacques, C. H.M.,331, 336, 351 Jaeckel, R., 340 Jagur-Grodzinski. J., 352 James, R. W., 152, 161(32), 179(37) Jamieson, R. T., 18 Jamroz, M.,214, 216 Jarzynski, J., 60, 72, 73, 74, 81, 84, 87(36), 88( 1% 89(36), 94, 100 Jenkins, F. A., 175 Jenkins, R. B., 351 Jobbins, R. M.,328 Jobling, A., 3 Jocteur, R., 467, 468(77) Johner, H.,138 Johnsen, U., 203, 204(76), 207(76), 212 Johnson, J. F., 47, 197 Johnson, K. L., 434

Johnson, R. N., 331 Johnston, W.G., 264 Jones, R. V., 29 Jones, R. W.,5 Jordan, I. B., 492,493 115) Jorgensen, L. A., 310 Jouan, G., 303 Joye, D. D., 50 Joyner, K., 494 Jung, R. H., 306 JYO,Y.,292, 293, 294(97), 310(97, 98)

K Kaess, G., 366 Kako, Y.,453, 471(30) Kakudo, M., 184 Kalkner, N., 476 Kalmanson, A. E., 195, 201(26) Kambour, R. P., 237, 238(9), 240, 241, 242, 243,252,253,255,256,257,258,264,265. 271(9), 272, 273(9), 297 Kammermeyer, K., 315(5, 7). 316, 362(5), 363. 364, 373(7) Kampf, G., 283, 294, 295, 2%(29, 101, 106), 298(101), 302(106), 305 Kanazashi, M.,481 Kane, E. O., 435, 436 Kanitz, P. J. F.,328 Kao, K. C., 452, 465(20) Kaplan, D. S., 275 Karasz, F. E., 275, 305(2), 308, 309 Kardos, J. L., 91, 100 Kasai, N., 184 Kasbekov, E. N., 195. 199(24, 25). 201(24, 25) Kashincheva, K. N., 226 Kashiwabara, H.,195(52), 1%, 201(52) Kastelic, J. R., 257 Kato, H.,483 Kato, K., 290, 292(75) Katz, C.,449, 462, 482(65), 488(75) Kausch, H. H.,186, 195(1), I%, 203, 206, 210(4), 211(4), 217(102, 1031, 218, 219(4), 223, 231( 124) Kawabata, S., 217(101), 218 Kawai, H.,33, 38, 281, 283(21), 288, 290, 296, 299(80), 300, 302(61), 304 Kawashima, T., 195(52), 1%

AUTHOR INDEX FOR PART C

Kaye, A., 49 Kayser, H., 318 Kearsley, E. A., 21 Kee, B. F., 236 Keedy, D. A , , 164, 167 Kelen, A., 485 Keller, A., 140, 141, 145, 156(22), 165, 181(18), 183, 297, 298, 299(131), 300, 301(138), 302, 304(131), 305 Kernp, D. R., 330, 333, 334(87, 105) Kendall, K., 434 Kennedy, G. C., 94 Kennedy, J. W., 490 Kenney, J. F., 310 Keskkula, H., 289. 292(74) Kesten, Y., 297 Keusch, P., 292 Keyes, R. W., 327 Keyte, D. N., 289 Kim, H., 301 Kim, K. Y., 3, 28(9), 45(9), 46(9), 47(9), 48P) Kirnrnel, R. M.,94 Kimmerly, G. K., 350 Kind, D., 460(59), 461, 481(59) King, A. O., 302. 313(148) King, G., 322, 350 King, W. H., 355 Kirkwood, J. G., 391 Kirkpatrick, M. L., 307, 308(176) Kirshenbaurn. A. D., 366 Kirske, R. G., 305, 306 Kitamura, S., 26 Kitanashi, Y., 138 Kitchin, D. W., 449, 482(10) Kleinert, T. N., 195, 201(28) Kleintjens, L. A., 277, 283(10) Kline, D. E.,38 Klinkenberg, D., 203. 204(76), 207(76), 212 Klopffer, W., 431 Klug, H. P., 175 Klute, C. H., 329 Khambatta, F. B., 290, 299(77), 300(77), 3 I l(77) Kherasov, L. N., 471 Knappe, W., 51, 94 Knauss, W. G., 18 Knight, A. C.,246, 247(26) Knox, E. O., 27 Kobayashi, 477.478

507

Kodama, K., 311 Kohnlein, W.,188 Kokes, R. J., 346 Kolbanev, I. V., 195, 197(31) Koleske, J. V., 311 Kolesov, S. N.,454,471 Kollinsky, F., 307(178) Komiyama, J., 334 Kondo, T., 246 Kondrup, M.,371, 372 Kong, J. M.,355 Kongarov, G. S., 289 Konig, D., 460(59), 461, 481(59) Koningsveld, R., 277, 283(10) Kono, M.,428 Kono, R., 61, 87 Kopp, R. W., 252, 264, 265 Koppelmann, J., 3, 27(16), 74 Kornfeld, M.I., 432 Koros, W.J., 324, 341, 348, 352(109, 127), 353(109), 361, 363(109) Korsukov, V. E.,220, 226, 227, 230 Kosaki, M.,452, 490, 495(107) Kosfeld, R., 106 Kosiyama, K., 55 Kosobukin, V. A., 226 Koster, W., 108 Koutsky, J. A., 298, 302(127) Kovacs, A. J., 40, 290, 298, 299(131), 304(131), 350 Koziowski, K., 214 Kramer, E.J., 244,247(32,35), 248(32), 253, 256(21), 271 Kramer, H.,366 Kratky, O., 176 Krause, S., 277, 278, 279, 280, 283, 286(8), 287, 288(52), 307, 308(177), 312(35), 3 13(35) Krenz, H.G., 244,246,247(32.35), 248(32), 253(32, 33), 256(21), 271(32) Krieger, I. M.,36 Krigbaum, W. R., 283, 294(28), 2%(28), 297(28) Kromer, H., 283, 294, 295, 2%(29, 101, 1061, 298(101), 302(106), 305 Kraus, G., 311, 312 Krupp, H.,431 Kruse, W.A., 305, 306 Kryszewski, M.,456,457(48,49), 464 Kuhn, R., 285

508

AUTHOR INDEX FOR PART C

Kuksenko, V. S., 220, 223. 230 Kuleznev, V. N., 31 1 Kumins, C. A,, 326 Kunz, W. B., 366 Kurata, M.,26, 27, 30 Kuroishi, M.,224 Kusy, R. P., 236 Kuvshinskii, E. V., 253, 265(55), 312 Kwei. T. K., 250, 283, 285(34), 286. 287(43, 45). 290(34), 291(34), 309(43), 31 1(45), 326, 349, 352 Kwolek, S., 137

L LaFlair, R. T., 281, 283(20), 294(20), 295(20), 297(20) Lally, T. P., 297, 298, 302(116), 303 118) Lamb, J., 77 Larnberson, L., 74, 75, 77(16, 22), 78(22) La Mori, P. N., 94 Landau, L. D., 117 Landel, R. F., 27, 33 Lando, J. B., 91 Landrock, A. H., 366 LaPelle, R. R., 340 Larkins, P. L., 372 Larsson, L.-E., 485 Lasoski, S. W., 330 Laven, J., 285 Lawrence, K., 272, 273 Lawrence, W. L., 490 Lawson, A. W., 327 Lawson, J., 449 Lawson, W. G., 452, 465(19), 474, 475(19), 491(85). 492(85). 493(85), 494(85) Lay, F. M.,435 Lazar, M., 191 Leaderman. H., 4, 5 , 11, 17(18),27 Leary, D. J., 283 Lebedinskaya, M. L., 212 Lee, G. F., 106 Lee, G. H., 119 Leffingwell, J., 283, 309 Legge, W. R., 7 LeGrand, D. G., 301, 453, 464, 475(29), 477(29) Lehmann, P., 94 LeitBo, D. M.,327 Lelie, H. J., 366

Less, K. J., 435, 436 Lethersich, W., 29 Levene, A., 86 Levin, B. Ya., 203 Levy, G. M.,253, 254, 255, 265(51) Levy, R. L., 228 Lewis, A. F., 41 Lewis, P. R., 294, 297 Lewis, T. J., 425, 439, 490, 492(103), 494( 103). 4% Li, N. N., 316, 325, 333(49) Lifshitz, E. M.,117 Lilaonitkul, A., 290, 299 Lilly, A. C., 490, 492, 494(104) Linde, Y.Y., 366 Linden, P., 18 Lindner, W. L., 150 Lindsey, S.. 297 Lines, G. O., 322 Lingle, R., 74 Linowitski, V., 366 Lippincott, E. R., 92, 93(15), 101(15), 104 Liska, E., 145 Lissner, H. R., 120 Litting, C. N . W., 438 Liu, C. H., 246 Livingston, D. I., 289, 308(70), 309(70) Llewellyn-Jones, F., 485 Lloyd, B. A., 207(80, 81), 210, 211(80),213, 220(80, 81). 221(80, 81, 83) Lloyd, W. O., 336 Lock, P. J., 442 Lodge, A. S., 47,49 Loeb, H.W., 418 Loh, S . C., 445 Long, F. A., 337, 346, 347, 349, 350(119, 136) Lord Rayleigh, 320 Loshaek, S., 45 Lotz, B., 290, 298, 299(131), 304(131) Love, A. E. H., 117 Loveland, R. J., 442 Lowell, J., 431, 432 Luk, W. H., 137 Lundberg, J. L., 352, 354 Lundberg, R. D., 311 Lundstrom. J. E., 327 Luy, H., 454 Lyons, J. W., 3, 28(9), 45(9), 46(9), 47(9), 48(9)

AUTHOR INDEX F O R PART C

Lyons, W. J., 217(100), 218 Lyssey, G. P., 369

509

Marci, A. I., 312 Marcircin, K., 310 Marikhin, V. A., 230 Mark, H.. 363 M Marker, L., 295, 300(110), 311(110) Ma, C.-C., 345,368(133), 371(133), 376(133), Markert, G., 307(178) Markovitz, H., 5, 14, 18(21), 40, 42(%), 377( 133) 43( 109), 47, 49 McBain, J. W., 322, 350 Marsh, P. A., 289 McCall, D. W., 350 Marshall, G. P., 253.254(53), 262, 271 McCandless, F. P., 330 Martin, E. H., 442 McCloren, A. D., 350 Martin, G. M.,92, 99 McCrum, N . G., 13, 38, 39, 89, 386,404 Marvin, R. S., 3, 21, 52, 75 McCubbin, M. L., 490 Mashimo, S., 393 McDowell, J. R., 490, 492(104), 494(104) Mason, J. H., 445, 449,450, 453(9), 481(2) McFee, R., 477 Mason, W. P., 80 McGarry, F. J., 87 Massa, D. J., 35 McGrath, J. E., 214, 309, 313(186), 331 Mast, W. C., 288, 293(59) McGrath, P. B., 481 Matsubara, H., 453 McGregor, R., 315(6), 316, 327 Matsui, M., 428 Machin, M. J., 145, 156(22) McIntyre, D., 56, 283, 284, 295, 2%(107), Matsuo, M., 250, 292, 293, 294(97), 295, 298(104), 310(97, 98) 297( 107) Matsuoka, S.,91, 92(5), 94, 95, 105, 109 McKay, B. H., 350, 356 Matsushige, K., 93, 249, 250(4l), 260, 261 McKean, A. L., 462, 46366).482(65) Matthes, A., 319 McKennel, R., 49 Matton, R. W., 313 McKeown, J. J., 455, 469(44), 475(44) Matzner, M., 309, 313(186), 331 Mackie, P., 50 Maxwell, B., 36, 91, 92(5), 94,95, 105, 109 McKinley, M. D., 362(230), 363 Mayburg, S., 490 McKinney, J. E., 3, 52, 75, 92, 99 Mayer, R., 2%, 298, 303( 112) Mackley, M. R., 145 Mead, W. T., 214, 216(91). 217 MacKnight, W. J., 275, 305(2), 308, 309 Meares, P., 360, 363 McLachlan, A. D., 186 McLachlan, R. J., 52 Mears, D. R., 91, 115 Mehta, R. E., 229 McLoughlin, J. R., 33, 55 Meier, D. J., 275,280,281(4, 18), 282,297(4) McMahon, E. I., 453 McMaster, L. P., 278, 283(14), 290, 291 Meissner, J., 50, 51(153) Melillo, L., 138 McMillan, J. A,, 188 Mellon, A. F., 350 McNulty, J. A., 372 Menefee, E., 33 McSkimin, H. J . , 61, 70, 78 Merrill, E. W.. 47 Madorsky, S . L., 350 Maeda, Y., 61, 86(5) Merz, E. H., 289 Maekawa. N., 483 Metzger, B. D., 243, 257, 265(71, 72). Mqjor, C. S . , 363(251), 364 266(72), 268, 271 Malkin, A. Ya., 51, 311 Meyer, J. A., 350, 362(228), 363 Mandlekern, L., 92, 99 Michaels, A. S., 327, 329, 330(59), 333(81), Mann. J., 116 352 Manning, R. E., 45 Middleman, S., 3, 48 Manson, J. A., 273 Mie, G., 138 Manuel, A. J., 175 Mile, B., 197 Mapleson, W. W., 425 Miles, D. O., 42

5 10

AUTHOR INDEX FOR PART C

Miller, D. B., 228 Mills, W. H., 26 Misrov, S., 1% Mita, S., 471, 473 Mitchell, F. R. G . , 428, 432(17), 433(17) Mitchell, J . K., 316, 320, 322 Miyamoto, T., 3 I 1 Miyashita, T.. 488 Mizutani, T., 492, 495(115) Moavenzadeh, F., 18 Mogens, J., 371, 372 Mohler, H., 369 Mohr, P. H., 363(252), 364, 365(252) Molau, G . E., 289, 292, 308 Moore, J. D., 292 Moore, N. B., 214, 215(87) Morbitzer, L., 302 Morgan, G. P., 271 Morgan, H. M., 168 Moritani, M., 300 Momson, T. E., 41,42(98), 43 Morrow, D. R., 115 Morton, J. R., 195, 201(28) Morton, M., 297 Morton, V. M., 454, 470(34) Mopsik, F. I., 85 Moseley, W. W., Jr., 169, 170 Mrowca, G. A., 61 Muenger, J. R., 47 Muinov, T. M., 227 Mullens, T. J., 289 Miiller, A., 188 Miiller, E. H., 300, 301(141), 313(141) Muller, F. A., 490 Muller, F. H., 363 Mullhaupt, J. J., 328 Munir, Z. A., 362 Miinstedt, H., 26, 27(54) Murasaki, K., 428 Muskhelishvili, N. I., 246 Mustacchi, H., 301 Myers, A. W., 329, 350 Myers, F. A., 249 Myrat, C., 374

N Nabb, S. V., 437 Nagai, S., 75, 86(23) Nagamatzu, K., 55

Nagamura, T., 192, 1%. 197(53), 203(22), 204(22), 206(22), 207(1 IS), 208, 209(76a), 210(76a), 212(76a), 211(118), 222, 223, 224(118) Nagata, S., 483 Nakajima, T., I I4 Nakamura, K., 137 Nakasima, H., 351 Nakatani, M., 38 Nagatoshi. K., 304 Nakayasu, H., 40,42(%), 483 Narisawa, I., 246 Natarajan, R . , 214 Nawata, M., 453, 482(26) Nederveen, C. J., 108 Negami, S., 48 Nelson, J. K., 481 Nemoto, N., 26, 27 Newitt, D. M., 323 Newns, A. C., 323, 350 Newton, R. G., 214 Nicholson, J . P., 283, 285 Nicholas, T., 42 Nielsen, L. E., 3, 28(12), 35(12), 38, 41(95), 84, 109, 245, 309 Nimis, A., 332 Ninomiya, K., 5, 55 Ninomiya, N., 33, 55 Nishi, M., 138 Nishi, T., 75, 86(23), 283, 285, 286, 287(43, 44),290(34), 291(34), 309(43). 311 Nissen, K. W., 452 Nitta, Y., 488 Niu, T.-F., 36 Noether, H. D., 178 Noland, J. S.,311 Noll, W., 47 Nolle, A. W., 75 Nordhage, F., 428,432,433(32), 434(32) Norldin, H. G., 490 Noto, F., 483 Novak, I. I. , 226. 227, 229(139), 230 Novak, R. C., 109 Nozaki, C., 292, 293, 294(97), 310(97, 98) 0

Oakes, W. G., 542, 468(17), 469 Oberst, H., 42 Ochiai, H., 351

AUTHOR INDEX FOR PART C Odani, H., 26, 27 Odintsova, R. R., 195 O'Dwyer, J. J., 444,448 Ogihara, S., 42 Ohno, M., 428 Oka, S.. 3, 28(11) Okamoto, H . , 481 Okamura, S., 138 Olabisi, O., 286, 287(46) Olf, H. G., 257 Olyphant, M . , Jr., 460,482(58), 484 O'Malley. J . J., 290, 298, 299(132), 314(132) Ong, P. H., 440 Ongchin, L., 249 Onogi, S.,42, 164 Oota, T., 483 Opoku, R. R., 445,481 O'Reilly, J. M., 35, 91, 94, 99 Orman, S., 460 O'Rourke, V. M . , 14, 15, 41(33), 50(33), 54(33), 55(39) Onvoll, R. A., 277, 278 Osaki, K . , 30 Os'kin, V. N . , 311 Ostler, M. I., 331 Oswald, F., 454 Ott, R. L., 195, 201(29) Oxborough, R. J., 249

P Padden, Jr., F. J., 49 Pae. K. D., 91, 115, 116, 131 Paik Sung, C. S., 313 Pake, G. E., 186 Palin, M. J . , 352 Palmer, D. G., 327, 358, 359(204) Palmer, R. P., 167 Pampillo, C. A., 203, 204(75), 206(75), 207(75) Pampus, G . . 295, 2%(106), 302(106) Papadakis, E. P., 87 Parish, M., 132 Parker, R., 329 Parkrnan, N., 452, 490(21), 491(21), 492(21), 493(21) Parks, G. S., 315, 326, 334, 347, 350, 351, 352(129), 356 Parrish, M. F.,'237, 253(7a), 257, 261(7a), 262(7a)

51 I

Parsons, C. A., 94 Parvar, M., 372 Pastemak, R. A., 370, 372 Pastine, D. J., 83, 85(33), 97, 98, 103 Patterson, D., 287 Patterson, G. D., 286, 287(45), 311(45) Paul, D. R., 311, 330, 331, 333, 334, 341, 352(109, 127), 353(109), 358, 360, 361, 363(109, 202), 374 Paul, F., 51 Payvar, P., 36 Pedemonte, E., 295, 297. 298, 302, 305(117) Pelekis, L. L., 366 Pelzer, H., 452 Penn, R. W., 21, 92,99 Pennings, A. J., 137, 145 Penwell, R. C., 48 Perepechko, I. I., 76, 77(25) Perkins, J. R., 453 Perlman, M. M., 440,441 Perret, J., 467, 468 Pemer, M., 463, 477(70), 478(70) Perry, C. C . , 120 Perry, E., 297 Peterlin. A., 175, 203, 204(66, 69, 70), 206, 207, 218(78), 220, 222, 223, 259, 329, 330(79), 349, 351, 352 Peterson, T. L., 246 Petler, P. J., 366 Petree, M. C., 83, 114 Petropoulos, J. H., 315. 332(2), 337, 349, 359, 361, 362, 374 Philippoff, W., 3, 28(13), 56 Phillips, D. L., 18 Pickering, S. F., 322 Piermarini, G., 101 Pierson, R. M., 288, 293(59) Piet, P., 294 Pilaf, J., 195(55), I%, 201(55) Pillai, P. s., 298 Pipkin, A. C., 49 Pistorius. C. W. F. T., 91,98 PlaCek, J., 195(54), I%, 201(54) Plazek, D. J., 5 6 , 14. 15. 18(21), 27(32), 29, 30(32. 6% 31(32), 40, 41(32, 33, 65). 42(24, %I, 45, 50(24, 33), 54(32, 33), 55(39) Poehlein, W.,50 Point, J. J., 141, 181(17) Pollack, H. O., 359

512

AUTHOR INDEX FOR PART C

Pollard, I. E., 425, 426 Pooley, C. M., 433 Poos, s., 355 Porod, G., 300 Porter, C. H.,380, 404, 405, 408, 415(22), 416, 420(22) Porter, G. B., 352 Porter, R. S., 47.48. 137, 140, 197, 202(62), 214, 217(92c), 276. 308 Powell, B. D. W., 276. 312(5) Powers, D. A., 455, 464(42), 490(42) Powers, J., 104, 167 Pozdnyakov, 0. F., 227, 230 Prager, S., 347, 350(136), 359 Pratt, 0. S., 449, 482(10) Prest, W. M.,35, 308 Prevorset, D., 217(100), 218 Price, C., 294, 297, 298, 302(116), 305(115, 116) Price, L. D., 289 Prigent, M.,471 Proctor,B. E., 366, 372 Ptaszynski, B., 303 Pugh, H. L. D., 116 Pulfrey. D. L.,470, 471 Pullen, W. J., 86 Pye, D. G., 372

Q Quach, A., 91, 99, lOO(7) Quickenden, P. A., 418 Quig, A., 362(232). 363 Quynn, R. G., 145

R Rabenhorst, H.,431 Rabinowitsch, B., 48 Rabinowitz, S., 129, 237, 252 Radcliffe, S. V., 93. 94, 249, 250(41), 260, 26 l(73) Radtsig, V. A., 195, 197(31), 199(38) Radushkevich, B. V., 50 Raghupathi, N., 5 , 18(21), 57 Rahalkar, R. R., 394, 395(14) Ramo, S., 410 Rgnby, B., 309, 310 Ranson, R. J., 286 Randle, K. J., 440 Rao, A. K., 362(229), 363

Rayner, M. G., 305, 313(170) Razgon, D. R., 195 Read, B. E., 3, 13, 27(15), 38(15), 39(31), 42(15), 89, 386, 395, 404 Reddish, W., 493 Reed, P. E., 186, 214, 216, 217(92c) Regel, V. R., 219, 227 Rehage, G., 52 Reid, D. R., I5 Reid, W.T., 37 Reihl, N., 464 Reynolds, E. H., 488 Reynolds, J., 226 Rhi-Sausi, J., 50 Rhodes, M. B., 299 Riande, E.,5 , 18(21) Richman, D., 337, 349 Riddlestone, H.G., 452, 468(22), 469 Riess, G., 288 Rigby, S. J., 462 Rivlin, R. S., 218 Roark, R. G., 117(5), 118 Roberts, A. D., 434 Roberts, J., 86 Roberts, J. E.,3 Roberts, M. W., 435 Roberts, R. F.,286, 287(43), 309(43) Roberts, S.,410 Robeson, L. M.,292,293,309,313(186), 331 Robinson, C., 323 Robinson, C. N., 218, 220(108) Robinson, D. W., 38 Robinson, R. A., 312 Robles, E. G., 445, 481(2) Rocha, A. A., 324, 348, 352(109), 353(109), 361(109), 363(109) Rodes, C. E., 368, 371 Rogers, C. E., 325, 329, 331, 333, 350, 362(228), 363 Rogers, W. A., 359, 370 Rollmann, K. W., 312 Romagosa, E. E., 256, 257(61), 258(62) Roman, N., 307, 308( 177) Romano, G., 298 Romanov, A., 310 Rongone, R. L., 289, 308(70), 309(70) Rooney, M. L., 372 Roovers, J. E. L., 285, 307(36). 312(36) Ropte, E., 295 Rosen, B., 354, 355, 363

.

AUTHOR lNDEX FOR PART C Rosen, S . L., 288 Rosengren, K. J., 366, 369(263), 371, 372(263) Rosenhain, 321 Rossi, J., 296, 303(112) Roth, L., 366 Roth, R. L., 27 Rounds, N., 283, 284(31) Rouse, P. E., 350, 363 Roussis (118), 349, 359, 361, 362 Rowson, C. H., 454 Roylance, D. K., 186, 195(47, 48, 49), 196, 201(48,49), 203,204(47), 207(67), 213(47), 214(48, 49), 220(67), 226, 229 Russell, R. R., 243 Russell, T., 290, 299(77), 300(77), 311(77) Rusu, M., 310 Rutherford. J. L., 121, 133, 134(21) S

Saam, J. C., 297 Sacher, E., 490, 494(106) Sadron, C., 303 Sagae, S., 295, 298(104) Saito, S . , 114 Saito, Y., 455 Sakaguchi, M., 186, 191, 195(5), 197(5), 199(20),201(20), 202 Sakurada, I., 351 Salama, M. M., 250, 251(48), 252(48), 253(48), 255(48), 264(48), 265(48) Salloum, R. J., 195(51), 196, 203(51) Salvage, B., 461, 481(61, 62) Saminskii, E. M., 195, 199(24, 25), 201(24, 25) Samuels, R. S . , 165, 167, 171 Sarnuels, S . L., 290, 299(83) Samulon, H.A., 418 Sapieha, S., 456, 457(48, 49), 464 Sapse, A. M., 281, 282 Sardar, D., 94 Sarge, T. W.,363(246), 364 Sasabe, H.,1 I4 Sauber. W. J., 363 Sauer, J. A., 91, 115, 116, 131, 250, 253(46), 255, 265(46) Savitskii, A. V.,203 Savostin, A. Ya., 203, 204(63, 73), 207(73) Sawa, G., 441

513

Saxon, R.,311 Sayre, J. A., 392 Schatzki, T. F., 14 Scheiber, D. J., 401 Schelten, J., 305, 313(170) Schimscheimer, J. F., 370 Schmidt, P. G., 142, 181(21) Schmitt, B. J., 305, 306(171), 311 Schneider, N. S., 302, 313 Schober, D. L., 331 Schoffeleers, H. M.,277, 283(10) Schremp, F. W., 33 Schuster, H.,294 Schuyer, J., 83, 86(34) Schultz, G. V., 352 Schwarzl, F.,3, 38(1), 42 Scott, C. E., 289 Scott, L. S., 140 Scott, R. L., 276 Seaman, L., 220 Seanor, D. A,, 113 Sears, G. R., 366 Secker, P. E., 425, 426,428 Seibel, D. R., 330 Seiki, T., 197 Seki, H.,63 Selb, J.. 303 Semenchenko, V. K., 89 Semjonow, V., 51, 52 Sen, S . K., 362(229), 363 Senecal. G., 459,460,490(54) Senshu, K., 33 Seow, P. K.,290, 304(79) Sessler, G. M.,435, 439, 440 Seymour, R. W., 313 Shakespear, G. A., 318, 321, 366(29), 372 Shannon, C., 418 Sharp, P. S., 492 Sheer, A., 332 Shen, M.,18, 312 Shepherd, L., 288, 293(59) Shibayama, K., 311 Shimada, S., I%, 201(52) Shimura, Y.,304, 314(167) Shinohara, U.,441 Shockey, D. A., 220 Shpak, G. V., I08 Shultz. A. R., 308, 309(183a) Shuman, A. C.,363(247), 364 Shur, Y. J., 309, 310

5 14

AUTHOR INDEX FOR PART C

Sibilia, J. P., 203, 204(74), 206(74), 207(74) Sidorovich, Ye. A., 312 Sieck, P. W., 75 Sieglaff, C. L., 11 Sieminski, M. A., 165 Sieniakowski, M.. 214, 216(92a) Sigelko, W. J., 350 Signer, R., 138 Silverman, S., 168 Simankov, V. V., 108 Simha, R., 91, 99, 1W7) Simmons, J. M., 50 Simonson, E. R., 214, 215(85) Sinclair, T. F., 363(252), 364, 365(252) Singh, H., 75 Singleton, J. H., 363 Singleton, R., 298 Skirrow, G., 327, 362 Skoulios, A,, 290, 293, 302, 303, 304 Sladek, K., 333, 334(103), 348, 352(103) Slagowski, E., 56 Slater, J., 333 Stater, I. C . , 86, 409 Slattery, J. C., 49 Slichter, C. P., 186 Slichter, W. P., 350 Slocombe, R. J., 288 Slonimskii, G. L., 285 Slupkowski, T., 214, 216(92b) Slutsker, A. I., 219, 223, 230 Smith, A. L., 277, 286(8) Smith, R. E., 61 Smith, R. G., 5 Smith, R. R., 273 Smith, T. L., 12, 27 Smyth, C. P., 381 Sneddon, I. N . , 246 Soar, S., 460 Soen, T., 281, 283(21), 288, 290, 295(21), 2%. 299(80), 303(61) Sofer, G. A., 87 Sohma, J., 186, 191, 195(5, 52), 196, 197(5), 199(20), 201(20, 521, 202 Sollner, K., 316 Sonnonstine, T. J., 441 Sorokin, V. E., 76, 77(25) Sourirajan, S., 316 Soussou, J. E., 18 Southern, J. H., 137, 140(5) Spannring, W., 464

Spegt, P., 304 Spence, R. D., 325. 332(50), 345, 376(132) Spencer, H. G., 362(232), 363 Spencer, R. S., 94 Sperling, L. H . , 273, 307, 308 Spingler, E., 352 Spivey, J. J., 362, 372(225) Sprague, B. S., 145 Stamm, A. J . , 350 Stani, A., 75, 86(23) Stanley, H. E., 285 Stannett, A. W., 454, 470(34) Stannett, V., 315,325, 326(47), 327(47), 329, 330(47), 331, 332, 333, 336, 337, 349, 350, 351, 352(95), 356(95), 359, 362(228), 363, 374(95), 375 Stark, K. H., 447, 469(5) Statton, W. O., 226 Staudinger, H., 138 Staverman, A. J., 3, 38(1) Stedry, P. J., 33 Steger, T. R., 245 Stein, D. J., 305, 306 Stein, R. S . , 163, 164, 165, 167, 290, 299, 300(77, 136). 31 l(77) Stern, S . A., 326, 328, 362(229, 231). 363, 364, 365(252) Sternberg, S . , 331 Sternstein, S. S., 226, 249 Stiel, L. I . , 328 Stratton, R. A., 40, 49, 50(132), 51(132) Stefan, 317 Stiskin, H., 288, 293(58) Stockmayer, W. H., 285, 393, 394( 1 I), 395, 418 Stoeckel, T. M., 229 Stoelking, J.. 275, 305(2), 308, 309(2, 184) Stone, F. T., 477 St.-Onge, H., 490, 492(105), 495(105) St. Pierre, J. A., 441 Streng, A. G., 366 Stuart, M., 456 Stubbersfield, R., 298 Suganuma, A., 38 Suggett, A,, 418 Sugiyama, K . , 452, 490(24) Surkova, N. S., 312 Sutherland, H. J., 74 Sutterby, J. L., 46 Suzuki, S., 304, 448

AUTHOR INDEX FOR PART C Svanson, S. E., 310 Swan, D. W., 464.4% Sweeting, 0. J., 325. 329, 330(46), 352(48), 362(46), 364, 365 Swindells, J. F., 21 Szwarc, M., 362(228), 363 Szware, M., 329, 333 Szocs, F., 191, 195, 196, 199(46), 201(54)

T

515

Tolbert. W. R.. 283, 294(28), 2%(28), 297(28) Tolstopyatov, G. M., 312 Tomashevskii, E. E., 189, 195, I%, 199(16, 24, 31), 201(24, 36, 39, 41, 44), 202(36), 203,204(65, 73). 205(39), 207(65,73), 21b, 230 Tomczyk, J., 456, 457(49) Tompa, H., 277 Toms, B. A., 3 Toporowski, P. M., 285, 307(36), 312(36) Toren, P. E., 372 Toriyama, Y.. 481 Toureille, A., 454 Traylor, P. A., 289 Treece, L. C., 330 Treloar, L. R. G., 45 Truell, R., 63 Tschoegl, N. W., 42, 52, 312 Tshudy, J. A., 361 Tsouladze, G., 303 Tsuji, K., 197 Tsutsumi, Y., 453,471(30) Tun, E. A., 138 Turner, D. T., 236 Turner, S., 3, 26, 108 Tutorskii, I. A., 311 Tuxworth, R. W., 421 Twiss, D. E., 352 Tydings, J. E., 98

Tabor, D.. 431, 432, 433, 434 Tagami, Y., 282 Takahashi, A., 290 Takayanagi, M., 35, 38, 137, 192, 195(53), 196, 197(53), 203(22), 204(22), 206(22), 207(118), 211(118), 222, 223(118), 224(118), 309 Takebe, H., 477,478 Tamura, M., 26, 30 Tanaka, T., 490 Tan, V., 15, 5339) Tanner, R. I., 18, 35, 36, 49, 50 Tanza. V. C., 350 Tarn, P. M., 352 Taure, I: Y., 366 Taylor, D. M., 425, 439, 490, 492(103), 494( 103). 4% Taylor, D. W., 307, 308(176) Taylor, N. W., 27 Tenisse, J., 303 Thierry, A., 304 U Thies, C., 283 Uchida, T., 2% Thirion, P., 45 Uhlmann, D. R., 94 Thomas, A,, 197 Ulbert, K., 195, I%, 201(27, 55) Thomas, A. G., 218 Unger, S., 440, 441 Thomas, A. M., 323 United Detector Technology, 30 Thomas, D. A., 3, 308 Urbanski, T., 195, 201(40) Thomas, E. L., 244, 245, 270(20) Urick, R. J., 171 Thomas, J. M., 437 Timoskenko, S.. 117 Urquart, A. R., 350,366(160) U. S. National Bureau of Standards, 353 Tido, J., 195, I%, 199(46), 201(54) Utracki. L., 50 Tkachenko, G. T., 312 Tkacik, J. J., 309 Tobolsky, A. V., 7, 33, 55,85,220,313,403 v Todd, H. R . , 363(249), 364 Vahlstrom, W., 449 Todo, A., 304 Vail, C. R., 463, 465(69), 470(69) Toh, H. K., 41 Toi, K., 354 Vale, D. G., 49 Tolanski, S., 459 Van Aartsen, J. J., 291

5 16

A U T H O R INDEX F O R PART C

Van Amerongen, G . J., 327. 330, 362(227), 363 Van Cakenberghe, J., 439 Vance, D. W., 439 Van de Weerd, J. M.,425 Van den Esker, M.W. J., 285 Van der Boogaart, A., 235 Van Duzer, T., 410 Van Gemert, M. J. C., 416,418(27), 420 Van Holde, K. E., 5 Van Krevelen, D. W., 287 Van Oort,W. P., 42 Van Turnhout, J., 425, 439, 440 Van Valkenburg, A., 92,93(15), 101(15), 104 Van Wazer, J. R., 3, 28(9), 45, 46, 47(9), 48(9) Vanzo, E., 2% Varlow, B., 445, 448(4), 454, 455, 465(4), 466(39,43), 467(4), 468(4), 469(4), 471(40), 472(4, 39, 40), 473(4), 483(4), 490(4), 491(4), 492, 493(4), 494(4, 40, 116), 495 Vaubel, G.,464 Verhaegen, J. P., 439 Verheulpen-Heymans, N., 246, 247(30), 255 Veliev, S. I., 227, 229(139) Verma, G. S. P., 203, 204(69, 70). 206, 222 Vettegren, V. I., 226, 227, 229(139) Vieth, W., 329, 333, 334, 347, 348, 352(103) Villforth, Jr., F. J., 47 Vinogradov, G. V., 50, 51 Voet, A., 289 Vofsi, D., 352 Von Frankenburg. C., 361 Von Hippel, A. R.,408,410,416 Vrancken, N. M.,29, 30(65), 40(65), 41(65) Vrentas, J. L., 326, 336, 347, 350 Vrij, A.,277, 285

W Wada, Y.,75, 86(23), 87, 89 Wagner, E. R.,292, 293 Wahlin, A., 432, 433 Wahlstrom, E. E., 163 Walker, J. S., 298 Wallach, M. L., 311 Walley, C. A., 465, 481(74) Walling, C., 457

Walters, K.. 3, 35, 36(6), 37(6), 47 Walters, M. H., 289 Wang, C. H.,394, 39314) Wang, S. S., 334 Wang, T. T., 250, 283, 28334). 286, 287(44, 45), 290(34), 291(34), 311(44,45), 349,352 Ward, I. M.,3, 35(8), 130, 137, 140(4), 150, 157(34), 169, 170, 175, 240, 241, 270, 271 Warfield, R. W., 83, 92, 98, 100, 103. 104, 105, 106, 107(32), 108, 109(35), 111, 113(47), 114 Wargin, R. V., 48 Wareham, W. M.,35 Warner, A. J., 490 Warner, F., 290, 299(77), 300(77), 311(77) Waterman, H.A., 61. 171 Waterman, H. I., 86 Waters, R. T., 425 Watson, A., 438 Watson, A. G., 297, 298(115), 302(116), 305(1 15, 116) Watson, D. B., 452, 453, 454, 465(20), 467, 468(27) Watson, W. F., 197 Weale, K. E., 323 Weast, R. C., 72, 73(14) Weedy, B. M.,462 Weeks, N. E., 140 Weiner, J., 366 Weir, C. E., 91, 92, 93(15), 94,98. lOO(15) Weisler, A., 66 Wellinghoff, S. T., 244, 245 West, C. J., 366 West, J., 435, 439, 440 West, J. C., 290, 299 Westerdijk, J. B., 86 Whinnery, J. R.,410 White, E. F. T., 312 White, H. E., 175 White, H. S., 21 White, J. P., 454, 455, 466(39, 43). 471(40), 472(39, 40). 494(40), 495(40) White, R. A., 288, 293(58) Whitehead, S., 476 Whitlock, W., 425 Wiersma, A. K., 195, 201(30) Wilczynski, A. P., 246 Wilde, A. F., 288, 293(58) Wilde, T. B., 214, 216(92) Wilk, M. B., 352, 354(185, 186)

AUTHOR INDEX FOR PART C

517

Wilkes. G. L., 290, 299 Yamamoto, K.,87 Wilkinson, S.,366 Yamamura, H., 351 Wilkinson, W. L., 47 Yamanouchi, S., 453 Williams, A. M.,350, 366(160) Yamashita, Y., 290 Williams, C. W., 13, 38, 39(31), 50, 89, 386, Yano, S., 394,395 394, 395, 401, 402, 404, 421 Yanovskii, Yu. G., 311 Williams, D. J., 292 Yarlykov, B. V., 51 Williams, J., 77 Yasuda, H.,325, 329, 332, 352(95), 356(95), Williams, J. G., 117(6), 118, 253, 254(53), 366, 369(263), 371, 372(263), 374(95), 375 262(53), 271 Yazgan, S., 283, 294(28), 2%(28), 297(28) Williams, J. L., 351 Yoda, M.,490,495(107) Williams, J. W., 5 Yokobori, T.,217 Williams, M.C., 283 Yoon, H.N., 131 Williams, M. L., 193, 195, I%, 201(48, 49), Yoshida, T.,483 203, 204(45, 47), 207(67, 80, 81), 210, Yoshimura, N.,289, 483 211(80), 212(23), 213, 214, 215(85, 87), Yoshino, K., 442 216(88, 89, 92), 218, 220(67, 80, 81), Yoshitomi, T., 55 221(80, 81, 83), 229 Yoshizaki, H., 87 Williams, R. J., 292 Young, G. M.,418 Willis, P. B., 288, 293(58) Young, W. C., 117(5), 118 Willmouth, F. M.,302 Yu, A. J., 275, 276(1) Wilson, E. G., 435,436 Wilson, J. A., 322 Z Windle, J. J., 195, 201(30) Zak, J., 366 Winkelnkemper, H., 476 Zaks, Yu. B., 212 Winkle, H. H., 427 Zakrevskii, V. A., 189, 195, 1%, 197(41), Wituschek, E. P., 429 199(16. 39, 41, 44),201(39, 41). 202(36), Woy, C. J., 228 205(39), 220, 230 Wong, C.-P., 30, 50, 51(69) Zakrzewski, G. A., 309 Wong, P. S. L., 328 Zapas, L. J., 41,42(98, 109) Wood, L. A., 107, 306 Zaslavskii, N. N., 229 Wood, M. H., 437 J., 309 Zelinger, Woods, D., 297, 298, 302( 116), 305(116) L., 287 Zeman, Wool, R. P., 145, 226 W., 117(4), 118 Zerna, Worsfold, D. J., 312, 313(217) Zhulin, V. M.,104 Wossnessensky, S., 322 Zhurkov, S. N., 189, 195, 199(16, 24), Wroblewski, S. von, 318, 320 201(24), 203(63,64,65), 207(65), 219, 220, Wu, J. B. C., 269 223, 226, 227, 230 Wunderlich, B., 54, 91, 94, 115, 138 Ziegel, K. D., 326, 370 Ziegler, E. E., 251 Zimm, B. H.,45 Y Zitek, P., 309 Yada, K., 114 Zoledziowski, S.,460 Yager, W. A., 388 Zosel, A., 52, 61, 75(3) Yahagi, K., 471,473, 492 Zupko, H. M.,349 Yamada, S., 331 Zwanzig, R. W.,350 Yamakawa, H., 351 Zwijnenburg, A., 137. 145

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SUBJECT INDEX FOR PART C A Absorption of ultrasonic waves definition, 70 in various polymers, 73 Acoustical densitometer, 100 Activation energy, for diffusion, 324 Adhesion, role of electrostatic charge, 433 Adhesives microstrain tensile test for, 128 shear testing of, 128 testing in joints, 126- 128 effect of bond-line thickness, 133 Alloys, polymeric, phase morphology, 287299 Andrade creep, 6, 15 Arrhenius law, of sorption and diffusion, 324 Arrhenius plot, of ultrasonic data, 89 ASTM test methods compresion test, 124 electric breakdown, 488 flexural test, 125 lap-shear adhesive test, 127 permeability, 363, 365 cup method, 366, 371 tensile test. 123

B Barometric technique for permeability, 362 for sorption, 352-354 Biaxially stretched films, 141- 142 Binodal, 277 Birefringence, 149, 161- 167 , form, 165 intrinsic, of polymer chain, 162 measurement of, 163- 165 positive and negative, 162 Blending methods, 276 Block copolymers crystalline, 290, 298 morphology, 294-299 thermodynamic theory, 278-283 519

Blocking factor, 329 Boltzmann superposition creep equation, 9 definition, 7 in dielectric theory, 385 stress-relaxation equation, 10 B r a e equation, 151, 178, 301 Breakdown, see Electric breakdown Bridge methods, in dielectric measurements, 397-400 Bulk modulus definition, 97 measurement of. 112

C Cable samples, for electric breakdown, 461 Capacitance, cylindrical geometry, 384 definition, 383 measured by bridge method, 398 measurement in resonant circuit, 406 Capillary viscometer, 45. 47 Casting of polymer films, from solution, 57, 457 Cellulose, water absorption, 319 Charge, see Electrostatic charge Chemical potential, 286 Chemorheology, 7 Cohesive energy density, 286 Cold-drawing, 138- I40 Cole-Cole plot, 390-392 Compatibility, between polymers, 275 Compressive strength, 125 Cone-and-plate geometry, in shear viscosity measurements, 49 Constant rate of strain, 12 radical generation during, 207-2 10 Contact area, real, 431 Contact electrification, 428-435 Contact potential, relation to charge density, 430 Corona discharge, 482 Couette geometry, in shear measurements, 27, 49, 52

520

SUBJECT INDEX FOR PART C

Crack, definition, 234 Craze definition, 233-235 growth, 252-255 internal structure, 243-245 optical methods of study, 237-242 porosity, 240 retracted thickness, 243 thickness, 235, 240-245 velocity measurement, 253-254 Craze front, in solvent penetration, 323 Crazing creep equation, 265 criteria for, 249 effect of pressure on, 116 environmental, 255 -262 molecular weight dependence, 236 Craze-opening displacement, 243, 248 Creep, in crazing polymers, 265-268 Creep compliance, 4 Creep rupture, molecular model, 219-220 Creep velocity, terminal, 5 Critical strain, for environmental crazing, 256-258 Crystallinity effect on electrical conduction, 495 on gas transport properties, 328-329

D Davidson-Cole function, 392 Deborah number, diffusional, 336 Debye equations, 388 Debye-Scherrer rings, 301 Density effect on electric strength, 473 various polymers, 72 Density gradient column, 473 Diamond anvil cell, 101 Dichroic ratio, 173- 174 Dielectric breakdown, see Electric breakdown Dielectric constant complex, 385 relaxation theory, 386-393 definition, 383 derivation from transmission line equations, 414 Dielectric loss factor conductance contribution, 404 definition. 385

derivation from transmission line equation, 414 measurement in resonant circuit, 407 temperature dependence, 394 Dielectric measurements distributed circuits, 408-421 lumped circuits, 395-408 Dielectric test cell, 400 Diffusion calculated from permeability time lag method, 357-361 concentration dependent, Fickian, 332, 339 dual mode, 333 measurement by chromatographic method, 355 non-Fickian, 334-337, 348 of gases, theory, 325 Diffusion coeficient definition, 318 of gases in polymers, empirical relations, 327 Dilatometer, in high-pressure measurements, 99-100 Discharges, see Electric breakdown Dissipation factor definition, 386 measurement by bridge method, 398 Domain dimensions in block copolymers, 280 morphology, 280-281, 295-2% Dow cell, for permeability, 363-364 Draw ratio, 140 Dual-mode sorption and diffusion in glassy polymers, 333 Dynamic mechanical properties definition, 10- 1 1 frequency limitations in measurement, 42 instrumentation, 35-44 isochronal measurement, 38 of NBS nonlinear test fluid No. 1, 21

E Elastic constants by high-pressure measurements, 108- 112 by ultrasonic measurements, 79-84 of various polymers, 81 Elastomers (rubbers) fluorosilicone, permeability, 368

52 1

SUBJECT lNDEX FOR PART C fracture studies by ESR, 213-217 low temperature ductility, 216 permeation of gases, historic, 316-320 Elongational flow, 143 Elongational viscosity definition, 12 measurement of, 25 Electret, 439, 468 Electrical conduction in polymers, highfield measurements, 489-497 Electrical resistivity, pressure dependence, 1 I3 Electrical breakdown by discharges artificial void specimens, 460 experimental, 481 theory, 450-451 cryogenic methods, 476-478 electrode application, 462-464 industrial tests, 488 in nonuniform fields, 448 in single crystals, 459 intrinsic, 444-445 prestressing effect, 466-469 space charge effects, 467-469 spark development, phototechniques, 478-481 specimen preparation, 45 1-462 thermal experimental, 475 impulse, 447 steady state, 446 theory, 446-447 time lag measurements, 470-471 voltage modes, 464-470 Electric breakdown field, in air at STP, 429 Electric displacement field, 381 Electric field at an interface, 433 definition, 381 time dependent, 384 Electric strength, see Electric breakdown Electrodes, application, 453, 462-464 Electron bombardment, 437 Electron microscopy of polymeric alloys. 291-299 to study crazes, 242-245 Electron spin resonance (ESR) spectrosCOPY apparatus, 188-191

limitations, applied to fracture studies, 225 principles, 186- 187 sensitivity threshold, 189 Electrolytic electrodes, 464,4% Electromechanical breakdown experimental, 473-475 suppression of, 475 theory, 447 Electrometer probe, for electrostatic charge measurement, 426 Electrooptical shutter, 478 Electrostatic charge decay, 440-442 density, 424 generated with ionizing radiation, 435 methods of measurement, 424-428 migration, 439 surface, maximum in air, 429 End-group analysis, 226 Energy, to form a crack and a craze, 271 Environmental crazing critical strain rate, 260 effect of hydrostatic pressure, 260 in gases, 257-259 in liquids, 256 Epoxy resin, see Polyepoxide Equation of state, of a polymer, 84-86 ESR, see Electron spin resonance spectrosCOPY Ewald construction, 153 Ewald sphere of reflection, 154 for oriented fiber, 156 Extensometer, capacitance-type, 121, 128

F Faraday cup method, for electrostatic charge measurement, 424 Fibers hard-elastic, 145 natural, 138 Fiber stress, maximum in flexural test, 125 Fibrils, in a craze, 233 Fickian diffusion, concentration-dependent, 332, 346 Fick's law of diffusion, 317-318 Fiducial marks, in strain measurement, 119 Field, see Electric field Field mill, for electrostatic charge measurement, 425

522

SUBJECT INDEX FOR PART C

Film, ultrathin preparation, 456-459 thickness measurement, 458 Film specimen, for electric breakdown, 454-456

Flashover, 445 Folded chains, 219 Four-point X-ray pattern, 180 Fracture as related to crazing, 270 molecular mechanisms, 217-225 Free energy of mixing, 276 segmental, 278 Free volume, theory of gas diffusion, 325 Fuoss-Kirkwood function, 391

I Impedence, complex, 386 Impedence transformation equation, for a transmission line, 412 Impulse voltages, in electric breakdown, 465

generators, 466, 467 Induced charges, 382 Infrared dichroism, 173- 175 Infrared spectroscopy, in polymer deformation studies, 226-227 Interaction parameter, 277 determination, 283-287 Interferometer for measuring craze thickness profile, 240-242

G Glass transition and anomalous transport of vapors, 334337

and ultrasonic measurements, 88 by compressibility, 106 of block copolymers, 311-314 of polymeric alloys, 306-3 11 viscosity at, 6 Glassy compliance, 4 Glow discharge polymerization, 456-457 Gravimetric techniques for permeability, 366 for sorption, 350-352 Griineisen parameter, 85, 112 Guard ring, 399,400,491

H Hencky strain, 13 Henry's law of sorption, 318 High-pressure measurements belt apparatus, I 0 0 diamond anvil cell, 101-104 calibration, 103 dilatometer, 99 piston-cylinder device, 94-99 calibration, 98 types of equipment, 92-104 High-speed photography, 478 Holographic interferometer, 246 Hysteresis loop, in stress-strain cycling, 118

holographic, for craze displacement field, 246

Interphase. 282 Intrinsic electric strength, 444 specimen preparation, 452 Ion bombardment, 439

K Kerr cell, 478

L Langmuir isotherm, 319 Leaderman technique, for shear viscosity, 5 Linde cell, for permeability, 365 Linear variable differential transformer, in strain measurement, 121 Logarithmic decrement, 41 Longitudinal wave absorption of, 70 definition, 59 Lorentz equation, 162 Loss factor, dielectric, see Dielectric loss factor Loss tangent, 10 dielectric, see Dissipation factor

M Mass spectroscopy, in polymer fracture studies, 227-228 McBain balance, in sorption measurements, 350

McLeod gauge, 340

523

SUBJECT INDEX FOR PART C

Mechanical dispersions, origin of, 13- 16 Mechanical properties, effect of pressure on, 114-116 Melt fracture, 48 Melting point, effect of pressure on, 107, I15 Miller indices, 152 Mixtures of polymers. 276-278 Mobility, of carriers in polymers, 441 determination by ”time of flight” methods, 442 Modulus, tangent and secant, 118 Molding, vacuum, 52-56 in electric breakdown specimen preparation, 452 Molecular entanglement, effect on creep, I5 Molecular network, creep of, 6 Molecular weight, changes during mechanical fracture, 229 Molecular weight distribution, effect on creep compliance, 15 Morphology effect on electrical conduction, 495 effect on electric breakdown, 471-473 effect on gas transport properties, 32833 1 of a craze, 245 of block copolymers, 294-299

N NBS nonlinear test fluid No. 1, creep properties of, 18-24 Neck region, deformation in, 139 Neutron coherent scattering length, 305 Newtonian liquid, 2 Non-Newtonian shear flow, 47 Normal stresses, 49 Nuclear magnetic resonance spectroscopy, for sorption studies, 356 Nucleation, of crazes, 251 Nylon 6, see also Polycaprolactam radicals generated during fracture, 197, 200, 202, 204, 205, 207 stress relaxation, effect of moisture, 55 Nylon 6, 6 contact electrification in, 430 radicals generated during fracture, 204, 207 work function, 431 yield stress and Young’s modulus, 131

Nylon 6, 10 effect of crystallinity on gas diffusion, 329 dielectric constant and loss, 380 yield stress and Young’s modulus, 131 Nylon 12, radicals generated during fracture, 204, 207 0

Optical lever, 474 Orientation by cold drawing, 138-140 by melt flow, 143-145 during polymerization, 138 effect on electric strength, 473 in craze fibrils, 265 in partially crystalline polymers, 146, 148 measurement by birefringence, 165- 167 infrared dichroism, 173- 175 small-angle X-ray diffraction, 175- 184 sonic modulus, 169-172 wide-angle X-ray diffraction, 158- 161 of films, 141 planar, 156 production of, 138- 146 ultimate, in polymers, 140 Orientation function, 148- 150, 166 Oriented crystalline polymers morphological model, 219 radicals generated during deformation, 203-2 13 Oriented crystallization, 143- 145 Ozone stress cracking, 214-215

P Pascal, unit of stress, 122 Permeability, to gases and vapors closed receiving volume methods apparatus, 362-367 data analysis, 356-362 continuous flow methods apparatus, 366-367 data analysis, 369-371 measurement of penetrant concentration, 372 cup method, 366-367, 369 membrane-type sample installation, 373 sources of measurement error, 372-377 historic perspective, 316-324 utility, 339-340

524

SUBJECT INDEX FOR PART C

Permeability coefficient definition, 318 of gases in polymers, empirical relations, 326 units, 324-325 Permittivity of free space, 381 Phase contrast microscopy, of polymeric alloys, 288-291 Phenolic resin density, 72 elastic constants, 81 ultrasonic properties, 72, 73, 74 Photoelectric emission, 134 Photothreshold voltage, 436. 437 Piezoelectric transducer, 61, 62, 168 Plasticizer, effect on gas transport, 330 Poisson's ratio definition, 80 during crazing deformation, 234 measurement of, 108 of various polymers, 81 Polarizability, 162, 166 Polarization, electric, 382 Pole figure, 157- 161 Poly(acry1ic acid), replica for electron microscopy, 242 Poly(acrylonitrile-co-butadieneco-styrene), (ABS) density, 72 elastic constants, 81 ultrasonic properties, 72, 74 Polybutadiene, radicals generated during fracture, 200-201 Polycaprolactam, see also Nylon 6 density, 72 elastic constants, 81 ultrasonic properties, 72 Poly(r-caprolactone), alloy with PVC, 290, 299,300 Pol ycarbonate radicals generated during fracture, 198199 work function, 43 1 yield stress and Young's modulus, 131 Pol y(carborane siloxane) density, 72 ultrasonic properties, 72, 74 Polychlorotrifluoroethylene crazing-type creep, 266, 268 stress-strain curves, 130

yield stress and elastic modulus, 131 Poly(2,6-dimethyl phenylene oxide) alloy with polystyrene, 308-309, 331 glass transition, 308 radicals generated during fracture, I%, 197, 200-201 Poly(dimethy1siloxane) as ultrasonic medium, 64 density, 72 ultrasonic properties, 72 Polyepoxide density, 72 elastic constants, 81 secondary transition, 105 Polyethylene crystal structure, 151 effect of crystallinity on gas diffusion, 330 electric breakdown in, 467-469, 472 by discharges, 450 fiber diffraction pattern, lS2, 158 high density density, 72 elastic constants, 81 electric breakdown in single crystal, 459 ultrasonic properties, 72, 73, 74 lamellae, 176, 177 low density density, 72 ultrasonic properties, 72 photoelectric emission, 435-436 radicals generated during fracture, 198199,204,207 small-angle X-ray pattern, 18I, 182 ultraoriented, 145 Poly(ethy1ene oxide) bulk modulus, 112 density, 72 radicals generated during fracture, 198199 ultrasonic properties, 72, 73 Poly(ethy1ene terephthalate) effect of crystallinity on gas diffusion, 329 intrinsic birefringence, 162 oriented, 145- 147 radicals generated during fracture, 204, 207 small-angle X-ray pattern, 182 work function, 43 1

SUBJECT INDEX FOR PART C Poly(hexamethylene adipate) density, 72 elastic constants, 81 ultrasonic properties, 72 Poly@-(2-hydroxyethoxy)-benzoic acid), (PEOB), stable primary free radicals, 204, 205, 207,208, 209 Polyimide, work function, 431 Polyisobutylene, radicals generated during fracture, 198-199 Polyisoprene, radicals generated during fracture, 200- 201 Polymer single crystals, 459 Poly(methy1 acrylate), radicals generated during fracture, 198-199 Poly(methy1methacrylate) craze opening displacement, 248 creep, effects of moisture, 55 density, 72 elastic constants, 81 electrical treeing in, 482 radicals generated during fracture, 198199 tensile strength in various environments, 259 ultrasonic properties, 68,71, 72,73, 74 Poly(ar-methyl styrene), radicals generated during fracture, 198-199 Polyoxymethylene density, 72 elastic constants, 81 radicals generated during fracture, 198199 ultrasonic properties, 72 Poly(pheny1 quinoxaline) density, 72 elastic constants, 81 ultrasonic properties, 72, 73, 74 Polypropylene density, 72 elastic constants, 81,82 hard-elastic fiber, 144 intrinsic birefringence, 162 radicals generated during fracture, 198199 small-angle X-ray pattern, 182 ultrasonic properties, 72, 74 yield stress, 131 Poly(propy1ene oxide) dielectric loss data, 395

525

radicals generated during fracture, 198199 Polystyrene bulk compressibility, 104, 105 critical strain, in environmental crazing, 257 critical stress for crazing, 251 density, 72 elastic constants, 81 glass transition, in block copolymer, 312 high impact crazing in, 271-273 morphology, 289, 292-293 radicals generated during fracture, 198-

199 ultrasonic properties, 72, 74 unperturbed end-to-end distance, 297 work function, 431 domain size vs. molecular weight, 2% glass transition, 31 I, 312 spherulite patterns, 299 Poly(styrene-6-ethylene oxide), morphology. 298 Poly(styrene-b-isoprene) glass transition, 312 interphase thickness, 304 small-angle X-ray pattern, 302 Polytetrafluoroethylene density, 72 radicals generated during fracture, 200201 thermally stimulated current spectrum, 441 ultrasonic properties, 72 work function, 431 Polyurethane density, 72 radicals generated during fracture, 20020I segmented block, 299, 301, 302, 313 ultrasonic properties, 72, 73 Poly(viny1alchohol), radicals generated during fracture, 200-201 Poly(viny1butyral) density, 72 ultrasonic properties, 72 Poly(viny1chloride) alloy with poly(ethy1ene-m-vinyl acetate), 293, 310

5 26

SUBJECT INDEX FOR PART C

radicals

generated

during

fracture,

204

work function, 431 Poly(viny1idenefluoride) alloy with poly(methy1methacrylate), 31 1 density, 72 elastic constants, 81 ultrasonic properties,, 72 Poly(viny1 methyl ether), alloy with polystyrene, 291, 309 Ponderomotive effect, 425 Pressure measuring devices, in permeation and sorption studies, 340-341 Primary electron emission, 437 Propagation constant, complex, of electromagnetic waves, 409

Q Quartz spring, in sorption measurements, 35 I

R Radical generation, effect of temperature, 210212

observation by ESR, 191-194 production from mechanical fracture, 189 reactions of, 197. 202 reactivity, 205 species, various polymers, 195-201 stable, 188 Radical concentration determined by ESR, 187- 188 of various polymers, 206-207 Radius of gyration, 291, 305 Recessed specimens, for electric breakdown, 452 Reciprocal lattice, 152 Refractive index, 162 Relaxation map, dielectric, 394 Relaxation strength, dielectric, 388 Relaxation time and ultrasonic measurements, 88 dielectric distribution in polymers, 403 model for, 387-388 spectrum, 389 temperature dependence, 393 in creep equation, 16

of polymer-penetrant system, 336 Resonant circuits, for high-frequency dielectric measurements, 404-408 Retardation time, in creep equation, 16 Rheology, see under specific viscoelastic functions Rheovibron instrument, 35 ROW Structure, 143-145, 156 Rubbers, see Elastomers Rubber-toughened plastics, 271 Rubbing contact, charge generation by, 432-433

S Schering (transformer arm ratio) bridge, 399-400

Secondary electron emission, 437 Secondary transitions, by high-pressure measurements, 105 Shear bands, 233 Shear compliance complex, 10 measurement of, 27 Shear flow, and crazing, 269 Shear thinning, 47 Shear viscosity dynamic, complex, 10 measurement, 45 Shear wave, 57 Shift factor, 14 Shish-kebob crystals, 143 Solid-state extrusion, 140-141 Solubility coefficient definition, 318 of gases in polymers, empirical relations, 327

Solution mixing, of high polymers, 56 Sonic modulus, 167-172 measurement of, 167-169 Sorption apparatus, 349-356 data analysis, 343-349 dual-mode, 333, 347, 361 measurements, calculating the diffusion coefficient, 345 types of methods, 342 utility, 339 Sound velocity, 172 Space charge, 448, 467-469 Sphere of reflection, 154

SUBJECT INDEX FOR PART C

Sphere of position, 154 Spherulites, 471 in polymeric alloys, 299 Spinodal, 277 Small-angle light scattering, in polymeric alloys, 299-300 Small-angle neutron scattering, in polymeric alloys, 305 Small-angle X-ray scattering, see X-ray diffraction, Standing wave method for dielectric measurements, using transmission lines, 409-416 for sonic modulus determination, 168 Star-shaped block polymers, 298 Static electricity generated by contact, 428-432 generated by rubbing, 432-433 role in adhesion, 434 Steady-state response, in rheology, 44-46 Strain definition, 118 in flexural test, 125 measurement of, 119-121 Strain gauge bonded resistance-type, 120 mechanical, 119 mechanical-optical, 120 Stress criteria, for crazing, 249-252 Stress field, of a craze, 245-248 Stress relaxation in crazed polymers, 268 in penetrant-swelling polymers, 335, 337338, 349

measurement of, 32 modulus, 11 Stress-strain curve effect of temperature on, 129, 132 effect of strain rate on, 132 of crazing polymers, 263-265 Submicrocracks, 230 Sulfur-iodine eutectic, for impregnation of crazes, 243-244 Surface discharges, elimination of, 455 Surface states, 434 Susceptibility, electric, 383

T Temperature control, 341 Tensile deformation

527

ASTM test method, 123 orientation by, 138-140 radical species, 203-205 studies by mass spectroscopy, 228 study of radical generation, 191-194 Thermal breakdown, see Electric breakdown Thermal impulse breakdown strength, 447 Thermally stimulated currents, 439-440 Thermodynamics, of polymer mixtures, 276-278

Thermoluminescence, 440 Thin film method of crazing study, 244 Tie molecules, role in polymer fracture, 2 19- 224

Time domain reflectometry, in high-frequency dielectric measurements, 41642 1

Time lag, in electric breakdown, 470 Time-lag method, for permeability, 357-362 sources of error, 374, 376 Time-temperature reduction, 14 Torsion angle detectors, 29 Couette geometry, 31 torque production, 30 Torsional braid instrument, 41 Torsional pendulum, 38-40 Tortuosity factor, 329 Transient currents, 494 Transient methods in dielectric measurements high frequency, 416-421 low frequency, 401-404 in high-field conduction, 492-495 Transition moment, of infrared absorption, 173

Transport, of gases and vapors, see also, Permeability, Diffusion, Sorption anomalous, 334-337, 348 basic theory, 316-319 effect of polymer morphology, 328-331 historic experiments, 319-324 Transmission line in high-frequency dielectric measurements standing waves, 408-416 transient methods, 416-421 system in electric breakdown studies, 470- 47 1

528

SUBJECT INDEX FOR PART C

Trees artificially produced, 461, 485 electrical, 449, 471, 482-488 electrochemical, 449,488 water, 449. 488 Triboelectric series, 422-423 TSC, see Thermally stimulated currents Two-point X-ray pattern, 179

U Ultrasonic frequencies, 60 Ultrasonic measurements delay rod technique, 75 immersion technique, 61-75 multiple echo technique, 77 Ultrasonic pulse, 65 Ultrasonic speed frequency dependence, 74 in various polymers, 72 measurement of, 66 temperature dependence.73 Ultraviolet irradiation, charge production, 436 Unit cell, 151 Upper and lower critical solution temperatures, 283-284

Viscosity, shear, 4, 5 Volumetric technique for permeability, 363 for sorption, 354

W Water trees, 449, 488 Water vapor transport, 331-332 experimental precautions, 375 Waveguide, 410 Weissenberg rheogoniometer, 35 Wheatstone bridge, 397 Wide-angle X-ray diffraction, see X-ray diffraction Work function, various polymers, 431 X

Xerography, 424 X-ray diffraction small-angle craze studies, 245 of alloys and brock copolymers, 300305 polymer fracture studies, 230 wide-angle, 150- 161 X-ray photoelectron spectroscopy, 437

V Viscoelastic behavior definition, 1 dielectric analogy, 403 effect of pressure on, 51 linear, 3-44 nonlinear, 46-51 Viscoelastic liquid, creep in, 3 Viscoelastic solid, creep in, 6

Y Yield strength, effect of pressure on, 116 Yield stress (yield point) definition, 129 effect of pressure on, 131 in crazing polymers, 264 micro-, 119 various polymers, 131